Quantum realizations of the superanomally in two-dimensional models

Nuclear Physics B261 (1985) 731-749 © North-Holland Pubhshmg Company

Q U A N T U M R E A L I Z A T I O N S OF T H E S U P E R A N O M A L Y IN T W O - D I M E N S I O N A L M O D E L S Paolo ROSSI1 ETH, Zurich, Switzerland

Received 31 January 1985 We show that the supersymmetnc versmn of the Wess-Zummo acUon m two d~menslons (SWZ) can be coupled m a supersymmetricand gauge-mvanantway to external gauge superfields We compute the generating functional for the Green functmns of the current supermult~plet and show that tt possesses a superchlral anomaly propomonal to the superfield strength We show that this funcUonal is also the superdetermmant for a gauged fermlomc theory, chiral SQCD2, eqmvalent to the gauged SWZ model at the quantum level

1. Introduction Effective lagrangians are a valuable tool in the process of understanding the low-energy phenomenology in many theories of elementary particles. Recently special attention has been devoted to the constraints imposed by chiral anomalies on the structure of effective lagrangians, and a very interesting extension of this approach deals with the problem of constructing supersymmetric effective theories. In this context, our attention was attracted by two-dimensional models because of the property often shared by effective lagrangians of being completely equivalent quantum mechanically to the original theories; this phenomenon has sometimes led to the solution of the models examined. This is also the case for the two-dimensional non-abelian Wess-Zumlno (WZ) model and its counterpart, the free fermionic theory, related to each other by Witten's bosonization rules [1]; this equivalence can be easily extended to the case when interaction with an external gauge field is introduced [2]. In a recent paper [3] the supersymmetric extension of this quantum mechanical equivalence has been shown to exist for the case when no gauge fields are present. The resulting super-WZ model is the obvious candidate for the explicit realization of a theory possessing a superanomaly or, otherwise stated, for the parametrization of the superdeterminant resulting from the mtegratlon over the matter fields in some properly defined (i.e. chirally invariant, in contrast with the standard one) version of N = 1 super-QCD in two dimensions. On leave of absence fromthe Scuola Normale Supenore and the Istltuto dl Flslca dell' Umverslt~--Plsa, Italy Permanent address after February 8, 1985 SNS, 56100 l:hsa, Italy 731

732

P Rossz / Superanomaly

The aim of this paper is therefore twofold: (1) We want to construct a gauged super-WZ model by introducing an interaction with an external N = 1 gauge supermultlplet in such a way as to preserve all the interesting symmetries of the model (ii) We want to exhibit an eqmvalent fermlonic theory that can play the rrle of SQCD in two dimensions. The generating functional for both theories will be exactly the superdeterminant we have been talking about. The symmetry properties of the models must be manifest in the expression of this functional. This p a p e r is orgamzed as follows. Sects 1-3 are of a rather formal nature, they are devoted respectively to the parametnzation of chlral superfields in two dimensions, to the parametnzation of gauge superfields m terms of chiral superfields and finally to the WZ gauge, the evaluation of the superanomaly and the construction of compensating gauge rotations restricting the supersymmetry transformations to the WZ gauge. In sect. 4 we construct the supersymmetric WZ lagranglan in the presence of gauge superfields, reahzing the superanomaly at the classical level. In sect. 5 we construct the quantum version of the same theory, evaluating exphcitly the superdetermmant and showing that when properly defined it is supersymmetry and supergauge invariant. In sect. 6 we discuss the properties of the free fermiomc supersymmetnc theory, extending the results of ref. [3] and finally in sect. 7 we construct the gauged fermlonic model eqmvalent to the super-WZ model previously exhibited and defining SQCD. Appen&ces are devoted to more detailed proofs of some of our statements. For all conventions not explicitly stated m this p a p e r we refer to ref. [3].

2. Chiral superfields in 2 dimensions and their parametrization The standard expression for an N = 1 scalar superfield in two dimensions is G = g + tO~b+ lzOOF.

(2.1)

However the (super)unitanty condition G+G = 1

(2.2)

implies a number of nonlinear constraints on the component fields making it very inconvenient and unphysical to work with these variables. We found that by introducing antihermitian unconstrained fields X and q~ (a spinor and a scalar) one can solve the constraints with the replacements ~0+= gx+,

O- = x - g ,

F = g4~ + ~x-gx+.

