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Vertex operator construction of the SO(2n+1) Kac-Moody algebra and its spinor representation
Nuclear Physics B277 (1986) 317-331 North-Holland, Amsterdam
VERTEX OPERATOR CONSTRUCTION OF THE SO(2n + 1) KAC-MOODY ALGEBRA AND ITS SPINOR REPRESENTATION Orlando ALVAREZ 1"2"3and Paul WINDEY 3
Department of Physics and Lawrence Berkeley Laboratory, Universi(v of California, Berkeley, California 94720, USA Michelangelo MANGANO 4
Department of Physics, Princeton UniversiO,, Princeton, NJ 08544, USA Received 2 December 1985
An explicit representation of the B,(,l) affine Lie algebra (Kac-Moody algebra) is constructed in terms of vertex operators associated with the Chevalley basis of the underlying finite-dimensional Lie algebra. This construction, contrary to the simpler current algebra one, gives a concrete realization of the spinor representation of the algebra. The key feature is a partial bosonization of two-dimensional Weyl-Majorana free fermions. The vertex operators associated with the long and short roots of the B,, algebra have fermion number zero and one, respectively.
1. Introduction Explicit representations of Kac-Moody algebras [1] can be constructed in two different ways. The first one amounts to the construction of a two-dimensional field theory with a set of conserved currents obeying the commutation relations of the Kac-Moody algebra. The central charge is no other than the traditional Schwinger term. The Kac-Moody algebra is then easily seen as the loop algebra associated with the Lie algebra f# of the symmetry group G of the theory. Indeed the currents are analytic functions of one complex variable and the infinite set of generators are the coefficients of their Laurent expansion. From this point of view the treatment of all semi-simple Lie algebras is similar. The simply-laced algebra* play no particular role. In principle, each different representation (different value of the central 1 Alfred P. Sloan Foundation fellow. 2 This work was supported by the National Science Foundation under grant PHY-81-18547. 3 This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contracts DE-AC03-76SF00098. 4 Supported by an INFN Fellowship. * A Lie algebra is said to be simply laced if all the roots have the same length. 0550-3213/86/$03.50 :~'Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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charge) is associated to a different model. A second approach is based on the construction of the vertex operators [2, 3] originally introduced in the context of dual string theories. Here one first reconstructs the underlying finite-dimensional Lie algebra and builds operators corresponding to the roots (raising and lowering operators) and to the Cartan subalgebra. The full affine algebra is then constructed either by taking the moments of these operators or by building along the same lines new operators corresponding directly to the extra roots (so-called imaginary roots) of the KacMoody algebra. These two constructions are of course completely equivalent although the latter one is closer in spirit to the original formulation of Kac-Moody algebra in terms of generalized root systems and Cartan matrices. Unfortunately, its apparent drawback was the difficulty in building vertex operators for non-simply laced algebras. The purpose of this article is to repair this apparent lopsidedness at least for the B,~1) algebras*. In principle, once one has the currents in the field theory the problem is solved. In practice, more information is required. For example, what is the operator that transforms under the spinor representation? It may be difficult to express the operator in question in terms of the elementary fields of the field theory. A representation of the operator may be found in terms of an equivalent but different field theory. These are the type of questions we address. Some of the results in this paper are apparently known to mathematicians [4]. In particular, the importance of having a fermion number associated with the different vertex operators was pointed out to us by Jim Lepowsky. The paper is organized as follows. In sect. 2 we discuss a two-dimensional fermionic field theory with a B2 symmetry. The construction of the associated vertex operator is exhibited. In sect. 3 the vector representation is constructed and the relation to the original fermions is given. The spinor representation is constructed in sect. 4. The relationship of the Ising model and fermions is seen to play a central role. The first use of the Ising model in a vertex operator construction is due to Friedan, Martinec and Shenker [5]. In sect. 5 we generalize the results to an arbitrary Bn algebra. Finally in sect. 6 we make some comments about a hidden supersymmetry.
2. T h e B 2 scenario
It is very simple to construct a two-dimensional field theory with a symmetry SO(2n + 1). Consider 2n + 1 Majorana-Weyl fermions ~V, i = 1, 2 . . . . . 2n + 1 with a lagrangian given by L = ~Jro~tpr.
* B~1) is the centrally extended loop algebra Bn, a.k.a. SO(2n + 1).
(2.1)
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The symmetry SO(2n + 1) is generated by the currents jrs = : ~r~ks:. These currents define a Kac-Moody algebra with an enveloping Virasoro algebra. The spinors q~r are the operators which correspond to the highest weight "vector type" representation of the B~~) affine algebra. We would also like to know what are the operators which correspond to the spinor representation of this algebra. This question is not easily answered by using the q~r but finds a simple solution through an argument of partial bosonization. Many of the main ideas involved in this construction are contained already in the B2(1) example. We first discuss this case in detail and afterwards generalize the ideas to the other odd orthogonal groups. Since SO(5) is a rank-two group the root diagram is easily drawn as seen in fig. 1. The roots a and fl are the simple positive roots. The roots are normalized such that
a.a=2,
/~./3= 1,
a./~= -1.
(2.2)
The remaining inner products are easily obtained from the figure. The bosonization [7] proceeds in the following way. The two generators of the Cartan subalgebra may be taken to be j12= :q~2: and j 3 4 = :~3~b4: . Bosonize kl and ~2 into a right-moving scalar ~1, and bosonize ~3 and ~b4 into another right-moving scalar ~2. In terms of the scalar fields, the generators of the Cartan subalgebra become 0z~ ~ and 0z~ 2. Let us denote ~p5 as ~/'. With a suitable definition the propagators of the free fields may be taken to be
(¢'(z)O,(w))
= -,'qog(z
- w),
(2.3)
1
(2.4)
w
a+2~
a Fig. 1. Root diagram for SO(5).
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The key formula in our derivation is the operator product expansion (OPE)* exp[i/~- ~ ( z ) ] e x p [ i v . ~ ( w ) ] = (z - w ) " % x p [ i ( / * - ~ ( z )
+ v. ~ ( w ) ) ]
- (z - w ) " % x p [ i ( / * + v ) . ~ ( w ) ] × [1 + i(z - w)lt. aw~(W)].
(2.5)
Non-vanishing commutators are determined by the pole singularities in the operator product expansion [8]. With our conventions /~-v is an integer. A singularity r e q u i r e s / , , v < - 1. There is another important result to bear in mind, namely that the operator exp[i/~. ~ ( z ) ] has conformal weight !2# . #. The current that generates the K a c - M o o d y algebra must be an operator of weight one. 2.1. THE CHEVALLEY BASIS We will now construct the B2°) algebra using • and ~/'. A very convenient basis for the study of the Lie algebra is the Weyl basis. If a is a root then let H~ be a suitably normalized corresponding element of the Cartan algebra. Also, let E~ be the associated raising operator. The commutation relations for the algebra may be written as:
[<, H,,] =0, [<, =
[E,,,E_~]=H~, [E,~,EI~ ] = N~./~E~+/~.
