Kac-Moody algebras and exact solvability in hadronic physics

KAC—MOODY ALGEBRAS AND EXACT SOLVABILITY IN HADRONIC PHYSICS

L. DOLAN

The Rockefeller University, New York, N. Y. 10021, US.A.

I

NORTH-HOLLAND PHYSICS PUBLISHING-AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 109, No. 1 (1984) 1—94. North-Holland, Amsterdam

KAC-MOODY ALGEBRAS AND EXACT SOLVABILITY IN HADRONIC PHYSICS* L. DOLAN The Rockefeller University, New York, N.Y. 10021, U.S.A. Received 1 October 1983

Contents: 1. Introduction 2. The role of symmetries in field theories 2.1. Continuous symmetries of the action and equations of motion 2.2. Algebra of the symmetry generators, classical and quantum 3. Kac—Moody and affine Lie algebras 3.1. Definition and other infinite parameter algebras 4. Exact solvability and infinite parameter symmetry groups 4.1. Action-angle variables — an infinite parameter Abelian algebra 4.2. Construction of an infinite set of commuting charges from Kramers—Wannier self-duality 4.3. NLo’M S-matrix from the infinite set of Lüscher— Pohlmeyer non-local charges 5. The principal chiral models 5.1. Non-local currents as Noether currents 5.2. The affine generators 5.3. Symmetry of ~‘(x) 5.4. A generating function 5.5. An extended algebra 5.6. Poisson brackets, canonical transformations

*

3 6 6 9 12 12 14 14 19 20 22 22 25 30 31 33 36

6. Loop space Yang—Mills theory 6.1. Yang—Mills functional formulation and the chiral models 6.2. Polyakov string theory 6.3. ‘t Hooft dual operators A(c) and B(c) 7. Affine algebra in self-dual SU(N) Yang—Mills theory 7.1. The symmetry generators 7.2. The algebra 7.3. Associated Noether-like conserved currents 7.4. A second infinite set of transformations 7.5. Affine transformations on the Wilson loop 7.6. Parameters of the instanton solutions 7.7. Complex Yang—Mills in the J-formulation 8. Reasons for a new symmetry of the strong interactions 8.1. Vertex operator of the dual string model 8.2. Kramers—Wannier self-duality of 4-dimensional SU(N) gauge theory 8.3. Loop space formulation is not self-dual 8.4. Kaluza—Klein to generalize ~ to full Yang—Mills 8.5. Affine connections for Kaluza—Klein 9. Kac—Moody algebras in other connections with integrable and physical systems References

40 40 43 45 45 45 49 63 67 73 75 76 77 77 80 84 84 91 92 93

Work supported under the Department of Energy under Contract Grant Number DE-ACO2-81ER40033B.000. Single ordersfor this issue PHYSICS REPORTS (Review Section of Physics Letters) 109, No. 1(1984)1—94. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 57.00, postage included.

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L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

3

Abstract: The appearance of infinite parameter symmetry transformations in particle theories is described. This invariance forms an afilne algebra, a class of Kac—Moody Lie algebras, whose representations can be given by the dual string model. The symmetries exist in the two-dimensional principal chiral or sigma models, in the three-dimensional loop space formulation of Yang—Mills theory, and, so far, in a restricted class of four-dimensional local Yang—Mills gauge field theory, the self-dual set. The use of such infinite parameter invariance in finding exact non-perturbative solutions is discussed in terms of action-angle variables, the inverse scattering problem method and exact S-matrix calculations. Both statistical mechanical spin models and continuum quantum field theories are analyzed. Evidence and suggestions for the extension of the affine symmetry to the complete non-Abelian gauge theory are given.

1. Introduction Gauge theories appear to be fundamental to the four interactions observed in particle physics: the familiar forces of electromagnetism and gravity which act over all distances; and the two more anti-intuitive forces of the weak and strong interactions which occur only at very short distances. Each of these interactions can be described by a gauge field theory containing infinities, on the quantum level, that can be consistently removed (except for gravity) in a weak coupling, i.e. perturbative expansion. The strong interactions, distinguished by their large coupling at distances of the order of the size of elementary particles, so far have no controlled approximation in which to calculate non-perturbative phenomena such as quark confinement and spontaneous symmetry breakdown, and to nail down a real quantitative connection with experimental hadronic physics. Lattice gauge theories have provided to date the most quantitative non-perturbative approximation [1]. These calculations are essentially numerical in character, however, and it is difficult to estimate systematically the corrections as the lattice is enlarged. Although quark confinement is calculated for strong (lattice) coupling, the finite lattice spacing destroys Lorentz invariance; so in the end it must be the continuum limit of the theory which is relevant for physics. Both Padé extrapolation techniques to weak lattice coupling (the continuum) and Monte Carlo simulations, valid for all couplings greater than zero, are suggestive that the lattice is useful to describe non-perturbative continuum theory, even though it is also difficult to calculate in this limit exactly. A second non-perturbative scheme is the large-N expansion of SU(N) gauge theory. Although all graphs which contribute to the leading order of 1/N have been identified in the planar diagrams [2], an explicit summation of these graphs in four dimensions has remained elusive. A third approach is the semi-classical approximation provided by instanton physics. Although this expansion is useful in solving [3] the U(1)-problem by admitting the non-perturbative form exp(—1/g2), it has been understood that so far it still is valid only for weak coupling g and does not reach to the confinement region [4]. 2 + 1-dimensional non-perturbative Yang—Mills theory is discussed in the literature and provides a view towards its implication for the 3 + 1-dimensional model [5]. An intuitive understanding of non-perturbative effects can be obtained using gauge fields with an effective mass; and semi-quantitative estimates of various dimensional quantities such as the string tension can be made [6]. Confinement and glue ball masses have also been calculated numerically in various other approximation schemes [7]. Nevertheless, it seems it would be useful to establish an analytic non-perturbative controlled approximation in continuum hadronic physics. This review describes a possible new approach to this problem by beginning with an analysis of additional continuous symmetry present in various for-

4

L. Dolan. Kac—Moody algebras and exact solvability in hadronic physics

mulations of Yang—Mills field theory. Symmetry in a theory has always been a tool which gives information about the solution independent of the approximation scheme. Sometimes, a symmetry provides enough restriction to solve the theory exactly. In any case, we are all agreed that a new continuous symmetry in the four-dimensional Yang—Mills theory would be extremely valuable information. There exists a candidate for such a symmetry. So far it has been realized explicitly on the self-dual solutions [8], a restricted class of non-Abelian gauge fields. In this report, reasons why this extra symmetry could be found in the complete gauge theory, and the connection between the symmetry and non-perturbative solvability are discussed. The new symmetry as it appears on the self-dual potentials is a global symmetry and forms a non-linear, non-local realization of a subalgebra of an affine Lie algebra, a class of Kac—Moody algebras [9], and a sophisticated mathematical structure already known to mathematicians in connection with representations of the dual string model [10]. This model is an alternative description to non-Abelian gauge theory of the strong interactions, and the fact that the same symmetry occurs is intriguing. In some sense, the symmetry is more fundamental to the physics than is the particular model we choose to describe it. Affine Lie algebras are infinite parameter, i.e. infinite-dimensional non-Abelian algebras. They have infinite parameter Abelian subalgebras, a hallmark of exactly integrable systems [11].Therefore, if this invariance can be formulated for the full Yang—Mills theory (not just on the self-dual set), it may provide a tool for non-perturbative solvability in the spirit of the stunning successes of one-dimensional mathematical physics. The affine algebras were first identified [12] in particle theories in the two-dimensional principal chiral models, a special case of which is the 0(N) non-linear sigma model NLo-M, containing the LUscher—Pohlmeyer infinite set of non-local charges [13]. These signalled the existence of an infinite parameter symmetry algebra. A second class of local charges in this model discovered earlier by Pohlmeyer [14] were known to be associated with the infinite parameter conformal group in two dimensions [15]. The symmetry group responsible for the non-local charges was subsequently shown to be a subalgebra of a type 1 affine algebra [12].In the quantum theory, the conservation of these charges can be used to calculate the exact on-shell matrix [16]by proving that 1) the n-particle S-matrix factorizes into a product of two-particle S-matrices; 2) there is no particle production but the S-matrix is non-trivial due to time delay (phase shift) and interchange of internal symmetry quantum numbers. These two statements were originally the assumptions made by the Zamolodchikov’s to compute the scattering amplitudes [17]. The two-dimensional chiral models, and in particular the 0(N) NLo-M, have many features in common with four-dimensional SU(N) gauge theory. They are both asymptotically free and renormalizable as quantum field theories. On the lattice the 0(4) NLoM has the same Migdal recursion relations as the SU(2) gauge theory. The 0(3) NLo-M has instanton solutions. A loop space (functional) formulation of the Yang—Mills equations looks like the local chiral model equations of motion when ordinary derivatives are replaced by functional derivatives [18]. Thus, when it was understood that the NLo-M and the principal chiral models had an infinite parameter symmetry, it was natural to ask if the four-dimensional NAGT also possessed one [12].Since there is no systematic procedure to translate properties from one theory to the other, the approach was to identify the abstract algebra responsible for the “hidden” symmetry transformations of the twodimensional models, with the hunch that the same algebra would occur in the four-dimensional case. That is to say, once the algebra itself was unhidden in the chiral theory, one could ask what representation of it is carried by the non-Abelian gauge field. This has proved to be difficult; yet guided

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

5

by the self-dual expressions, it may be possible to identify the new affine global internal symmetry as particular (space-time) coordinate transformations in a multi-dimensional non-Abelian Kaluza—Klein theory, much in the same way that local gauge invariance of the four-dimensional gauge theory is identified as a special choice of the general coordinate invariance of a 4+ K-dimensional Einstein— Hilbert action [19]. When the four-dimensional gauge configurations are restricted to self-dual fields, the characteristic curves of certain partial differential equations appearing in the Kaluza—Klein analysis are exactly the Yang [20] D and D functions, the detailed non-local functions which occur in the affine algebra transformations. This program specifically employs dimensional reduction: the gauge fields are assumed to be independent of the extra dimensions. A more general treatment of Kaluza—Klein theories uses spontaneous compactification, in which the gauge fields are allowed to be periodic functions of the internal (compactified) dimensions (with very small period 1/rn planck). The metric tensor is expanded in terms of periodic eigenfunctions determined by the choice of internal space. The (four-dimensional space-time dependent) coefficients of the zero-modes are the zero-mass gauge fields established by dimensional reduction. In addition, the theory now includes a “tower” of massive states. If the general coordinate transformations in the multi-dimensional theory are also expanded in terms of these eigenfunctions, then those infinite number of coefficients induce an infinite number of symmetry transformations, on the four-dimensional fields, which mix the massless and massive fields. These also form an infinite parameter Kac—Moody-like algebra, and may be the symmetry which leads to ultraviolet finiteness in some of these compactified models [21].In the self-dual gauge field system, the affine symmetry transformations transform real self-dual solutions to real self-dual solutions with the same action infinitesimally. Presumably the instanton solutions with their 8n 3 parameters reflect the infinite-parameter symmetry algebra [8]. This review will also discuss the existence of infinite parameter Abelian symmetry in the exactly solvable spin models of statistical physics: the Baxter and Ising models, and its relationship to the Kramer—Wannier self-duality of these partition functions [22].This is another property which is thought to be shared by four-dimensional SU(N) gauge theory [23]. The quantum inverse scattering method, applicable to both these lattice systems as well as to 1+ 1-dimensional continuum field theories provides the exact solution and is closely tied to the infinite algebra. In some cases, such as the NLoM, the inverse scattering problem has not been formulated. Nevertheless, the existence of the conserved charges leads to a solution of the S-matrix, essentially by imposing enough constraints on it. Thus, from theory to theory, the new symmetry is useful in different ways for a non-perturbative solution. Clearly the approach applicable to Yang—Mills will depend on how the algebra appears. In still another context, the affine algebras have arisen in the mathematical literature on integrable systems. For the Korteweg—de Vries (K-dy) equation, the Lax pair L = [L, M] takes place in a Kac—Moody algebra, and this can be used to prove exact solvability. The algebra plays a central role in the explicit construction of a general class of solutions for K-dy and in the linearization of the periodic Toda lattice. In the case of finite parameter algebras, the orbit method in the representation theory of Lie groups has led to the quantization of the integrable generalized Toda chain [24]. Also, it has shown up in transformation theory and a Riemann—Hilbert problem [25]. How these various subjects are related if at all is not yet totally clear, but it seems that the algebraic structure is sufficiently rich to bring together the gauge theories of hadronic interactions and non-perturbative solvability. —

6

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

2. The role of symmetries in field theories 2.1. Continuous symmetries of the action and equations of motion In a field theory, a symmetry is a transformation on the field 4’(x)—~q5’(x) which leaves the action I invariant. In n dimensions, I = f &x L(x) where L(x) the Lagrangian density is a functional of fields q~(x)and their derivatives: L[4(x), a~4(x)].If the transformations form a group, it is called a symmetry group of the theory. Noether’s theorem states that for a continuous symmetry transformation there is an associated constant of motion. For continuous symmetry, there must exist an infinitesimal transformation (e~*~ 1) 4(x)—~qY(x)=4(x)+ EazYfr which shifts the Lagrangian density by a total divergence without use of the equations of motion (off-shell): L-~~L+ 9KaS*

(2.1)

On-shell, the shift for any z.1’~/ is given by ~YL= a~(~~i).

(2.2)

Eq. (2.2) holds quite generally, since the equations of motion are defined by the requirement that the action is invariant under small variations 4 4 + S4 for which ~4.(x) vanishes on the surface at x: —~

J

I[~ + 3~] I[~] -

—

d~x~L[~ +

1d~ f?iL~

—J =

=

Since

~,

J f

X.t~ ~

d~x~

(~L

3~

+

~)]

-

L[~]}

?iL ~

~) (~a~ } J (~_a.~

dS”’ ~ (aL~)

+

—

~L) ~

d~x= 0.

+

(2.3)

0, the equations of motion are ~iL/&,b= ö

((~L/~3s4~)a4~)sudace

5. ~L/&8~çb,

and thus on-shell the

variation carried out in (2.3) for 4i O4~instead of ~i4 yields (2.2). The conserved current J~(x)is constructed from (2.1) and (2.2), since on-shell 5’

—

.

5’

Then (25 ~ja~

=

0.

(2.6)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

7

In n dimensions the charge is =

J

d~xJo(x~,.., x~, t).

(2.7)

.

From (2.6), Q~is conserved if J~(±oo,t) =

0

(2.8)

since then, =

J

d’~x3OJ~ =

—

J

= d”1x 9~J~’

—

J

dS’~n, J~’= 0.

(2.9)

Note that the existence of a transformation (2.1) insures the current conservation (2.6). We emphasize that the charge (2.7) is conserved only when the fields in .10 are on-shell and (2.8) satisfied. One must of course check further that (1) Q” itself on-shell is not identically zero and (2) the validity of (2.8) on-shell; for the existence of a non-trivial constant of motion. The expressions for the charges are more fundamental than the field transformations themselves, in the sense that different field transformations can give rise to the same charge on-shell. For example, in the case of the two-dimensional chiral models discussed in section 5, the three different transformations zl1g, 5’g and zi1g give rise to the same 0 [26,27]:

J

~1g = —(g)

=

~1g

0=

zl tg

=

J

(-

+

dx’ [Ao(x’, t),

~J

J

g

dx’ [Ao(x’, t),

dx’ [Ao(x’, t),

dx’ [—A 1+ ~[A0,

J

(2.lOa)

T]

(2.lOb)

T]

(2.lOc)

T])g

dx” e(x’

—

x”)Ao(x”, t)]].

(2.11)

As we will see in section 5, these different transformations and their respective families are found to form slightly different symmetry algebras. This report will take the point of view that what is important are (1) the charges, and (2) the simplest way to organize the algebra of transformations which lead to those charges. Furthermore, we see that L~L= 85’K,. off-shell is the starting point for the existence of the conserved charge. In ref. [26],this is referred to as a symmetry of the entire space of fields. That is to say, without

8

L. Dolan. Kac—Moody algebras and exact solvability in hadronic physics

= ~ off-shell, for example if the field transformations could only be expressed on-shell, then there would be no systematic way to find the Noether current. Strictly speaking [27]for a real symmetry of the action, ~XI= f d~xÔMK/. = 0, i.e. KM must vanish on the surface at infinity for the entire space of fields. But clearly, the Noether construction can be made when K,. vanishes merely for on-shell fields on the surface, which is the case in the chiral models. We emphasize that this is still a stronger condition than a symmetry of the solutions, since it admits the Noether construction. In the following, a proof is given that if ~L = a,.K,., and if ~I = 0 either on-shell or off-shell, then ~4 is a symmetry of the equations of motion, i.e. if /, is a solution to ~L/~i~= a,. ~ so is 4 + ~ to first order in ~/.

Proof. Let

=

~‘

~

+

is some given function of ç~,~/[/] and

where ~

~,

(2.12)

L[4’]—L[~]= a,.K5’[~]

for any

~/.

Let L[4] +

~]

—

=

~9,.4c9Mçb

L[4’]

—

+ V(4).

L[q~ + sq’]

Then, for arbitrary ~4,and ~4,

+

(—‘+~)~ — a5’(a5’4~

9,.(a,.Ø~)+

=

(—~i~ +~)s~

~)—

=

L[qY

+

(2.13)

~q~] L[q~ + ~q~] —

—

L[q~ + z~q5+ ~] Now choose =

5’ [~]. —

L[4

+

(2.14)

~çb] a,.K —

(2.15)

i.e. ~/ is now given in terms of the arbitrary function L[~ +

~

+

~[q5

+

~q~]] L[4 —

+

~]

—

8,.K5’[çb]

~lt/.

=

Then eq. (2.14) is from (2.12)

a,.K~[~+ ~]

—

a,.K~[4].

(2.16)

Then the integrals over all space-time of (2.13) and (2.14) are zero for arbitrary ~/ since (.1 d”x ~eq. (2.14)}) is the integral of a total divergence (see 2.16); and (2.14) = (2.13). Then 0

=

=

J J

d~x{eq. (2.13)} d~x{(-L~’+ ~V/~q5’) ~

-

(-L1~+ ~V/~çb)~}.

(2.17)

Therefore, if —IIlfr

+ ~V/8~

=

0

(2.18)

9

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

then

J

d~x(—El~’+ ~V/~4’)~~ = 0

(2.19)

which implies —L1t~’+ ~ vi~’ = 0 since ~çb,although chosen to be a specific function of

(2.20) ~,

is still arbitrary since ~4 is arbitrary.

QED

In general the inverse is not true: a symmetry of just the equations of motion does not imply t~L= off-shell and L~I= 0 on-shell; and it does not automatically lead to a conserved current. In this review, the chiral models are shown to have such an off-shell symmetry, whereas the self-dual gauge field transformations are still restricted to be (because of the self-dual condition) on-shell. Since the conserved currents were already known in the chiral case [13,28] the real advantage of the derivation of the off-shell transformations was that it was then possible to identify the simple algebra involved since one did not always have to take into account the imposition of the equations of motion. In the gauge theory, although there is no systematic Noether way to identify conserved currents, since the transformations are always on shell, it is possible due to the specific nature of the transformations to show that they shift L by c9,.K5’ where K,., though on-shell, is different from (~L/b85’A~)AA~, and .1,. from (2.5) is not trivial. As a last comment on new symmetries, we remark that all the above is true for continuous transformations, those that can be made arbitrarily close to the identity. New discrete symmetries, while also quite interesting for gauge theories, will not have the signal of a conserved current and charge. Though a charge can be defined for a discrete symmetry, it is not conserved. So it is expected that symmetries associated with conservation laws and solvable systems will form continuous groups. 2.2. Algebra of the symmetly generators, classical and quantum The field transformations and charges completely define the symmetry of a theory. It may be useful however to ask what is the algebra or the group associated with the invariance. This provides a concise way to discuss the symmetry, it may lead to new invariance if the known transformations do not close an algebra, and it may allow analogies to be drawn between systems which share the same abstract group analysis. In mathematics, the study of algebras and groups is often quite separate. Field theorists, on the other hand, discuss Lie algebras and Lie groups almost interchangeably. What is interesting to the physicists is the symmetry. Of course the mathematicians are interested in the structure of the algebras and groups and rarely identify them as a symmetry of some theory. This section reviews the relationship between infinitesimal field transformations and Lie algebra generators, finite field transformations and group elements, and the difference between the classical and quantum representations of an invariance. In Lie theory, the algebra of infinitesimal transformations 4, (x)—*4, (x) + ~j ~4,(x), has generators [29] Ma

=

—

Jdnxz~4,(x)6~).

(2.21)

This is defined so that with the standard property of the functional derivative &4,(y)/&4,(x) = ~‘~(x y), —

[M”, 4,(y)] = ~a4,(y).

(2.22)

10

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

The Lie algebra of the transformations can then be easily identified, i.e. the structure constants found, by calculating the commutator [M’~,M”]. From (2.21) and (2.22), it is given by [Ma, Mb] =

=

J J (J

J

(X)~4,~)~

dnxib4,(x) ~x)

J

d~y~4,(y) ~q5(y)

5(x)

d~x

J -

dny 4,(y)~)L1bcb(x)) ~q d~y(J dnx (x)~ 4,~~)z1aq5(y)) ~4,(x)

J

-

=

J

d~y 4,(y)~4,~~)dnx

d~x(~b(4,(x)+ ~çb(x))- ~b4,(x))~x)

J

&y (~a(4,(y)+ ~b4,(y))

~a4,(y))~~.

(2.23)

The last line of (2.23) is again of the form (2.21)4,.where by LY’(4, +transformations ~a~/,)_ ~jb4, — To seeLIa4, thatisa replaced set of infinitesimal i”(4, the + ~i”4,)+ii”4, which can to be generate defined asa Lie _Cabc~C meet necessary conditions group, the Lie integrability condition must be checked as follows. Assume that = p~~a4, is the infinitesimal form of a finite transition law T~(4,).Then T 2). Since T 0(çb) = 4, + z1,4, + O(p 0 is a group element we can define its inverse as Ta-’, and it must be that ~

~

=

J(4,)

(2.24)

where J is also an element of the group. The infinitesimal form of (2.24) is the integrability condition ~&,(4,+ ~4,)— ~t~q5 ~~(4, + ~4,)+ ~ —

=

d,~(4,)

(2.25)

where d,7~ must be an infinitesimal transformation define dffp(4,) = cabccrbpa~4,and 5C4, is included in the set ~ then the algebra ofin the 4”4 group. is said Iftowe close. Since (2.25) appears as Paffb t times the brackets in (2.23), the structure constants are defined as c abc such that (2.23) is [M’~,Mi’] = cabcM~.

(2.26)

When the integrability condition is satisfied, (2.25) can be integrated to give the finite transformations. These are expressions involving finite group parameters Pa. For example, for global 0(3) isospin transformations, ~i4,a

=

(2.27)

EabcPb4,c

[M”, M”]

= E

(2.28)

abcMc.

The finite transformation [30] is q5~(y)= =

exp(

J

d~x~4,~(x)~~c(x))

(exp(Eabcpb))

4,c(y).

4,~(y) (2.29)

11

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

Such finite transformations form a group, whereas the infinitesimal transformations generate an algebra. In a field theory, where the transformations are symmetries, the charges ~a defined by (2.7) can also be thought of as generators of an algebra, similarly to the M~of (2.21). Again, as in (2.22) {Q~4,(~,)}=

(2.30)

~ja4,(~,)

But now, the bracket of (2.30) is defined to be the Poisson or possibly a Dirac bracket of classical field theory. The Poisson bracket, used when the fields have no constraints, is defined by {F[4,(x)], G[4,(y)J}

=

J

~4,(~) ~()

d~z(~

(2.31)

b4,(z))

where i~(z) ~L/&4(z), and F and G are arbitrary functions of 4,. This definition is constructed so that {IT(x), 4,(y)} =

(2.32)

ô~(x— y)

and the (Lagrange) equation of motion =

3~.&L/~ô,.4,

(2.33)

can be expressed as Hamilton’s equations: (2.34) H=Jd~1x~r

(2.35)

{H, qS(x)} = ~H/~ir(x) = 4(x)

(2.35a)

and {H, 1r(x)} =

_____—

S9~ ~4,()

= —

For internal symmetry charges, K,.