(2.3a) (2.3b)

Substitution of eqs. (2.3) in eq. (2.1) leads to the reparametnzation G = (1 + tO+x_)g(1 - sO_x+)+ iO+O_gdp,

(2.4a)

P Ross~/ Superanomaly G + = (1 + IO_x+)g(1 - tO+O-) - iO+O-q~g+ •

733 (2.4b)

The (linear) supersymmetry transformations

8g = t~O , 60 = t~ga + Fa, 8F = ~6t~O ,

(2.5)

turn into 6g = ~(a+x-g - a - g x + ) ,

(2.6a)

8X+ = (g+0+g + 'X+x+)a- + ~ a + ,

6)(- = (ga-g ++/X-X-)a+ + g~g +a_, 6d~ = -we+O_X+ - ~a_g +O+x_g + za-[x+, ~]

(2.6b) (2.6c)

Eqs. (2.6b) in particular are related to the existence of the two light-cone components of the "current":

J+ = G+D+G = 'X+ + iO-(g+O+g + ~X+X+)+ iO+~ + 0+ O-(g+O+x-g - [X+, ~ ] ) , (2.7a) J_ = GD_ G + = - ~X- - iO+(gO-g ++ tX-X-) - ~Og-~g ++ 0 +O-(gO_x+g +- [X-, g~bg+]) • (2.7b) In ref. [3] we found that a supersymmetric Kac-Moody lnvariant lagrangian can be defined for the fields (3, and its expression is 1

SS(G)=]-~

I d2xd2OTr{½DG+DG+ f d t G + ~ t DG+y, DG}

(2.8)

(a factor ½was inadvertently omitted in ref. [3]). The variation of this lagrangian is

6SS( G) =8-81f d2x d2O Tr { D ( G+ l--+-/--2G ) G+SG} ,

(2.9)

leading to the classical equivalent equations of motion

-D_J+ =0,

D+J_ = 0 .

(2.10)

In component language

1 f d2xTr { O~,g+O~,g - 2 Idtg+~te~'~O~,g+O~g } SS(g, X, ~b)=~-~

'I

167r

d2x Tr {,I~X + 14~4~},

(2.11)

and using the equation of motion for the field 4~ we can reduce eq. (2.11) to

SS(g, X) = S(g) - i - ~

d2x Tr,f'hX,

(2.12)

734

P Rossl / Superanomaly

invariant under 8g = t( a + x _ g - a - g x + ) ,

(2.13a) (2.13b)

8X± = j ± a ~ ,

where we have defined j+ = g+O+g + iX+X+,

J-= gO-g++ tX-X- .

(2.14)

More about this model can be found m ref. [3].

3. N = 1 gauge superfields in two dimensions and their parametrization

N = 1 gauge superfields in two dimensions are special in the context of supersymmetric gauge theories. Indeed they do not form as usual a scalar superfield, and they must be introduced m the form of a spmor superfield [4]: V,~ --- ~:,~+ y5%,B~'0 + ( M + ysN)O+½zO0~,~

(3.1)

(a rescaling factor of 2 has been introduced in the definition of some components for computational convenience and the couphng constant is absorbed in V~). The light-cone components of the spmor superfield are therefore V+ = ~+ + B+O_ + ( M + N ) O + + ~0+0_~+,

(3.2a)

V_ = ~_ + B_O+ + ( M - N ) O _ + tO+O_~_.

(3.2b)

The gauge-covariant derivative takes the form (3.3)

V ,~ = D,~ + tys V,~ ,

and we can form the covanant field strength VysV = t(Oa V~ -1Vy5 V) = t~.

(3 4)

Eq. (3.3) indicates that a pure gauge superfield ( ~ = 0) can be given the form V,~ = - z G + y s D , ~ G ,

(3.5)

and suggests the following parametrizatmn of two-dimensional gauge superfields in terms of a couple of chiral superfields" V+ = - z A + D + A ,

(3.6a)

V _ = IB+ D _ B

(3.6b)

The parametrizatlon of components is shown in appendix A. Supergauge and superch~ral transformations are easily defined in terms of transformations of the chiral superfields. A supergauge rotaUon amounts to A ~- A ~ ,

B ~ B~,

(3.7)

P Rossz/ Superanomaly

735

and a superchiral transformation is simply A~AC¢,

B ~ B q ¢ +.

(3 8)

The complete analogy with the corresponding formulation of the standard chiral Wess-Zumino model is self-evident.

4. The Wess-Zumino gauge and the superanomaly It is always possible to remove some components of the gauge supermultiplet by means of a gauge transformation like those described m eq. (3 7). In particular one can always choose the following supergauge conditions. ~=0,

N=0,

(4.1)

known as the WZ gauge condition [5] As a consequence it is interesting to focus our attention on the class of chiral superfields A, B such that eq. (4 1) is satisfied. Using the component formulation given in appendix A it is easy to recognize that X+A=X_n=0,

~bA=--~B = M ,

(4.2)

satisfy the above constraint and the following WZ gauge parametrization holds: B+ = a+a+a, ~t+ --- ~+ = -a+0+~o_a,

B_ = b+O_b, )t_ --- ~_ = b+a_co+b,

(4.3a) (4.3b)

where we have redefined the WZ gauge matrices: A = (1 + zO+oJ_)a+ iO+O_aM,

(4.4a)

B = (1 + iO_oJ+)b- zO+O_bM.

