String representations and hidden symmetries for gauge fields

Volume 82B, number 2

PHYSICS LETTERS

26 March 1979

STRING REPRESENTATIONS AND HIDDEN SYMMETRIES FOR GAUGE FIELDS A.M. POLYAKOV Landau Institute of Theoretical Physics, Moscow, USSR Received 29 November 1978

We show that gauge fields can be considered as chiral fields defined on the space of all possible contours. That leads to an infinite number of functionally conserved currents. A possible application of this result is discussed. Also a one-dimensional approximation is proposed.

In this paper I shall present results of a new formulation of non-abelian gauge theory, motivated by the hope that this theory, being closely analogous to the two-dimensional chiral fields [1], possesses some hidden symmetry which will permit some day to find an exact solution to the gluon dynamics. The main idea o f the approach is to use equations for the fields defined on the space of all possible contours. The main result is that there exists, at least in the case of threedimensional s p a c e - t i m e , an infinite set o f conserved currents in the above-mentioned contour space. Another result is that in the large D limit (where D is the number o f s p a c e - t i m e dimensions) gauge theories are described by the dual string equations, except perhaps some unconventional solutions of these equations, one o f which is explicitly given. Finally, we shall discuss a discrete gauge group Z(2) which is o f great interest (in three dimensions) for the usual phase transition theory and also bears certain interesting consequences in four dimensions, illuminating, perhaps, the nature of fermions. In this paper only some hints about the derivations will be given. I intend to publish detailed proofs elsewhere. Let us stress the analogy with a two-dimensional chiral field on the lattice. It is described by the action:

s = (1/e 2)

X,6

Tr

(1)

Here e 0 is a coupling constant, x is a lattice point and 6 are two possible unit lattice vectors. For reasons to

be clarified below, let us consider the chiral fieldsg x which are matrices o f the general real linear group GL(n, R). The equations o f motion for the action (1) take the form

6

(gx+6gx 1 - gx g x l ~ ) = 0 .

(2)

Then we have our first statement: there exists an infinite set of currents Jx(~ which are functions Ofgx+ s g x 1 satisfying the continuity relation as a consequence of eqs. (2):

6

-

) = 0.

(3)

The explicit form o f the f r s t two Jx(,k6) is: j(1) x,6 = A x,6 , j (x,1 2 ) =Ax,1 Ax+l , 2 - A x , 1 Xx+l

(4)

J(x2,)2= - A x,2 Ax+2,1 - A x,2 Xx+ 2 , where

Xx+2-Xx=-Ax-l,1

,

Xx+l-Xx=Ax-2,2

(here Ax, ~ =gx+ngxl). (The function X exists due to eq. (2).) So, the system (1) is completely integrable. Now let us explain why GL(n, R). In the formal limit, if we have initial data lying, say, in O(n) C GL(n, R), our fields will remain in'O(n) forever. So, the discovered integrability applies to O(n) chiral fields in the continuous limit [2,3]. However, in the lattice version O(n) fields are not integrable. That means that in the quantum 247

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mechanical treatment of action (1) we may encounter topological anomalies in the quantum continuous limit for O(n), which are absent in the GL(n, R) quantum mechanics. Implications of these anomalies are not completely investigated yet, but in most cases they are quite harmless. So the answer to the question "Why GL(n, R)?" is that the GL-group is completely integrable in the lattice quantum mechanics and its explicit integrability most probably implies integrability for the compact subgroups of GL. In any case, GL is the simplest non-abelian Lie group to deal with. Now let us consider the case of the gauge fields. The action given by [4,5] : S = (1/e 2) ~

Tr (Bx,aBx+a,~Bxlg,aBx, l~).

(5)

X~OG~

Here again x is a lattice point, ~, ~ are unit lattice vectors and e 2 is the coupling constant. The equations of motion for action (3) take the form:

5S/SBx, a = 0 .

(6)

It is important that eq. (6) can be rewritten in the following way. Let us introduce the field ~ (Cy), where is a matrix of the gauge group and Cy is a certain closed contour starting at point y :

~(Cy) = c ~ Bz,a '

0

(7)

8Fu(s, C)

~x~(s)

=0,

tu(s)Fu(s ,C) = 0 ,

~(c)

Fu(C, s) - 5xu(s) ~ - I ( c ) ,

sg (s)

(9)

SFAs')

- + [Fu(s),F~(s')I = O. 5x.(s') 8x.(s) Here the curve xu(s) represents the contour C, t~(s) = (dxu/ds)l dxu/ds[ -1 is the tangent vector at point s. The condition tuF, = 0 is a consequence of the reparametrization independence of ~k(C) [71. Now, there exist an infinite number of currents J(uk)(s, C) satisfying the equation:

f d, 6J.(,. c)/sx .(,) = o

(lo)

The first two currents are given by: J(l)(s, C) = Fu(s, C), (11)

+ [Fu(s, C), X(s, C)I , where

6 X(s, C)/Sx v(s ) = euvx tv(s) Fx (s , C ) .

+

(8)

(here the contour Cy + Ilxo¢ is obtained from Cy by replacing the link (xa) by the letter II forwarding in the/3-direction, or, in other words, by gluing the unit square to the link (xa) in the direction/3). Eq. (8) can be verified by direct calculation of the left-hand side of eq. (6) and by substituting the resulting identity into eq. (7). It is not necessary now to comment on the similarity of eqs. (8) and (2). An important idea which we get from this similarity is that the gauge fields can be considered as the chiral fields on the contour space and, roughly speaking, the curvature in the usual space defined by the Yang-Mills field 248

strength defines the trivial connection (with zero curvature) on the contour space. Correspondingly, conserve( currents exist on this space. We shall give their explicit form in the classical continuous limit of the threedimensional Yang-Mills theory. The continuous limit ofeq. (8) is:

42)(s, C) = euvx tv(s)Fx (s , C)

which is of course a very well known quantity [ 4 - 6 ] . A less known statement is that eqs. (6) are equivalent to the following equation for ~:

-- ¢ ( C y ) ~ - l ( c - Ilxa/3)} = 0 ,

26 March 1979

(The functional X exists, and can be written down, due to the first of eqs. (9).) These considerations are a consequence of the compatibility of the equations:

c)

c) t

6x u (s)- + 7euvxtv(s) 8~x(s ) J = Fu(s'C)¢' (12) which are the functional analogues of the Lax pair for the chiral fields [2,3]. (Here 3, is an arbitrary parameter.) The physical meaning of these conserved currents has yet to be clarified, but there are no doubts that their existence is an important property of the gauge theories. These considerations can be generalized for the fourdimensional case and also for the lattice theory with the GL(n, R) gauge group. The last statement means that these conservations hold in the quantum theory of

Volume 82B, number 2

PHYSICS LETTERS

continuous SU(n) up to possible topological anomalies. Another application of the above results is given by the Z(2) gauge groups. In the case of three dimensions this theory is just the Ising model (due to the KramersWannier duality). In this case it is possible to construct explicit variables v~hich linearize the equations of motion. These variables describe some fermionic string, and are just as in the two-dimensional Ising model, given by the product of the contour order parameter and a sequence of disorder parameters adjacent to the contour. Detailed formulae will be published elsewhere. They imply the existence of contour currents in the three-dimensional Ising model but their physical significance has not so far been fully explored. Another set of results which I would like to report here is connected with the mean field approximation to the gauge theory. This approximation has validity of order 1/D where D is a space-time dimension. Let us consider the functional integral:

I I I dB

e -S(B) Tr $(C)

O(C) -

(13)

f H dBx,c~ e -S(B) Then the following exact equation is true:

G(C) = =
By,~).

(14)

Here the contour C is obtained from C' by cutting out the link (xa) and/~(H) is the function defined by:

f dB exp e~-2 Tr (B+H + H+B ) /3(H) = - f dB exp e0-2 Yr (B+ H+ H+ B ) '

to expand/~:

B (H) ~ (o#e2)H + O(H 2)

(17)

(a is a certain group-independent number). Then eq. (14) takes the form:

G(C) = (ale 2) ~ G ( C

+ Flxa~).

(181

In the formal continuous limit this equation is replaced by:

6x~u(s) dxu

\ as ! ) G(C): o, (19)

6G

ds ax.(s)

--0

.

Here:

m 4=l-2a(D-l)/e

2<1

.

(20)

The second of these equations expresses the parametrization independence of G. It is clear that G plays the role of bare propagator for contour averages. Eqs. (19) coincide with the dual string equations but we are interested in a different sort of solution than the one explored in the dual string theory. I have found some solution of eqs. (19) which, most probably, is not a correct propagator, at least not for all contours. Nevertheless, I would like to present this solution since it may be a good starting point for the search of a better one. It is given by the following ansatz: let us assume that G(C) depends only on the tensors

(15)

a,~ = f x,~ dx~ .

and

[Ix, ~ = ~_1 I-I By,u. t3 ( n x ~ )

26 March 1979

(16)

Here [Ixa ~ is the contour represented by the letter H attached to the link (xa) in the direction/3. Eq. (14) and similar equations for:

GN(C 1 ... CN) = (Tr ~(C1) ... Tr $(CN)) , form the analogue of the Schwinger equations in the usual field theories. It is possible to derive a perturbation expansion from these equations in which we have Feynman diagrams formed by the surfaces instead of lines. In the leading lID approximation it is possible

(21 )

Surprisingly enough this simple assumption reproduces itself, when substituted into eqs. (19). The resulting solution is:

G(C) = F(u+ ) + F ( u _ ) ,

(22)

1

(23)

U+ = (020 +-~ eex(3~60 ,~ Oy 6 )1/2 and

F"(u) + (1/2u)F'(u) - rn4F = O .

(24)

The higher 1/D corrections to eq. (19) can be expressed in terms of the surface Feynman diagrams with G(C) as a bare propagator.

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Most part of this work has been done during my visit to the California Institute o f Technology in spring 1978. I am deeply grateful to the CalTech theory group for their hospitality and important comments on my work. I would also like to thank my friends from the Landau Institute, A. Belavin, A. Migdal and A. Zamolodchikov for much valuable advice. In a futile attempt to complete my list of references I wish to point out that contour variables appeared before in refs. [ 6 - 8 ] . Also, eq. (18) in a different context has been independently derived in ref. [9].

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26 March 1979

References [1 ] A. Polyakov, Phys. Lett. 59B (1975) 80. [2] K. Pohlmeyer, Commun. Math. Phys. 46 (1976) 207. [3] V. Zacharov and A. Mikhailov, Zh. Eksp. Teor. Fiz., to be published. [4] K. Wilson, Phys. Rev. D10 (1974) 2445. [5] A. Polyakov (1974)i unpublished. [6] S. Mandelstam, Phys. Rev. 175 (1968) 1580. [7] R. Minlos and Ya. Sinai, Mat. Sbor. 18 (1967). [8] C. Marshall and P. Ramond, Nucl. Phys. B23 (1974) 582. [9] D. Foerster, Cornell preprint (1978).