=

ir(x).

(2.35b)

0, J~a= (~L/ba,.4,)~i4,’~, and it is trivial to check for

1j~4,

independent of 4, that {Q~4,(y)} = 8Qa/~.(y)= zi~4,(y). Also {Qa, Qb} =

J J

d~z d’~’xd’’y(~’~4a4,(x) ~(y)

—

1T(x) 64,(z) ~ja4,(x) 6ir(z) 7T(y) 4b4,(y))

(2.36)

12

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

=

=

J J

d~x d~y (ir(y) ~a4,(x)

~4,~x)~4,(y)

d~y~(y) cabc~4,(y) =

-

CabcQC,

7r(x) ~b4,(y) (2.37)

so that, in this case the algebra of the M~ is identical to that of the charge Poisson bracket. When the transformations are non-local and if the theory involves constraints so that Dirac brackets must be used, then the bracket of two charges may be singular and difficult to regulate in such a way that the Jacobi identity is satisfied. For this reason, it is often more convenient to analyze the algebra in terms of the generators defined in (2.21). In the quantum theory, the Poisson brackets are replaced by the quantum commutators. The fields 4 and ir are operators; and the charge, defined for classical fields in (2.7), involves operators multiplied together at the same space-time point and must be regulated. It must be checked that anomalies do not develop, i.e. that quantum corrections do not spoil the charge conservation. Work on the quantum analogues of the affine symmetries has been done for several two-dimensional theories. The quantum analysis of the gauge theory, however, has much less in common with the two-dimensional theories than the classical symmetry analysis. 3. Kac—Moody and affine Lie algebras 3.1. Definition and other infinite parameter algebras A Lie algebra is completely determined by its structure constants cabc. Familiar Lie algebras have a finite number of generators T”, with the relation that [T~r~,Tb] = cabcT . For example, SU(2) has three generators a = 1,2,3 and the structure constants are Cabc = Eabc. The defining or spinor representation is T” = Ua/2i where a-” are the Pauli matrices. An infinite parameter, i.e. infinite-dimensional algebra has an infinite number of generators. The Kac—Moody type 1 affine algebra associated with a finite parameter semi-simple Lie algebra G is The generators are M”~and P with commutation relations [9]: G®C[t, r1]~jC~. [M~, M~]= ca&M~+m+

n8n._mc(Tr(TaTb))P

(3.la) (3.lb)

[P,M~]=0.

Here n, m = . —1,0,2,. . . Ta are matrix generators of G, Cab~are structure constants of G, and c is some constant number. The term cn~~ _,~(Tr(T”Tt’))Pis called the central extension and vanishes identically for n, m = 0, 1, 2, . . . ~. Eq. (3.lb) is trivially satisfied for P = 0. The commutation relations (3.1) then become —~,.

.

,

, ~,

,

[Ma”,M~]=

CabcMCn±m.

(3.2)

The algebra defined by the generators in eq. (3.2) is called the loop algebra: G®C[t, r’]. A defining

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

13

representation of the (3.2) generators is M”~=T°®t~.

(3.3)

Here t is just a variable. For example, when G = SU(2) and T~= o~/2i,then from (3.3), (3.4) The fact that t can be a complex variable and n is both positive and negative integers leads to the notation C[t, ~1] in the symbol for the loop algebra. For t of unit modulus, t = &°,this describes a circle or loop and (3.3) describes a mapping from S1 to elements of G, and the elements M~of (3.3) are called loops in G [31]. Clearly for P ~ 0, the Kac—Moody algebra of (3.1) is more complicated than (3.2). In general, different theories give different representations or realizations of M~.For P = I, a representation for M~was first written down in the mathematics literature for G = SL(2, c). Note that (3.lb) is again trivially satisfied. For an orthonormal basis of T”, then Tr T”T~’= tSab, and the commutators (3.1) are now given by [Mg, Mt,]

=

CabcMcn+m

+ nô~,_mCöab.

(3.5)

The central extension is present, denoted by the ~ C~,and defines a structure much less trivial than the loop algebra. Indeed (3.3) is not a representation for (3.5); and the one found by Lepowsky and Wilson in ref. [101 is quite complicated. It was subsequently observed by Kac and Frenkel that similar representations were exactly the coefficients found in an expansion of the vertex operator of the dual resonance model [10]. This is discussed explicitly in section 8.1. It may be useful to point out here, however, that the Kac—Moody algebra of (3.1) is different from the infinite parameter yirasoro algebra which is well known in the physics literature on the dual string. The Virasoro algebra [32] is defined by generators L~,n = . . ., 1,0, 1,2, - .., where —~,

11 1 1— 1 t~n,L~mJ~fl

—

\I

j

,~ 2_

m,~n÷m-1-cn~,n

j\~

l)Un._m.

Leaving aside the central extension, we see that (n — rn) is not of the form c~&.The yirasoro algebra is not associated with a finite parameter algebra in the same sense as is the affine algebra. If P = I, but n, m are restricted to be non-negative integers, then again the central extension disappears. This case is written G ® C[t] and is sometimes loosely called “half” of a Kac—Moody affine algebra. Its commutation relations are again [Mg, Mt,]

=

c~M~+m.

(3.7)

But now n, m = 0, 1, 2,. . - , Eq. (3.7) is the algebra which occurs naturally in the field theories discussed in this report; the different theories giving different, non-local, non-linear realizations of M~.We speculate, in the case of the four-dimensional Yang—Mills theory, that whereas eq. (3.7) is the algebra of self-dual transformations (modulo gauge transformations), the extension of this symmetry to D,.F,.~= 0, classically,

14

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

will result in (3.2) the loop algebra; and perhaps the full quantum SU(N) gauge theory will carry a representation of (3.1): SU(N)®C[t, t’~(~C~. This is similar to the case of the two-dimensional Liouville equation LII p(x, t) = e”° ‘~ where the classical theory admits an infinite symmetry algebra which only in the quantum theory becomes the full Virasoro algebra (3.6). Some physicists may more easily understand (3.1) as a current algebra. The remarkable observation by Gell—Mann that not only charges but also the currents themselves, which arise in the electromagnetic and weak interactions of hadrons, satisfy a set of equal time commutation relations led to a whole set of predictions relating j3 decay parameters and pion—nucleon scattering data. The idea is that the commutators can be derived from the canonical formalism without any reference to specific dynamics and therefore must always be true. The time components of the currents satisfy 3(x — y) (3.8) J~(x,t) 6 The brackets are now defined by the quantum commutation relations of the fields. The moments J°~ close the algebra (3.7): [J~(x, t),

ja

J

.

J

0b(y

t)]

= Cabc

d3x Ix~J 0”(x, t)

[J~, J~:1i=

cabcfn±m.

(3.9)

(3.10)

The appearance of the central extension in (3.1) is similar to Schwinger terms. Although the affine algebra is thus already familiar to particle physicists, it should be understood that in this report the affine algebras appear as symmetry algebras of a particular theory: infinitesimal transformations associated with the infinite number of generators leave the action invariant. This is not the case with current algebras. And thus the existence and the use of the algebra is different. In general, Kac—Moody algebras include a much larger set than the affine algebras. For example, a second class of Kac—Moody algebras is the hyperbolic Lie algebras, and there are infinitely many other classes. The particular algebras described in (3.1) are called type 1 affine algebras.

4. Exact solvability and infinite parameter symmetry groups 4.1. Action-angle variables — an infinite parameter Abelian algebra Exactly solvable systems are described most often by the inverse scattering problem method [11].This method has unified many of the techniques of one-dimensional mathematical physics: namely 1) the transfer matrix formalism 2) the exact solution of the Ising and Baxter models 3) infinite sets of conserved charges 4) the Bethe Ansatz solution of quantum continuous field theories 5) the diagonalization of quantum Hamiltonians. The method solves classical field theories, quantum field theories, and lattice spin models. It can be thought of as a canonical transformation to action-angle variables, and it has as a hallmark an infinite

15

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

set of commuting charges [Q~, Qm] = 0. These form an infinite parameter commuting algebra, a subalgebra of a Kac—Moody algebra where for example Q~=M~, n0. Then [Qn, Qm]

=

[A’I~°,M~m)]=

E

11cM(cul+m) =

0

since the structure constants are antisymmetric. Therefore, theories which have charges in an affine algebra may be solvable by the inverse method. Depending on which inverse method is used, the questions being answered are slightly different and therefore the role of the infinite set of charges is different [22]. The Classical Inverse Method solves the initial value problem of the classical non-linear field equations: given ~(x, 0), what is q~(x,t)? The way it does this is to map the initial data ~‘(x,0) into the scattering data a(k, 0) of an associated linear eigenvalue problem. Then a(k, 0) is evolved to a(k, t) which is mapped back to p(x, t). The non-linear Schrödinger equation is a 1 + 1-dimensional classical field theory of one complex scalar non-relativistic field: i9~ ~

(4.1)

The linear scattering problem is L0~çli II

/ iö~ \V~*

=

—

k~/i II

\/cço~

/~‘i

~)

“t/12

The Lax pair is [L00,M0~]= L,~,,where 9~4i= M0~i/iand M00 is defined such that k = 0. The integrability condition 3X3~i/J= ö~.~/i is a condition on ~(x, t) given by the NLSE. The solution to the linear scattering problem has the form 1

(~i1(x,0; k)\ — (e~ k. ~/i2(x, 0; k)) ~ —

a(k, 0) b(k, 0)

42 -

(

Here a(k, 0), b(k, 0) are functions of ~,(x,0), w*(x, 0). Since k from ç(x, 0): 2t) b(k, 0). a(k, t) = a(k, 0), b(k, t) = exp(—ik The action-angle variables are P(k,

t) =

~3=

lnIa(k, t)I,

Q(k, t)

arg b(k, t)

=

-

)

0, a(k, t) and b(k, t) are determined (4.3)

16

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

since the NLSE Hamiltonian can be written as

J

H=

dx

f

(Ia~ 2+cI~I4)_~=

dkk~P(k,t).

That is to say, in terms of P and Q, the Hamiltonian is only a function of P. Thus it is always true that ?iH/6Q(k, t) —P(k, t) = 0 and an infinite set of conserved charges M1 arise from the expansion of P(k, t) in a power series in k, or lna(k,t)= —ic~M1~ir. Then

J

4-~= MI=—

all !‘~!~=0.

dkk’P(k,t),

To derive the NLSE as Hamilton’s equations, define H(x, t) = i ~,*(x,t) and the Poisson bracket {H(x, t), ~(y, t)} = ~(x — y). The ~ = {H, ~}, H = {H, H} are the NLSE. We see that the transformation from H(x, t), ~(x, t) to P(k, t), Q(k, t) is canonical, i.e. the two sets of variables obey the same commutation relations: {P(k, t), Q(k’, t)} Since {P(k, t), P(k’, t)}

=

=

~(k — k’).

(4.4)

0, then the conserved charges M, satisfy {M,, M~}= 0. They form an infinite

parameter Abelian algebra. The quantum inverse method The fields p(x, t) are now quantum Schrodinger picture operators, and solving the quantum theory means to diagonalize the quantum Hamiltonian, i.e. to find its eigenvalues and eigenfunctions. For the example above, the quantum Hamiltonian is made finite by normal ordering

H=

J dx:~a~~I2+cI~4:.

(4.5)

The associated linear operator problem is 3~ ~I’=

O=

—

i : Q~I’:where

(

1k

Vcp*

k

and

~

(~‘ t/12

(4.6)

~

~/1i’

The Poisson brackets are replaced by commutators. The scattering data a, b are now operators which, although complicated functions of ~ and

~‘

have

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

17

simple commutation relations between themselves and the Hamiltonian: [H, a] = 0 and [H, b] -~b. Therefore a(k, t) generates an infinite set of conserved quantum charges and b(k, t) creates energy eigenstates which coincide with Bethe’s ansatz. The NLSE is the second-quantized form of an N-body quantum mechanics problem with a s-function potential: N

H=~-~--~-+c~ô(x1—xj). Bethe’s ansatz for the N-particle wave function ifr(x1, ~i(x1,. .

.

,

(4.7)

j
I

j=1

XN) -~exp(—

. . . , x~)where

Ht/i = E/i was

~ x1 ~i). —

(4.8)

i
In the NLSE, the solution to the associated linear problem can be written as a path-ordered exponential

~(x, t) = :P exp(—i

fa ‘~b

J

dz O(z, t; k)) ~(y, t):.

b*\ a*)

(4.9)

Eq. (4.9) is the scattering data. To make contact with a finite lattice, observe that ‘P can be written as N+1

‘P(x, t) = :lim

[~

L~(k)‘P(y, t):

Ne = x — y = length of lattice = L where L~(k)= exp(—ieQ(y

(n — l)e, t; k)). Then define

~

N+1

TL(k)=

—

i:i L~(k)=

~

On the lattice, the trace of the scattering data is the transfer matrix operator: e.g., for the Baxter model [33] t(V)=tr

[T

>

w1d1(n)oy—A+D.

(4.10)

n’l j1

The Baxter model or symmetric eight vertex model is a two-dimensional theory of classical statistical mechanics. A comprehensive treatment of an infinite-dimensional algebra of commuting charges,

18

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

quantum lattice integrability, the transfer matrix method and Kramers—Wannier duality transformations can be made in this model. The partition function is ~ exp(- ~ ciNi).

(4.11)

allN

1

j=1

Each site (n0, n1) of a rectangular lattice has an associated energy ~ depending on the vertex configuration and ~ is the number of vertices of type j in an allowed set of configurations. The site configurations are shown in fig. 1, and the weights e1 of the first four are set equal to those of the last four respectively (hence the symmetric 8-vertex model). If we define a,,,, = (a nol, a n~2,- . . fl~N) where a,,.,.,, = ±1 represents a single vertical lattice link, the partition function Z can be expanded in terms of direct-product vector states ,

a,,,,)

a nol) ®~

=

®

~~

IJ5

where

a~

or

=

For periodic boundary conditions,

z=

(aiITla2Xa2IT~as)”~‘~ITI~~)

~ al-am

=

Tr

(4.12)

f~m

where I’, the transfer matrix, is given by eq. (4.10) and w1 are functions of e1. The transfer matrix T is the unifying concept between statistical mechanics in d dimensions and quantum field theory on the lattice in d — 1 dimensions. If there is some quantum Hamiltonian H which commutes with T, then diagonalizing T is equivalent to solving the quantum theory. On the other hand, since Z = Tr T, this solves Z. Baxter found a parameterization of w3 —~ ~ V, I, such that for fixed and 1, ~‘

[i’(V),

TI’(V’)] =

0.

(4.13)

This one-parameter family of commuting transfer matrices led to the diagonalization of T, a set of conserved commuting charges 0,~,and the diagonalization of ~ since In T(V)

=

~fl±~

= —

J3 sn(2~1)

and 00

-1

W

+

~

~—

~

constant

~

(4.14)

-~

—~

~

—a--

a

*

Fig. 1. Configurations for the symmetric 8-vertex model.

.*_

~

-~

~

19

L. Dolan, Kac—Moody algebras and exact solvability in hadmnic physics

where Jaó~~(fl)&(n+1)

~ n1

a=1

and J1/J3 = cn(2~,I);

J2/J3 =

~

[On,Om]~r0.

sn(2~I).

Then (4.15)

4.2. Construction of an infinite set of commuting charges from Kramers—Wannier self-duality Although the inverse scattering method has been associated with the transfer-matrix formalism of statistical mechanics, we cannot apply either of these techniques to the gluon theory because we have no clue as to what is the relevant linear problem, such as the form of the commuting charges. We now give a theorem [22]which uses another property of the spin models, i.e. Kramers—Wannier self-duality, to construct an infinite commuting algebra. Consider any quantum Hamiltonian H = KB + [‘B, where B is the dual of B such that B = B, B’A = B + A and BA = BA. F and K are coupling constants. If [B, [B, [B, ~j] = 16[B, B], then for 02,, = K(W2,, — 1’~~2~_2) + F(1’~í2,,— W2,,_2) where W2,,+2 = —~[B, [B, W2,,1] — W2,,, W0 B there exists a set of conserved commuting charges: -

102,,, I~]= 0,

[Q2n,

Q2m] =

0.

(4.16)

Notice the power of this result. It does not refer to 1) dimension of space-time 2) lattice or continuum local or loop space. To make it more familiar, some examples are: a) Ising model: —

B= ~â3(n)â3(n+1). The fundamental variable is defined on a lattice site, see fig. 2. The eigenvalues take on 2 discrete values ±1.The dual transformation is o3(n) = êi(1).

n~12

N

Fig. 2. One-dimensional lattice.

.

tJ-i(n) and

20

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

to make sure B = B then ó~1(n)= d-3(n) ô-3(n

+

1). Then

B=

‘i’

F\

F~~_ 1

Under this dual transformation, the theory for weak coupling F/K 4 1 is related to the same theory for strong coupling 1I(FIK) 4 1. It is a powerful non-perturbative tool, since information about weak coupling can be used to calculate in the strong sector. In the underlying 2-dimensional Ising partition function, this operator transformation corresponds to the original Kramers—Wannier transformation which they used to calculate the critical temperature. b) X — Z model: B= ~á3(n)â3(n+l). The dual transformation is ö3(n) = d-1(n); ó1(n) = ê3(n). Then

In both cases [B, [B, [B, A]]] = 16[B, A] and the commuting charges constructed from the theorem are those generated by a specific choice of Baxter’s transfer matrix. 4.3. NLuM S-matrix from the infinite set of Lüscher—Pohlmeyer non-local charges From the affine symmetry transformations, (5.6) an associated linear problem can be constructed. For example, for ‘/i~l”~~”~’,

~~°~=1

and on-shell =

—e,.~(a~ +

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physicS

21

then = [~.(1)

T]

and 1

The linear problem can be written as —(1 +

=

l2)~ (A,. +

le,.~A~)t/i.

(4.17)

In Lax form, o&/i=—M~, so the consistency condition for eq. (4.17) is [L,M]=Ii.

(4.18)

Here 1 M=—1--~-~(A0+ IA1) and eq. (4.18) is valid for ~A

,.

_~(—1.~

—

,.~g ,.g — When one attempts to solve the initial value problem (for example for the NLa-M) with the inverse scattering method, the scattering data remain constant for all time. The model is not yet exactly integrated classically or quantum mechanically, i.e. the correct linear problem has not yet been found [34]. Nevertheless, in the quantum theory, some progress has been made using a quantum version of these non-local charges. The exact on-shell matrix can be calculated using the non-local charges [16, 17]. The n-particle S-matrix factorizes into a product of 2-particle S-matrices. There is no particle production but the S-matrix is non-trivial due to time delay (phase shift) and interchange of internal symmetry quantum numbers. 2Wa’Pb with ~~‘_ 1 ‘P = 1) carry isospin c = a and For example: the NLo-M fields ‘Pa (from g~ = &~b have rapidity 0 = ln((p°+ pt)/m). ,.—

—

22

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

For elastic scattering, the non-local conservation laws Q”~are used to show that (0~c~, 0~c~OUT~Q~01c1, 02c2IN) 0~c~ OUT(01c1, 02c2 IN)

=

~

=

(0~c~, 0~c~ OUTI0ic~’,02c’~IN)(Q~LN)c;e~cc,

(4.19)

which are equivalent to the factorization equations after some algebra [16].

5.

The principal chiral models

51. Non-local currents as Noether currents The Lagrangian density for the general class of principal chiral models is ~(x)

tr 8.~g(x)~,.g’(x).

=

(5.1)

The matrix field g(x) is an element of some t Lie group generated by a finite parameter algebra G. = g~)or G = O(N)(gT = g’). Here g(x) are N X N Simple examples correspond to G = U(N)(g Hermitian or orthogonal matrices respectively, and the Lagrangian density is invariant under the finite parameter global transformation G ® G: g(x)—* g 1 g(x) g2.

(5.2)

These are sometimes called matrix models. The8abfield2 4,a(X) g(x) may also be restricted to a non-singular 4,b(X) with 4,a(X) = 1, for real fields subclass of matrices group. If g~b(x) 4, T = g~=in gsome and (5.1) is the 0(N) =non-linear sigma model (NLa-M): 0(x), then g ~‘=i

~,.4,a(x)a,.4,a(x).

(5.3)

Since 4~,is a vector under the internal global “isospin” symmetry G, (5.3) is called a vector model. For gab(x) = ~3ab —2 4,~(x)4,b(X) with ~ 4,~(x)4,a(X) = 1, then (5.1) is the CPN sigma model: ~(x)

=

~

((4,~a,.4,~)2+ ~

ä,.4,a).

(5.4)

Other interesting subclasses are discussed in ref. [28]. For compact Lie groups, ft 0,.g = A,. is an element of the Lie algebra G, i.e. if T” are the N X N generators of G, then A,. = A~(x)T”. The equations of motion for (5.1) are a,.(g5 ~,.g) = 0.

(5.5)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

23

The following calculations are performed in Euclidean space, but can be extended without difficulty to Minkowski space. The model (5.1) admits an infinite set of global infinitesimal symmetry transformations [12] whose Noether charges in Minkowski space in the case of the NLa-M are the LUscher—Pohlmeyer non-local charges [13].These transformations for n = 0, 1, 2,... are =

_gfl(fl)

A °‘(x, t)=

=

J

—gA~:’~p~ = paz.1~g ~I)g

dy D0A ~(y,

J

t)

dy (80A ~(y, t) + [Ao(y, t), A

A’°~== T = T”p~

A”~=[x~ T]=

(y, t)])

(5.7)

(5.8a)

dyAo(y, t), T],

[J

(5.6)

etc.

(5.8b)

Here Pa are infinitesimal constant parameters and Ta are example, for 1~ = the ~ matrix generators where D,.of G.3,.For + [A,.,. the1)0(3) = e~.On-shell, (5.7) is o,.A~ TheNLcrM, ~i~g are(T”)~ symmetries of the equations of motion: if g is a solution to (5.5), so is g + z.l~gto first order in Pa. Since g—~.g+.~.1g implies g ~—*g1—g~ ~igg’, then 3,.(g~a,.g)-+ 3,.((g’

—

g1 z.lggt) 9,.(g + dig)).

(5.9)

For z1~g= _gfl(fl), ~~)(3,.(g_l 0,.g)) =

Since A,.

=

f13,.g,

—

~

=

0.

(5.10)

then e,.

1.D,.D1,A’~”~’= ~e,.1,[F,.1., A’~”~] = 0 where F,.1. = 3,.A1. — 3~,A,.+ [A,., A1.] = 0. Therefore, if a,.D,.A~”~= 0 for n = N, then D,.A~”~= e,.1.3~4”~’~ and a,.D,.A~’~= ~ = 0. Also 3,.D,.A~°~ = 0. 2) The z.l°”galso shift .9~(x)by a total divergence without using the equations of motion. For ~i~”~g= _gfl(fl), (5.11) For n

=

1, 2 we find explicitly, without 3,.A,. =

3,.(~e,.1.tr((~[31.X, x] + A1.)T))

=

3,.(~e,.1.tr(([31.~,~

=

0, that (5.12)

and +

k[E31.x, x]~xl + [A1.,~])T))

(5.13)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

24

where

x(x, t)~

J

(5.14)

dyAo(y, t)

and

~

J

t)

t) + ~[Ao(y,t), dy (8~x(y~

x(y, t)]),

(5.15)

since from (5.7) and (5.15), T] + ~[x,[,y,T]]

A~2~ = [,~2),

(5.16)

-

The shift L1~”~5~ for general n is discussed in section 5.3. Using the equations of motion, one finds the change in ~ for ~

=

—gA” to be

(5.17)

3,.~trA,.A’~~~’.