(4.4b)

There is still a residual gauge invanance corresponding to standard gauge rotations:

a~ag,

b~bg.

(4.5)

Within the WZ gauge it is easy to compute the superfield strength and find [4] ~=-M+z0h

1 - ~ F,~. +~z00e

(4.6)

We recognize that the last component of ) is the standard anomaly term and we are therefore tempted to identify (1/4~-)) with the anomaly supermultiplet of some supersymmetric chiral model in two dimensions We also recognize immediately that the supersymmetric functional - S S ( A B ÷) defined by eq. (2 8) enjoys the property

-,3SS(AB ÷) = - ~

d2x d20 Tr {-½(A÷SA- B+SBL¢},

(4.7)

736

P Rosst / Superanomaly

and it is therefore a supersymmetric, supergauge-invariant functional whose anomaly under superchiral transformations C = 1 + z0c is exactly given by eq. (4 6). The WZ gauge is very convenient for the analysis of the physical content of the model. However manifest supersymmetry invariance is lost because a supersymmetry transformation of the gauge supermultiplet will not in general respect the constraint equations (4.1). It is however obvious that we can always find a gauge rotation of the transformed gauge field such that the new field will fulfill the WZ gauge condition. The composition of a supersymmetry variation and the corresponding gauge rotation bringing back the WZ gauge defines a new kind of supersymmetry transformation respecting the WZ gauge condition. The explicit determination of this transformation is therefore particularly important in view of possible checks of supersymmetry made without giving up the simplicity of the WZ gauge. Working directly with the A-, B-matrices and making use of the multiplication formulas for unitary superfields given in appendix B we easily find out that the gauge rotation accompanying a supersymmetry transformation is characterized by X ~ = -,ga - ysMot,

(4.8)

and after some algebraic effort we recognize that ~b~ = ½z~ysA •

(4.9)

Some details of the derivation are presented in appendix C. We can now evaluate the composite transformation and obtain the following invariance: 6a = za+o~_a,

6b = za_oJ+b ,

6oJ_ = 2 a M a +a_ + t w _ w _ a + + ab +O_( ba +)a+ ,

(4.10a) (4.lOb)

&o+ = 2 b M b +a+ + too+o~+a_ + ba +O+( ab + ) a _ , 6M

= -½1~A

(4.10c)

Substituting eqs. (4.10a) and (4.10b) into the defining equations (4.3) we obtain the set of transformations 6B~, = -½~fft%, ysA ,

(4.1 la)

= -½,aA,

(4.1 lb)

6M

8A = e~'~ F ~ a - 2YJ M a .

(4.1 lc)

As a consequence of eqs. (4 11) one can check directly, without introducing a superfield formulation, the lnvariance of the kinetic gauge lagrangian Tr { - z F1 ~ , , F

,tLV

1 ~ +~V~,MV M - ~ z1 A- -~ A - ] t £ y s A M } .

(4.12)

Moreover, by making use of eqs. (4.8) and (4.9), it is quite easy to find out the

P Rossl / Superanomaly

737

effect of the accompanying gauge rotation on the supersymmetric transformation of an arbitrary unitary superfield, as defined in eq. (2.1). The gauge-covariant supersymmetry transformations replacing eqs. (2.5) are 6g = i 6 ~ ,

(4.13a)

6q~ = Ylgot + [ M, g] ysot + Fo~ ,

(4.13b)

6 F = zFtYl~b+ ~6tys[ M , q,] + ½Zays[g, X],

(4.13c)

and we can deduce from eqs. (4.13) the transformation properties of the component fields (g, X, q~): Sg = za+ x _ g - la_gx + ,

(4.14a)

6?(+ = a _ f + - 2 M a + + [6 + M + g+Mg)a+ ,

(4.14b)

6)(- = a+f_ - 2 M a _ + g( qb + M + g + Mg)g+ot_ ,

(4.14c)

B( c~ + M + g+ M g ) = - 1 a+(V_X+ + A_) - ~a_g+ (V +x_ - A+)g -

ta_[cb + M + g+Mg, X+],

where we have defined f+ = g+V+g + iX+X+,

(4.15a)

f - = gV-g+ + IX-X-

(4.15b)

5. The gauged super-Wess-Zumino model in two dimensions In the previous sections we have set up the formalism that will allow us to exhibit a simple construction of the gauged super-Wess-Zummo (SWZ) model in two dimensions. Thanks to the parametrlzation of gauge superfields in terms of unitary superfields in our construction we can mimick the procedure that led to the WZ model in two dimensions, starting from the free WZ lagrangian. Let us briefly recall that, defining S(g)-~-~

1

f d2xTr{O~,g+O~'g-2 f dtg+-~te~'~O~,g+O~g}

(5.1)

we can prove that WZ (g, B~) =- S( agb +) - S( ab ) ,

(5.2)

where B+ = a+~+a, B_ = b+O_b, is a gauge-mvariant lagrangian. For further reference let us also recall that, because of the property [2] S(ag) = S(a)+S(g)--~

d2x Tr (a+O+agO_g+),

(5.3)