(2.6)
The integers N~,/~ can be chosen such that N~,~ = + ( p + 1) where p is the greatest integer for which fl - p a is a root. N~,~ = 0 if a + fl is not a root. Taking et and fl to be simple roots ai, the above commutation relations reduce to the defining relations of the Chevalley basis:
[U,, U,] =0, [H,, g] = [H.C] =
-:,+F+.
[ Ei, Fj] = SijHj.
(2.7)
* To interpret the OPE (2.5) as commutation relations of the Kac-Moody algebra one has to expand it in modes and introduce appropriate cocyclesmultiplying the vertex operators. A consistent choice of these cocycles for our construction will be given in the last chapter where we will analyze the general case. In the meantime, we will only worry about the OPE.
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We have used the obvious notation that H i = Ha,, E i = E~, and F, = E ~ . are the entries of the Cartan matrix given by
The aij
2 0 / i • £ltj
A = (a/j) - - -
(2.8)
Ot i • O~i
Our problem reduces to building a representation of the associated current algebra in the field theory. Since we work in the formalism of conformal field theory, we will build operators which obey in the Chevalley basis an algebra equivalent to: [J~(z), Jb(w)] =fabcjc(w)8(z
-- w) + k S ' ( z - w ) 8 "b,
(2.9)
where J a ( z ) is the current which generates the appropriate group transformation and the f~h,. are the structure constants of the algebra (2.6). (i) First we will consider the case where the two roots in (2.5) are long ones. In this case exp[i/t- ~(z)] is an operator of weight one. The inner product /~. v can only take the value 0 or + 2. If /x. p = 2 then /~ = u and there is no singularity in the OPE. The operators commute and (2.5) corresponds to [E,, E~] = 0. If /t. 7, = 0 then /~ + ~, is not a root. Once again the OPE is non-singular and corresponds to [E~ E,] = 0. If ~- 1, = - 2 then /~ = - p . The OPE gives 1
exp[i/t. ~ ( z ) ] e x p [ i ( - / x ) .
~(w)]
(Z
-- W) 2
[1 + i ( z - w)l~" Ow~(W)],
(2.10)
which is the equivalent of [E,, E ~ ] = / t . H. Note also that we got the central charge. All other commutators involving only long roots vanish. (ii) When p is a short root exp[io. ~(z)] has dimension ½. Since g" also has weight ½ we can combine them to get an operator e x p [ i p - ~ ( z ) ] ' / ' ( z ) with the correct conformal weight of one. Consider the product of two such operators: e x p [ i p . ~ ( z ) ] 'q'(z)exp[io. ~ ( w ) ] ~ ( w )
- (z-
w)°°exp[i(p
x [(z-w)
-1 @
+ o). (z-w)(aoj
2t)
2t(w)
] . (2.11)
In the above vI'(w)vl"(w) = 0 since the g' are Grassmann variables. From fig. 1 it is clear that p- o can take three possible values: (a) If p . o = + 1 then p = o and there is no singularity in the OPE. This corresponds to [E o, Ep] = O.
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If p- o = 0 then p + o is a long root. The singular part of the OPE is given by exp[io- ~ ( z ) ] ~P(zlexp[io. ~ ( w ) ] ~/'(w) 1 - --exp[i(p
+ o)-~lt(w)],
Z--W
(2.12)
and is equivalent to [Ep, E,] - Ep+o. (c) If to" o = - 1 then P = - o. Eq. (2.11) reduces to exp[ip, q~(z)] q S ( z ) e x p [ i ( - P ) " ~ ( w ) ] ff'(w) 1 ( z - w) 2 [1 +
i ( z - w)p" 0wq~(w)],
(2.13)
which is the equivalent of [Ep, E_e] = p. H with the required central charge. (iii) Finally we have to verify the commutation relations for a long root )~ and a short root o. Note that the sum of a long and a short root is either a short root or not a root. The relevant terms in the OPE are
~(z)]exp[io.
exp[i?~-
- (z-
°[1 +
~ ( w ) ] g'(w) i(z
-
w)X. aw (W)]
×exp[i(h + o).
(2.14)
If X. o _ 0, then their sum is not a root. This is reflected by the absence of a singularity in (2.14) and the zero commutator in (2.6). If X. o = - 1 , which is the only remaining possibility, their sum is a short root and we have exp[iX-
~(z)lexp[io.
1 - --exp[i(h Z--W
~ ( w ) ] qS(w) + o)" ~(w)] ~(w),
(2.15)
which is [E x, Eo] - Ex+ o. We are only left with the task of checking the commutators of the raising and lowering operators with the Cartan generators o. 0 ~. This computation is entirely analogous to the corresponding one for simply-laced groups since the qp field plays a passive role. The key formula is
O" Oz~(z)exp[i~.
~(w)]
--ip" l~
- -
Z--W
exp[i/~- t~(w)],
(2.16)
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which states that the exponential transforms as an element of a representation containing the root It. T w o comments are now in order. Firstly, the infinite set of operators of the affine Lie algebra are defined from the operators we just built by taking the coefficients of their Laurent expansion. They are of the following type*:
1 -~dzz"exp[iit. ~(z)]
2~ri
1
2vri ~ dzz~t~" 0 ~ .
(2.17)
Secondly, the normalizations of the operators should be treated with some care in order to reproduce (2.6). In particular the generator of the Cartan subalgebra corresponding to the simple root a, should be multiplied by eg where the e, is defined in the following way. The symmetric matrix of inner products B may be written in the form A = DB where D = diag(e~) and the e i are the set of minimal positive numbers such that B is symmetric. These remarks will be equally valid in the treatment of the general case in sect. 5.
3. The vector representation In this section we will show how to reconstruct the original spinors ~bi, i = 1 . . . . . 5 in terms of the • and q" operators. The straightforward method is to construct operators with the correct SO(5) transformation properties. The associated weight diagram is given in fig. 2. Define an operator 12A(Z) = exp[iA • ~(z)]. If/~ is a root, then the O P E of the Cartan algebra generator It. a ~ with 12A(Z) immediately verifies that A plays the role of a weight in the operator 12A(Z). Choose A to be the highest weight of the vector representation A = a + ft. The OPE with an operator corresponding to the root It is given by
exp[iit- ~ ( z ) ] 12a(w ) - (z - w)~'aexp[i(it + A ). ~ ( w ) ]
x [1 + i ( z - w)it.