~(n)~~

Equate (5.17) for n = 1, 2 with (5.12) and (5.13) respectively, to derive the conserved Noether currents [26]: 0,

Pao,.ktr(o(n)Ta) =

(5.18)

where =

[A,., xI

—

e,. 1.A1.

—

~e,.1.[31.~, xI

2’]+ ~[A,., xl~ xl

=

—

e,.

(5. 19a)

t2i + ~[0

1.([31.~, x

[A,., x~

1.~,xl~ x] + [A1.,x]).

(5.19b)

Since A,. is an element of the Lie algebra G, the matrix currents J~I) are also conserved: 3J~=0.

(5.20)

The matrix charges ~ space):

Q(l) =

2) = Q(

J J

dy {—A1(y,

=

f dy~~(y,t) formed from

t) +

~[Ao(y,t), x(y,

dy{—[A 1(y, r), ;~ (y,

t)]

—

(5.19) are the non-local charges (in Euclidean

t)l}

~[~y(y, t), [A0(y, t), x(y~t)]]}

(5.21a)

(5.21b)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

25

and

dt

— —

“

0

(5.22)

-

As mentioned in section 2.1, the current conservation (5.20) is automatically assured from the Noether construction. The charge conservation (5.22) follows from the boundary condition for on-shell fields: A,.(±oc,t) =

0.

(5.23)

Without this boundary condition the charges are not conserved. We remark here that this boundary condition (5.23) is preserved by the transformations ~ since as g g — gfl~~), then A,. A,. — D,.A”~so -+

= —D,.A”°= —3,.A’~~—[A,.,A(n)] =

e,.1.D1.A “~

—

[A,., A~~)].

(5.24)

Therefore using ~ = —e,.1.D1.A’~~~ which is true on-shell, one can easily show that for any integer n 0 each term in (5.24) contains at least one A,., so that if we assume A,.(±ca,t) = 0, then z~A,.(±ao,t) = 0 also. 5.2. The affine generators For current conservation (5.20), the fields g(x, t) must satisfy 9,.A,. = 0. The expressions ~1(~~)g in (5.6)—(5.8), however, are defined off-shell and shift the Lagrangian density by a total divergence for arbitrary field configurations. (This was shown for n = 1, 2 above and will be proved for integer n > 2 in section 5.3 below.) For these infinitesimal transformations ii~°g,we now check the Lie integrability condition to see if they generate finite values which form a group. This calculation will also provide the structure constants of the corresponding Lie algebra. The generators Me,, are constructed from the non-local infinitesimal transformations in standard fashion [29].Following (2.21), 2x ~ ~g(x) ~g(x)~ M”,,

=

(5.25)

d

— J

We compute the commutators [M~, Mt,]. If the integrability condition (2.25) is satisfied, the structure constants are defined in general as f”,,~such that Frt,ra

11 1 fl,

A,fb

1

mJ

—

—

~abc AA’C I .

/ nm11

From (5.6)—(5.8) and (5.25), we find explicitly that [Ms, M~]= ~

(5.27)

26

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

and [Mr,

Mb~l

=

(5.28)

Ca&MCn+i

where Cabc are the structure constants of 0: [Ta, Tb] follows for n, m = 0, 1,2, that - .

1M0a,

=

CabcTC.

From (5.28) and the Jacobi identity it

- , ~

(5.29)

Mbm] = CabcMCn±m.

This is the infinite parameter algebra G ® C[t]. The proof (by induction) of (5.27), (5.28) and (5.29) is as follows. From (5.6)—(5.8), (5.25), (2.23) and (2.25), to order Pa~b, [Ma t n, M~,lPaUb=

J

d~x{~i¶~‘°[g(x)+ i ~°g(x)] —

~

~‘~g(x) ~i~[g(x) —

+

i ~“‘~g(x)] + zi ~“~g(x)} (5.30a)

=

—

—

J

d2x g{ub[A $~(g— gp aA a) ~

—

A]

—

[~ ~(g—

Pa”a

grbA ~2))

A aJ (~)1

(5.30b)

~ ~,m)]} ~

PaOb[ht (fl)

—

First, we now prove (5.28): assume [Mr, Mt,] = Since as

g g —~

for

CabcM~n±l,

m

=

(5.31)

M.

gil ~,m)O.b then

—

x

x—J

~

where x =

—

J

x dyDoA~

=

—

J

dy

~ 1fl~m±l)

=

(5.32)

_A$,m±l)

and then A~(g—g’°)—A~V=

o.b[~i~m)x,Ta]= .[A~m±l) Ta]O.b.

(5.33)

So, from (5.30b) and (5.31), ~

Tal_pa[A~),A~m)]=

CacpaA~m±l).

(5.34)

27

L. Dolan, Kac—Moody algebras andexact solvability in hadronic physics

In general, since A ~

=

f~dy D0A S,m’ and z~~A0

—D0A ~

=

then

x

A ~,m+l)(g

—

gpafl ~“~‘

=

a)

dy {D0(A ~,m)(g— gpafl ~t)))

J

3iX

=

J

[D0A~Pa, A ~,m)]}.

(5.35)

A

Then using (5.35), (5.34), (5.32) and

Pa~b[M~,M~+1]

—

0, we have

2xg(x){~~[A~l)(g — gpaA d

~)-Are]—

pa[A~(g

—

g~4~m+l))_ A

~)]

_Pa~b[A~,1~m~l}

~g(x)

x

=

—

d2x g(x)~b

J

+ Pa0b[A

~,m+2)

Ta]

J —

dy (D

0[A ~~~)(g — gpail ~) — A ~] A ~,m+1)]} ~g(x)

PatTb[A,, x

=—J

1~T’—C aJ abc A(m±l)\ c j ,

d2xg(x)ubp4(J dyDo(—[A~

+ [A

D

~,

0A ~,m)])+ [A ~,m+2) Ta] — [A ~

A~m+h]}~g(x)

x

2x g(x)u~pa — Ca~cJD =

[D0A~Pa, A tm)])

d

—

J

—

[A

0A ~~~±1) + 1),

[3’x,7”]] + [,~,

J

dy ([—D0A

Ta], D0A ~“~])+ [A ~m+2) Ta]

Ta]

~l),

—

[A (1) A (m+1)1 a,

x

=

— J

d2x g(x) trbp~{_Ca~A~n±2L

a —[A(1)

=

‘

J

1 ~

11&g(x)

n

dy (3 1[A

~1),

[~,

Ta]]

A~m+1)])}~g(x) ~

clbpaCabc

J

2x g(x) A ~“‘~2~(x) ~g(x) d

(5.36)

or equivalently, [Mr,

M~,I+ 1] = CQ.t,CMCnI+2.

(5.37)

28

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

Also, [Mr, M~]= CabcM~.Note that (5.31) is true for M A~(g—gpaAa(1)) — A (1) b — =

=

1 because (5.34) is true for M

=

1: (5.38a)

T~’]p~

~ {[A

~,

T”]

+

[A (1) A (1)1 a b ] — CabcA (2)1 c JPa-

(5.38b)

Eq. (5.38a) follows from (5.33), and (5.38b) follows from (5.16): —[A

a,1 J (2) ~rb1

+

[A

[xt2~Tb], T”] ~[x,[x,Tall, T”] + ~[x~ Ix, Tbl], T~l — Cabe([Xt2~, TC] + ~[x~ [x~T’~]])+ [x’ 7”], [x~Tb]] (5.39)

T~]= —[x~2~, T~]Ti’] +

~,

=

—

-

Therefore, from (5.37), we have proved (5.28). The proof of (5.27) is similar. Now assume [Ma,

Mbm] =

CabcM~n,

for

m

(5.40)

M.

=

Then, from (5.30b), (5.41)

~

We find from (5.41) that 7’~)

Pa~b[MO,Mrn±l]_Jd2xg(x){Ub[A~)(ggPaT)A~)]Pa(T

Pa~b[T,A~1~1} x

=

- J

d2x g(x){crb

J

dy (Do[A~(g

Pa~b[T,A~t)

=

—

J

~1

—

gp~T”)—A~]- [D

0T”pa, A ~])

~ ~ig(x)

x

d~xg(x) ~bPa

{ J

dy (D0([T”, A sm)] — ~

~)

-

[D0Ta, A tm)])

1~ ~i~g(x) —[7”’ A~~~1 x

dyDoA(m~ ~ C J~g(x)

— Jd2xg(x)crbpa

=

C~crbP~ d2x~(x)A~(x)~g(~)

J

{_Cabc

J

=

(5.42)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

29

or equivalently, [Ms, M~,,÷1] = C,,,.,,,M~,,+1.

(5.43)

Also, [Ms, M~]= Ca&MCO and [Ms, M~]= ~ —

[T”, Ti’]

=

CabCTC

since

Ca&A ~

=

(5.44a)

and A~(g— gp~Ta)_A ~

—

pa[Ta, A ~)]

Tb] +

[T”,A ~i)pa

=

—([A

=

—([Ix~ T”], T”] + [Ta, Ix~T”]])p~

=

—Ix~[T”, TtI]]p~

~,1)

=

(5.44b)

~C~A~)pa.

Therefore, from (5.43), we have proved (5.27). To prove (5.29), assume for n = N and [M, Mbm] = CaI,cMCn+m.

m

=

M that (5.45)

Eq. (5.45) is true for n = 1 and all integers m 0 (from (5.28)). Therefore, [M”,,, M~,+1]= C~M~,+,,+1 for n = 1 and all integers m 0. Now we prove, using the Jacobi identity, and given (5.45), that FrtIa [AVI

n+1,

AA’bl_r’ Ts,fc 1 mJ — ‘-abc ,,+m+1~

C~~[M~+1, Mt,]

=

[M~, M~],Mr,,]

=

[Mt,, Mfl, M~]+ [M~, Ms,], M~]

= Cj,~ja[M~n+t,M~]+ Cct,a[M~+m, Mfl =

(CMaCacg — CbcaCwjg)M~+m+l

—c’

(5.46a)

AA’g 1~1 n+m+1

I.’

— ‘~~~dca’’bag

—r r —

cda

Iufg

abg’

,,+m+1 -

Eqs. (5.46a, c) follow from the antisymmetry of Ca~ in the first two indices. (5.46b) holds since the structure constants are also generators of the algebra: if Tb = ~ then FTd , TC1 I Jbg [1

— 1.’ 1.’ — ‘—‘bda’-’acg

—

1.’ 1.’adg ‘—bca

— (‘ —

tTa\ )bg dca~

— —

‘-dca’-bag.

That is to say, since C,,.,,~ is defined by [T”, TC] = C&aT”, the Jacobi identity, which is satisfied by this bracket, implies [Ta, Tdl, Tb] = C&aCabgT~= =

[r’,

T”], TC] + [T”,

(Co.baCacg + CbcaCadg)T5.

TC], T”] (5.48)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

30

Therefore (5.29) is proved. In conclusion of this section, eqs. (5.27), (5.28) and (5.29) are an explicit construction of the subalgebra 0 ® C[t] of the affine Kac—Moody algebras. The M”~defined by (5.25), (5.6) are of course much less trivial than M” = T~0 t~.Since the infinitesimal transformations (5.6) form an algebra, they generate finite transformations which form a group. In comparison with (2.29), one such law is given by the formal expression

g’(y)=exp{_(pa Jd2xg(x)A~(x)~))}g(y).

(5.49)

For global isospin symmetry the analogous formula is

J

~(y) = exp{—(pa

d~xg(x)T” ~g(x))}~~

(5.50)

which due to the linearity of the transformation is = g(y)

exp(—paT”).

(5.51)

53. Symmetry of ~E(x) That LI ~g shifts .2~(x)by a total divergence for all integers transformations (5.6) shift the Lagrangian density by ..9?(x)]

7 Pa[M”n,

paLI~’~$~

Therefore, to first order in Pa0bCabe~~ (n+1)

$f~(x)=

PaUb,

—

n

0 is now proved by induction. The

=

and from (5.28) and Jacobi’s identity, we have

Cabe [M’~,,”’ ‘~,

~[(x)]p~o-b

Mr], ~(x)]

=

—[M~,

=

{[M~, LI S,’~(x)]— [M~,°,LI ~‘~/](x)]}PaO~b.

PaUb

(5.52)

From (5.12), LI ~1/?(x) = ~3,.e,. 1.tr((A1. + ~[31.~, x])Tb).

(5.53)

Therefore, from (5.52) ~

LI ~~(x)]

~

=

~3,.e,.

tr((D1.A (n) + ~[a~.A (n±1) xl + ~

=

1.tr(([A1., A ~]

+

~

A

~1)Tb).

t~])Tb)

A ~,“~ (5.54)

That is, (5.54) is also a total divergence. We assume that LI ~‘~(x) can be written as a total divergence off-shell: LI~1(x)= 3,.K~([g],x).

(5.55)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

(This is true for n

=

31

1.) Then

[Mv, LI~°..~’(x)] = —3,.K,.~’([g gA~~], x)+ 3,.K~”([g], x). —

(5.56)

So (5.56) is also a total divergence and from (5.52), then so is LI “~.,9~’(x). 54. A generating function A compact way to discuss the transformations ziI~fl)gof sections 5.1—5.3 in terms of a generating by the function has been given in refs. [35] and [36]. Define the generating function S(t) of the At~~ power series, S(t) =

~

Then, since 3

(5.57)

t’°, 1A~~~”= D0A

3 1S— t 3~S= t[A0, 5].

(5.58)

Since S(t) is an element of the algebra, a consistent form for it is 1(t) S(t) = U(t) T U where

(5.59)

U(3 1— t30)U~= —tA0 or (5.60)

31U= t(30-I-A0)U.

From (5.57), LIg= ~LI~”~gt”= —gS.

(5.61)

This infinitesimal transformation shifts ~(x) without use of the equations of motion by LI~=~trA,.3,.S =

~3,.e,.1.tr(tA,~S+ (t

+

~)

U_13,.UT).

Eq. (5.62) is derived using (5.59) and (5.60).

(5.62)

32

L. Do/an, Kac—Moody algebras and exact solvability in hadronic physics

The commutators of the generators of the algebra now can he rederived by considering Ma (t) =

—

J

d2x LI ~g(x) ~g(x)

(5.63)

where LI ~g

—g Sa(t) = —g ~

(5.64)

A ¶~t”.

n = t)

Then [Ma(t),

Mb(r)l

=

=

J

d2x g(x)[S~(t), Sb(r)l ~g(x)

~abc

-

d~x(t~t~~

J

With the expansion of M”(t) [M”n, Mbn] = CabCMC,,+m

=

~

J

d2x g(x) (LI ~Sb(r)LI ~Sa(t))

~g(x) (5.65)

~

M~ta,then by equating powers of t”r’~, eq. (5.65) reduces to

(5.66)

.

The derivation of (5.65) follows from LI~Sb(r)~ ~ r~(A~(g—gS~(t))A~)

(5.67)

n=0

where this satisfies the differential equation for the variation of (5.58): 0 1LI ‘~Sb(r)= t{30LI ~Sb(r) [D0Sa(t), Sbfr)l + ~ —

LI ‘aSb(T)1}

(5.68a)

-

The solution to (5.68) is LI~Sb(r)= —r [Sa(t)_S~(r)Sb(r)]

—

~

([Sa(t), Sb(r)]

—

CabcSc(r))

(5.68b)

.

Eq. (5.68b) has the appropriate boundary conditions which follow from the definition of A Subsequently, different sets of infinitesimal transformations, also defined off-shell, were considered in refs. [36] and [37]. Both these sets become equivalent to zi~”~g on-shell. Some of the calculations in the literature are performed in Minkowski space. Without exception, these can be straightforwardly changed for a Euclidean metric. The transformations of [37] are shown also to close GOC[t]. The usefulness of all these different sets of off-shell transformations is to show that there is no unique way to go off-shell. ~.

L. Dolan, Kac—Moody algebras and exact solvability in had ronic physics 55

33

An extended algebra

In the derivation of zi(fl)g = —gA~° via a Dirac bracket {Q~,g(x)} which will be discussed in section 5.6, we were naturally led to a new second set of symmetry transformations corresponding to “left-multiplications” [26]. In fact, already from (5.2), for g2 = ~ gi = I, then LI ~g

=

gT”p~

—

whereas for g2 = I, g~= e~°~ then =

(5.69)

~

We define the new set to be A~g=A~”~g forn=0,1,2,...,cc.

(5.70)

Here

~

t)=

—

dyboA~(y, t)

J

J

(30A~+[A0,A~])

dy

(5.71)

and

A0 gö0g~ ~4a(0)~ T~

(5.72) where [,f~a 1 = g so A

~f’b]

=

CabCTC.

(5.73)

In the NLoM, g 0 = A0, and the Noether charges associated with A~gare proportional to those associated with LI ~°g thus leadingto no new information about constantsof the motion. For example, for LI ~)g= [~, Ta]g where 3~= —A0, then without the equations of motion, 0]

= =

~ tr A,. ~,. —

~

Ix~T

tr((A

1. + ~[31.k, k]Yt”).

(5.74)

With the equations of motion, =

since 3,.A,.

=

~t9,.

tr([A,.,

,~]a),

0 implies 3,.A,.

=

0.

(5.75)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

34

Therefore the Noether current is =

[A,.,

~]

+

e,.~,A1.+ E,.1.2[3vx,

For the NLo-M, g = g~,A0 = =

—[A,.,

A0,

(5.76)

~].

then ~ =

—x and (5.76) becomes

xl + ç~.A1.+ ~E,.1.[31.x, xl =

(5.77)

_5(,.1).

Compare (5.77) with (5.19a). Furthermore, in the general chiral models and for T” = T~,(5.29) can be rederived for LI~”~g, so that even in the case when the second set of charges is different, they just form another G®C[tl among themselves. The commutation relations between the two sets is in general complicated giving rise to symmetry transformations which on-shell differ from LI~’gand ~ by T” replaced with another constant but g-dependent generators of 0 and no new conserved charges result. Clearly what is fundamental to the physics of the chiral charges is the 0 0 C[t] structure. For mathematical purists, we note that by a clever choice -

=

-

g(—x, t)Tag~(~’~, t)

(5.78)

constant

(5.79)

where g(—c~,

t) =

and a redefinition

~ ~~1)g

for

n

=

1,2,...

for

n

=

—1, —2,.

(5.80a) .

(5.8Db)

-

LI ~)g+ A~g

(5.80c)

then, on-shell the two sets can be combined to give for all integers n and

m;

(5.81)

[M~, M~1]= CabcM~+m.

In terms of generating functions,

LIg = —gUTU~ =

Ag

=

gWTW~=

—g ~

t”A~”~

t~A~g

(5.82a)

(5.82b)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

35

Here U= t(0

~1

0+A0)U

U=

~

~

~

oix(n+D

=

(3~+ ~

D1 W

—

t30 W

=

I

(5.83) for n

=

0, 1,

- .

.

and

w

=

(5.84a)

~ ~

=

(5.84b)

1~(x, t) —g~(x,t) =

x~~

J

dy g(y, t) ~

t)

for n

=

1,2,...

(5.84c)

so (3~+ Ai)x~~~ = ~oxt~’,

(5.85)

x~°(x, t)

(5.86)

and g~(x,t) g(—~,t)

so x~(_c0,t) = 1.

(5.87)

Therefore =

g(—~,t) Tg1(—cc, t)

(5.88)

This is the origin of (5.78) [38]. On-shell,

o

0U= —t(31+A1)U

(5.89a)

and D0W = ta1W. We then use these generating functions to compute the algebra (5.81). Define 1(t’) = — gUFTbUF_I. LI~g= g U(t’) Tb U —

(5.89b) (5.90)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

36

Then Lla(g + LI~g)—LIag

—

LI~(g+ iag)±i~g

=g(U1TbU~lUTaU~_U[V, —

UTa U’ U’T”U’~+ U’[Va,

T”]U 7’b]

U’~)

(5.91a)

{tLI~g t’LI~g}.

=

(5.91b)

—

Here Va,(U1~UTaUtU~

T”)

V~=~t(U1UfTbU~1U_

Tb).

and

Similarly, we compute Aa(g +

A~g)-A~g- A~(g+ A~g)+A~g (~-~-~) ~~bC(t’ACgtA~g)

LIa(g +

A~g) LIag

=

-

(5.92)

and —

—

A~(g+ LI~g)+ A~g =

C~b~(1l

+ tt’LICg).

(5.93)

The on-shell expressions (5.89) were used in the derivation of (5.93). By equating powers of t”t” in each of (5.91), (5.92) and (5.93), one derives the affine algebra (5.81). In light-cone coordinates, a different off-shell set of transformations for all integers n was defined in ref. [39]. 0ff-shell these transformations are not at fixed time. But they provide a mathematical mechanism to give the algebra of (5.81) off-shell. 56. Poisson brackets, canonical transformations The transformations (5.6) are given for the general class of chiral models. Their form was derived [26] from the following considerations restricted case of 0(N) NLcrM. In this theory gab = 6ab 24~a~bwhere /~ is constrained ~ of the = 1, 50 g’ = g. In order to discuss the Noether charge as a generator of the infinitesimal transformation (see section 2), Poisson brackets must be defined. But since there~are constraints, one must use the modified Dirac brackets for consistency. The brackets are defined in the following way: —

=

H

J

~3A~a

3,.4~a —

A (4~2—

1)

dx{~HaHa+~(3iq~a)2+A(çb21)}.

(5.94a)

(5.94b)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

37

Treat A(x) as a canonical field. Then the ordinary Poisson brackets would be {11a(x, t), 4~~(y, t)}

~ôabô(X

=

—

y)

(5.95a)

{HA(x, t), A(y, t)} = —S(x — y)

(5.95b)

where Ha

=

0oçba

(5.95c)

HA

=

&~f/~n9oA = 0.

and (5.95d)

But (5.95d) is not consistent with (5.95b), so we write HA

=-

0 and list the first constraint:

Xj=HA(x).

(5.96)

Then define

HT = H +

J

dy v(y)HA(y).

(5.97)

Now from (5.95b) {HA(x), HT} = Again since HA

=-

(~2(x)..1).

(5.98)

0, the second constraint is

X2= —(~(x)—1).

(5.99)

Then from (5.95a), {(4~2(x)— 1), HT} =

(5.100)

çba(x)[Ia(x)

so =

(5.101)

4~a(X)11a(X)

and 2(x) {4)a(X)11a(1), HT}

=

_ir2(x) + 2A(x)4i

&(x) 3~a(x)

(5.102)

38

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

so =

—

~.2

2A~2

(5.103)

&(3i)2&

—

+

Finally, {X 2v + 20i(Hac9i4~a) (8 2Ha4)a is a consistency condition for v. 4, HT} = 24 The Dirac brackets are set up so that 1) {IJA (x, t), A (y, t)}D

0

=

(5.104)

which unlike (5.95b) is consistent with (5.95d). The Dirac brackets are defined from the constraints X,:

{~(x.t), ~(y,

t)}

{~(x,t), ~(y, t)}

0

-

J

dz dz’ ({a(x, t), X1(z, t)} C~(z,z’){~(z’,t), ~(y, t)}) (5.105)

where C,,’ is the inverse of the matrix of constraints C’~~: C~1(z,z’) = {X,(z, t), X1(z’, t)}. Also f dz’ C11(z, z’) C~(z’,z”) {Ha(X, t), llb(Y, {q)a(x, t), ~b(Y,

=

t)}D

t)}D =

4~b(Y,t)}D

=

~(z — z”)

(5.106) We find

&k.