P Rossz / Superanomaly

738

one obtains WZ

1 I d2x Tr [B+gO_g++ B_g +O+g+ B+gB_g+- B+B_] .

(g, B.) = S(g)--~--~

(5.4) Let us now define the following lagrangian: SWZ

((3, V,) =--SS(AGB +) - SS(AB+) ,

(5.5)

where SS(G) is defined in eq. (2.8). SWG (G, V~) is supersymmetric and supergauge invariant by construction and we can interpret eq. (5.5) as the gauged super-WZ model in two dimensions. Superchiral invariance is manifestly broken by the term -SS(AB) and therefore the (classical) superanomaly of the model we have constructed is exactly (1/4~)~. Eqs. (5.3) and (5.4) can be easily generalized to the supersymmetric case. Indeed one can prove

SS(AG)=SS(A)+SS(G)+I-~-f 8zr

d2x d20Tr

[ GI~,,G+ l +~ ys A+D,,A ]

(5 6)

and as a consequence SWZ

(G, V,~)= SS(G)+~--~ f

d2x d20 Tr [ V+J_ +

V_J+- ,V+GV_G++ ,V+V_]. (5.7)

In order to extract the physical content of eq. (5.7) we can use the WZ gauge conditions and substitute eqs. (2.7) for J±. We can also perform explicitly the integration over anticommuting variables and obtain the following component lagrangian: SWZ (g, X; B•, M, A) : WZ (g, B~,) 1

16~r

f d2xTr[$~X-$A-AX-4tM 2]

(5.8)

where we have also made use of the equations of motion for the field ~b:

~b+ M + g +Mg = 0.

(5.9)

An easy way to rederlve eq. (5.8) makes use of the definition of SWZ, eq (5.5), and of the composition rules for unitary superfields presented in appendix B. We want to present it here because it will be relevant in the discussion of the quantum properties of the model. We noUce that, having defined SS(g, X+, X-, ~b) in eq (2.11) we can prove the following relationships:

SS(AGB+) =-SS(agb +, bx+b++to+, ax_a++w_, qb+M+g+Mg) SS(AB ÷)=-SS(ab÷ ; w+, w_; 2 M ) . Then eq. (5.3) follows trivially from eqs. (2.11) and (2.12).

(5 10) (5.11)

739

P Rossl / Superanomaly

It is interesting to consider the variations of the functional SWZ (G, V~) with respect to the fields. Standard mampulations lead to

l I d2xd2OTr{G+SG[-V_]++~]+rV+]_+rV_]+},

8 SWZ (G, V,) =~--~

(5.12) where we have defined the covariant currents .~+ = G+V+ G = 6 SWZ, 8V_ ~_ = GV_G+ = 6 SWZ 8V+ '

(5.13)

and the following relationship holds: Tr { G + r G [ - V _ . L

+ ,~]} = Tr { S G G + [ - V + ] _

+ t~]}.

(5.14)

The resulting classical equations of motion V,,.~ = 0 ,

(5.15)

V~Ys-~ = - 2 , ~ ,

(5.16)

are equivalent to the superanomaly equation. In component language the equations of motion turn out to be V_(g+V+g) + F_+ ~ -g+[V+(gV_g +) + F+_]g = 0,

(4.17a)

~X - A = 0.

(5.17b)

By making use of the results of sect. 4 we can write down the supersymmetry transformations that leave eq. (5 8) invariant: (5.18a)

8 g = ~ot+x_g - l a _ g x + ,

8X+ = ot_f+-2Mot+ , (5 18b)

8)(- = o z + f _ - 2 M o t _ ,

accompanied by the transformations of the gauge fields, eqs. (4.11). Moreover one may check that, because of the complete decoupling of the field M in SWZ and of eq. (4.11b), all the terms proportional to M in the variation equations can be removed together with the term M 2 in the lagrangian. We therefore define a restricted ( M = 0) lagrangian !

I

SWZ = WZ (g, B~,) - 1 - ~ )?~X + ~--~ SA,

(5.19)

enjoying the restricted invariance 6g = ~a+x_g - ia_gx+ ,

(5.203)

740

P Rossz / Superanomaly

6X± = a:~f ± ,

(5.20b)

6B~ = -½16y~ysh,

(5.21a)

6A = e ~ F ~ o L .

(5 21b)

This observation wdl simplify our discussion of the quantized version of the SWZ model

6. Quantum mechanical version of the S W Z model

So far we have only considered the properties of the classical langrangian. However in order to define a quantum theory we must be able to perform a functional mtegratlon over the variables g, X. Keeping m mind that the restriction M = 0 does not really lead to any loss of information, let us focus on the functional integration of SWZ (g, X, B~, A) = SS(agb+; bx+b++ w+, a x _ a + + c o _ ) - S S ( a b +, to+, co_).