(3.1)
* The interested reader should consult ref. [3] for more details. In particular we will not rewrite the correct Chevalley basis corresponding to the extended Cartan matrix [1] nor dwell upon the construction of the operators corresponding to the imaginary roots. Given the above construction of the algebra these extensions from the simply-laced to the non-simply-lacedcase are straightforward.
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v~v
~
Fig. 2. Weight diagram for the vector representation.
If g is a positive root then g . A > 0 and there is no singularity in the operator product. This means that the two operators c o m m u t e as required since A is a highest weight. If g = - a then # . A = - 1 and the c o m m u t a t o r is [exp( - i a . ~ ), I2a] - ~2~.
(3.2)
In other words the root - a takes the weight A = a + fl to the weight ft. If g = _+fl, t h e n / ~ . A = 0 and there is no singularity in the OPE, therefore (3.3)
[exp( +_ifl . ~ )vl", ~A] = O.
O n e easily verifies also that if one chooses g = _+(a + 2fl) then one gets the correct c o m m u t a t i o n relations. T h e only case left to check is when g = +_(ct + fl). N o t e that the inner product /s- A equals _+ 1. The c o m m u t a t o r vanishes if/~ = a + ft. If g = - ( a + fl) then the O P E gives exp[i(--a--fl)-~(z)]~(z)~2A(W
)-(z-w)
X~(w).
(3.4)
Therefore the central weight corresponds to the operator ~/'. To verify this we note that [ g . 0 ~ , xO]- 0 since the O P E has no singularity. If /~ is a long root then exp[i/~ • ~ ( z ) ] g ' ( w ) - 0. If g is a short root then e x p [ i ~ , q~(z)] x t ' ( z ) ' t ' ( w )
- ( z - w)-~exp[i/~ • ~ ( w ) ] .
(3.5)
This is the correct c o m m u t a t i o n relation with a short root. N a m e l y the short root m o v e s us from the origin to one of the corners of the square.
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We have constructed explicitly the five operators which transform according to the vector representation of S0(5): exp[i(a + fl). ~ ( z ) ] , exp[i(-fl). ~(z)], exp[i(-a
- ~ ) . q~( z)] ,
exp[i/3. ~ ( z ) ] , '/'(z).
(3.6)
Note that the non-zero weights have length one, hence all the above operators have conformal weight ½ as required since the Virasoro algebra is the enveloping algebra of the Kac-Moody algebra. This proves that the above are essentially the five-spinors q~i. To get the correct anticommutation relations one should multiply the operators given above by the appropriate cocycle.
4. The spinor representation The four-dimensional spinor representation of SO(5) corresponds to the weight diagram shown in fig. 3. The four weights are equidistant from the origin forming the corners of a square. One moves along the diagonal with a long root while a displacement across an edge is produced by a short root. If tt is a weight of the spinor representation then the operator ~2~ has the same conformal weight for the four values of ~. This would be perfectly alright if one only had to consider the long roots. But the vertex operator for the short roots, which contains a fermion, will change the fermion numbers when acting on $2 and will cause havoc by changing
Fig. 3. Weightdiagram for the spinor representation.
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the conformal weight of the operator. To remedy this situation it appears that one needs an operator with indefinite fermion number which will behave correctly under the action of the short roots. Such an operator exists, it is the celebrated spin operator of the Ising model. The highest weight for the spinor representation is A = I a +/3,
(4.1)
with A ,a
-
1
(4.2)
One will need the vertex operators
exp[ ila
. dp( z
)],
exp[ i( l a + /3 ) " exp[ i(-
z )] , (4.3)
la - /3 ) . ¢( z )] .
Let us begin with the highest weight A and study the action of the long roots. This is summarized in table 1. Next, let us see what happens to ~a under the action of a short root - / 3 . Note t h a t / 3 - A = I and therefore the OPE behaves as
exp[i(-fl),
rb(z)] g ' ( z ) e x p [ i ( l a
+ fl ) . ~ ( w ) ]
1
vfz - w e x p [ i l a " ~ ( w ) ] ~ ( w ) .
(4.4)
Notice that ~ has the same conformal weight a s f~a/2, therefore, the short root has not only produced an operator with the wrong weight but also with a different fermion number. More significantly one should get rid of the square root and recover an integral power to obtain the desired commutator. It is clear that one
TABLE 1
Action of roots on highest spinorial weight Long root p +a
a+2fl - (a + 2fl)
OPE power p •A
Commutator
0
0
1 - 1
0 exp[i(- 12a- fl). ~]
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needs an additional operator with indefinite fermion number or a "coherent" state of fermions. Such a composite operator would have the necessary properties under the action of any of the members of the algebra. This operator is the spin operator of the Ising model we alluded to previously. In fact the Majorana-Weyl fermion g' is equivalent to an Ising model. Let ~ be this spin operator. Its conformal weight is ~ ; Kadanoff and Ceva [9] demonstrated years ago that a
qt(z)2~(w)
~/z - w Y.(w),
(4.5)
where a is an irrelevant constant. It is straightforward to verify that the required operators are exp[i½a + / 3 - @ ( z ) ] ~ ( z ) , exp[i~a-dp(z)l,~(z),
exp[ i ( - ½a - /3 ) . ~ ( z )]~( z ), exp[i(-½a). ~(z)l,~(z).
(4.6)
The spin operator is innocuous as far as the long roots are concerned, but it provides the necessary correction for the OPE with the short ones. Again to obtain the correct anticommutation relations one should multiply the operators with the appropriate cocycle. Notice that each of these operators has conformal weight 1 ~ "q- 1 - -
76.
5. The general case
The discussion of the previous sections generalizes to the case of B. without additional complications. The algebra B. has the following Dynkin diagram: A
A
A
A
~
There are n - 1 simple positive roots a x, a 2. . . . . an-1 with length 2 and one simple positive root fl of length 1. It is useful to note that the root system of B, can be interpreted as the union of the root system of D n with the weights of the 2n-dimensional vector representation of D,. This corresponds to the decomposition adj(SO(2n + 1)) ~ adj(SO(2n)) • vect(SO(2n)),
(5.1)
under the reduction SO(2n + 1) ~ SO(2n).