(eI~aHb+ ç~Ha)6(x

—

y)

0 (‘~ab

+ 4’a4~b)~(x

—

y)

{H~(x,t), {HA(x, y), A(y, t)}D

=

(x, t), A (y, t)}D

0

{A

=

0

etc. Therefore in

(5.107) gab

notation, where

{A,.~’(x),gra(y)}D

=

gab = ~ab — 24’a~,bb,~2

—4{(4a(x) 3,.4’b(x) — 4’b(x)

=

1, (A,.)~,=

t9,.4’a(X)),

2(4~aô,.~b— 4~b0,.4)a), 50

~c(Y)~a(y)},

(5.108)

i.e. {A~”(x),gcd(y)} 0

=

0 and 4)a4)Ct~bd— 4~b4~d~~aC — ~b~/’~c’~ad)

{A~”(x),g~(y)}~== 4—23(x ~(x ——y)y)(&4aö,,~ + (g,s.’SbC + g~~’5bd — gbaôaC

—

g&’5aa).

(5.109)

L. Dolan,

Kac—Moody algebras and exact solvability in hadronic physics

Therefore, for the 0(3) NLuM, (T”)b~= {Q~,g(y)}~= ~

=

J

Ebac = —(T”)Cb,

39

and

dx {tr Ta(A(x)+ ~[A0(x),x(x)]), g(y)}~

—~[,~(y), T”], g(y)]

(5.llOa) (5.lIOb)

where ~(y)= x(~) — 1Q~°>, Q~) f~ dx Ao(x, t). Eq. (5.llOa) is eq. (2.12) of ref. [26]. Thus ,~g,and not LI ~g = ~ T~] is the transformation generated by the Noether charge Q~P.(Q~is the trace of ((eq. 5.21a) times T”).) Note that both ~ and LI ~g give rise to the same Noether charge Q~. The condition that the (5.llOa) LIg = [A, g] is a symmetry of the equations of motion is =

LIA + [A,,, 0,,A] = 0

(5.llla)

and

LIlA

+

[g3,,g~, 3,.A] = 0.

(5.lllb)

Eq. (5.llla) is the condition when zig = —gil and (5.lllb) corresponds to LIg = Ag. Equations (5.111) are equivalent when g~ = g. For the general chiral models, we chose the transformation zl~”g = with ~ given in (5.7) and (5.8) and found that 1) it is a symmetry of the solutions 2) it shifts ..~‘(x)by a total divergence and can thus be used to construct conserved Noether currents and 3) it can be simply described by the algebra G®C[t]. Of course, since in the particular case of the 0(3) NLuM, the LI”~g —{Q”, g},~,it is not surprising that {g(x)—zig(x),g(y)—z.I~g(y)}D 0.

(5.112)

Therefore in general, the charge algebra JQ(n) 1

a,

(‘nil

b

— ;(n)(mXl)

fDJabc

(1) C

may be more complicated than the simple algebra of LI ~°g. Since constraints differ for various choices of g(x), it is impossible to discuss the charge algebra, i.e. the bracket definition, in a general way. For example, a careful treatment of the 0(3) NLrM model above gives {Qa, g} = —~[x~ T], g] whereas a standard Poisson bracket for g GL(N, C) gives {Qa, g} = —[xe T]g. See ref. [27]. An interesting calculation would be to compute {Qa, Qb}D for the 0(3) NLrM using the exact definitions (5.107)—(5.110). The chiral models have so far been relevant via a classical connection to the functional Yang—Mills fields and the local self-dual Yang—Mills fields. It is in the effort to maintain a

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

40

g(x) which is then replaced by = P exp(~ A d~)or J(x) = D(x) D’(x) in the study of the symmetry that an emphasis is made on checking the algebra of M”~rather than Q~. Clearly, a discussion of the symmetry in the quantum theory must be carried out using quantum brackets. Calculation of the quantum charge algebra also involves a regularization procedure for the charges [13, 16]. general matrix

~i[~]

6. Loop space Yang—Mills theory 6.1. Yang—Mills functional formulation and the chiral models

The first hint that strong interaction gauge theory might contain a hidden symmetry came from an observation by Polyakov [18] that a functional formulation of the non-Abelian theory was similar to the local equations of motion of the two-dimensional principal chiral models. In this approach, the role of the fundamental local variables A~(x)is taken on by the path dependent field ~‘[~]= P exp(~A d~)= P exp{~ds ~,. (s) A,. (~(s))},which is an element of the holonomy group, i.e. a path-dependent element of the gauge group. Here, we consider SU(2) and A,. = Aa,.o.~I2iwhere o~”are the Pauli matrices. The line integration is done around a cIu~~°d path in four-dimensional Euclidean space parameterized by four functions ~,(s), where 0< s ~ I and ~,.(0)= ,(1) = x,.. The path ordering P is defined by N+l

P exp~A. d~)= liii P exp[

=

(~

~

-

c;’) A~(~’)}

lirn exp{(~1— x)A(x)} exp{(~2— ~‘) A(~)}-- . exp{(x — ~) A(~)} (6.1)

In (6.1), the path described by ,(s) has been made discrete in order to exhibit that path ordering is necessary to express the product of exponentials of non-commuting classical matrix fields A,,(x) as the exponential of the sum of these fields. The gauge-covariant functional derivative ~i/~,,(s) is given [40]as follows:

+

6~]- ~[fl =

=

8x,. [A,.(x), ~[fl] +

-&ç[A,,(x),

J

ds 6~,.(s)~x:~(s) F,.~(s))b(s)

~]+J

ds~~(s)~).

~s)~x

(6.2a)

(6.2b)

Therefore, ~x~/iIa~,.(s)= IIJx:~(s)F,.Il(~(5))~I 3(5) ‘4’~s):x.

(6.3)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

41

In (6.3), i(s)

c~x:e(s)=PexP(J d~’.A). Also, N+1

=

urn i5 exp{—

=

lirn exp{—(x —

A,, (~i_1)}

(~, — ~i_1)

~

A(r)}.

~N)

. .

exp{—(~ x) A(x)}.

(6.4)

—

~r’ is defined by going in the opposite direction around the path C, l/r1[~] ~/,[~] = I. With all these definitions, a functional differential equation for l~I[s] can be derived:

~

(~~_i~

=

lim [O(s t) hI1x:~(t)F,,a(~(t))IIIt:s F~~~(~(s)) lIJ~(S):X~~(t) ~,~(s) —

—

O(s — t)~J;’~(s)F,.a(~(s)) çl’;~ F~qs(~(t)) ~/~c~t):xs~ia(t) ~,~(5)

+

(~

+

6(t— s) ~I1x:~(t)D,.F,.a(4(t)) ~a(t) ~P~(r):x}

~(t

—

s))

II1x:~(t)F,,,.(~(t)) ~/‘e(t):x

=0,

(6.5)

when A,. satisfies the Yang—Mills equations of motion: D,.F,,a These calssical loop space equations

~ ( ~~s)

‘ ____

~

—0

=

0. (66)

—

look like the chiral model equations:

3,,(g’ 3,.g)=0.

(6.7)

In three dimensions, eq. (6.6) can be used to generate an infinite set of functionally conserved currents in analogy [18,28] with the nonlocal charges of section 5. Since eq. (6.6) is true, we can let

F,,(s, [i]) =

E~1.A

~1.~5)

~

x(s, [f]).

(6.8)

L. Dolan, Kac—Moody algebras

42

and exact solvability in hadronic physics

The appearance of in (6.8) occurs so that x(s, [f]) is reparametrization invariant. (A functional or path dependent object is reparametrization invariant if it is invariant when ~,.(s)—*~,.(t)~~,.(s(t)) and s t if there is explicit s-dependence. That is to say, f ~ A(4) is reparametrization inyariant since f d~ A(~)= f ds (d~,.(s)/ds) A,.(~(s))= f dt (dsldt) (dtlds) (d~,,(s(t))Idt)A,.(~(s(t)))= f dtsc,,(t)A,,(~(t)) = f d~_A(~).Note that other reparametrization invariant quantities are ds V~2(s) and (1!V~2(s))~/~,. (s).) Eq. (6.8) implies —*

-

x(s,

__

[f])

-

—

EA,,$

___

,-----—

V~2(s)

~A(5)

,~(s,[f])

(

.

)

since we assume ~p(s) ~x(s,[fl)/~~(s) = 0 from the reparametrization invariance. That a solution x(s, [f]) of eq. (6.9) exists can be proved [41].Then

x(s, [i])+ ~[F,,(s, [f]),x(s, [f])]

J,, (s, [f])=

is also functionally conserved (&1,.(s, [fl)/~,.(s)

[fl)-

=

0) since

[~])+[F,.(s’, [fl), F

~~F,,(s’,

(6.10)

1.(s, [f])]= 0.

(6.11)

(We remark that all this is valid only for s not equal to the endpoints 0 or 1. This restriction occurs because functional derivatives only commute away from the endpoints: (~/,,(5’))(~I~1.(5))— (~I1.(5))(~/~,.(5’)) = 0 only for s 0, 1.) In this way, a whole family of functionally conserved currents can be constructed. Note that eq. (6.10) has exactly the same form as the two-dimensional Noether current when the chiral fields are solutions. In the two-dimensional model, the Noether currents for the first two non-trivial charges on shell are [26]

fr~2=3,,x”~+~[A,.,x”i

where A,.

=

x~’i+ ~[A,,, x°~l, x”~]

~

=

=

~

(6.12a) (6.12b)

g~’3,~,g

J

If we replace

dx E(y x) Ao(x, t).

x”~by

(6.12c)

—

x(s,

[f]) and A,. by F,,(s, [~9)then eq. (6.12a) is (6.10) and it is now trivial to

construct the next current in functional space from (6.12b). It is the result [40] J~)(s,[~:]) = [axI~,.(s), x]

+

~[F,.(s, [~:]), xl, xl

-

(6.13)

Since there are an infinite number of chiral model currents, this procedure generates an infinite

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

43

family of functional currents for paths in three Euclidean dimensions. Since the formulae are so similar, it is reasonable that the same symmetry algebra should be responsible for the transformations in both theories. 6.2. Polyakov string theory Another use of loop space is Polyakov string theory. Now the functional field çb[~] is thought of as a string whose dynamics are to be determined. The simplest object which plays the role of the free propagator G(x) = (01T4(x) ~(0)I0) in particle language is G[~] = (01 tr ~r[~]l0).

(6.14)

Just as the interacting n-point function in local field theory (0IT~(xi)~(x2)~-

.

~(x,,)Io) is expressed in

terms of G(x), so is (01 tr

Ip[~1]

~‘[~2]I0)

(6.15)

given in terms of G[~].The program is to (1) describe free string theory, (2) find hidden symmetries of free strings, and (3) build interactions for strings so that the interacting string theory also has these hidden symmetries and is therefore exactly integrable or solvable. In integrable models, the number of constants of the motion is equal to the number of degrees of freedom. In free theories there is always an infinite number of conserved quantities. The fact that only certain interacting theories are exactly integrable means that only certain interaction terms preserve an infinite class of symmetries. That is to say, to preserve integrability in an interacting system, the interactions must be such that the symmetry of the free system is not destroyed. Even if the interacting theory is not exactly integrable, symmetries could be used to set up Ward identities which may restrict loop space Green’s functions enough so that they are completely determined. This is what happens in the two-dimensional non-linear sigma model, where the existence of the affine symmetry algebra implies S-matrix factorizability and no particle production and a calculation of the exact S-matrix. In analogy with the proper time representation of the free particle propagator which expresses G(x) as a sum over paths of free string theory, i.e. G[~]for non-interacting i/i[fl, can be defined as a sum over surfaces. It then reduces to the two-dimensional Liouville model. For free particle amplitudes, in Euclidean space, we have used the states of the zero-dimensional operators ~,. and j3,,, 2) G(x (—LI + m

G(x

—

xl)

=

—

x1) = ~4(x — x1);

(0IT~(x)~(x’)l0)

1,.Ix)= x,.jx);

j3,.Ip)=p,,Ip);

(xIOlx’);

(xIx1)= 54(x—x’);

(6.16)

(xIj3,.Ixt)= Jd4q~4(q_x)(_i~.~4(q_x1)) —i~r~(x—x’). =

Then (j3,.jS,.

+

m2)G

=

I.

(6.17)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

44

Equation (6.17) follows from +

m2)O~x1)= =

=

J J

d~y(xI~2+ m2~y)G(y - x1) d~y(—8,,8,. + m2) ~4(x

(—L1~+ m2) G(x

—

—

y) G(y x’)

x’) = 54(x

—

—

x’).

(6.18)

From (6.17)

O=~Jdrexp[_~(j3+m2)]

(6.19)

so that

(xIO~y)= G(x

-

y) = ~

=

=

~

J J

dT (xIe~y~

dT em2~2 N

J

~ ~

D~,,(t)exp(— ~

J

dt~)

(6.20)

paths

p

where J~’= 132/2+ m212 and ~,,(t) is a path through four-dimensional space-time from x to y, ~,.(0)= x,., = y,.. Eq. (6.20) can be thought of as the definition of a sum, over paths, of the exponent of minus the length of the path. For gauge systems, it has been suggested that the analogy of this proper time expression of “the sum over paths” be used to define the free string propagator [42]: G[fl

=~

~

(6.21a)

e_A~~

surfaces Sc

=

=

J J

d~(Se)

(6.21b)

D~(u,v)exp{—(26 41f~)S[4]}

(6.21c)

where S[cb]

=

J

2u d

{~()2

+

~t2

e~}.

(6.21d)

L. Dolan, Kac—Moody algebras

and exact solvability in hadronic physics

45

Eq. (6.21d) is the Liouville action, (6.21a) defines G[~] formally as a sum over all surfaces of the exponent of minus the area enclosed by that surface. This symbol ~(su~a~ssc)e_A~c) is then explicitly defined in (6.21c). The motivation for (6.21c) comes from a two-dimensional parametrization of the surface S~and a two-dimensional coordinate transformation to a conformally flat space [42]. In this program, a solution to Liouville equation will then give information via (6.21c) about the non-interacting string propagator. Classical Liouville theory [43] itself is an exactly integrable system. The equations of motion of (6.21d) are LI~= _,22 e~.Classically the most general solution is known from the inverse scattering method or via a Bãcklund transformation which connects the Liouville equation to the wave equation flçb = 0. Recently there has been much activity in quantizing Liouville theory so that it has the boundary conditions imposed by its connection to the string. The infinite parameter group of conformal transformations is a symmetry of Liouville theory and serves as a guide in defining consistent commutation relations [44]. Once the free string (6.21c) is solved, one can investigate hidden symmetries for and attempt to construct interactions which preserve these symmetries. So far, the only example of hidden symmetry in an interacting string theory are the functional currents of section 6.1. In this case the interaction is built into *[~]since ~s[~] = P exp(~A. d~)where An is a solution to the (interacting) Yang—Mills equations. ~i[~]

6.3. ‘t Hooft dual operators A(c) and B(c) ‘t Hooft’s operators [23]A(c) and B(c) are also loop space objects

A(c)

~tr P exp{i

~

A dx} .

A(c) B(c’) = B(c’) A(c) exp(2lTin/N).

(6.22a) (6.22b)

Their definitions are discussed in section 8.2. Their form is also suggestive of an infinite number of conservation laws.

7. Afllne algebra in self-dual SU(N) Yang—Mills theory 7.1. The symmetry generators An infinite set of infinitesimal symmetry transformations on real self-dual SU(N) gauge fields are defined [8]. The self-dual equations of motion for Euclidean Yang—Mills theory are F,.~(x)=

~ F~(x).

(7.1)

The field strength is F,,~= a,,A~— 0~A,.+ [A,,, Ar], where A,,(x) = A~(x)T”and T” are the antiHermitian generators of SU(N), with structure constants [T”, Ti’] = CabCTC. For example, 1’~~ = ua/2j for SU(2) where ~.a are the Pauli matrices. Real solutions of (7.1) for SU(N) correspond to real functions Aa,,(x). The symbol e,,,,~is totally antisymmetric, with e~= 1. An infinite set of infinitesimal

L. Dolan, Kac—Moody algebras and exact solvability in hadron/c physics

46

transformations which leave (7.1) invariant is ~

=

iD2Q~

=

—iD1f2~~~ i~A4 = iD3Q~~~,

LI~~~A3 (7.2)

where 1A~~~~D),

a,. + [A,.,

D,.

Q~’~ = ~

for n

=

0, 1,.

(7.3)

.

+ D

To define D, D, and ~ we use the Yang [20] formulation and change variables from x~, 1, 2, 3,4 to two complex variables y and z:

=

V2y=x~—ix~, V~=x~+ix~ (7.4) V2z=x2_ixt,

V22=x2+ixl.

The gauge potentials in these coordinates are then given by An

=

~A,.(x),

a = y,

~,

z, 2, w~ y, w2

z, wZ

2

W~=

V2A~A 4+iA3, =

V2A2

A2+iA,,

Eq. (7.2) becomes 1AY = ~ LI~~ LI~’°Ag = —D

V2AgA4iA3

(7.5)

A2—iA,.

=

LI1~~A~ = —

,

1A 9Q~,

LI~~

2~

(7.6)

=

Eq. (7.1) is Fgy+Fz~=0,

(7.7)

FYZOFZY.

Since in these complex coordinates some of the field-strength components are zero, we can define local, i.e. not path-dependent, functions from P exp(.f d~A) in the following way. In general i/i[~] = P exp(f di,. A,,(~))is path dependent. Here the path ordering is defined as in (6.1) (latest on right). This path dependence can be exhibited explicitly by

+

~]

-

~s[~]= ~

A,, (y) - ox,. A,. (x)

~J +

ds

~

:~(s)F,,~(s))~(s):

~

n(s) + (7.8)

0

Therefore for F,,~= 0, ~(‘ depends only on the endpoints x, y: the non-integrable phase factor becomes integrable. For self-dual fields, F9~and F~are zero; so if a path is chosen (through four-dimensional ~(i

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

complex space so that fixing z, 9 does not fix 2, y) with z and 9 fixed, then ~ and

D[~] = P exp{

+

J

47

=

0=

~,

and O~= 0=

(d2’ A~+ dy’ A~)}

34~]— D[~] = OP,. D A,,(P)— OQ,.A,.(Q)D +

J

ds

~/1X

:~(s){F,.4~3~,,} ~I~(s):x

(7.9)

where ~

0k,,

=

F,.~5~2 0~,.+ ~

=

F~ 0~+

0~,. (7.10)

FZ~Y0~

=0. The metric g,.~(x)= 0,.. becomes in complex coordinates: g~g=ggy=g2s=gsz=l,

allothergas=0.

(7.11)

D[~] depends only on the endpoints. Fix Q = and let P,, = (y, 9, z, 2) = x,., where 9 = y~and 2 = z~, i.e. a point in real four-dimensional space, then —~

D(x) = P exp(

J

(path has fixed 9, z).

(7.12a)

(dy’ A~+ dz’ A~)}~ (path has fixed y, 2).

(7.12b)

(d2’ A2 + dy’Ay)},

Similarly, from F2~= 0,

b(x) = P exp{

J

So a~D= DAN, a~D= DA5, ägD = DAg, a~D= DA~.Since A,. and A~= —A2. Therefore Dt = D~since t

=

=

—A~,for

j.t

=

1,2, 3,4, A

=

~ exp~J(d5’ A + dy’ A~)}

D

=

~

exp~

—

J

(dz’ A~+ d9’ Ag)}

=

i~,

(7.13)

where P is opposite direction path ordering (latest on the left). Yang observed that one could always

48

L. Dolan, Kac—Moody algebras and exact solvability in hadron/c physics

find a gauge so that any self-dual potential could be written in the form AY=D~aYD -

AgD’agD

A5=D’a2D

-

-

(7.14)

-

A~=D’a~D,

for some D and D. The emphasis for our discussion is that for any self-dual gauge potentials An, we can always construct the functions D and D from the definition (7.12). They depend only on one space-time point x,,, and are non-linear (because of the exponential) and non-local (because of J~)in the fields An. In this respect the L1~~~An are similar to the LI~’°g. The functions A (n) are defined iteratively: a~A(n+i) (n+i)

=

=

a2n (n+1)t

~l~A (n)

—~A~”~ a~A(n+l)t

Tepe,

A “~=

=

—~~A (n)t

(‘flt

(7.15)

a ~+ [JagJ~

=

=

(0)

9l~A

1,

a~+ [J-’a~J,, J—=DD~, A

=

(7.16)

T]

~,

J

=

dz’ [J’

a~J,

T], etc.

(7.17)

Here Pc are infinitesimal constants. Also, define LI ~p”~AaPezi ~e~Aa,

ç~(n)

(7.18)

pa[l~a’°.

First we check the consistency conditions for (7.15). They are ôg~yA~~~+ a~

2A”~ = 0

(7.19a)

a~~9A~~+ a~A”~= 0.

(7.19b)

To prove (7.19), note that since 13g(J~3)J)D,

1a

Fz

F9~= D

2

=

D’a5(7

2J)D,

then the self-dual equations (7.7) are

1a 1a a~(J 5J)+ a~(J 5J)

=

0.

(7.20)

Also, J (eq. (7.20)) J~is equal to

a2(Ja~J’)+a~(Ja9J~)= 0.

(7.21)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

49

Therefore (7.19) is satisfied for n = 0 with use of the self-dual equations. Assume (7.19a) for n = N~then

—[~, 2~

3g~yA~’~”~1+

(7.22a)

2]fl~

=0.

(7.22b)

Eq. (7.22a) follows from (7.19) and (7.21); eq. (7.22b) follows since

[~

13),J)+ [J’a~,J,J1a

~]fl(N)

=

[(a~(J~a2J)—32(J

=

[17F~217~,

2J]), A~”~]

A(”~’)]= 0.

(7.23)

We now prove that LI ~ is a symmetry of the equations of motion (7.7) by checking that if An is a solution to (7.7), then so is An + LI ~ Since under the transformation (7.6) ~

=~

(7.24)

LI°’~D = —Th2~

and 4°’~J = ~JA~’°—A(~~)tJ then

~~(J~LI~J),

=

LI(J0gJ~) =

~2(J~LI~’°J), so the symmetry condition is

(n)t

~

+

1),

zI~’°(JO~J~) = —~5((LI~’°J)J

=

—

_~g((LI(~t)J)J_1)

a

5(—~2A~~) — J~a2A(f)tJ) = 0

+ J~ ~n(~~J~) + o 2(~A(n)t

(7.26a)

+ .ThZA (~~)J~) = 0.

(7.26b)

With use of (7.19)—(7.21), eqs. (7.26) are easily verified:

a9(J~a5,A(f)tJ)+a~(J~35A(f)tJ) = J_l(3~fl(n)t + a5~5A(f)t)J= 0. The transformations (7.6) leave the gauge potential A,, real: since ~ (z1~~)A5)~ = ~

(7.27) =

=~

and

(LI~’°A~)~ = —D5f1~”~ = —LI~’°A2 .

(7.28)

Z2. The algebra The operators Me(fl) associated with the infinitesimal transformations LI ~A,, are given by 4x LI ~ Aa(X) &Aa(X) M~’~ =—

J

d

(7.29)

50

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

The commutator is computed using the definitions (7.15) and the particular choice of boundary

conditions discussed in (7.42a). It is found to be [M~m),M~] =

f~’M~ — J d~y(_D,,[f1~m), ~

(7.30a)

~A,,(y)

where tmnl

J

C

—

~or n, m —

cba — cebaulo

ccba(Ol,n±m+2(—l)O,o)

for n

=

m

1

(7.30c)

ccba(fSl,n+’n + (_1)m5,,~_~)

for n

>

m

1.