(6 1)

If we could perform safely the variable changes (6.2a)

g -->agb + , X+--> bx+b+ + co+, X_-> ax_a + + to_ ,

(6.2b)

corresponding to the classical chlral symmetry of the fields, the result would be straightforward. In I dg dx exp (~ SWZ) = - ~ S S ( a b +, to)

(6.3)

However in obtaining eq. (6 3) we have ignored the chiral anomaly that in two dimensions affects also the adjoint representation. Taking into account this anomaly we immediately recognize that the supersymmetry transformations equations (5.20) are not an invanance of the functional measure because, apart from trivial translations, they correspond essentially to an infinitesimal chiral rotation with parameters OL = a + X - ,

OR = a - X + ,

(6.4)

V+(~X-X-)c,

(6.5)

affecting also the fermionlc fields PAs a consequence the classical operators V-(zX+X+)c,

should be replaced quantum mechanically by V_(tX+X+)q+ CAF_+,

V+(zX-X-)q + CAF+_,

(6.6)

P Rosst / Superanomaly

741

where m the adjoint representation the generators obey Tr (TaT b) = CA6ab,

T~T ~ = CA1,

(6.7)

and for O(N) groups CA = N - 2 . Therefore in the limit A = 0 lnf dgdxexp[iSWZ(g,x,B~,A=O)]=-z(l+CA)S(ab+),

(6.8)

and if we are to suppose that supersymmetry can be actually realized at the quantum mechanical level by a proper redefinition of the measure (that is by a lagrangian counterterm) we are then bound to expect the following result for the functional mtegration In J dg dx exp z (SWZ)q(0, X, B,, A) = -t(1 + CA)SS(ab +, to),

(6.9)

where (SWZ)q must incorporate the anomaly effects due to the measure. It is actually possible to give a formal expression for (SWZ)q which, together with a quantum (operator) version of the supersymmetry transformations, realizes the request that eq. (6.9) be saUsfied. Let us consider indeed L ~A, (SWZ)q = WZ (g, B~,)- 1 -i- ~ T X + _8~r

(6.10)

S=x/1 + CAX

(6.11)

where is the quantum mechanical operator such that 6S± = a~f±.

(6.12)

One can compare eqs. (6 11), (6.12) with the definmon of S introduced in ref. [3] and find agreement up to an irrelevant x/2 scale factor depending on the different normahzation of the free fermiomc field lagrangian. The quantum mechanical supersymmetry transformations are eq. (5.12) and 1 ~/1----~A[la+x_g-ta_gx+],

~g

6X±

1 . l +~Aa~J± .

(6.13a) (6.13b)

Now when we take a variation of the functional (SWZ)q according to eqs. (6.12) and (6.13) we find after some algebra that it reduces to: t

6 (SWZ)q=

1

87r 4 1 ~ A Tr{[v-(zx+x+)q+CAF-+]a-x+ + [V+(tX-X-)q+ CAF+_]a+X_}.

(6.14)

P Rossz/ Superanomaly

742 Notice that classically

V_(~X+X+)coe_X++ V+(zX_X_)coe+X_

(6.15)

would be an irrelevant total divergence. However we know that because of the anomaly the correct replacement for the operator equations (6.5) is given by eq. (6.6) and therefore eq. (6.14) is a total derivative at the quantum mechamcal level, i.e. all anomaly effects have now been reabsorbed. The conclusion we can draw from this analysis is twofold: (1) Supersymmetry is not broken by the anomaly. However the naive measure is not invariant and therefore the lagrangian must be changed in order to compensate for the measure effects. Notice that replacing eq. (6 10) in the 1.h.s. of eq. (6.9) and performing the integration with a proper evaluation of the integral over X reproduces exactly the r.h.s (11) The modified lagranglan is stdl supersymmetric in an operator sense, at the price of using the quantum anomaly equations and quantum operator supersymmetry transformations when performing the variation. The inclusion of the dependence on M is straightforward, as promised. Under the request that

In f dg dx exp [~ (SWZ)q (g, X, B~, A, M)] = -z(1 + CA)SS(ab +, to, Z M ) ,

(6 16)

we find that (SWZ)q =.(SWZ)q-~-~ (1 + CA) I d2x Tr M 2 ,

(6.17)

and the quantum mechanical supersymmetry transformation eq. (6.12) must be extended to 6S~ = a , f ± - 2 ( 1 + CA)MOt±, (6.18) with obvious consequences for eq. (6 13b). The general conclusion of this section is contained in eq. (6.16) and can be formulated in the following way. a quantum version of the SWZ model exists and it possesses supersymmetry and supergauge invariance. The generating functional of the connected Green functlons, i.e. the superdetermlnant resulting from the integration over the matter fields, is expressible in the form W(V~) = -(1 + CA)SS(AB+),