(5.2)
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Since D n is simply laced with roots # normalized in such a way that ~t-/~ = 2, the weights of the vector representation of D, will play the role of the short roots of Bn. In [3], it was shown that it is possible to identify the vertex operators constructed for vect(SO(2n)) with the 2n-dimensional Dirac gamma-matrices. If + ~tz for l = 1 . . . . . n are the weights of vect(SO(2n)) satisfying/~z" #k = S/k, then the operators r,(z)
= a.,(z)
+ ~
,,(z),
r,,+,(z) = - i [ a . , ( z ) - ~ .,(z)],
(5.3)
and an appropriate cocycle will generate a Clifford algebra. In fact it is straightforward to see that:
r,(z)r.+,(w)-0, r,(z)r~(w)
28tk - - ZmW
ro÷,(z)r.+k(w)
28lk - - -
Z--W
(5.4)
In [3] it was proven that there exist cocycles which turn the OPE in (5.4) into the anticommutation relations of a Clifford algebra when we go into the mode expansion. With this in mind we return to our construction. A representation of the Kac-Moody algebra is furnished by 2n + 1 Majorana-Weyl fermions with lagrangian t = @ros@r
(5.5)
and currents j i j = : i~t~j:. One bosonizes the ~,k and ~kk+" for (k = 1 . . . . . n) into a scalar @k. The last Majorana-Weyl fermion will play exactly the same role as in the SO(5) case, and we keep the same notation q,2,+1_ ~. The Cartan subalgebra generators become 0fl) k. If X is a long root and o a short one then the collection of operators exp(ih. ~ ) , exp(io- ~ ) ~
(5.6)
are the raising and lowering operators for the Kac-Moody algebra. This may be verified by choosing the standard representation for the roots of the B~ algebra:
+ei+%, ± ei ,
for i :~j roots of D,,, for i ¢ j weights of v e c t ( D , ) ,
(5.7)
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where ( e i } is the standard basis for R n, and explicitly computing the commutators. It is clear that in the case of two long roots there are cocycles which will turn the O P E into the correct commutation relations. The reason is that we are only probing the D, subalgebra of B, and we know that such cocycles exist. In the case of a long root with a short root one requires two observations. Firstly, the decomposition described in eq. (5.1). Secondly, the existence of a cocycle which transforms Fl(z ) into a gamma matrix makes exp(io • ~ ) ' / ' transform as a vector under D,. In fact, if F~ are the gamma matrices and if M u is a generator of the D, algebra, then one immediately obtains
[ui,, rk] =
<], rk]--
(5.8)
If both roots are short ones then the only subtlety arises from the consistency of the choice already made for the cocycles. Naively, it appears as if one would get anticommutation relations instead of commutation relations. One has to remember the presence of the anticommuting fermion xo which resolves the apparent paradox. Due to the presence of the fermion '/', the anticommutation relations between the vertex operators in vect(D,) become ordinary commutation relations. This defines the remaining part of the algebra as a straightforward calculation will show. The vector representation is obtained by considering the operators exp[ +__iej. eb] and g'. The Kac-Moody spinor representation with dimension 2" is obtained by first specifying the spinorial weights for the B, representation: ½(+_e l___e 2___ . . . +_e,).
(5.9)
The vertex operators are of the form exp[iA. ~ ] X ( z ) ,
(5.10)
where A is a spinorial weight. The dimension of the vertex operator is 1 1 $" a n + ~6 = ~ ( 2 n + 1).
(5.11)
This construction also works for C 2 which is isomorphic with B2 but alas not for C,, n > 2 . 6. Hidden supersymmetry It is not generally appreciated that some two-dimensional field theories have a hidden supersymmetry [10]. This is most obvious if we apply our construction to the SO(3) situation. This is an interesting case since there are two ways of constructing SO(3) Kac-Moody algebras. We can use a single scalar and think of the algebra as being an A 1 algebra or we can use three Majorana-Weyl fermions and think of the algebra as being a B1 algebra. In the A 1 case, one is constructing the representation* * The n u m b e r k is the central extension of the affine algebra and the number c is the central extension of the Virasoro algebra. Thsre are normalized such that k = 1 and c = 1 of a single scalar field.
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with k = 1 and c = 1. In the B 1 case, one is constructing the representation with k=2and c=3. The B 1 construction is quite interesting since partial bosonization leads to a system with a right-moving scalar • and a Majorana-Weyl fermion xo. This system is obviously supersymmetric. The supersymmetry current is given by either of the following two expressions: 3 ,
,9"= (Oz~) ~/'.
(6.1)
The supersymmetry interchanges the operators ~/i and g'. In the purely fermionic theory, the supersymmetry respectively interchanges Majorana-Weyl fermions ffa, g,2 and q,3 with SO(3) currents j23, j31 and ja2. This theory is actually an example of a super-Kac-Moody algebra [11]. There is some type of supersymmetry in many of these scenarios even though one does not have a super-Kac-Moody algebra in general. For example, consider the theory with five Majorana-Weyl fermions where one bosonizes four of the fermions. This is the scenario discussed in the SO(5) example of a previous section. The partially bosonized theory is described by a lagrangian which has a variety of symmetries. For example, there is an obvious SO(2) symmetry which rotates the scalars. There is also a supersymmetry which transforms ~x into q" and vice-versa. In fact, the combined SO(2) and supersymmetry algebra will close if in addition one has the following symmetry:
~2
=
__ 0 z l ~ l '
8 q ' = 0.
(6.2)
One immediately sees that given some scalars and some fermions one can always play such a game. The full symmetry group of such a theory is always larger than what one thinks. It contains a hidden supersymmetry but not a super-Kac-Moody symmetry. This is a special case of the work that has been discussed in ref. [11]. We are greatly indebted to Jim Lepowsky for his patient ear and his useful comments. We would also like to thank Tim Killingback, Neil Marcus and Emil Martinec for conversations. Orlando Alvarez and Michelangelo Mangano would like to thank respectively the theory group at Fermilab and the theory group at LBL for their warm hospitality. Fermilab is operated by Universities Research Association, Inc. under contract with the Department of Energy. We would like to thank the coffee brewers of the cafrs in the State of California.