(7.30d)

The proof of (7.30) will be given below. We point out here that linear combinations (L~”1or equivalently LI ~2A,,) of M~”~ and LI ~ respectively for N n can be chosen so that >-

[L~’n~, L~]=

ccbal}a

~— J

d~y(—D,,[u1~, ~‘1~A,,(y)

(7.31)

Here Q~is also a linear combination of Q~m)for m ~ M. Up to the second term in (7.31), this is the infinite parameter affine algebra C[t] 0 SU(N) [8]. We now make several comments which provide the analysis of the Lie algebra properties of real self-dual SU(N) gauge theory: (1) The second term in (7.31) spoils the integrability condition whenever it is non-vanishing. (2) The second term in (7.31) is itself the generator of a real (local) gauge transformation. (In general, infinitesimal gauge transformations on the gauge potentials can be written in either ,a = 1, 2, 3, 4 or a = y, 9, z, 2 coordinates as O°A,,= —D,,A(x), where At = —A. oGA~= _(3aAa + cabcAaA), A Aa(x)Tlz. Aa(x) is the local gauge function. Since Q(n)t = (1(n), then [fIl~, ~~ 3~)]t = —[(1~, Q~]and the second term in (7.31) is a particular gauge transformation with A = [(1~2,f1~A~].) (3) Therefore we enlarge our set LI ~f2A,,by adding all real local gauge transformations which vanish at x = —~ (D,,A where At(x)= —A(x)) to LI~A,,and check the new integrability condition to see if this new set 0,~AA,, = p~LI~A,, — D,,A can generate finite values which form a group: 8P,A(A,,

+ Off, KAn)

(Sp.AAa

—

Off.K(A,, +

OpAAa)+ 8,,,

KAn = (ST

sAn

=

T~LI~Aa— DaS

(7.32a)

where S(x)= —[A, K]— pZo~’[11~5~, Q~]—A[A

+

O~,KA]+A + K[A

+

OPAA]— K,

5,m+npbcrc Ta =

c&a4

n m

(7.32b)

(4) Therefore the larger set 5,~,AAa does close the algebra. The finite transformation law

A,,(x)-+ A,~(x)= [exp J d4y (8P.AA~)~A~(y)]An(x)

(7.33)

L. Dolan, Kac—Moody algebras andexact solvability in hadronic physics

51

provides a representation of the hidden symmetry group. We now return to the derivation of (7.30), (7.31) and (7.32). Proof of (7.30). From (7.29), the commutator (7.30) is given by C [M(m)

If~.’~]p~~b = J

—

LI(m) p ~A(x)’~_ / d4x{LI~°(A,,(x)+

A (n)

A

(x)

LI ~‘n~(A,,(x) + LI~°A,,(x)) + LI (m),,4(~)} 6A,,(x)

(7.34)

Define A nm A up ~a

=

A (n)(A +

— —.

‘-

LI (m) ) — ~A (n) A p ~ ff ~ A

—

A (m)( A p ~

~

+

1~a) +

A (n) A ff

A (m) A hJ p

(7.35)

Then, from (7.6), to order PcUb, LI nm up A A nm

Obp~(Dy[Q~, f)~i)]+

A

=

ffbpC(D~[(1~C)

D~K~’)

fJ~,n)] — D

9K!J,~) A ~‘

A

nm

A

nm

A

up P15

~ °P~

=

(7bPc(Ds[Q~, flS,’°]+ D2K~”)

=

ObpC(Dz[fl~,

11~”~] — DZK~~)

(7.36)

where Vmn =

— =

Q~(A+ LI (m)~4)_(1~,n)— (1(m)(A + LIS,~A)+fl(m) 1[A (m) A ~‘~]D — ~D’[A t(m) A ~7’~]D ~15 —~I51[AS,”~(A+~ —

1D~[A ~,n)t(A + LI (m)A) — A ~,n)t — fl (m)t(A + LI ~“~A)+ A (m)t]D

(7.37)

Eqs. (7.36) and (7.37) follow from LI~m)D= Df1~m), zi~’n~i7 = —Df2~’n~, LI~’1(A + A (m)A~)_LI~”~A~ = D~(fl~”~(A + LI ~‘n1A)— f2~”~)+ [D~[l~’n1, (1~2)] y

[1~,~~(A + LI (m)A)_ f1S,~>= —~D’(A~(A + LI ~m)A)_ A~)D —

—

etc.

~D~(A

t(n)(A +

LI (m)A) — A t(n))D — ~Q~m)D~A ~‘~i5+ ~D~A ~‘~I5L1~’n1

“~D~A ~“~tDQ~C”~ + ~D’A~tD[l~”~, (7.38)

52

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

Then ‘D’([TC,

(TOn =

1~cb

A ~)]— A $,‘~(A+ LI ~A)+ 4 ~D

+ ~D~([I~,

A(~t)t]— A~t(A +

LI ~,°~A)+A~’~~)D

2

—C

=

(7.39)

()(n)

cba~”a

The proof of (7.39) is as follows: Assume ~

for n

(7.40)

CCb,,A~’1

N.

=

[TC,

(Eq. (7.40) is true for n

=

0.) Then

A~,”~]— A~”~~(A + i~9~A)+ ~ z

dz’ ~~(A~(A

+

i~°~A)— A~)+

J

dz’ [a~A~’~,A~]+ [Ta, ~

(7.41a)

now with (7.40) z = -

2

dz’ ~

J

A~] —

2

CCb,,

=

J

CCbaA~)+

=

dz’ [a~A~, A~~] + [To,A~]

2

dz’~~A~)_J dz’[TC,

z

—J

J

a

2A~’~]

2

dz’[J’ô~

Tc],A~]+ (J

dz’[a

~C ~ A~N)])+[T ~, A(N~)] b

2 ,~(1)

(7.41b)

~

In the above, we use the definitions (7.15) for A~f~~” with the boundary condition A~f~l~)(y, 9, z = 2 = —cc) = 0 for N 0, i.e.

—~,

2

A(N+i)(x)_

11,,

J

1a

dz’~ A(N) ~y’

-

(7.42a)

Note that in general, the solution to (7.15) is A (N±i)(y,

a

9, z, 2) =

J

zo

dz’ ~

~N)+ A ~N+l)(y,

9, z = zo, 2 =

2~).

(7.42b)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

This follows from (a) 82A ~“

~f’~ and (b) ~4

~

=

(N+1) =

~~2fl

~.

53

Then from (a)

z ~ ~l+l)(y,

9, z, 2) =

J

dz’ ~A ~

+ f(y,

9, z).

(7.42c)

zo

From (b) 2

a

11a

~

J 20

)

dz’(—a z ~ 2 A(N)\+ôgf(y,9,2)

= —~

A

(N) ~

+ a

A (N)I

z~zo

9f(y,

9, 2)

2=

(7.42d)

V’a

Then it

f(y,9,2)_Jd9’(~ A(N)\ a

+C(y,2)

211

)0

$7 J

—

d9’(8~A~’~’~’ a

-,

—

ito =

11a(N+l)IIz=zo ~

11a —

~4 (N+l)I

Iz=zo + y — yo

(7.42e)

C(y, 2).

Therefore, 2

A~”(y,9,z,2)=

J

dz’~A~f’~’ -t- A~ ,,‘~‘~(y, 9,

Z 0,

2o) A ~ —

9°,z0, 2~)+ C(y, 2).

(7.42f)

20

Then, from (7.42f), A ~fl+l)(y0,

9~,z0, 2~)= C(y, 2)

~7.42g)

and (7.42f) becomes z

A

k,y, 9, z, 2) =

J 20

dz’ ~,A ~ + A ~

9, z0, 2o),

~7.42h)

54

L. Dolan, Kac—Moody

algebras and exact solvability in hadron/c physics

which is (7.42b). We emphasize that different choices of A ~fJfl(y, 9, z0, 2~)will give different symmetry 1Aa and may lead to different algebras. Clearly, the choice of boundary conditions is transformations LI ~,‘space of fields A,.(x) which are being transformed. To carry out the calculations of determined by the this section, we have made the choice (7.42a). Now return to (7.41b). Also, assume [Ta, A ~,n)t]— A ~t(A + LI ~A) + A ~,n)t = C (n)t (7.43) cba1t a

for n = N. (Eq. (7.43) is true for n = 0.) Then, similarly to (7.41), [Ta,A~~t]_A~~)t(A + LI~A)+~ 2

=

— J

2

d2’ ~~(A ~J)t(A

+

LI ~°~A)— A

~.J)t)

— J

d2’ [a

2.A(1)t fl ~N)t]

+

[Ta, A ~+i)t]

1t a

Ccba (N+1)t Therefore, from (7.36), (7.37), (7.39) we find explicitly that

(7.44)

=

[M(0)

M~°] =

CCb,,M~”~ — J

d’ty (Da[fl~, (1~])aAa(y) a

(7.45a)

Eq. (7.45a) is (7.30b). We now prove (7.30c and d). First, it is shown that EM(1) and for n

!vI~,’>]= C (M~2~— 2M~)—J d~y(Dn[fl~, cba

a

‘11~~~~A— a

2

[M~°, M~]= Ccbas,(M~’~— M~t~)— J d~y(—Da [(1(1) a

~,

Q~1\ b]~U~A.

(7.45b)

From (7.37), K~ ~ A ~~)] — A ~‘~(A + LI~’~A) + A~ Cb = ‘D~{[A 2 +

~D’{—[A

(i)t

A ~,n)t]

— A ~~)t(A +

+

A ~C~(A + LI ~,n)A) A ~~}D

LI ~‘A)+ fl ~,n)t + fl (i)t(A

—

+

LI ~~A)— A ~)t}D

(7.46)

Then [TiO = 1~cb

(7.47a)

c’

~‘j(1) ~—cba~ a

K~= —C~~(fl~ — 2fl~) K~g= —C~~,,,(fl~’~— u1~’~) for n

(7.47b) 2.

(7.47c)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

55

Eq. (7.47a) follows since [Ar, AS,°i+A~(A+LI~,°~A) A~C~”CcbaA~a’~

from (7.40). Eq. (7.47b) follows since A ~(A

+

LI ~A)

A~

—

=

J

dz’ [LI~l)(Jla~J), Tb]

=

J

dz’ [(~~(—A ~ f’A ~)tJ)), Tb]

=

J

dz’ [(- a

-

2.A (~2)— J~3A ~

Tb]

=

J

~

=

J

dz’ [(—a2.A~~a2.(J~T~J)), Tb]

Tb]

2~,Tb]—[J’T~.J, Tb]+[T~, Tb]. —[A~ The last term in (7.48b) follows from the boundary conditions (7.42a), since A ~(y,9, z = but f1TcJIz..,s~_ff~0. In fact we here assume that J T~JIz 2...~= T~,i.e. that

(7.48a) (7.48b)

=

—cc

=

2)

=

0,

(7.49) 1 and a boundary condition on the potentials: Eq. (7.49) follows from J A,,(y,

9,

z

=

—cc,

2

=

DIY —cc)

=

g~3~g.

(7.50)

Since (~.it. z,2)

D(y,

9, z, 2) =

P exp{

d~,.A,. }

(-ff,

and (y,y,z.5)

I~(y,9, z, 2) = P exp~

~

d~A,, }

(7.51a)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

56

then

D(y,

9, —cc, -cc) =

P exp{

dy’ A~(y’,9, -cc, _cc)}

(:1i1)

and

Pexp{

~(y,9,_cc,_cc)=

(Y~:-:-I)

(7.51b)

d9’A~(y,9’,_cc, _cc)}

so for An(y, 9 —cc, —cc) = g~’a,,gthen D(y,

9, —cc, —cc) = g =

D(y,

9,

—cc, —cc)

and (DI7_l)Iz~:::= 1

(7.51c)

which is eq. (7.49).

We remark here that if the last term [Ta,Tb] were not present in (7.48b) then the commutators of M~”~ would be given exactly by those of (7.31). Return to (7.48b). Then to compute K~(7.47b), we find from (7.48b) that

=

Tb]—[J’TJ, TC]—2[I~, Tb].

1TcJ,

CCbaA~+[J

(7.52a)

(7.52a) follows since A~

=

J

dz’ ~

~

J

=

dz’ {[a~x, Tb] + [a~x,

[,~(2) Tb] +

where A ~/) = [,y, Tb] so [A ~

8zx =

[x,Tb]]}

~[x,[x,Tb]]

f~ a~Jand x(z

=

—cc)

(7.52b) =

1. From the form (7.52b), then

A ~P]+ [A~2) Tb] — [AS~,T~]= [x~2~ [T Tb]] + ~[x~ft, [Tr~, Tb]]] =

using the Jacobi identity.

CCbaA ~

(7.52c)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

57

Then 1(C ba(A ~

(71n = 2i1.5 ~‘cb

—

2A ~“)+ [J~ T,J, Tb]

—

[J1 T~,J,T~])b+ Hermitian conjugate

Ccba(f12(1~°~ a)

=

(7.53)

since D~([J~T~J, Tb] [f~TbJ, TCJ)D is anti-Hermitian. This proves (7.47b). The proof of (7.47c) is by induction. Consider for n —

2,

[A~.’1,A~°] — A~~~(A + LI~A)+A~°+ A~C1~(A + LI~°A)— ~

We assume for n

=

N that

C~,,,,(A

=

(7.54)

[J~A ~

A ~1))+

~‘~—

T~]+ [JtT~.J, A~’~].

(7.55a)

Eq. (7.55a) is true for n = 2. Then 2

n+i

—

A cb — — J dz’ {~

c

5(A$,‘°(A+ LI ~A) — A $,~~)) + [‘ ,A (2)

p~ 1 +

a

(n)] +

[a2(J’T,,J),A $,‘~]}

1~A ~,n+i)]

‘ IC

[A~+2)

—

A

T~]+ [A tf, T~]+ [J~ T,J, A ~].

[J~A ~

11a

(7.55b)

(~~) + [J~A ~t) Therefore (7.55a) is true for all n 2. Eq. (7.55b) follows from (7.55a) and CC,,,,(A

=

(n+2)

—

A

2

dz’ ~~(A~,’°(A + LI ~A)— A S~~)

— J

z =

+ =

dz’ ~~(—Ccba

J

—

[A

~“,

11a(n+1)

(1

— ~A

(n-i)\

) — [f_iA t(n_i)j T~]+ [J’T~J, A ri)]

A ~:‘~][A~,n+1) T~]+ [f’A ~,n—i)tjTa])

CCba(A~ ~

—

~)+[A~,

T]

)

2 + J

dz’ {[J~ T,J, ~

z’JiC

2.A ~] —

[a

A (2)

A ~~)]

I.1A”> c ,

a A ut)] + [A~,n+i) aZ”1C A 0)]~ I-

(7.55c)

—

Then, from (7.55a), for n 2 (Tin = itji)~bD + Hermitian conjugate 1~cb

2 =

~

r” — A ~‘~)D

+

Hermitian conjugate

(7.56a)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

58

(7.56b)

a a 1). CCba(f1~~.u1~’

=

This is (7.47c). Eq. (7.56a) follows since D’([J1A~tf, T~]+ [J1T~J, A~])Dis anti-Hermitian. Therefore, we have shown (7.45b). Note that

a

J d~y(~Da[(1~(1$,n)])

aA,,(y) = [S~m),si”>]

C

(7.57a)

where

ç(m)

C

— J

a d~y(Dn(1~) aAa(yy

(7.57b)

‘~‘

From (7.57b), IS(m) IC,

S~]= J d~y(5~(A~ + (Stm A C P1a) \

(SflbP1a A —

a

~

An + t~c P1

8~A,,)+ 8~’Aa)~A

(7.57c)

where 8~’Aa= _D,,12~~~ Then 8~’(An+ 8~Aa) 8~A,, = _D,,((1~m)(A+ =

8~A)—(1(m))± [Dn(1~,

—D,,[Q~”), (1(m)] + [D,,(1~’), (1(m)]

since u1~m)(A+ OZA)— (1(m) = [(1~,n)(1(m)]

(1(m)]

[g~,n) D ~(1(m)1 C]

=

and 8~D= —D(1~,

8~1~ = —b(1~,

(7.57d) 8~J= 0, so

A~”~(A+8~A)—A~=0.

Therefore

a

[S~m~,S~”~] = J d~y([11~t), D,,(1~m)] — [(1(m) ~

=

—J

d4yD,,[fl~, (1~)].

(7.57e)

This is (7.57a).

We now prove, from (7.45b), that [W~m~,Mr)]

= CCb,,(Mrm)+

Eq. (7.58) is true for m ~

=

=

(_1)mM~1_m))_[S~m),S~]

1. It is also true for m

[M~, Me)] + ~

+

=

for n > m

1.

(7.58)

2 since from (7.45b),

[S~’~,S~],

(7.59)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

59

and then, for n >2, a, ~M’~fl~ eJ +1[S~~, 5~/)]p,4~(n)] M’~°] = [M’~, M’~/~], IvP~°] + 2C~b,,[M’~°~

~

=

+ =

(7.60a)

[/W’$P,JW’~’°],)W’”~]+ [/I,4~(i) p4(n)] f~4~,i)] 2CcbaCaefM’~°~ 2Ccba [~0) V(~~] + [S~~, S~/~], p4(n)l “a ,‘-‘e C j

(7.60b)

—

(Ct,egCcgi — CcegCbgj’)(M’~f~2~ + p,4~5n_2)) +

C~eg[(S(~t~~ — “g c(n—i)) ~‘-‘C c(i)] — ~ — St~’~ SS2] 2CCba[S~°~ ~‘~1 g ‘—ceg [(S(~~+i) g g a ‘-‘c

+

[S~,’~,S~~] A4”~ cJ

—

[RU) ç(”~],A1~°] + [S~,”,S~°], 14 ~,‘°] ‘—‘c,’-’e

—

= CcbaCaef(MY’ +2) +

,

(7.60c)

M~’2~) — C~[Sr, ~ C’.

(7.60d)

Eq. (7.60c) is true tor n > 2. Eq. (7.60d) follows from the structure constant identity CcegCbg.f = ~ and from (7.62). The commutators involving M~~”-~ in (7.60c) are computed as follows: M~]= - J d~y{8~(A,,+ l~Ie~~a) A(n) A

[Sr,

\

~m fl UCPIa

d~yD,,((1~(A + 3~A)- (1(m)) C

=

J

—

rS(n) [ e , p4r(m)] c j

(n)

C~gCC~ —

a

— LI~(A,,+ (S~’Aa)+LI~Aa}aA(y)

a

(7.61a)

Therefore [S(m) c

,

p4(n)] e j

=

J d4y ro

=

jpnm J d~ya’fl ~

((1(m)(A + C

LI ~‘°A)—(1(~m)_(1(n)(A + LI (m),4) + (1(n)\ a

a

aA,,(~)’

(7.61b)

_______

see (7.37). The commutators involving M~”~ in (7.60c) can be written as [([Sw, [si”)] C —

[~(i) L’’C

M-~/)]),S~”~]

+[([S”~ M’~0)]) S~~][([S~, C, p4(n)] e j — [S~,’°, —

p~r(n)] e j—

(7.62)

[S~”~lW’~~]), S~~] e

so that (7.61b) for n = 1 given in (7.47) can be used to evaluate (7.62) which with use of (7.57a) in (7.60c) gives —C~b,,[S~,S~°] in (7.60d). Eq. (7.60d) is (7.58) for m = 2, n> m. To prove (7.58) for general m, we assume (7.58) for m = M where M 2. Then for m = M + 1, CCb,,[lW’~~~ p4(n)] a e ,

= [p4(i)

p4’~/~4)][s4’~n)] + CCba[Mr(M_t) a

p4(,n)]

—

[JiM Cb~p4(n)l ej

for Al’

2, n

>

m.

(7.63a)

60

L. Dolan, Kac—Moody algebras and exactsolvability in hadronic physics

Here we have defined for convenience J~j’=

—[Sr, S~m)].

Then similarly to the derivation of (7.60), we find c rJ~4’~±~) 1vI~”~l = ~ (_l)M±hJJ5~~_M_fl) Cb~2I

a

C

+

Cb

=

C~,,C~f(M(f7+M+

M(n—i)

Ccc ~

(_1)MC begJl(n~ML cg Mn M~]+ [J~, ge + CCbgJ(M_l)P1 — Ff Ji(n±M)+

CCeg~Jbg

Alr~] Ft~cb riM M(n)] e

(7.63b)

CCba~’f(M±1)n ae for M

2, n > M. (7.63c)

—

+(_l)M~M 5P1_M_i))+

Eq. (7.63c) is found from (7.63b) similarly to the derivation of (7.62). We also use ipMn

— —

t-cba%~ a a ‘-‘ ((1(n+M)+(_l)M(1(n_M)\

)

for n >M

1,

(7.63d)

which follows from the assumption of (7.58) for m = M and (7.34)—(7.36). Therefore, [!vI(m) a ,

IvI~)]=

(n-m))—

1

e

~

[S(m)

Caet(!VI5m~~1)+ (1)M

a

for n

>

1.

m

(7.63e)

,

Eq.(7.63e) is (7.58) which corresponds to (7.30d).

To derive (7.30c), consider for n

1.

Eq. (7.64a) is true for n = 1, 2. Then, again by iteration, for n

2,

[M(n) a~ M~”~] e = Caei(M?”~+2(—1y’M~°~~+ / jac, Inn

(7.Ma)

e J — [~‘Tin Cb, CCbaLFA4’~’~’~ 1vI’~’~~] = [Al~°, PvI~’)],Al~’~’~] + CCba[IV1’~~’~IVI Tt~r(n±i)1 a e ,

— —

CCbaCaef(Al(n+2) + f

—

2( 1y’M$°~) + CC,,,,J~i)(n+ 1) +

Cceg~rn(n+2) — r~ inn bg ‘—~eg-’ gb — [J~

jyj’(n+i)l e J

Cbegj ~2~i+ 1)_

r rn(n+i) M”~] I~ be C j + [Ii(n±i) J Ce

CCba~ae T(n+i)(n+i)

= ~

(

1)nC~Jll

TvI~’)] (7.64b)

(7.64c)

Therefore [M~’~,M~”~] = C,,~~(M?”+ 2(—1)~M~°~’-(nn ) ~‘ ae

for n

1.

(7.64d)

Eq. (7.64d) corresponds to (7.30c). The derivation of (7.30) is completed. Proof of (7.31):

In order to recover the familiar affine algebra of (7.31), linear combinations For the general form

L(~N)of

M~”~ are defined.

N 1(N)

~ ~ n=0

(7.65a)

L. Dolan, Kac—Moody algebras

and exact solvability in hadronic physics

61

then for N 2, = [My, L~,t”~]

N

a [M’”~P4~,12)] C~

fl

n0 N =

CCbe(aoM~+ai(M~—2M~)+ ~ an(M~”~~— M~’~))—

N

~ ~ anS~] L n0

(7.65b)

Therefore it is consistent to define L~f’~ from L~”~ by [Mv), L~’’~] = CCbCLt1”~— [S~, S~’V] e

(7.65c)

or equivalently N

~ (N+1)

=

2~ — 2M’~°~) + ~ a~(Al~”~ — p4(11_i))

a 0A1~’~ + a1 (Al~

n=2

and N =

~ a,,SS~.

(7.65d)

n=0

Now define ~

=

~

and

~

~

=

(7.65e)

Then [L~”,L~]= ~

—

[S”~ c ~LbJ ,

for n

0

(7.65f)

where ~

=

S~

S~=S~,1~ N

forN2

~ n0

and the set {a 0,. . ., aN} is differently valued for each N For example, since ~ = M~’~ then, for N = 1, a0 = 0, a1 = 1 and from (7.65d) ~ S~= S~—2S~°~. So, for N= 2, a0= —2, a1 = 0, a1 = 1, etc. Also, [L~°~, L~]= ~

—

[S~°~, S~].