(6.17)

and the superanomaly resulting from the variation of W with respect to superchiral transformations is

8W(V,,) = 1+4¢rCA~ where @~= -½t(A+6A - B+SB).

d2x d20 Tr ~@c,

(6.18)

743

P Ross, / Superanomaly

7. Fermionization: the free theory Having in mind the purpose of constructing an interacting supersymmetric fermionic theory whose superdeterminant coincides with the generating functional of the SWZ model, eq. (6.17), let us first exhibit an interesting construction of the free supersymmetric fermionic model. In ref. [3] we have shown that S S ( G ) is quantum mechamcally equivalent to the fermionic theory ~ -- - ½ ~ A J ~bn - t~X1-'~J X a,

(7.1)

possessing an invariance parametrized by an anticommuting variable: 8~A ± =

2( ra) AaX'~ ~bB±Ot~:,

(7.2a) (7.2b)

8xa~ = -j~:a~_ ,

where [ r ~, r b] =f~bcrc and we have defined ja=/--

±

a

-- ±--,"¢ba

LIPA "ra~q~ * J

b

cx

(7.3)

X±X±).

Originally the only manifestly supersymmetric formulation of this model we could produce was given in terms of constrained chiral spinorial superfields: ~ a = ~ba + 2Xar~n~baO ,

(7.4a)

X ~ = X ~ -j~O.

(7.4b)

It is however possible to give an unconstrained superfield formulation showing that eqs. (7.2) are only a special case of a more general invariance of purely fermiomc chlral models in two dimensions. Let us define chiral spinorial superfields on the hght cone in terms of a onecomponent antlcommuting variable O: 1/3 = ~ba + toa0,

(7.5a)

X a = X a -jao,

(7.5b)

where the notation is chosen for convenience and all the components are off-shell independent. Let us introduce a covariant derivative 0 D =--00

~00.

(7.6)

We need not specify the + sign because the two light-cone components are completely independent at the classical level. The most general lagrangian involving dimensionless couplings in two dimensions IS

~ [

J

d20 d X { ~I A D ~ A + ~ X 1 . D X . + l ~. A r A.n ~ n X

+~'FbcX~XbXC},

(7.7)

744

P Rossz / Superanomaly

and performing the expansion of the integrand in components we obtain

Iq,A~,A-½Xdo + ~'A~B~'BX ° + '~fab~x°XbXC 1 a a 1 1 a a "~-O[20)AO)A "~- 21.~tOAT"An~bnX + ~t~AO~JA + ~IX OX

(7.8)

+ ~l,da - I.~bAr~a~nj ~ + 3 vf~b~XbXT~]

Integration over dO leads to a lagrangian where toA and jo appear only as auxiliary fields. Their equations of motion are t0A = 2/zz~nX~bn,

(7.9a)

Ja = ~'~IA'raAB~B dr- 3 vf~bcXb X" ,

(7.9b)

and after substitution in (7.7) using eq. (6.8) we obtain ~ I | ~ A O ~ A " ~ - I x a O x a "~- ~,L(~,L "~- 3 V)~bAZ:~bBfbcxbXc.

(7 10)

The lagrangian equation (7.10) is by construcuon invariant under the (supersymmetry) transformation 8~a = ~ A a , (7.11a) (7.1 lb)

8)( ~ = - - f ~ a ,

where 0~A and f~ are now defined by the constraint equations (7.9). Whenever the condition u = -l/z

(7.12)

holds, the lagrangian reduces to the free supersymmetric fermionic theory, eq (7.1), and the transformations reduce to eqs. (7.2) apart from an irrelevant multiplicative factor. The on-shell variations of oSa and j-, are both proportional to /z(/z+3u), and they therefore vanish whenever the theory is free, as expected from ref. [3]. It is obvious from our construction that the partner of Zf m the supermultiplet equation (7.8) is the supersymmetry generator ~. Its value after elimination of the auxiliary field is 1 X . .~JA'T . . ABffrl B --~l, 1 ij~,b~X ~X bX =--~].L

c,

(7 13)

and the condition (7 12) reduces ~ to the value found m ref [3]. 8. Fermionization: the gauged fermionic model Comparison of the results presented m the previous sections for the free and guaged WZ models and for the free fermionic model suggests that the fermionization rule might be applied to the SWZ lagrangian in a straightforward way. Let us notice indeed that the only term requiring fermionization in eq. (5.8) ~s WZ (g, B~,). However we already know that the functional integration over the

745

P Rossl / Superanomaly

g-variable [2] leads exactly to the same result as the lntegraUon over the ~ variable [6, 7] for a lagrangian 1

~'-

16~r

(8.1)

~A~I~tA,

where (8.2)

~1A = IJ ~1A + ,~ABI]tB .