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References [1] V.G. Kac, Math. USSR Izv. 2 (1968) 1271; R. Moody, J. Alg. 10 (1968) 211; V.G. Kac, Infinite dimensional Lie algebras (Birkhauser, Boston, 1983) [2] K. Bardakci and M.B. Halpern, Phys. Rev. D3 (1971) 2493; M.B. Halpern, Phys. Rev. D12 (1975) 337; T. Banks, D. Horn and H. Neuberger, Nucl. Phys. B108 (1975) 1976; I.B. Frenkel and V.G. Kac, Inv. Math. 62 (1980) 23; I.B. Frenkel, Proc. Nat. Acad. Sci. USA 77 (1980) 6303; G. Segal, Comm. Math. Phys. 80 (1981) 301 [3] P. Goddard and D. Olive, in Vertex operators in mathematics and physics, eds. J. Lepowsky, S. Mandelstam and I.M. Singer (Springer, New York, 1985); I.B. Frenkel, Representations of Kac-Moody algebra and dual resonance models, Chicago meeting, 1982 [4] J. Lepowsky and M. Primc, Standard modules for type-1 affine Lie algebras, in Number theory, New York, 1982, Springer Lecture Notes in Mathematics, vol. 1052 (1984), pp. 194-251; J. Lepowsky and M. Primc, in Vertex operators in mathematics and physics, eds. J. Lepowsky, S. Mandelstam and I.M. Singer (Springer, New York, 1985), p. 143; I.B. Frenkel, J. Funct. Anal. 44 (1981) 259; V.G. Kac and D.H. Petersen, Proc. Nat. Acad. USA 78 (1981) 3308 [5] D. Friedan, E. Martinec and S. Shenker, Phys. Lett. 160B (1985) 55 [6] M. Virasoro, Phys. Rev. D1 (1970) 2933 [7] S. Coleman, Phys. Rev. D l l (1975) 2088; S. Mandelstam, Phys. Rev. D l l (1975) 3026; J. Kogut and L. Susskind, Phys. Rev. D l l (1975) 3594 [8] D. Friedan, in Recent advances in field theory and statistical mechanics, Les Houches Session XXXIX, eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam, 1984); A.A. Belavin, A.M. Polyakov and A.M. Zamolodchikov, Nucl. Phys. B241 (1984) 1799 [9] L. Kadanoff and J. Ceva, Phys. Rev. B3 (1971) 3918 [10] E. Witten, Nucl. Phys. B142 (1978) 285 [11] P. di Vecchia, V.G. Knizhnik, J.L. Petersen and P. Rossi, Nucl. Phys. B252 (1985) 701; I. Antoniadis, C. Bachas, C. Kounnas and P. Windey, Physics Lett. B171 (1986) 51; P. Windey, Com. Math. Physics 105 (1986); P. Goddard, W. Nahm and D. Olive, Imperial preprint TP/84-85/25 (1985)
VERTEX OPERATOR CONSTRUCTION OF THE SO(2n + 1) KAC-MOODY ALGEBRA AND ITS SPINOR REPRESENTATION Orlando ALVAREZ 1"2"3and Paul WINDEY 3
Department of Physics and Lawrence Berkeley Laboratory, Universi(v of California, Berkeley, California 94720, USA Michelangelo MANGANO 4
Department of Physics, Princeton UniversiO,, Princeton, NJ 08544, USA Received 2 December 1985
An explicit representation of the B,(,l) affine Lie algebra (Kac-Moody algebra) is constructed in terms of vertex operators associated with the Chevalley basis of the underlying finite-dimensional Lie algebra. This construction, contrary to the simpler current algebra one, gives a concrete realization of the spinor representation of the algebra. The key feature is a partial bosonization of two-dimensional Weyl-Majorana free fermions. The vertex operators associated with the long and short roots of the B,, algebra have fermion number zero and one, respectively.
1. Introduction Explicit representations of Kac-Moody algebras [1] can be constructed in two different ways. The first one amounts to the construction of a two-dimensional field theory with a set of conserved currents obeying the commutation relations of the Kac-Moody algebra. The central charge is no other than the traditional Schwinger term. The Kac-Moody algebra is then easily seen as the loop algebra associated with the Lie algebra f# of the symmetry group G of the theory. Indeed the currents are analytic functions of one complex variable and the infinite set of generators are the coefficients of their Laurent expansion. From this point of view the treatment of all semi-simple Lie algebras is similar. The simply-laced algebra* play no particular role. In principle, each different representation (different value of the central 1 Alfred P. Sloan Foundation fellow. 2 This work was supported by the National Science Foundation under grant PHY-81-18547. 3 This work was supported by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the US Department of Energy under Contracts DE-AC03-76SF00098. 4 Supported by an INFN Fellowship. * A Lie algebra is said to be simply laced if all the roots have the same length. 0550-3213/86/$03.50 :~'Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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charge) is associated to a different model. A second approach is based on the construction of the vertex operators [2, 3] originally introduced in the context of dual string theories. Here one first reconstructs the underlying finite-dimensional Lie algebra and builds operators corresponding to the roots (raising and lowering operators) and to the Cartan subalgebra. The full affine algebra is then constructed either by taking the moments of these operators or by building along the same lines new operators corresponding directly to the extra roots (so-called imaginary roots) of the KacMoody algebra. These two constructions are of course completely equivalent although the latter one is closer in spirit to the original formulation of Kac-Moody algebra in terms of generalized root systems and Cartan matrices. Unfortunately, its apparent drawback was the difficulty in building vertex operators for non-simply laced algebras. The purpose of this article is to repair this apparent lopsidedness at least for the B,~1) algebras*. In principle, once one has the currents in the field theory the problem is solved. In practice, more information is required. For example, what is the operator that transforms under the spinor representation? It may be difficult to express the operator in question in terms of the elementary fields of the field theory. A representation of the operator may be found in terms of an equivalent but different field theory. These are the type of questions we address. Some of the results in this paper are apparently known to mathematicians [4]. In particular, the importance of having a fermion number associated with the different vertex operators was pointed out to us by Jim Lepowsky. The paper is organized as follows. In sect. 2 we discuss a two-dimensional fermionic field theory with a B2 symmetry. The construction of the associated vertex operator is exhibited. In sect. 3 the vector representation is constructed and the relation to the original fermions is given. The spinor representation is constructed in sect. 4. The relationship of the Ising model and fermions is seen to play a central role. The first use of the Ising model in a vertex operator construction is due to Friedan, Martinec and Shenker [5]. In sect. 5 we generalize the results to an arbitrary Bn algebra. Finally in sect. 6 we make some comments about a hidden supersymmetry.
2. T h e B 2 scenario
It is very simple to construct a two-dimensional field theory with a symmetry SO(2n + 1). Consider 2n + 1 Majorana-Weyl fermions ~V, i = 1, 2 . . . . . 2n + 1 with a lagrangian given by L = ~Jro~tpr.
* B~1) is the centrally extended loop algebra Bn, a.k.a. SO(2n + 1).
(2.1)
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The symmetry SO(2n + 1) is generated by the currents jrs = : ~r~ks:. These currents define a Kac-Moody algebra with an enveloping Virasoro algebra. The spinors q~r are the operators which correspond to the highest weight "vector type" representation of the B~~) affine algebra. We would also like to know what are the operators which correspond to the spinor representation of this algebra. This question is not easily answered by using the q~r but finds a simple solution through an argument of partial bosonization. Many of the main ideas involved in this construction are contained already in the B2(1) example. We first discuss this case in detail and afterwards generalize the ideas to the other odd orthogonal groups. Since SO(5) is a rank-two group the root diagram is easily drawn as seen in fig. 1. The roots a and fl are the simple positive roots. The roots are normalized such that
a.a=2,
/~./3= 1,
a./~= -1.