=

~

—

2M~and

(7.65g)

62

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

With use of the Jacobi identity, (7.6Sf and g) gives (7.31): [L~m),

L~]=

CCbaL(a~m)—

[Sw, St?]

for m, n ~ 0.

(7.65h)

Eq. (7.65h) is derived in an analogous (but much simpler) calculation to (7.63c). Instead of (7.61a) and (7.63d) we must consider [St), L~~~]—[S~), L~m)]=

f

d4yD11((1~2(A+ LI~2A) ~

(1~)(A+ LI~A)+

~

= J d~yDa(Ceca(1~a~~~) aA~(y)

(7.65i)

where f)(n)

fl(m) Lc

a~ Lc n=0

etc.

(7.65j)

Proof of (7.32): To compute (7.32), we find that + OP,AA11)

8~,K’4~=

o~’LI~(A,, + OP,AA,,)— D11K[A

= ff~’LIS~(A11 +p~LI~5~A11 — D11A) Da(K[A+

+

8P,AA] — [8p,A~4a,K] — cT~LI~A~

8pmAA]K)P~[LI~)Aa,K] +[D11A,K].

(7.66a)

Also LI~(A11+p~LI~A11-D,,A)-LI~2A11

=

~ 0 a,,(LI~(A,.+p~i~A11)-LI~A11).

(7.66b)

p=

For a

=

—

y, (7.66b) is equal to

[A,D~flS~’2] — pg[D

11(1~, (1~ff2]— D~[(1~(A+ p~LI~A) — (1~j].

(7.66c)

Then, + =

8~.AA~)— 8~,KA~

—D~(K[A + OPAA]

—

—

OP.A(AY +

K — A[A

+ O~.KA]+

~ =

8~,KA~)+ 3~,KA~

DyS_Dy(C&a(1Sm~)o~1p~

A

—

(1w])

[A, K] — p~o~’[(1~5~,

(7.66d) (7.66e)

L. Dolan,

Kac—Moody algebras and exact solvability in hadronic physics

63

where (7.66e) follows from (7.65i), and the definition of S in (7.32b). Similar calculations for a = 9, z, 2 lead to (7.32b) since LI ~1AY = D~(1~2 etc. from (7.65j). After all these computations, the reader may feel a bit overwhelmed at all the symmetry transformations. It is useful at this point to observe that the existence of an affine symmetry algebra such as (7.31) or (5.29) in a theory depends largely on the existence of just one non-trivial generator such as ~ By non-trivial, we mean that the commutator of two of them does not close, i.e. [L~, LIP] is not equal to a linear combination of ~ If such a generator L~C1~exists and its commutators can be written as [L~, L~]= ~ up to possible gauge terms, then an affine structure [L~’~, L~m)] CcbaL~m)follows immediately from the Jacobi identity (see eq. 13 in ref. [12]). In the remainder of section 7 we will discuss various properties and extensions of the new symmetries on the self-dual solutions. In the effort to generalize the transformations to D,,F,.,. = 0 however, the point is that one needs to focus on finding a single new (non-trivial) generator.

7.3. Associated Noether-like conserved currents Although the transformations of (7.6) are defined for (self-dual) solutions, it can be shown that LI”~A,,shifts the Lagrangian density of the full theory by a total divergence K,,, and K,. is different from —2 tr F,,,,LI ~ the usual shift using the equations of motion. In this way, conserved currents for all n 0 can be constructed using the Noether argument. Since different infinitesimal transformations can sometimes lead to the same conserved quantities (see e.g. (5.77)), these currents can be used to distinguish a set of “fundamental” transformations. Specifically, we find that although a second set of self-dual symmetry transformations LI~’~Aa can be defined (see section 7.4), these lead to the same Noether currents as the LI”~A,,.In the development of this section we will also prove that the n = 0 “hidden” symmetry transformation LI~°~A,, is a particular local gauge transformation, but that LI~1~A,, is not a gauge transformation. The extension of LI~Aaoff the self-dual sector is difficult because invariance and localness (LI~~~AC, should depend only on one space-time point x,,) are tied to (7.1). (From (7.9), D and D are path dependent for F 92, F~,2 0 respectively.) But, since the n = 0 affine transformation LI~°~A,, can be written as a particular gauge transformation, it represents an invariance which can be extended easily off the self-dual1~Aa setmay to the full theory. thorough of the the self-dual relationship therefore provideAa more clue for writingunderstanding down LI~1~A~ off set. between LI~°~A,, and LI~ The Lagrangian density of the full Yang—Mills theory in Euclidean space is ~x)

=

—~tr F,,~(x)F,,~(x).

(7.67)

It has the equations of motion D,,F,.,,

=

0.

(7.68)

From the Bianchi identity D,.F,. 11 + D,~F,,,,+ D,,F,,~= 0, which follows directly from the definition of F,,~, it is well known that all solutions to (7.1) are solutions of (7.68), since ~ = 0 identically. For an arbitrary infinitesimal shift A,. A,. + iSA,,, then ~ = D,Lt~AP D~ ~A,. and —~

=

—2 tr F,,~D,,~

—

(7.69)

64

Dolan,

L.

Kac—Moody algebras and exact solvability in hadron/c physics

With use of the equations of motion D,.F,,,,

=

0, (7.64) becomes

~~f= a,,(—2tr[F,.~ E~A~]).

(7.70)

For gauge transformations then ~x) is invariant since from (7.69), i~1=2trF,.~D,,D~A = trF,,~[F,,,,,A] =

tr[F,.~, F,,~]A= 0.

(7.71)

Conversely, if A.~ 0 for some L~Aa,then ~A,, is not a gauge transformation. From (7.6), the n = 0 affine transformation is LI~°~A = 2 —D2Q~°~

=

D~(1~°~ ,

=

—D9(1~°, LI~°~A2 = D5(1~°~

(7.72a)

where

u1~°~= —~(D’TD—D~TD).

(7.72b)

Since D~(D’A(n)tD)

D_i a~A(n)tD

=

D5(D’A (n)tD)

=

D’a2A (n)tD

and 1aA~~~D D D9(D_mA(n)D)

=

D—

2(I~’A~~~17) = D~a2A~D,

then D~(D’TD)=0,

D9(D’TD)=O, etc.

Therefore (7.72a) can be written as =

D,j1~°~

(7.72c)

where 1TD =

+

D’TD).

(7.72d)

~(D

Note that now

[

1(0)t = —u1~°~, so that (7.72a) is seen to be a particular local (real) gauge transformation with gauge function A = (1(0)~ Since any gauge transformation leaves ~‘(x) invariant, a conserved Noether current may be constructed for LI°~A,, from (7.70). It is

=

—2 tr(g~F,,~D,,11’°~)

(7.73)

L.

Dolan, Kac—Moody algebras and exact solvability in hadronic physics

corresponding to the conserved matrix current J~where Ja

a)J),

j(O) = ~j(J~i

fl)

=

~

=

=

65

—~tr(J~T):

L1(Ja2Ji)

L~(Ja9f~), J~°~= E(f~a2J).

(7.74)

Eq. (7.74) is derived from (7.73) in the following manner. From (7.73), =

—2 tr(F~9D~(1~°~ + F~2D5[l~°)).

(7.75a)

Then,

~

a9(f’ a),f) 1~[(15a~I7’+ DA~D_i),T])

tr F~9D~11~°~ = tr(D~ =

and

~tr([a9(f~ a~J),f_i a3J]T)

t0~= ~tr(D~~ tr F,2D5i’2

1 a 2(J

=

5J) 15[(17

1a

(7.75b)

a2b_i + DASD~),T])

~tr([a,(J~ a3,.!), f 2J]T).

(7.75c)

13),J~ V

5—f~a5f.Then a2V~—a3,V2+[V~,V5J=0 and, on the self-dual set a5V~=

Define V~,—f —a2v5, so =

tr(a2[ V2, V3,]T)= —tr(a2(a2V~— a3,V~)T) =

—tr((a2a2 + a~a~)V~T)=

—~tr(L~V3,T).

(7.75d)

Similarly, trF95D2f1~°~ = ~tr[a5V9,V2]T trF93,D9,ã~°~= ~tr[a3,V9,V9]T

trF22D2.ã~°~ =~tr[a5V2, V5]T trF23,D9f1~°~ =~tr[a3,V2, V~]T trF52D2t’?~°~ = ~tr[a2V2, V2]T

trF29D3,I?~°~ = ~tr[a9V2,VY]T

(7.76)

1 = _(V~)tand V. ~fa

where V9 asJa9f

1 2J

=

_(V

4t, a2v9+ a9v2 +[V9,

V2]= 0, and for self-dual

solutions a5v2 = —a3,v9. In the above derivations, we use that 1+Da~D’, V V~= DA3,D 9= DA9Di+Da9Di —

—

—

—

V2=DA2Di+Da2D~, and

V2=DA2Di~l~Da2Di

(7.77a)

66

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

F93, = D~a9 V3,D = —D1 a3,V~D F25

=

F92

=

Ff3,

=

D~-ia2V5D = —D’ a5V2D

1a D~a9 V2D = —D 2 V9D 1a 1 a~V D 2V3,D = —D 2D

and gYit = gZS = g$7Y Then, we find

= gSZ =

(7.77b) 3= 0.

1, all other g~’

~~~=—2a 2[V5, V3,]=~V3, =

—2 a5[V,, V9]

=

—

,~°~=2a~[V2, V~]=L1V2

(7.78)

~~~=2a9[V2, V3,]=LIIV2. From the integrability condition (7.19) for (7.15) and the Brezin et al. method [28], a set of conserved matrix currents can be identified: since from (7.19),

a9~3,A(11)+~2~j~5~(n) = 0 (n)t + a2~12)t = 0 so J~can be defined by ~lJ3,A(11~f~h1)

and

~2A(11~J~

(779a)

D9At=J~

and

~2A11=J~h1),

(7.79b)

where a11f~)= 0. These eqs. (7.79) are not derived as Noether currents and (7.74) does not appear naturally in this set. What is more natural is (7.80)

~

since from (7.79), ~ = T]. The currents ~ may serve as a guide to generalize (7.6) off-shell, since for n = 0 or (7.74) the full Yang—Mills analog exists. 1~A The n = 1 affine transformation LI~ 11does not leave ~x) invariant. From (7.69) one finds after use of (7.77) that LI”~f?=_2trF~~D11LIU)A~ = aaK,, where K,,

=

—tr{a,,([V2, V3,] + [V2, V9])T}.

(7.81)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

67

Since (7.81) is not zero, then LI~’~A,, is not a gauge transformation. Furthermore, from (7.70), with much use of (7.77), LI “~f= =

a~(—2 tr(g’~’F,,~LI “~A7)) —tr{LI([V2, V9]

+

[V5, V.1)T}— tr{EIIIa11V~T}

(7.82a)

where

V~=a3,X+~[V~,,~y], =

— ~(i)t

vi).

a2x+~[V2,x]

i07y1) =

—

(7.82b)

i)t

and a2x

=

V3,,

a9x =

Therefore, although

—

V2.

LI”~Aa is

(7.82c)

defined only on-shell, we still have found two different expressions for

LI”~[, namely (7.81) and (7.82a). Equate them to derive a conserved current:

a~(—trEV~)T)= 0 =

riV~.

(7.83)

The associated charge for (7.83) equals that of J~ from (7.79) up to a commutator with Ta and functions of Q~.Presumably the Noether-like procedure to compute (7.83) for higher n will generate currents $(,,~t) related to J~’~ of (7.79) in much the same way as the chiral Noether currents [26] are related to those derived via the Brezin et al. procedure [28]. Finally we examine the conservation of $~,,‘°for n = 0, 1,

a3,J~7”~+ a2~~= 0= aJ~)+aJ~r).

(7.84)

~

That is to say, ~ J~ are conserved and are conserved, separately. A second set of transformations LI~~Aa are defined in the next section. It has been checked for n = 0, 1 that they give 11A,,, that is to say rise to the same currents as LI ~(n)= i ,~‘(n) ~(n)= —i ~(n) ~

13,

=

iJ~’°,

19

~

~(n)

=

j~~)

(785)

_jJ~n)

7.4. A second infinite set of transformations By inspection of ~ in (7.6) and the definition of (1~~) in (7.3), it easily can be seen that there exists another Hermitian combination f1~~~: =

‘i(D~1A(n)j5

—

D~A~

(7.86)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

We show below that the two pieces D_IA(11)D and D_iA~tD separately lead to symmetries of the self-dual equations. So in fact there are two linearly independent infinite sets of infinitesimal transformations, namely LI~~~Aa defined in (7.6) and LI~~~A11 /t

A — (n)’’y — =

y

(n)

— — —

A’(n) A P12

LI~A

1,

ñ(n)

zA

2= D212~~~.

—D9fI~~

(7.87)

In this section we discuss (1) the invariance of D’A~~~D and DlAc~tD and (2) derive an equation satisfied by LI~~~A11 which is reminiscent of the conformal Killing equation, but where the affine connection of Riemannian geometry is replaced with the vector potential A11, i.e., the connection of a fiber bundle geometry. It is this equation which may serve as the bridge to LI~~~A11 in the full gauge theory. It has motivated the search for the new symmetry using a Kaluza—Klein approach [19] described in this review in section 8.4. The commutators of M~”~ and the generators M~’~ of the transformations (7.87) can be worked out using the techniques of the calculations of section 7.2. In general they are equal to linear combinations of the set [45].As stated in section 7.3, however, the conserved currents associated with the second set of transformations ~ are equivalent to those of ~ so no new information about conserved quantities is contained in this extended algebra. What is fundamental to the self-dual equations is the affine algebra C[t] ® SU(N). It may be that in the full non-self-dual classical theory, this fundamental algebra is enlarged to the loop algebra C[t, t’] ® SU(N), whereas the invariance of quantum NAGT may include the central extension C[t, t] ® SU(N) ~ C~,an affine algebra suggested by the dual string formulation. -

A discussion of the currents

-

Similarly to (7.72c), we find that ~ =

is a gauge transformation:

D,,f2~°~

where 1TD). =

(7.88)

~i(D~ TD — D

Here = _~(O) -

So its Noether current, from (7.70) i’,.

=

is

—2 tr g’3~F,,~D (7.89)

3,f1~°~

corresponding to the conserved matrix current j~ where .1,,

fl)

=

—iLl V9,

)~

=

iLl] V2.

=

—~trLf~T):

(7.90)

L.

Dolan, Kac—Moody algebras and exact solvability in hadronic physics

69

Eq. (7.90) is eq. (7.85) for n = 0. Eq. (7.90) is derived from

trF3,9D34’1~°~ = ~itr([a9V3,, V3,]T) tr F3,2D 5Q(O) = ~jtr([a2V3,, V5]T) tr F95DZ11~°~ = —~itr([a2 V9, V2]T) trF93,D9[l~°~ = —~itr([a3,V9, V9]T) tr F22Dj1~°~ = —~itr([a2 V2, V.] T)

(7.91)

tr FZ~D~i1~°~ = —~itr([a3,V2, V~] T)

trF52D5Q~°’ = ~itr([a2V1, V5]T) trF29D3,L2~°~= ~itr([a9V±, V.]T) as in section 7.3. Also, for LI”~A,,,from (7.69) =

—2 tr F~~D,,LIu)A~ = aRK,,

where I~,,=—itr{a,,([V5, V3,]+[V9, V2flT}.

(7.92)

From (7.70), another expression for LI”~9?is =

a~(—2 tr(gTh’F,,~z.1O)A7))

=

—i tr{fl([V5, V3,] — [V2, V9])T}— i tr{ll1(89V~+a~ VV~—a3, ~7~—a2 V~~)}.

(7.93)

Equate (7.92) and (7.93) to derive a conserved current: J~=—itrE~, J~= —itrLl~,

7~

J~=itrLl%

J~=itrE~~”.

Eqs. (7.94) correspond to (7.85) for n

=

(7.94)

1. No new currents are appearing since f

3,, f2, and f~,J2 are conserved separately (see (7.84)). That is to say, we see from (7.79) that all the currents J~which can be defined from the self-dual equations are already appearing as the Noether-like currents of LI ~

1A ~ TheUpon invariance of D~A”°D and D closer examination of (7.3) and (7.6), we see that they are the sum of two sets of infinitesimal symmetry transformations SAa and 15A,, which separately preserve (7.7). For simplicity of notation, define ~(n) D~A~D,

w~’°~ D_iA ~

(7.95)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

70

Then _D2oj(11)t

=

Dyw(n),

~

=

_D9w(n)t,

3A5 = D5w~~~

=

for n

0

(7.96a)

and 6A3, = D3,w(12)t =

—

=

D_la3,A(n1)tD = ~

—D-w~”~= —D’a9A~~~D = — 1aZA~~~D = ~D2w~”~ —D —

6A

=

—

—

for n

>

1.

(7.96b)

1a

2 = D2w~t= D

2A~~~D = a9(D~A(11~)tD)

For n = 0, the first column of definitions in (7.96b) is identically zero. For n 1, the third column in (7.96b) follows from (7.15). Note that both (7.96a) and (7.96b) have the correct reality properties for transformations in an SU(N) theory, i.e. (~A3,)t= —(8A~),etc. The fact that (7.96) are both invariances of F93,+F22=0,

F3,5=0=F9~

is seen from LIF,,~= D,, LIA~— D~LIA,,,so = (D,JI)Z

—

D5Dy)w(11)

(n)tlj — 3,2 — _FL’ [L ~ — .~L’ ~JI yz —1i’ I~-’yz~~(n)tl0 I ~

=

[F3,5,w(hl)]

=

0

0

(7.97a)

0~ I (n)1_O

~r’ 3,2 __FF I. yz,

and 6F 93, + 5F25 = (D9DY + D3,D9 + D2D5 + D5D2) ~(n) =

5F93, + oF22

=

=

~

0

=

(7.97b)

0.

(7.97c) t).

(7.97a) is true since F92 = 0 = F3,2. Eq. (7.97c) follows from (7.97b) since (D11DaW(11))t = D11Dn(w(~~) Eq. (7.97b) is shown below: DaD,,w(n) = [(F 93, + F25), =

w(”)] + 2(D5D2 + D3,Dg)w~”~

2(DsD2w~’° + D~D9~~”~) 1~

=

2D5(D

2A~

+

2D3,(D’ a 9A t~~)J~)

(7.98a) (7.98b) (7.98c)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

=

=

=

2{D’a5a2A~~~D + a2D~a2A~”~D + D_ia2A(11)a5D + [A2, j~_ia2Anj7] 18 1a 1a + fr 3,a9A~~~O + a3,D~a9A~~~D + D 9A~”~a3,D + [A3,,D 9A~’°D]} 2D’{a5a2A (n) + [(15a215_i + DA2D_i), A ~)]+ ~ 1), A(~)]}Ji + [(15a3,1.7_i + DA3,D

(n)

71

(7.98d) (7.98e)

2D1{~ 5a2A~~~+ ~3,a9A~~~}D 0.

(7.98f) (7.98g)

(7.98b) follows from F9~+ F22 = 0; (7.98c) from (7.14); (7.98f) from (7.77a) and (7.16); and (7.98g) from (7.19a) and (7.20). — It has been shown that both BA,, and 6A,, are symmetries of the self-dual equations. Therefore so are their sum and their difference. Their sum is the original set of affine transformations ~ We see

from (7.69a) that ÔA,,

+

OA,,

=

2LI (n)A

(7.99)

The difference is a real gauge transformation: —

OA,, = ~

(7.100)

— w(~~)t)

since W(n)_ ~(n)t is anti-Hermitian, At = —A. But we can also construct LI~A,,which in general is not a gauge transformation, from a consideration of the difference, i (6A3, — iSA3,) = i D3,(w~’° — ~(n)f)

=

2~

and

(7.lOla) i (iSA5 — iSAs)

=

i D5(w~’~~(n)t) = 2 D5f~(11). —

Given the two transformations (7.lOla), where D’~D11(w~~~— ~(n)t) = 0 (from (7.97)), then we can always find a real symmetry transformation from (7.lOla) where LIA3, and LIA5 are given by (7.lOla) and LIA5 and LIA2 are defined by t, LIA = —(A3,) 2 (7.101) is 2LI~”~A,,: =

t.

(7.lOlb)

—(A5)

~i(3A 3,— OA3,) = ~

=

~i(iSA5— 0A5) = D2Q~~~ t”~)= _D ~iD9(w~~)t — w 91~11)

=

~iD2(w(11)t — w~”~) = —D2f2~’°

=

7 102

72

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

Conformal Killing equation [19] The infinitesimal transformations (7.6) LI~~~Asatisfy 11 the background gauge condition 11LI “~A D 11= 0 -

(7.103)

(7.103) is true since for self-dual solutions,

Dnzl(h1)A,, = (D9DY — D3,D9 + D2D5 =

—

[(F93,+F25),(1~~~]=0.

(7.104)

Since i11)A,, is a symmetry of (7.1) and LIF11,-~= D11 iA~— D~ LIA,,, then ~ ó~11hiDaLI~Aii =

0 and

(all&h3 —

‘O.

Here

=

(7.105)

is the antisymmetric, anti-self-dual tensor o~”= (1/(2i))(a~ä’~8k”), where a~= (_~ff11,I),

I~’~”

—

(ioa, I) or in complex coordinates

=

aã~~-~(I—o~), 9

=

=

~(I

+

u3),

a2_ã2_~(oJ+io2)

a2

=

=

~(u’

—

i~r2).

(7.106)

a

Define

f,, =

D~[2~,

f

9=

D9(1~,

f~ = —D2(1~~~,f± = —D5(1~.

Then, a11&~(D11f~ + D~~fa ~ —

f) =

0.

(7.107)

Eq. (7.107) is reminiscent of the “conformal Killing equation” a,,j,, + a~f,,—~g,.~a -f= 0 which gives LIx,. = f,.(x), the fifteen four-dimensional space-time conformal transformations. Now however the covariant derivative with the affine connection which would appear for curved space is replaced by the gauge covariant derivative a11 + [A,,,. From (7.98), D11D,,11(h1)

0

(DIDZ + D3,D9)(1(n).

So (DSDZ + D3,D9)

=

(DZDS +

D9D3,)[l~°= 0.

(7.108)

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73

Therefore

.f

D,,f~+D~f,,~

0

(7.109)

for

a, /3

=

(y, 9), (z, 2), (y, 2), (9, z).

These calculations are made in an effort to get some handle on the nature of the new symmetry transformations. If an equation can be identified for which LI~~~Aa are solutions, such as (7.107), it may be useful in removing the self-dual condition. A Killing equation is a signal of a geometric symmetry or isometry. Section 8.4 is a brief review of the relevance of Kaluza—Klein fiber bundle geometry for this problem. 7.5. Affine transformations on the Wilson loop

We now ask does the existence of the new symmetry group lead to a choice of variable which reflects the symmetry more naturally than A,,,(x) and may therefore be more useful to discuss non-perturbative effects of Yang—Mills theory. The answer is that the gauge invariant quantity lI’~tr~=trPexp(~A.d~)

(7.110)

(restricted to the self-dual sector) carries a representation of the affine algebra C[t] ® SU(N). Under any transformation A,, A,. + LIA,,, ~/i + LIt/i, —*

LI~= ~ J ds ~

—* ~/i

LIA,.(~(s))~,,(s) ~s:x

(7.111)

where ~,.(0)= ~,,(1)= x,.. This follows from t/J(s) P exp( J ds’ d~,,A,.) ~— t/i(s) =

çfi(s)A,,(~(s))

LI i/i(s) = LIt/i(s) A,, + i/i(s) LIA,, =

d tK’(s)) i/I(s) + i/i(s) LIA,, —LIt/i(s) (~—

((LIi/i(s)) i/i_I(s)) = i/i(s) LIA,.i/i~1(s)

(7.112a)

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in hadron/c physics

so

(LI~(s))~~i(~)

=

J

ds’ ~(s’) LIA,. (~(s’))~‘(s’)

which is (7.111). Thus for LIA,,

(7. 112b)

=

LI~ç~ ~

and from (7.31), i~(~+LI 11)—LI 11—LI

P+LI~1’)+LI~1’= —CCbaLI~’”~1’.