Therefore the classical lagrangian t

t

1

Lv=----t~AT~bA16zr --1--~ Tr)?~X + 8-~ Tr )?A - 4-~ TrM2

(8.3)

is our natural candidate to the equivalence with SWZ (g, X, B., M, A). Again simple fermiomzation arguments suggest the structure of the supersymmetry transformations for the matter fields, starting from the corresponding equations (7 2) for the free case" 8 ~ =

a a 2 r ABX ± ~b±BOt=: ,

8X a = - j ~ a ,

(8.4a) (8.4b)

- 2 M %t± ,

where we have just replaced the covariant bosonic currents g+V+g, g V _ g + with their fermlonic counterparts that keep the same form as in the free case. However, and not unexpectedly, the classical lagrangian L F is not mvariant under the transformations (8.4) accompamed by eqs. (4.11) for the gauge fields. A formal invariance can be obtained by replacing eq. (3.11c) with (8.5)

8A = - 2 Y l M a ,

but in this way we spoil the supersymmetry properties of the gauge supermultiplet. This phenomenon is obviously related to the fact that it is only because of the anomaly that the equivalence between WZ (g, B,) and ~¢ could be established, and the anomaly is a purely quantum phenomenon. If we focus on the quantum mechanical version of the gauged fermiomc model, remove the irrelevant dependence on M and introduce the quantum lagrangian l

(/SF)q=

l

16~r t f f A ~ I ~ A - I - ~ T r x ~ I X + - ~

!

Tr~A

(8.6)

together with the quantum transformation property [3] 8S+ = - a _ j + ,

(8.7a)

8S_ = - a + j _ ,

(8.7b)

P Rosst / Superanomaly

746

we now easily find after some algebra that

g'tr l

1

a

- -

81r x/1 + CA

+

a

+

a

{~-x+(V-(~'A~AB~'~)o--F_+]

a

--

a

--

a

(8.8)

+ ,~+x_[(v+(q,~rA,q~)o- F+_]}, and one obtains 1

~A~ ~A=/1-4-~{ -x+[ -(~ArAB~B)q '~-X+[V+(CJAr~B~B)q)}. (8.9) By repeating the argument discussed in sect. 5 in the case of the fermionic field X, we notice that the correct replacement for +

a

+

v_Cq,A~a~%)q,

--

a

--

--

a

--

V+(~oarA,,~O~)q,

(8.10)

is respectively +

a

+

V--(OArAB~bB)¢-- F~-+,

a

V+(~baraB~bB)c- F + - .

(8.11)

Therefore substituting eq. (8.9) into eq. (8.8) we find 6(/~F)q = t~ (SWZ)q = 0

(8.12)

Otherwise stated, the extra term in the variation of LF e' 5d 2 x S L F

_~. e ( l / S , r t )

I

(°t-xaFa-++a+Xa-F~-) d 2 x

(8.13)

is exactly the inverse of the jacobian resulting from the chiral rotation of the variable qJA in the supersymmetry transformation equation (8 4a), computed by means of the anomaly. Therefore the fermionizaUon formula must hold automatically at the quantum level, and we can claim that the lagrangian (Lv)q-

1

16~

q~a~ ~0a

t _ t 1 -~--~TrxYIX+-~--~TrSA---4-~(I+CA)TrM2 (8.14)

is a supersymmetric gauge-mvariant fermiomc theory whose generating functional

WF(V,~)=lnf d~bdx exp[, f d2x(LF)q]

(8.15)

is again expressible in the form WE(V~) = - ( 1 +

CA)SS(AB+),

(8.16)

as one may check from direct integration of the r.h.s, of eq. (8.15). The theory described by eq. (8.14) is what we would like to call N = 1 super-QCD In two dimensions" it is gauge invariant and supersymmetric at the quantum level, while chiral invanance is broken by the superanomaly.

747

P Rossz / Superanomaly

9. Conclusions The main results of the present paper are in our opinion essentially two: (1) The posslblhty of realizing the superanomaly at the quantum level has been explicitly shown by exhibiting two equivalent models, both possessing a gaugemvariant, supersymmetry-invarlant and superanomalous generating functional for the connected Green functions of currents and fields in the adjoint representation. (li) The explicit construction of the superdeterminant may be relevant in such topics as the study of SQCD2 [when the kinetic term for the gauge fields eq. (4.12) is added to W( V~)] and possibly the integration of supersymmetric chiral models, by a generahzation of ref. [7]. Thanks are due to E. Guadagnmi, K. Konishi and M. Mmtchev for many useful discussions and especially to C. Destn for an illuminating conversation on twodimensional anomalies. The author also acknowledges the hospitality of the Theoretical Physics Department of the ETH in Ziirich, where this work was completed. Appendix A When we parametrize the gauge superfields in terms of chiral superfields, according to eq. (3.6) V+ = - i A +D + A , V_ = ~B+ D _ B .