(2.2)
The remaining inner products are easily obtained from the figure. The bosonization [7] proceeds in the following way. The two generators of the Cartan subalgebra may be taken to be j12= :q~2: and j 3 4 = :~3~b4: . Bosonize kl and ~2 into a right-moving scalar ~1, and bosonize ~3 and ~b4 into another right-moving scalar ~2. In terms of the scalar fields, the generators of the Cartan subalgebra become 0z~ ~ and 0z~ 2. Let us denote ~p5 as ~/'. With a suitable definition the propagators of the free fields may be taken to be
(¢'(z)O,(w))
= -,'qog(z
- w),
(2.3)
1
(2.4)
w
a+2~
a Fig. 1. Root diagram for SO(5).
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The key formula in our derivation is the operator product expansion (OPE)* exp[i/~- ~ ( z ) ] e x p [ i v . ~ ( w ) ] = (z - w ) " % x p [ i ( / * - ~ ( z )
+ v. ~ ( w ) ) ]
- (z - w ) " % x p [ i ( / * + v ) . ~ ( w ) ] × [1 + i(z - w)lt. aw~(W)].
(2.5)
Non-vanishing commutators are determined by the pole singularities in the operator product expansion [8]. With our conventions /~-v is an integer. A singularity r e q u i r e s / , , v < - 1. There is another important result to bear in mind, namely that the operator exp[i/~. ~ ( z ) ] has conformal weight !2# . #. The current that generates the K a c - M o o d y algebra must be an operator of weight one. 2.1. THE CHEVALLEY BASIS We will now construct the B2°) algebra using • and ~/'. A very convenient basis for the study of the Lie algebra is the Weyl basis. If a is a root then let H~ be a suitably normalized corresponding element of the Cartan algebra. Also, let E~ be the associated raising operator. The commutation relations for the algebra may be written as:
[<, H,,] =0, [<, =
[E,,,E_~]=H~, [E,~,EI~ ] = N~./~E~+/~.
(2.6)
The integers N~,/~ can be chosen such that N~,~ = + ( p + 1) where p is the greatest integer for which fl - p a is a root. N~,~ = 0 if a + fl is not a root. Taking et and fl to be simple roots ai, the above commutation relations reduce to the defining relations of the Chevalley basis:
[U,, U,] =0, [H,, g] = [H.C] =
-:,+F+.
[ Ei, Fj] = SijHj.
(2.7)
* To interpret the OPE (2.5) as commutation relations of the Kac-Moody algebra one has to expand it in modes and introduce appropriate cocyclesmultiplying the vertex operators. A consistent choice of these cocycles for our construction will be given in the last chapter where we will analyze the general case. In the meantime, we will only worry about the OPE.
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We have used the obvious notation that H i = Ha,, E i = E~, and F, = E ~ . are the entries of the Cartan matrix given by
The aij
2 0 / i • £ltj
A = (a/j) - - -
(2.8)
Ot i • O~i
Our problem reduces to building a representation of the associated current algebra in the field theory. Since we work in the formalism of conformal field theory, we will build operators which obey in the Chevalley basis an algebra equivalent to: [J~(z), Jb(w)] =fabcjc(w)8(z
-- w) + k S ' ( z - w ) 8 "b,
(2.9)
where J a ( z ) is the current which generates the appropriate group transformation and the f~h,. are the structure constants of the algebra (2.6). (i) First we will consider the case where the two roots in (2.5) are long ones. In this case exp[i/t- ~(z)] is an operator of weight one. The inner product /~. v can only take the value 0 or + 2. If /x. p = 2 then /~ = u and there is no singularity in the OPE. The operators commute and (2.5) corresponds to [E,, E~] = 0. If /t. 7, = 0 then /~ + ~, is not a root. Once again the OPE is non-singular and corresponds to [E~ E,] = 0. If ~- 1, = - 2 then /~ = - p . The OPE gives 1
exp[i/t. ~ ( z ) ] e x p [ i ( - / x ) .
~(w)]
(Z
-- W) 2
[1 + i ( z - w)l~" Ow~(W)],
(2.10)
which is the equivalent of [E,, E ~ ] = / t . H. Note also that we got the central charge. All other commutators involving only long roots vanish. (ii) When p is a short root exp[io. ~(z)] has dimension ½. Since g" also has weight ½ we can combine them to get an operator e x p [ i p - ~ ( z ) ] ' / ' ( z ) with the correct conformal weight of one. Consider the product of two such operators: e x p [ i p . ~ ( z ) ] 'q'(z)exp[io. ~ ( w ) ] ~ ( w )
- (z-
w)°°exp[i(p
x [(z-w)
-1 @
+ o). (z-w)(aoj
2t)
2t(w)
] . (2.11)
In the above vI'(w)vl"(w) = 0 since the g' are Grassmann variables. From fig. 1 it is clear that p- o can take three possible values: (a) If p . o = + 1 then p = o and there is no singularity in the OPE. This corresponds to [E o, Ep] = O.
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If p- o = 0 then p + o is a long root. The singular part of the OPE is given by exp[io- ~ ( z ) ] ~P(zlexp[io. ~ ( w ) ] ~/'(w) 1 - --exp[i(p
+ o)-~lt(w)],
Z--W
(2.12)
and is equivalent to [Ep, E,] - Ep+o. (c) If to" o = - 1 then P = - o. Eq. (2.11) reduces to exp[ip, q~(z)] q S ( z ) e x p [ i ( - P ) " ~ ( w ) ] ff'(w) 1 ( z - w) 2 [1 +
i ( z - w)p" 0wq~(w)],
(2.13)
which is the equivalent of [Ep, E_e] = p. H with the required central charge. (iii) Finally we have to verify the commutation relations for a long root )~ and a short root o. Note that the sum of a long and a short root is either a short root or not a root. The relevant terms in the OPE are
~(z)]exp[io.
exp[i?~-
- (z-
°[1 +
~ ( w ) ] g'(w) i(z
-
w)X. aw (W)]
×exp[i(h + o).
(2.14)
If X. o _ 0, then their sum is not a root. This is reflected by the absence of a singularity in (2.14) and the zero commutator in (2.6). If X. o = - 1 , which is the only remaining possibility, their sum is a short root and we have exp[iX-
~(z)lexp[io.
1 - --exp[i(h Z--W
~ ( w ) ] qS(w) + o)" ~(w)] ~(w),
(2.15)
which is [E x, Eo] - Ex+ o. We are only left with the task of checking the commutators of the raising and lowering operators with the Cartan generators o. 0 ~. This computation is entirely analogous to the corresponding one for simply-laced groups since the qp field plays a passive role. The key formula is
O" Oz~(z)exp[i~.
~(w)]
--ip" l~
- -
Z--W
exp[i/~- t~(w)],
(2.16)
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which states that the exponential transforms as an element of a representation containing the root It. T w o comments are now in order. Firstly, the infinite set of operators of the affine Lie algebra are defined from the operators we just built by taking the coefficients of their Laurent expansion. They are of the following type*:
1 -~dzz"exp[iit. ~(z)]
2~ri
1
2vri ~ dzz~t~" 0 ~ .