(7.113)

Since

i/i is gauge invariant we recover the algebra C[t]®G exactly [8]. The enlarging of this algebra to include all local gauge transformations (7.32a) which was necessary in the discussion of the gauge non-covariant field A,,(x), is not needed here, because gauge transformations on i/i are zero, i.e.

80i/i=0.

Also, LI~11)i/iprovide the construction of four-dimensional Polyakov loop space currents for self-dual

potentials: ~

( \ s,,” /

f(n)(F~1 /-‘

‘~—

0

\L~J’sJ—

-

a

where 11

J~)([~],s) =

=

and s

J

+

[~i

~s)’

~]‘

(7.114b)

A

ds’ i/’~LI(n)A,.(~(sl))~(s’)

i/’s’:x

(7.114c)

0, 1. Because of the self-dual restriction on A,.(x) in A11[~], (7.114c) is tied to four dimensions, and it is not possible with this expression to make contact with the three-dimensional Polyakov construction discussed in section 6. What has developed in our investigation of the hidden symmetry of Yang—Mills theory is a series of non-overlapping windows in which we can exhibit the new invariance. The loop space analysis by Polyakov of course makes no reference to the self-dual restriction. All these approaches are suggestive that the symmetry exists in the full four-dimensional theory. It is heartening to see that the invariance Polyakov found in his three-dimensional, loop space formulation of Yang—Mills, namely that of the chiral models and which is now known to be an affine algebra C[t] ® SU(N), is the same as the one present in the four-dimensional loop space formulation albeit restricted to the self-dual configurations.

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7.6. Parameters of the instanton solutions All finite action solutions of (7.1) for the gauge group SU(2) can be grouped into classes having I = 8ir2q where q is the Pontryagin index and takes values q = 0, 1, 2,. . . ~. For any given q 1, the solutions in that class are parameterized by (8q 3) constant (global) parameters. For q = 0, the F,.,. = 0 so A,, is a pure gauge. Since the transformations (7.6) are valid for self-dual solutions they can be applied to these q-instanton solutions. For q = 0 (pure gauges), then D = D and J = 1 so J~a~f = 0 etc. and —

fl(n±l)

dz’

J

~ 3,A(11)=

J

dz’

a3,A~~~= 0

since

a3,A~°~= 0.

Therefore, when evaluated on gauge potentials which are pure gauges LI(11)A,, = 0. For q 1, ~ will not be zero. Under any infinitesimal variation A11 A,, + LIAa, the change in the action is given by *

4x tr(F,,,,.D,. LIA,.) LII = —21 d =

(7.115)

—21 d4x {a,,(tr F,.,.LIA,.)+ tr D,.F,.vLIA,.}.

Therefore the infinitesimal change in the action for solutions (D,,F,,V LII = —2

J

d4x a,,(tr F,.,. LIA,.).

=

0) is (7.116)

Thus, if tr F,,,.LIA,. vanishes on the surface at so that (7.116) is zero, then the action I’ = 8ir2q’ for A’ A,, + LIA,, will not differ from the action I = 8ir2q of A,, infinitesimally. That is to say, the infinitesimal transformation will at most shift the 8q 3 parameters. The action and equations of motion D,.F,,,. and F,,,. = ~ are all invariant under local gauge transformations and the global fifteen parameter conformal group of transformations. For the q = 0, 1,2 instanton solutions all the parameters can be changed by these known transformations. For q >3, we need 21 or more infinitesimal transformations to move all the parameters. For arbitrary q, we need 8q 3 infinitesimal transformations. If in a given q-instanton solution, the centers of the “instantons” are far apart, then each “one” is described by four position parameters, one scale parameter, and three gauge orientation parameters. (Since the gauge transformations are local, and the q “instantons” are far apart, then the three local SU(2) gauge transformation functions induce 3q global gauge orientation parameters.) When the “instantons” are close together in a q-instanton solution, however for q 3 we know of no transformations which shift all the parameters. As q gets larger, more and more transformations are needed. The set LI ~ has an infinite number of transformations, and is therefore a candidate symmetry group which in addition to the conformal group in four dimensions and the gauge group could generate all the parameters. Whether or not the affine transformations do this, i.e. act transitively on the instanton solutions remains an open question. —

—

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L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

7.7. Complex Yang—Mills in the J-formulation The discovery of the transformations (7.6) was made possible by the work on the chiral models [12]and then by the old observation [46]that the self-dual Yang—Mills equations in the J-formulation (7.20) were similar to the two-dimensional (chiral) equations of motion (5.5). This is a completely different connection between the gauge theory and the chiral models from the one made by Polyakov [18]between loop space Yang—Mills and the chiral theory, which is discussed in section 6. The latter makes no reference to self-duality. In this section, we review how the transformations (7.6) were found [8] and summarize the simple form they take when the gauge group of the Yang—Mills fields is allowed to be complex [35, 45, 47], i.e. SL(N, C) instead of SU(N). Because of the similarity between (7.20):

a9(J’a5J)+a2(J’a2J)=0

(7.117)

and (5.5) a,,(g_ia,,g) = 0

(7.118)

the hidden symmetry transformations (5.6) form

~

=

—gA~~~ suggested

transformations for (7.117) of the

(7.119)

8(11)J= _fA11.

Then, from (7.25) t~1(J~’a =

8

_~3,fl(11),

(7.120a)

2f) = —~l12A (n)

so A~ in (7.13) is defined such that a9~~A(11)+ a2~5A’~”~ = 0,

(7.120b)

see (7.15) and (7.19). For real SU(N) gauge theory, (A~)t=_A9,

(A2)t=_A2,

so

Dt=D~I

J~DDl=Jt,

see (7.13). Therefore transformations which leave the gauge potentials in SU(N) must induce a Hermitian LI(11)J. Clearly (7.119) iS~”~f = _JA(11) is not Hermitian since JA(11) A11)tJ. In order to make it Hermitian, (7.119) is modified to be LI~~~f= _JA(~~_A111)tJ. Now (LI (n)f )t

=

LI~”~J, so

(7.121)

transformations

1—~--— (7.122) ~ LI~~~D=—~——17, LI~°D’=D LI~J which imply (7.121) will give rise to real transformations on the gauge potentials through (7.14). These —

77

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

result in LI~A,,of (7.6). (Note that for any matrix M, LIM’ = _M_i LIMM1.) The fact that —A (~~)tfis also a symmetry of (7.117) or its equivalent form (7.21) follows from (7.25) and (7.19b) [8]. For complex gauge groups however, the transformations Ô~’~J = —fit ~a’~ and (iS~J)t= —A ~tf may be considered separately. For the combinations O~J—iS~J=—fA~ for n1 iS~J+ (iS~)J)~ = JTa 3~J~(iS(;11)J)f=_A(;n)itJ

+

TaJ for n—1

the generators

Q~= Jd4x~(;)J(x)~J~X) —

are computed to be [45,47] [Q~i)

(7.123)

Q~,m)]= ~

This is the loop algebra C[t, t1] ® SL(N, C), (without a central extension).

8. Reasons for a new symmetry of the strong interactions

8.1. Vertex operator of the dual string model In the late 1960’s, a model to describe the scattering of hadrons was invented without reference to the non-Abelian gauge theory. The N-point functions GN(ki,. k 11) for N scalar particles are given in terms of a vertex operator V0(k, z). This is a fundamental object of the string model [48].The scattering amplitudes associated with these N-point functions satisfy a list of general properties such as Lorentz invariance, crossing symmetry, analyticity and factorization. General S-matrix elements are calculated using the factorization condition, and they provide an approximation to hadronic scattering in which all resonances are infinitely narrow and the Regge trajectories (the square mass versus spin plot) are linear. The spectrum of the dual model consists of an infinite number of states, which are conveniently described in a Fock space generated by an infinite number of creation operators a The commutation relations of the relevant operators in the theory are, for m, n = 0, ±1,. .

.

,

~.

[a~,, ar,,]

=

mg”~5m,_,,.

(8.1)

The annihilation and creation operators in these variables are a~= \/rn a~and a~_m= Vrn ~ for m = 1, 2 The momentum operator is defined by a~= V2 a’p~ and the position operator has [x~,p~]= ~ The vertex operator is then given by 2~~’k1~ exp(_V~ k,, ~ z~”). (8.2) V0(k, z) = exp(V~~k,, n1 ~ ~ 2”) exp(ik x)z ~ n .

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L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

The connection between the dual model and Kac—Moody affine algebras is seen as follows [10]. Consider the string moving in just one dimension (~x= 1) and with conformal spin a’k2 = 1. Then V 0(k, z) can be written as V0(y, z) = exp(~ ~— y(—n)) exp{y

+

in z

exp(— ~

-f}

~—

y(n))

-

(8.3)

Here, y(n)= V2a’k

y = ik ~x,

.~,

k .p(2~’).

~=

(8.4)

Define coefficients X11(±)from V0(y,z)=~z”X11(+), and V~(y,z) = ~

z” X~(—).

(8.5)

It is these coefficients together with y(n) which carry a representation of the affine algebra. From (8.1)—(8.5), then [y, ala7] —2 and the affine algebra sl(2, C) ® C[t, t l] ~ C2 ~l(2,C) appears as [y(n),

y(m)]

=

2niSn,_m I

[X~(+),Xm()] =

y(n +

(8.6a)

m)+ niS n-tn’

(8.6b)

[y(n), Xm(±)]= ±2Xn+m(±)

(8.6c)

and [X11(+),Xm(+)] = 0,

[X11(), Xm()]

=

0.

(8.6d)

A basis for sl(2R) or sl(2, C) is /0 e=~0

1\

o)’

/0 0\

~

/1

o)’

Then [h, e] = 2e, [h, f] = —2f, [e,J] commutators for sl(2, c) are

0

h=~0 ~ =

-

2 = 2. For x(0), y(O) generators of sl(2, C), the

h, and Tr ef = 1, Tr h

[x(m),y(n)] = [x(0),y(O)] ® t”~m+ m iS,,,,

_,,

(Tr x(0)y(0)) 1.

(8.7)

Therefore, from (8.6), y (n), X,,(+) and X,,(—) are the generators of §1(2, C) where y(0) = h, Xo(+) = e, X 0(—)=f.

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79

For higher Euclidean dimensions ~ > 1, ~ = ~ sl(2, c) is replaced by a complex semi-simple Lie algebra [10]. In the QCD picture of hadrons, the quarks are held together “inside” the hadron by the gluon color flux. Since Yang—Mills is a non-Abelian (self-interacting) gauge theory, these lines of color force do not spread out in space as in an electric dipole, but rather are confined in a narrow tube or thick string. It can be shown that in the N cc limit of an SU(N) non-Abelian color group that the tube of confining flux becomes infinitely thin thus resembling a real string. The specific dual model which corresponds to this field theory limit has not yet been made precise. Also, the dual model of eq. (8.3) has tachyons, and it cannot describe baryons. Nonetheless, the old string models provide an extremely useful intuitive or physical understanding of a possible mechanism of quark cofinement. More recently, supersymmetric string models have been formulated. These include fermions and do not have tachyons. Massless spin two states still appear in the spectrum, and the models are interpreted not as hadronic strings, but as a theory of the fundamental particles, quarks, gluons, gravitons, etc. They have the so far unique distinction among gravity theories of having controlled ultraviolet divergences. But they imply that space-time has dimension ten. In order to describe a four-dimensional string, a Kaluza—Klein ansatz is imposed to spontaneously compactify the theory to Minkowski space with six very small internal dimensions. The “local” field theory limit is regained as the Regge slope a’ and the size R of the internal dimensions go to zero, with Va’/R held fixed. Therefore, the field theory is a low energy limit of the string theory. The two supersymmetric dual models N = 1 and N = 2 reduce in this limit to N = 4 supersymmetric Yang—Mills and N = 8 supergravity field theories respectively. Whereas N = 4 Yang—Mills is ultraviolet finite [49],the N = 8 theory may contain divergences starting in three or seven ioop calculations. As a low energy approximation, however, it could be physically reasonable. A vertex operator of a supersymmetric string is given by -~

V(k, 2) = k,, H~(z)V0(k, z).

(8.8)

The operator H~(z) is defined below. Since the string is supersymmetric, the Fock space of states is enlarged by anticommuting creation and annihilation operators. In addition to (8.1), there is now also {br, b~,}=g~~iS,,_m

(8.9a)

{a~,br}0.

(8.9b)

Here b~i1= ~

for I

=

H~(Z)= ~b~z’.

~,

~

Then (8.10)

The coefficients of these supersymmetry vertex operators have not yet been analyzed in terms of an infinite parameter algebra. Their similarity to the original string, however, indicates the presence of a related affine structure presumably connected to the finite ultraviolet behavior. Although the supersymmetric strings and field theories have had much success as finite theories including gravity, their extra symmetry has not yet provided a guide to formulating a non-perturbative approximation. For example, armed with the stunning result that N = 4 Yang—Mills is ultraviolet finite

L. Dolan,

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Kac—Moody algebras and exact solvability in hadronic physics

[49],we are still unable to calculate quantities outside of perturbation theory. For hadronic interactions at present day energies, as well as for gravity at energies of the order of the Planck mass, the coupling constant is large and perturbative estimates cannot give a quantitative analysis. But, motivated now by the physical concept of confining strings of color flux, we are led to another property of strong interaction theory a Kramers—Wannier-like self-duality between color electric and magnetic confinement. —

8.2. Kramers— Wannier self-duality of 4-dimensional SU(N) gauge theory Since one is interested in calculating in the non-perturbative region which controls quark confinement, it is useful to recall a similar phenomenon also present in gauge theories namely superconductivity and magnetic monopole confinement. This effect is most simply illustrated in electromagnetism. It can be generalized to the non-Abelian theory where monopoles occur naturally. With the introduction of monopoles, the gauge theory becomes invariant under the transformation E~—~E~—=B and 1 B1—~.B~=—E1 on the fields and e—*—g, J~fm and g-+e, J,,,-’~Jon the coupling —

constants. Except for the coupling transformation, this would be a symmetry of the theory. Because the field transformations result in the same theory with the coupling constants exchanged, this is similar to a Kramers—Wannier self-duality transformation discussed in section 4. Because of the existence of this transformation, once the confinement of magnetic monopoles is illustrated in the theory, a dual transformation of the result will describe electric, i.e. quark confinement in the same theory [50].In this section, following ref. [50],we review (1) superconductivity, i.e. magnetic confinement in an Abelian theory, (2) the dual “symmetry” of the equations of motion, (3) the ‘t Hooft language of dual operators in which the dual transformation is most naturally discussed, and (4) how these dual operators are suggestive [22] that the theory then possesses an infinite set of symmetries. We stress that this formulation of the four-dimensional non-Abelian gauge theory makes no reference to the F,,,. = ~ self-duality condition discussed in section 7. Kramers—Wannier self-duality and F,,,. = ~ self-duality are two distinct concepts which unfortunately are described by the same name self-duality. One similarity in the two concepts is F,.,. = ~ means that E~—F01 = ~EjjkF~k B, and the Kramers—Wannier transformation involves E, B,. But the existence of the K—W transformation in the gauge theory is in no way restricted to the F,,,. = ~ sector of the theory. And therefore the implication of an infinite class of symmetries by the K—W transformation is evidence that these symmetries should be present in the full Yang—Mills theory, not just the self-dual set of section 7. In the absence of magnetic charges and currents, Maxwell’s equations of electromagnetism are —~

a,,F,.,.=—J,.

(8.lla)

i.e. V~E=J°,

VxE=-B

V•B=0,

VxB=E+J.

(8.llb)

The Meissner effect inside an Abelian superconductor is that B = 0. Classically one says that inside a superconductor J = crE, where o-, the electrical conductivity, is infinite, so since J can be measured to be finite, it must be that E = 0 so —v x E = B = 0. If no B field is present in the superconductor to start

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

81

with, then E = 0 implies B = 0 and all magnetic flux is excluded from this region. Microscopically, J = o-E is replaced by J = nqv for n, q, v the number, charge and velocity of the electrons present. So it is J = (nq2/m)E, since my = qE in an electric field. J instead of J is proportional to E. For Bext outside the superconductor large enough, the finite number of electrons inside does not keep B = 0, but rather narrow tubes of quantized magnetic flux appear, each one having cbB = ~ A . dx = f B . da = 1r/e. A heuristic quantum mechanical derivation of irle can be given in terms of spontaneous symmetry breaking. An effective Hamiltonian inside the superconductor is given by ~~/C = ~(E2+ B2)+ ~I4I2 +

(a,. — 2ieA,,)~I2+ V(I~)

(8.12)

where 4(x) is a complex scalar field representing a bound state of two electrons with opposite spin, the Cooper pair, formed by the phonon interactions between electrons. It has charge —2e which appears in (8.12). Under gauge transformations, a symmetry of (8.12),

4, —~e’~4, A,,

(8.13a)

a,,A (x).

(8.13b)

—~

The potential V(~frI)has the form V(I4,I)= A(4,4,*

—

(8.14)

v2)2

where v is the value of I4,(x)I where V(I4,I) is at a minimum. In the superconducting phase, i.e. for T the temperature less than some critical temperature T~,then v 0. Therefore, quantum mechanically, the observable fields will be a shifted 4,(x), v exp{if(x)}

(8.15)

where f(x) is some phase. If exp{if(x)} is chosen to be exp{if(x)} = (x + iy)lr

away from the origin (and constant in

(8.16) 2),

then the contribution of this shift to f d2x ~‘ is

f (2rrlr) dr v2.

Thus for a ground state configuration, (8.17) so that E = f d3x ~‘ conductor is

=

0. Then from (8.17), the magnetic flux through the x—y plane inside the super-

cbB=JBda=rrIe.

(8.18)

Furthermore the flux is channelled in a tube around the origin, since the shift (8.15) induces a photon

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mass term m2 = 8e2v2 in ~‘, so that the solution for the gauge fields is E~B e_mH. Thus it is gauge invariance which is the spontaneously broken symmetry. The beginning and end of such a flux tube must absorb irle magnetic flux. Thus we are led to

postulating the existence of monopoles, which have such flux, and can thereby be attracted to each other or confined by the flux tube (its energy is proportional to its length). Dirac introduced monopoles into the Abelian theory, rather artificially, by showing that singular gauge field configurations could be made physical, since the singularity can be gauge rotated away from any given point. His monopole has

B=~

(~)

(8.19a)

and

g

(

z+r

A=~—(

2~2) ~ 2 + y2) —~0, z which is singular at (x

—~

r, (r2

.19 x~+ y2 + z2). This singularity can be removed by a gauge

transformation A-~A’=A+a

1A

(8.20a) (8.20b)

2~’tl/J, =

e

where A = [(tan~’ylx) + 2m1r] nl2e. With this transformation, (x2 + y2).—* 0, z —~r:

i/i

(8.20c)

and A,. are single-valued and the gauge potential A’ is regular as

A’=(~-~) 2 2 2~r 2e (x +y) ~ 0 =0

(8.21)

when gli-r=n/e

or

g=irn/e.

(8.22)

From (8.22), we see that a monopole of strength g can absorb exactly the magnetic flux at the end of a flux tube inside a superconductor. Kramers—Wannier-like self-duality can now be exhibited in the gauge field equations (8.11). With the

L. Dolan, Kac—Moody algebras and exact solvability in hadron/c physics

83

assumption that monopoles exist, (8.11) is modified to read V.E=eiS(x),

VXE~i3Jm

V.B=giS(x),

VxB=E+J.

(8.23)

Eq. (8.23) has the property that a theory with (E, B, e, J, g, Jm) is equal to the theory with (B, —E, —g, ~Jm,e, J). Therefore, since magnetic monopoles are confined by the magnetic flux tubes which appear due to spontaneous symmetry breaking driven by the Cooper pair bound state electrons, in the dual picture electric monopoles (quarks in the non-Abelian theory) will be confined by electric (color) flux tubes which arise from a magnetic Higgs mechanism, i.e. the Cooper pair is replaced by a bound state of monopoles. Due to the quantization condition (8.22), it necessarily follows that a perturbative treatment of the electrons (quarks) as elementary fields e < 1, requires a non-perturbative treatment of the monopoles g> 1, etc. So it is difficult to proceed in the quantum theory, due to the lack of a non-perturbative approximation scheme. ‘t Hooft’s idea was to concentrate directly on the formation of electric and magnetic flux tubes. He introduced path dependent operators A(c) and B(c) which measure the magnetic and electric flux respectively:

A(c) = ~tr P exp(i ~ A dx) = exp(i4,8)

(8.24a)

B(c) = exp(i4,E).

(8.24b)

The dual operator B(c) is defined by its commutation relation with A(c’)

A(c) B(c’) = B(c’) A(c) exp(2lTin/N),

(8.25)

where N is from SU(N) and n = 1, 2,... is the number of times the closed path c’ encircles c. Furthermore, the operator B(c) can be derived from A(c) also by a partial Kramers—Wannier Z(N) dual transformation. That is to say the operator A(c), an element of SU(N), can be expressed in terms of matrices of the factor group SU(N)/Z(N) and of the group Z(N). On the lattice, a Kramers—Wannier duality transformation on the Z(N) piece then changes A(c) to B(c) [51]. Also, in a treatment of the whole theory, the four-dimensional SU(N)-gauge theory partition function has been noted to reflect a self-duality property similar to the two-dimensional Ising model [50]. In order to generate conserved charges with the result of (4.16), the fundamental operators must satisfy the extra condition [B, [B, [B, B]]] = 16[B, B]. If we define B = A(c) and B = B(c’), then 2(c)(1 exp(2iriN/n))2[A(c), B(c’)] —

[A(c),[A(c),[A(c),B(c’)]]] = A

.

(8.26)

That is to say, with this choice of B and B, the first charge is not conserved and we cannot generate the infinite set. This may be due to the fact that B(c’) is only a partial dual of A(c). In this way, we can think of the extra condition as a guide in choosing the correct duality transformation. It of course must also have the property that A = A.

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The self-dual formulation of four-dimensional SU(N) gauge theories in terms of A(c) and B(c’) operators does not yet fit exactly into the theorem of (4.16) which relates self-duality to an infinite set of conserved charges. But the existence of this relationship is suggestive that an infinite parameter symmetry group may also be present in Yang—Mills theory. 8.3.