Now recalling the parametrizatlon of chiral superfields obtained in eq. (2.4), G = (1 + zO+x_)g(1 - iO_x+) + zO+O_g&, we obtain by comparison the following parametrization of components:

~+ =

x +~ ,

~_ =

-~?_~,

B+ = a +O+a + iXA+X A ,

B_ = b+O_b + ~a_~a_, M=I(c~A

q~B)

1 B-B -B B -~i(X+X- +X-X+),

1 A N=~(~b +& B ) + ~1t •( X +B X"-B+ X - "XB+ ) ,B

~+ = _ a + a+xAa + [XA+, q~A] ,

~_ = - b

+

~B

B

~B

B

O_x+b + [6 , X-] + t[X+, )~B)~],

where we have defined ~8_ = b+xa_b,

~a+ = bx+b" + .

748

P Rossl / Superanomaly

It is therefore obvious that the condition ~. = 0 amounts to the request that

x+~=x_~=0, and as a consequence

M=½(~A-~), N=½(~A+~"), and the condition N = 0 leads to dpA=--dpB=M.

Appendix B Let us consider the multiphcatlon of two unitary superfields: G = G ( 1 ) G (2). The resulting field is still unitary, and it can therefore be still p a r a m e t n z e d according to eq. (2 4). In component notation the result of the composition is g = g(1)g(2), X+ = g ( 2 ) + x ~ ) g(2) + X(+2) , X - = X ~) + g(1)X(-2) g(1)+ ,

~b = g(2)+~O)g(2) + (])(2)

, ~ ( 2 ) + / . (1), (2) ..L ,,(2),,(1)) or(2)

--t~;

~/(+

A[-

"A-

/(+

1~5

•

Appendix C Assuming the WZ gauge condiUon on the fields A, B we obtain lmmedmtely gA=a, g B÷---b+,

X _A=to_,

g A=0, X+B+ = t o + ,

X -8÷=0,

4)A = M , dPB ÷ = b M b +

and performing an infinitesimal supersymmetry transformation we violate the WZ gauge condition since g t A .= a + la+to_ , X'+A = M a + + ( a + O ÷ a ) a _ , X 'A = to_ + a M a + a _

+ (aO_a + + uo_to_)a+,

qb 'AA = M - l a _ ( a ÷a+to_a ) , grl~+ = b + _ l a _ b + t o + , X'+8+ = t o t + b M b +a+ + ( bO+b + + ito+to+) o~_, X'_~+ = M a _ + ( b + O _ b ) a + , ¢b 'e+ = b M b + - t a _ ( b M b +to+ - t o + b M b +) - la+O_to+ .

749

P Ross' / Superanomaly

If we now perform a compensating gauge rotation ~d = (1 + iO+x~_)(1 - zO_x~+) + tO+O_c~ ~ , it is rather trivial to recognize, by making use of the composition laws presented in appendix B, that in order to fulfill the WZ gauge condition one must have

x~+= -x'+A '

x~_=x,y +"

leading to the relationship X ~= -ga

- ysMa

But we must also satisfy the relationship g,, 8+~b ,, ~+g,,+ B+ = ~b"A '

implying the equation ¢b ~=½g,B+q~,B+g,+~+_½~b,A

1,

,a

,~--

-~t~X+ X-

,B

,a,

- t - X - X+ ) •

Direct substitution now leads to ~b~ = ½itiTsA,

and some more algebra based on the results of appendix B leads to the transformation equations (4.10), implying immediately eq (4.11).

References [1] E Wltten, Comm Math Phys 92 (1984)455 [2] P D1 Vecchla and P Rossl, Phys Lett 140B (1984) 344, P DI Vecchla, B Durhuus and J L Petersen, Phys Lett 144B (1984) 245, A M Polyakov and P B Wlegmann, Phys Lett 141B (1984) 223, D Gonzales and A N Redllch, MIT prepnnt CTP 1155 (1984), Y Fnshman, Phys Lett 146B (1984) 204, D Gepner, Nucl Phys B252 (1985) 481, E Abdalla and M-C B Abdaila, Nucl Phys B255 (1985) 392 [3] P Dt Vecchla, V G Kmzhmk, J L Petersen and P Rossl, Nucl Phys B253 (1985) 701, R Rohm, Pnnceton Umverslty prepnnt (Oct 1984), C Abdalla and E Abdalla, Nlels Bohr Institute prepnnt (1984) [4] S Ferrara, Lett Nuovo Clm 13 (1975) 629 [5] J Wess and B Zummo, Nucl Phys B78 (1974) 1 [6] A D'Adda, A Davis and P D1 Vecchla, Phys Lett 121B (1983) 335, O Alvarez, Nucl Phys B238 (1984) 61 [7] A Polyakov and P B Wlegmann, Phys Lett 131B (1983) 121, PB Wlegmann, Phys Lett 141B (1984) 217