(2.17)
Secondly, the normalizations of the operators should be treated with some care in order to reproduce (2.6). In particular the generator of the Cartan subalgebra corresponding to the simple root a, should be multiplied by eg where the e, is defined in the following way. The symmetric matrix of inner products B may be written in the form A = DB where D = diag(e~) and the e i are the set of minimal positive numbers such that B is symmetric. These remarks will be equally valid in the treatment of the general case in sect. 5.
3. The vector representation In this section we will show how to reconstruct the original spinors ~bi, i = 1 . . . . . 5 in terms of the • and q" operators. The straightforward method is to construct operators with the correct SO(5) transformation properties. The associated weight diagram is given in fig. 2. Define an operator 12A(Z) = exp[iA • ~(z)]. If/~ is a root, then the O P E of the Cartan algebra generator It. a ~ with 12A(Z) immediately verifies that A plays the role of a weight in the operator 12A(Z). Choose A to be the highest weight of the vector representation A = a + ft. The OPE with an operator corresponding to the root It is given by
exp[iit- ~ ( z ) ] 12a(w ) - (z - w)~'aexp[i(it + A ). ~ ( w ) ]
x [1 + i ( z - w)it.
(3.1)
* The interested reader should consult ref. [3] for more details. In particular we will not rewrite the correct Chevalley basis corresponding to the extended Cartan matrix [1] nor dwell upon the construction of the operators corresponding to the imaginary roots. Given the above construction of the algebra these extensions from the simply-laced to the non-simply-lacedcase are straightforward.
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v~v
~
Fig. 2. Weight diagram for the vector representation.
If g is a positive root then g . A > 0 and there is no singularity in the operator product. This means that the two operators c o m m u t e as required since A is a highest weight. If g = - a then # . A = - 1 and the c o m m u t a t o r is [exp( - i a . ~ ), I2a] - ~2~.
(3.2)
In other words the root - a takes the weight A = a + fl to the weight ft. If g = _+fl, t h e n / ~ . A = 0 and there is no singularity in the OPE, therefore (3.3)
[exp( +_ifl . ~ )vl", ~A] = O.
O n e easily verifies also that if one chooses g = _+(a + 2fl) then one gets the correct c o m m u t a t i o n relations. T h e only case left to check is when g = +_(ct + fl). N o t e that the inner product /s- A equals _+ 1. The c o m m u t a t o r vanishes if/~ = a + ft. If g = - ( a + fl) then the O P E gives exp[i(--a--fl)-~(z)]~(z)~2A(W
)-(z-w)
X~(w).
(3.4)
Therefore the central weight corresponds to the operator ~/'. To verify this we note that [ g . 0 ~ , xO]- 0 since the O P E has no singularity. If /~ is a long root then exp[i/~ • ~ ( z ) ] g ' ( w ) - 0. If g is a short root then e x p [ i ~ , q~(z)] x t ' ( z ) ' t ' ( w )
- ( z - w)-~exp[i/~ • ~ ( w ) ] .
(3.5)
This is the correct c o m m u t a t i o n relation with a short root. N a m e l y the short root m o v e s us from the origin to one of the corners of the square.
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We have constructed explicitly the five operators which transform according to the vector representation of S0(5): exp[i(a + fl). ~ ( z ) ] , exp[i(-fl). ~(z)], exp[i(-a
- ~ ) . q~( z)] ,
exp[i/3. ~ ( z ) ] , '/'(z).
(3.6)
Note that the non-zero weights have length one, hence all the above operators have conformal weight ½ as required since the Virasoro algebra is the enveloping algebra of the Kac-Moody algebra. This proves that the above are essentially the five-spinors q~i. To get the correct anticommutation relations one should multiply the operators given above by the appropriate cocycle.
4. The spinor representation The four-dimensional spinor representation of SO(5) corresponds to the weight diagram shown in fig. 3. The four weights are equidistant from the origin forming the corners of a square. One moves along the diagonal with a long root while a displacement across an edge is produced by a short root. If tt is a weight of the spinor representation then the operator ~2~ has the same conformal weight for the four values of ~. This would be perfectly alright if one only had to consider the long roots. But the vertex operator for the short roots, which contains a fermion, will change the fermion numbers when acting on $2 and will cause havoc by changing
Fig. 3. Weightdiagram for the spinor representation.
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the conformal weight of the operator. To remedy this situation it appears that one needs an operator with indefinite fermion number which will behave correctly under the action of the short roots. Such an operator exists, it is the celebrated spin operator of the Ising model. The highest weight for the spinor representation is A = I a +/3,
(4.1)
with A ,a
-
1
(4.2)
One will need the vertex operators
exp[ ila
. dp( z
)],
exp[ i( l a + /3 ) " exp[ i(-
z )] , (4.3)
la - /3 ) . ¢( z )] .
Let us begin with the highest weight A and study the action of the long roots. This is summarized in table 1. Next, let us see what happens to ~a under the action of a short root - / 3 . Note t h a t / 3 - A = I and therefore the OPE behaves as
exp[i(-fl),
rb(z)] g ' ( z ) e x p [ i ( l a
+ fl ) . ~ ( w ) ]
1
vfz - w e x p [ i l a " ~ ( w ) ] ~ ( w ) .
(4.4)
Notice that ~ has the same conformal weight a s f~a/2, therefore, the short root has not only produced an operator with the wrong weight but also with a different fermion number. More significantly one should get rid of the square root and recover an integral power to obtain the desired commutator. It is clear that one
TABLE 1
Action of roots on highest spinorial weight Long root p +a
a+2fl - (a + 2fl)
OPE power p •A
Commutator
0
0
1 - 1
0 exp[i(- 12a- fl). ~]
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needs an additional operator with indefinite fermion number or a "coherent" state of fermions. Such a composite operator would have the necessary properties under the action of any of the members of the algebra. This operator is the spin operator of the Ising model we alluded to previously. In fact the Majorana-Weyl fermion g' is equivalent to an Ising model. Let ~ be this spin operator. Its conformal weight is ~ ; Kadanoff and Ceva [9] demonstrated years ago that a
qt(z)2~(w)
~/z - w Y.(w),
(4.5)
where a is an irrelevant constant. It is straightforward to verify that the required operators are exp[i½a + / 3 - @ ( z ) ] ~ ( z ) , exp[i~a-dp(z)l,~(z),
exp[ i ( - ½a - /3 ) . ~ ( z )]~( z ), exp[i(-½a). ~(z)l,~(z).
(4.6)
The spin operator is innocuous as far as the long roots are concerned, but it provides the necessary correction for the OPE with the short ones. Again to obtain the correct anticommutation relations one should multiply the operators with the appropriate cocycle. Notice that each of these operators has conformal weight 1 ~ "q- 1 - -
76.