Loop space formulation is not self-dual

We recall from section 6 that the evidence found in Yang—Mills 1oop space for the existence of an infinite parameter hidden symmetry made no reference to the self-dual restriction of section 7. The purpose of section 8 has been to list three independent reasons which suggest that the new symmetry is not just a reflection of the special properties of F,,,. = ~ but rather is an invariance of the complete Yang—Mills theory which reduces, in the case of self-dual fields, to the special form of LI derived in (7.6). A suggestion [52] has been made that the four-dimensional Yang—Mills equations do not have a complete set of integrals, since there is no complete set for the subsystem of equations obtained from = iS”~’.This choice results in the ansatz A~= 0, a,A~= 0, A~= O~f11,f’= x(t),f2 = y (t),f3 = 0 and O~O~ all fields dependent at most on t only. Therefore conserved currents would have a 0J0 = a~J,= 0. Clearly the non-local set of hidden symmetries of Yang—Mills suggested in this section do not have this property, that is to say the hidden symmetries will rotate gauge potentials out of the above ansatz. Also the infinite set of hidden symmetries suggested in this section may be a smaller infinity than is needed for a complete set of integrals. In this case the invariance would not lead to exact integrability in the standard sense; but nonetheless it is always useful to know the symmetries of a theory, to restrict S-matrix elements, etc. With this philosophy, it seems reasonable to use (7.6) as to find ~ mayfor theory. To 1A,, were pointed outa guide in (7.8) which bethe thefullkey for the this end, several properties of LI~~ generalization. This will be briefly reviewed in section 8.4. 8.4. Kaluza—Klein to generalize LI ~

for full Yang—Mills

Non-Abelian Kaluza—Klein theories are studied with respect to using the invariances of multidimensional general relativity to investigate hidden affine symmetry of the four-dimensional Yang—Mills theory. In this program, a system of first order partial differential equations is derived for the new symmetry transformations. The complicated non-linear non-local structure of D and D which appears in (7.6) is seen to arise naturally in the solution of the characteristic curves of the following system of equations. This method may thus be the key in extending ~

Kaluza—Klein theory starts from a multi-dimensional Einstein—Hilbert action

J

d~”
(8.27)

In any number of dimensions, the fundamental field is the symmetric metric tensor g,~,.(z);and g(z) = det g,,,.(z). R(z) is the scalar curvature R(z) = g”~R,~,.(z), the Ricci tensor is

R,,,.

=

~

—

a

0F~,.+ r’~,,f’;— ~

(8.28)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

85

and the affine connection is defined to be the Christoffel symbols ro

1 —

2g

po(~

+

~v,.g,,,.

— ~

u,,,g,.,.

(We will be able to work in a coordinate basis.) Eq. (8.27) is invariant under general coordinate transformations: for arbitrary infinitesimal transformations f~(z), z~’—~ z’~+7(z).

(8.30)

LIg,.,.

(8.31)

Then ~

2)g(z’) The is 1— given f,.;,. ~,.f,. — 1J,,. Since gis a The tensorscalar density, wherecovariant D detderivative az~laz’~ a .f.byTherefore LIVIg~ = a,.(f~V~gI). R g(z)—* has LIR(1/D = 7 a,.R. Under general coordinate transformations,

LI(\/~g~R) = a,,(f~V~jR).

(8.32)

For simplicity, we will restrict the discussion to the study of the SU(2) four-dimensional gauge theory, so that the internal dimensions K = 2, 3 or K >3 depending on whether the K-dimensional “internal space” d’~yis (1) the manifold of the coset space SU(2)/U(1), (2) the group manifold of SU(2) itself or (3) the manifold of some n > 3-dimensional space whose set of Killing vectors contain those of SU(2) as a subset. That is to say, each of these spaces has (at least) three Killing vectors K~(y),a = 1, 2, 3, described below. (Let z~= {xm, y”} where ~x= 1,. . .4+ K; m = 1,.. .4; a = 1, . . . K.) The variables y” parameterize the internal manifold which is assigned a metric y~$(y). The Killing vectors are three infinitesimal transformations on ya y” + cK~(y)which close the SU(2) algebra. They are defined as solutions to —~

K”~a,,K~— K~a,,K~= ~

(8.33)

Kaa;~+K~a;a0.

(8.34)

and

c is a constant which has the dimension of length. The covariant derivative of (8.34) is with respect to the metric -y”’~(,y).The metric y”’9(y) is restricted by the requirement that solutions to (8.33) and (8.34) exist. One choice is the “natural” metric y~$= ~ Eq. (8.34) is then automatically satisfied for any K,,a. For any K~,y~ = iS~= iS~ = K11,,K, 5ab since K,,~= y”~K~a = K~(K~K~a) 3,,. If K= 3, then K”,,K,,b = and K”~is square. A non-Abelian Kaluza—Klein ansatz is g,,,.(z)- (~mn(x)C2A~’n(X)A~@~)hab(Y) ~cK~(y)A~(x) —

cK~(y)A~(x) _y,,p(y)

~835

86

L. Dolan,

Here hab

Kac—Moody algebras and exact solvability in hadronic physics

and gmn(x) is the metric of the four-dimensional space-time manifold.

K~(y)K,,b(y)

Substitution of (8.35) into (8.27) using (8.28) and (8.29) gives =

\/~(x)IY(Y)I {~(X)_~F~fl(X)Fmnb(X)hab(y)_~(y)},

(8.36)

see section 8.5. R is the four-dimensional curvature tensor of ,~mn(x),R is the K-dimensional curvature 3ab and the Einstein—Hilbert tensor of y,,~(y), and F~’mn= a~A~ — ~ + EabcA,nA,,. For K = 3, hab = Lagrangian density itself is related to the four-dimensional Yang—Mills theory: V~R=Vy{V~(R+c2~yM)_VI~R}.

(8.37)

Under arbitrary coordinate transformations z° —~z’~+ f~’ (z), then xm

y11 +f11(x, y). Note that f,, LIya~=

LIgmn

=

g,.,.f~.Since LIg,.,. = f,.,. +f,.;,,,

-+

xm

+

ftm (x, y)

a~j,,+ a11f,,, —

LI(CKaaA~m)=

CK 11aFm””fn +

aajm

+

+

y11

—~

(8.35) implies

—(a,,f~+ a~f,,— 2F~f

7) 2E~inJk+ cK~A~,, a,,f,,

and

cK~A~,, a,,j,,,

af,, — cA ~,{ y~f7a~K,,11+ an(’~~fy)Ksa}.

(8.38a) (8.38b) (8.38c)

Here fm ~f,,, + ~ That is, we find that only fa and the combination Tm appear in these expressions so that instead of fa and fn, we can equally well consider fr, and fn as independent functions. To study the symmetry transformations of Yang—Mills theory in Euclidean space, we set g,,,,, = and look for particular coordinate transformations fm and f,, which are solutions to (8.38a, b) when LIgmn

=

and

0

LIy,r,~= 0.

Since any coordinate transformation is a symmetry of

LI(\/~g~ R)

= \/~g~ c2

~

=

a,.(f~VIgIR).

(8.39)

R, solutions to (8.39) imply from (8.37)

\/~g~

(8.40)

One solution of (8.38a, b) and (8.39) is

Tm(x, y) =

Jm(X) =

f,,(x, y)

CK11a(y) E”(X).

am + W,nnX”

(8.41a)

and =

(8.41b)

Here am and Wmn = t0nm are constants, and ~(x) are arbitrary functions of x. With this choice, (3.88b) becomes the Killing equation for translations and 0(4) rotations: LI~mn=

amfn(x) + anfm(x) = 0.

(8.42)

L. Dolan, Kac—Moody algebras and exact solvability in hadron/c physics

87

And from (8.38c), the associated transformation on A~(x)is found: LI(K,,,,(y)A~(x))=

K,,aFm””Jn — cA~(K~11~K,,,, + K,gaô,,Kfl~’+ Kaa8mg”

(8.43)

or LIA~m=

Fm”fn +

am~’+ E,r,,,CA~,,,EC.

(8.44)

Eq. (8.43) follows from (8.33) and (8.34). Eq. (8.44) gives the standard space-time translations and rotations and the internal local gauge symmetry of SU(2). Special conformal transformations and dilitations can easily be added to (8.44) by noting that the Weyl transformation is also a symmetry of ‘~‘YM in curved space: LIwgmn~A(x)gmn(x);

~

(8.45)

Therefore, since (8.42) is a symmetry of LIgm,, + LIw~mn=

—

an invariance of

~

in flat space is given by

amj,,(x)+ anfm(x)+ ,~mnA(x)= 0.

The trace of (8.46) implies A

amjn + oJ,,,

\/~g~ ~

= —~t9. f,

(8.46)

so (8.46) is

~gmna J= 0.

(8.47)

Eq. (8.47) is the “conformal Killing equation”. It has more solutions than (8.42) since zero. They are =

a~+ wnmxm + cx,, + c,,x2 — 2x,,c x.

a .f need not be (8.48)

Now, (8.48) together with (8.44) gives the most general known symmetry transformations of SU(2). Note that (8.41b) implies y”

ya

—

cK”,,(y) (E”(x)— ! A’~(x)).

(8.49)

For consistency, i.e. when we consider these transformations, (8.49) means that function of xm in the K—K ansatz (8.35). This dependence is given by [53]

ay”/ax”

=

c K~(y)(g’am,~)°.

y”

must already be a

(8.50)

The integrability condition is satisfied since (a,,am ama,,)y” = cK”,,F~,,,[f1ak~]= 0. g(x) is a group element of SU(2): ~ = exp((o~/2i)A’3(x)), ~1ak~is an element of the algebra, infinitesimally ~~1akg (~,.aJ2~)3kAa ~o (g1amgy’ akA’. From (8.49), —

amy”

-+

amy” — CK”a3m(E” J. A”) —

which is consistent with (8.50) for A”

=

~

—

T A”).

L. Dolan, Kac—Moody

88

algebras and exact solvability in hadronic physics

New transformations We now look for new solutions of (8.38) and (8.39). Instead of (8.41), let fm(X, and y” and f,(x, y) = 0. With this choice, (8.38b) becomes LI,~mn=

amfn(x, y)+ cK”aA~,a,,f,,+ a~f~(x, y)+ cK~,,A,,aa,,f,,,

y)

be a function of x”

0.

(8.51a)

/+ cK~A’”’ajk) = 0.

(8.51b)

=

In analogy with (8.45)—(8.47), we can modify (8.51a) to be am/n +

cK~Aa,,,a,,f,, + an/rn

+

cKnaAana,r,frn

— ~gmn(a

-

This is a system of first order partial differential equations. Given a new solution of (8.51b), we can find from (8.38c), the associated transformation on KaaAa,,,: LI (cK~A~,) = ~cKaaFrn”fn + an/rn. Eq. (8.51b) is a system of ten first order partial differential equations which can be solved by the method of characteristics. First look at the four diagonal equations m = n in (8.51b). In order to establish contact with what is known for self-dual potentials, we relabel the above notation such that xm —~(y, z, ~), the coordinates of the four-dimensional space-time base manifold and y~ —~w” the coordinates of the internal space. Then the four diagonal equations of (8.51b) are ~,

a~f~ + K~(w)A~a,,f~

0

=

a~fg+ K~(w)A; a,,fg a~f

2+ K~A~ a,,f2

0

(8.52b)

U

=

a5f5 + K~A~ a,,f5

=

(8.52a)

(8.52c)

(8.52d) 2. The isometry group of C2 contains S0(4) which has a subgroup For thethe internal space we choose SU(2), isotropy group, whose C Killing vectors can be expressed linearly in w”. Since we are on C2, the index a which labels the internal space is written ~ = (w~= wz = Z, w~= Y, w2 = Z), and the symmetric metric y”~= = = 1 = y~ = 7z2, with all other components zero. The subgroup of Killing vectors which form SU(2) are given by the complex two-vectors =

0.

‘~/,

K~(w)= ~(Ta)~pW~,

where a, /3

=

Y, Z; a

=

1, 2, 3.

(8.53)

From (8.53), K~(w) (Kna(w))*

=

Ta)n~(ijI3

where a, /3

=

Y, Z; a

=

1, 2, 3.

(8.54)

The notation in (8.54) is as follows: Ta = = (a.~~/2i)* ~ = w’3 for /3 = Y, Z. The method of characteristics solves a partial differential equation by the following method. For a function of two variables f(y, w) satisfying a partial differential equation

a~f+B(y, w) ~

=

0,

(8.55)

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

89

one must solve the ordinary differential equation dw/dy

B(y, w).

=

(8.56)

Then co = w(y, c) where c is one constant since (8.56) is first order. Now invert the solution of (8.56) to c = c(y, co). This is called the characteristic curve. The solution to (8.55) in terms of one arbitrary function of one variable is

w) = g(c(y,

f(y,

(8.57)

to))

where g is an arbitrary function. That (8.57) is a solution to (8.55) is checked: a~f=g’ a~c

a~f g’ 3~c =

(8.58)

dc=0= a~cdy+a~cdw dw/dy = —a~cIa~c = B(y,

to).

We now return to (8.52) and solve each equation separately by the above method. In analogy with (8.56), the ordinary differential equation for (8.52a) is dw”/dy

=

K~(w)A”~(x),

(8.59)

so we have dw~’/dy=

{(Ta)~Y+(Ta)~Z}A”y

dwzldy = —{(T~)~Y+ (T~)~Z}A

(8.60)

dw~/dy {(Ta)~Y+(TaflZ}A dw5/dy=

~

Therefore dw”ldy = —(A

(8.61a)

3 5)~w’

dw&/dy

=

—(A~)~w~

(8.61b)

where

(A~)~A~”(x) Ta. Eqs. (8.61) can be rewritten for the independent solutions w~’,w~,w~’,w~:

Wi

~2I

and

~~=((~, \Wj

(~) W2

L. Dolan, Kac—Moody algebras and exact solvability in hadron/c physics

90

to be dW/dy = —A~W

(8.62a)

dW/dy

(8.62b)

and =

~

The solution to (8.62) is

=

P exp(

dy’ A~)

(8.63a)

=

P exp(_J dy’ A5)

(8.63b)

J

where P~the opposite path ordering to P, is like time ordering in that it involves only one variable dy’. Therefore ‘111 and W are not path dependent. The most general solution of (8.61) in terms of four arbitrary constants is then 3. (8.64) = ~rWnpcP and w’~= ‘IYc’ Thus the four constants are given by =

(D~1~’)~w~ and

c~= (D~))ãw~~

(8.65)

where

D”~=

= p exp(

J

dy’ A

5)

(8.66a)

and T =

=

p exp(

J

dy’ A5) = (D”~)

since =

u2D”~o2= ~

~

=

(D”~)”.

(8.66b)

L. Dolan, Kac—Moody algebras and exact solvability in hadron/c physics

91

The solution of (8.52a) can now be written in terms of one arbitrary function of four variables: =

5ç~a~,~

3, g~(DS~w’~

(8.67a)

The notation is ii~= w’~and a = 1 means a = y; a = 2 means a = z. Note the appearance of the Dt1~ and D~1~ functions. They are similar to the Yang D function which occurs in the self-dual expressions for LI ~ So far here however, no restriction is made on the gauge potentials. Similarly, from (8.52b, c, d), we find the solutions for f~,,f~ and f~ to be =

g 9(D~w~, ~

~

f~= g~(D?~w~, ~

f5

=

I5~ii~)

(8.67b)

L5?~’~, ~

g1(D~w~, ~

(8.67c)

5~5$)

4~$,

(8.67d)

where D~= p exp(

J

d9’ A9),

etc.

1~ = D~1~A Note that from (8.66), D and .15 have the property that a3,D~ and 5 a~L5(1)= IY’~A5.Therefore, if we now pause in the calculation, to see1~ what happens for self-dual fields then from (8.67d) = Dt4~= D and D~2~ = D~3~ = D. Then IYDsince = (D~)_l)T= J5* a~D~ = D~A5etc., we can identify D~ etc. Next the six cross equations of (8.51b) must be imposed on (8.67). There is a certain degree of choice involved in satisfying the cross equations, and so far it has been difficult to find the appropriate choice to carry out the calculation. It seems reasonable however to emphasize here that if we collapse the Kaluza—Klein equations back on the_self-dual potentials, we have generated the particular special non-linear non-local functions D and D as the characteristic curves of (8.51b). This is rather remarkable and would appear to suggest that perhaps the Kaluza—Klein approach will generalize the affine algebra [19]. 8.5. Affine connections for Kaluza—Klein With eq. (8.35), we calculate the affine connection using (8.28). We have tacitly assumed we are working in a coordinate basis and with this procedure we get the standard answer. (See eq. (8.36).) The components are (a, /3,~’= 1,... K; i,j, k, m, n = 1,... 4): rma$

— —

=

=

~y’~’(8,,y,,~+

a~

71,.,,

—

2AmdK~K~~F, ~, cA7K~ F~j= ICF7’Kaa F~= ~C = —cK~{A~ + ~F~} ~ + A~F~,I} + C2AfK”dA~K~a —

—

~rk_fk_.121. — ~ ij 2~ ~

5~bj’kc,~~bg’kc j

i

)

,r,

92

L.

Dolan, Kac—Moody algebras and exactsolvability in hadronic physics

where

K~,, ~ j’k

_

A~’;j

+

~

1~krn

~9JA— ~

+

aj~~mt — a,,,~) -

9. Kac—Moody algebras in other connections with integrable and physical systems From the mathematics literature, since 1978, various relationships have been established between exactly integrable systems and generators of a Kac—Moody algebra. The Lax pair L = [L,M] takes place in the algebra; for the K-dy equation this can be used to prove exact integrability of the system. Also, the affine algebra has been used to construct explicitly a general class of solutions for K-dy and to linearize the periodic Toda lattice. In the case of a finite parameter algebra, the method of orbits of Lie groups has led to the quantization of the integrable generalized Toda chain [24]. A transformation theory for the self-dual Yang—Mills equations can be presented as a Riemann—Hilbert problem, and the affine algebra derived [25]. Infinite algebras also occur in twodimensional theories of -gravity [54,55], and in supersymmetry and supergravity [54,55, 56]. Affine algebras can be associated with non-Euclidean lattices. These ideas can be applied to relativistic strings and magnetic monopoles [57]. In a different approach, Kac—Moody-like symmetries have been identified in the four-dimensional Lagrangian including the massive states, obtained via Kaluza—Klein from pure gravity in five dimensions with ground-state M4 x S1. In this theory all the generators are spontaneously broken save for Poincaré 0 U(1) with the vector and scalar Goldstone bosons providing masses for the spin 2 states. A non-Abelian generalization including supersymmetry will involve super-Kac—Moody algebras, and the symmetries in these models will no doubt be important in amilyzing the question of ultraviolet divergences [21]. In the discussion of five-dimensional Kaluza—Klein, the fundamental fields in ~(x) transform linearly under the Virasoro generators. The perturbative spectrum is thus given by a representation of this algebra. Since the masses are of order Mplanck, there will be non-perturbative corrections to the physical spectrum. In a confining theory, such as QCD, however, we are only interested in the non-perturbative spectrum. In Yang—Mills, the physical states are bound states. The theory is ultraviolet free and should have a mass gap. If the new affine symmetry exists in the full (non-self-dual) theory, it will probably be realized non-linearly on the fundamental fields A,,” of the Lagrangian (this is suggested by the self-dual expressions). The physical particles will be composites (of gluons and eventually quarks) which are speculated to transform linearly in an infinite-dimensional representation of the new symmetry. Since this is an infinite parameter algebra, the representation will be infinite-dimensional and there will be an infinite number of bound states, i.e. hadrons. Since the observed hadronic spectrum is not degenerate, most of the affine symmetry will be spontaneously broken. The addition of quark matter mayalso explicitly but weakly break the affine symmetry. In this way, a direct analysis of infinite-dimensional representations of Kac—Moody algebras may give information about the hadronic spectrum [58].

L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

References [1] For a concise review and other references, see M. Creutz, Physica Scripta 23 (1981) 973; Also, E. Tomboulis, Phys. Rev. Lett. 50 (1983) 885. [2] G. ‘t Hooft, Nuci. Phys. B72 (1974) 461; See also E. witten, NucI. Phys. B149 (1979) 285. [3] G. ‘t Hooft, Phys. Rev D14 (1976) 3432. [41C. Callan, R. Dashen and D. Gross, Phys. Rev. D17 (1978) 2717; 19 (1979) 1826. [51R.P. Feynman, NucI. Phys. B188 (1981) 479; 1.M. Singer, unpublished. [6] J.M. Cornwall, Phys. Rev. D26 (1982) 1453. [71S. Adler, Phys. Rev. D23 (1981) 2905; K. Johnson, Proc. Non-perturbative QCD Conf., 05U, March 1983. [8] L. Dolan, Phys. Lett. 113B (1982) 387. [9] V.G. Kac, Izv. Akad. Nauk SSSR, 5cr. Mat. 32 (1968) 1323 [Math.USSR Isv. 2 (1968) 1271]; R. Moody, J. Algebra 10 (1968) 211; V. Kac, Infinite Dimensional Lie Algebras (Birkhäuser, Cambridge, Mass., 1983). [10] lB. Frenkel and V.G. Kac, mv. Math. 62 (1980) 23; J. Lepowsky and R.L. wilson, Corn. Math. Phys. 62 (1978) 43; See also lectures by I. Frenkel and J. Lepowsky, in: Proc. AMS Summer Seminar in Application of Group Theory in Physics at University of Chicago, July 1982; and lB. Frenkel, Representations of Kac—Moody algebras and dual resonance models, lAS, Princeton preprint. [11] For excellent reviews of integrable systems, see L.A. Takhtadzhan and L.D. Faddeev, Russian Math. Surveys 34:5 (1979) 11-68; and H.B. Thacker, Rev. Mod. Phys. 53 (1981) 253. [121L. Dolan, Phys. Rev. Lett. 47 (1981) 1371. [13] M. Luscher and K. Pohlmeyer, Nuci. Phys. B137 (1978) 46. [14]K. Pohlmeyer, Corn. Math. Phys. 46 (1976) 207. [15] F. Gursey and H.C. Tze, Ann. Phys. (NY) 128 (1980) 29. [16] M. Luscher, NucI. Phys. B135 (1978)1. [17] A.B. Zamolodchikov and A.B. Zarnolodchikov, NucI. Phys. B133 (1978) 525. [18] A.M. Polyakov, Phys. Lett. 82B (1979) 247; NucI. Phys. B164 (1980) 1971. [19] L. Dolan, Rockefeller preprint RU83/B/55, to be published in Proc. Orbis Scientiae, Coral Gables, Jan. 1983; and Proc. Kaluza—Klein workshop, Chalk River, Ontario, Aug. 1983, RU83/B/49. [20] C.N. Yang, Phys. Rev. Lett. 38 (1979) 1377. [21] L. Dolan and M.J. Duff, Phys. Rev. Lett. 52 (1984) 14; L. Dolan, Rockefeller Univ. preprint RU 84/B/89, May 1984. [22] L. Dolan and M. Grady, Phys. Rev. D25 (1982) 1587; see also L. Dolan, Lectures in Applied Mathematics, Applications of Group Theory in Physics, Vol. 21(1984). [23] G. ‘t Hooft, Nuci. Phys. B138 (1978)1 and B153 (1979) 141. [24] For early references see B. Kostant, mv. Math. 48 (1978) 101; M. Adler and P.V. Moerbeke, Adv. in Math. 38 (1980) 267; RIMS preprint 362 (1981) Kyoto. [25] K. Ueno and Y. Nakarnura, Phys. Lett. 109B (1982) 273. [26] L. Dolan and A. Roos, Phys. Rev. D22 (1980) 2018. [27]M.C. Davies et al., Phys. Lett. 119B (1982) 187. [28]E. Brezin et al., Phys. Lett. 82B (1979) 442. [29] See for example, R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (wiley, New York, 1974) p. 89. [30] See ref. [29]p. 90. [31]H. Garland, J. Algebra 53 (1978) 480. [32] M.A. Virasoro, Phys. Rev. Dl (1970) 2933. [33] R. Baxter, Ann. Phys. (NY) 70 (1972) 193, 323 and 76 (1973)1, 25, 48. [34] In a private communication, S. Michailov has stated that the classical initial value problem for the NLo’M can be solved using a Riemann—Hilbert analysis. [35]C. Devchand and D.B. Fairlie, NucI. Phys. B194 (1982) 232. [36] BY. Hou, M.L. Ge and Y.S. Phys. Rev. D24 (1981) 2238. [371 M.L. Ge and Y.5. Wu, Phys. Lett. 108B (1982) 411.

wu,

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L. Dolan, Kac—Moody algebras and exact solvability in hadronic physics

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