5. The general case
The discussion of the previous sections generalizes to the case of B. without additional complications. The algebra B. has the following Dynkin diagram: A
A
A
A
~
There are n - 1 simple positive roots a x, a 2. . . . . an-1 with length 2 and one simple positive root fl of length 1. It is useful to note that the root system of B, can be interpreted as the union of the root system of D n with the weights of the 2n-dimensional vector representation of D,. This corresponds to the decomposition adj(SO(2n + 1)) ~ adj(SO(2n)) • vect(SO(2n)),
(5.1)
under the reduction SO(2n + 1) ~ SO(2n).
(5.2)
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Since D n is simply laced with roots # normalized in such a way that ~t-/~ = 2, the weights of the vector representation of D, will play the role of the short roots of Bn. In [3], it was shown that it is possible to identify the vertex operators constructed for vect(SO(2n)) with the 2n-dimensional Dirac gamma-matrices. If + ~tz for l = 1 . . . . . n are the weights of vect(SO(2n)) satisfying/~z" #k = S/k, then the operators r,(z)
= a.,(z)
+ ~
,,(z),
r,,+,(z) = - i [ a . , ( z ) - ~ .,(z)],
(5.3)
and an appropriate cocycle will generate a Clifford algebra. In fact it is straightforward to see that:
r,(z)r.+,(w)-0, r,(z)r~(w)
28tk - - ZmW
ro÷,(z)r.+k(w)
28lk - - -
Z--W
(5.4)
In [3] it was proven that there exist cocycles which turn the OPE in (5.4) into the anticommutation relations of a Clifford algebra when we go into the mode expansion. With this in mind we return to our construction. A representation of the Kac-Moody algebra is furnished by 2n + 1 Majorana-Weyl fermions with lagrangian t = @ros@r
(5.5)
and currents j i j = : i~t~j:. One bosonizes the ~,k and ~kk+" for (k = 1 . . . . . n) into a scalar @k. The last Majorana-Weyl fermion will play exactly the same role as in the SO(5) case, and we keep the same notation q,2,+1_ ~. The Cartan subalgebra generators become 0fl) k. If X is a long root and o a short one then the collection of operators exp(ih. ~ ) , exp(io- ~ ) ~
(5.6)
are the raising and lowering operators for the Kac-Moody algebra. This may be verified by choosing the standard representation for the roots of the B~ algebra:
+ei+%, ± ei ,
for i :~j roots of D,,, for i ¢ j weights of v e c t ( D , ) ,
(5.7)
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where ( e i } is the standard basis for R n, and explicitly computing the commutators. It is clear that in the case of two long roots there are cocycles which will turn the O P E into the correct commutation relations. The reason is that we are only probing the D, subalgebra of B, and we know that such cocycles exist. In the case of a long root with a short root one requires two observations. Firstly, the decomposition described in eq. (5.1). Secondly, the existence of a cocycle which transforms Fl(z ) into a gamma matrix makes exp(io • ~ ) ' / ' transform as a vector under D,. In fact, if F~ are the gamma matrices and if M u is a generator of the D, algebra, then one immediately obtains
[ui,, rk] =
<], rk]--
(5.8)
If both roots are short ones then the only subtlety arises from the consistency of the choice already made for the cocycles. Naively, it appears as if one would get anticommutation relations instead of commutation relations. One has to remember the presence of the anticommuting fermion xo which resolves the apparent paradox. Due to the presence of the fermion '/', the anticommutation relations between the vertex operators in vect(D,) become ordinary commutation relations. This defines the remaining part of the algebra as a straightforward calculation will show. The vector representation is obtained by considering the operators exp[ +__iej. eb] and g'. The Kac-Moody spinor representation with dimension 2" is obtained by first specifying the spinorial weights for the B, representation: ½(+_e l___e 2___ . . . +_e,).
(5.9)
The vertex operators are of the form exp[iA. ~ ] X ( z ) ,
(5.10)
where A is a spinorial weight. The dimension of the vertex operator is 1 1 $" a n + ~6 = ~ ( 2 n + 1).
(5.11)
This construction also works for C 2 which is isomorphic with B2 but alas not for C,, n > 2 . 6. Hidden supersymmetry It is not generally appreciated that some two-dimensional field theories have a hidden supersymmetry [10]. This is most obvious if we apply our construction to the SO(3) situation. This is an interesting case since there are two ways of constructing SO(3) Kac-Moody algebras. We can use a single scalar and think of the algebra as being an A 1 algebra or we can use three Majorana-Weyl fermions and think of the algebra as being a B1 algebra. In the A 1 case, one is constructing the representation* * The n u m b e r k is the central extension of the affine algebra and the number c is the central extension of the Virasoro algebra. Thsre are normalized such that k = 1 and c = 1 of a single scalar field.
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with k = 1 and c = 1. In the B 1 case, one is constructing the representation with k=2and c=3. The B 1 construction is quite interesting since partial bosonization leads to a system with a right-moving scalar • and a Majorana-Weyl fermion xo. This system is obviously supersymmetric. The supersymmetry current is given by either of the following two expressions: 3 ,
,9"= (Oz~) ~/'.
(6.1)
The supersymmetry interchanges the operators ~/i and g'. In the purely fermionic theory, the supersymmetry respectively interchanges Majorana-Weyl fermions ffa, g,2 and q,3 with SO(3) currents j23, j31 and ja2. This theory is actually an example of a super-Kac-Moody algebra [11]. There is some type of supersymmetry in many of these scenarios even though one does not have a super-Kac-Moody algebra in general. For example, consider the theory with five Majorana-Weyl fermions where one bosonizes four of the fermions. This is the scenario discussed in the SO(5) example of a previous section. The partially bosonized theory is described by a lagrangian which has a variety of symmetries. For example, there is an obvious SO(2) symmetry which rotates the scalars. There is also a supersymmetry which transforms ~x into q" and vice-versa. In fact, the combined SO(2) and supersymmetry algebra will close if in addition one has the following symmetry:
~2
=
__ 0 z l ~ l '
8 q ' = 0.
(6.2)
One immediately sees that given some scalars and some fermions one can always play such a game. The full symmetry group of such a theory is always larger than what one thinks. It contains a hidden supersymmetry but not a super-Kac-Moody symmetry. This is a special case of the work that has been discussed in ref. [11]. We are greatly indebted to Jim Lepowsky for his patient ear and his useful comments. We would also like to thank Tim Killingback, Neil Marcus and Emil Martinec for conversations. Orlando Alvarez and Michelangelo Mangano would like to thank respectively the theory group at Fermilab and the theory group at LBL for their warm hospitality. Fermilab is operated by Universities Research Association, Inc. under contract with the Department of Energy. We would like to thank the coffee brewers of the cafrs in the State of California.
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