Supersymmetry between interacting fermions

Volume 190, number 1,2

PHYSICSLETTERSB

21 May 1987

SUPERSYMMETRY BETWEEN INTERACTING F E R M I O N S H. FALOMIR and E.M. SANTANGELO Departamento de Fisica, Universidad Nacional de la Plata, C. C. 67, 1900 La Plata, Argentina

Received 3 November 1986

The well-known supersymmetrybetween two species of free fermions in two dimensions is shown to survive when they are coupled to a non-abelian gauge field. After integrating out one of the fermion species, the supersymmetryis reinterpreted as a symmetrybetween bosons and fermions. The Kac-Moodyinvariances of the effective theory are studied and the connection of its strong-couplinglimit with the supersymmetricWess-Zuminomodel is discussed.

The ability of two-dimensional free fermionic theories to provide a representation of Kac-Moody algebras was crucial in the proof by Witten [ 1 ] of their equivalence to non-linear sigma models with a Wess-Zumino term. Because of their conformal invariance [ 1,2], the latter have been considered in relation to string theories, thus allowing for a reduction of the critical dimension of Minkowski space [ 3 ]. Recently, it has been shown that the addition of free fermions in the adjoint representation leads to a supersymmetric version of the nonlinear sigma model with Wess-Zumino term, invariant under super Kac-Moody transformations [4-6]. At the same time, it has been noticed that free fermionic theories in two dimensions provide a non-linear realization of supersymmetry [ 4 ]. In a previous paper [ 7 ], we concluded that bosonic non-linear sigma models with Wess-Zumino terms can be obtained as the strong coupling limit of two-dimensional non-abelian gauge theories with fermions. In this work we generalize this approach to the supersymmetric case, by exploring the effect of the addition of a new set of fermions. We show that the already mentioned non-linear supersymmetry remains unaltered when a gauge field is coupled to both species of fermions with the same strength. After eliminating the gauge field and one of the fermionic species in favor of new bosonic degrees of freedom, we reinterpret the supersymmetry from the point of view of the resulting theory. We study its symmetries and relate them to the supersymmetric Wess-Zumino model [ 4,5], which has been studied in relation to superstring theories [ 8,9 ]. We consider two types of Majorana fermions Z = (z~) in the adjoint representation and ~ = (¢¢+) in the fundamental representation of SO(N). It is well known [4] that the free action remains unchanged under the (local) supersymmetric transformation ~± --,exp(ot~z_+ ) ~_+ , Xa ta=X+_--*X+

q-~OC~:~+_@~t+_,

(la)

(lb)

with O+a_ =O_a+ =0,

(2)

and ta the hermitian generators of SO (N), satisfying tr(ta tb) = 26ab • 117

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It is easy to check that this invariance survives the introduction of a non-abelian gauge field coupled to both fermions with the same strength:

S[A,~,Z] = ~-

l tr( ~dZxFuvFU~)+fd2x ~D~ +½tr(f d2x~[D,z]) .

(3)

We choose the A_ = 0 gauge, which allows us to write

A+ -A+ta=iU-'(O+ U) --

(4)

a

with

U(x + =0, x-) = ~ ( x - ) ,

(5)

where ~ ( x - ) is an arbitrarily chosen matrix in the group. We then get for the vacuum functional [7]

ZJNU O[U(O,x- ) -,,~(x- )] exp{ (i/g 2) V[ U] }ze[ U]zz[U] ,

(6)

where

V[U]=@tr(~d2x [O_(U-IO+U)]2)

•

(7)

U is the invarint measure over the group, and the fermionic path integrals

z¢[U]=fN~ N~ exp(iv/-2fd2x

(~+ D+ ~+ +i~_ 0_ ~ _ ) ) = (det I0) m

zx[U]=f~2Nxexp[~22tr(Id2x(x+[D+,x+]+ix_O_x_))l

(8)

(9)

are defined in a gauge invariant way [ 10 ], being De=(Do+D~)/x/~,

D+[U]=i0++A+=iU

-~0+U,D

=i0

.

(10)

The functional z, is given by

-ilnz¢=W[U]=~61fd2xtr(OuU-lOuU)-~4tr(f(U -1 d U ) 3 )

,

(11)

where the last integral is performed on a ball, whose boundary is the compactified two-dimensional spacetime. The resulting effective theory is described by the vacuum functional Z = J N U 0[ U(0, x - ) - r e ( x - )] J@X ~Z exp{i[(

l/gz) V[U]+ W[U]+X[ U, Zl]},

(12)

with

X[U,z]=~2 tr(fd2x{x+[D+,x+]+x_iO_x_}).

(13)

We proceed now to study the symmetries of the effective theory. The action in (12) remains invariant under the transformation of the remaining fermions given by Z- ~ Z - + f l - ( x +) (where fl_ is an arbitrary grassmannian function of x + ), related to the "conservation equation" 118

(14)

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o_z_ =o

(15)

and under X+ ~)~+ + U - ' f l + ( x - ) U ,

(16)

giving the "conservation law" O+ ( Ux + U - ' ) = O .

(17)

Eqs. (14) and (16) are nothing but the equations of motion for fermions. These symmetries being simple translations, they leave the fermionic measure in (12) invariant and so both currents continue to be conserved at the quantum level. Since we are working in the axial gauge, there is still a residual gauge freedom, which corresponds to the simultaneous transformations [ 7 ] U--.Uff2(x+ ),

X ~ f f 2 - ' ( x + )zff2(x+ ) .

(18)

This leaves the effective action in (12) and also the quantum measure invariant, because of the gauge invariant definition o f z z [eq. (9)] and of the property ~(U~2(x+))=~U

(19)

enjoyed by the invariant measure over the group. This Kac-Moody symmetry has an associated quantumconserved current: O_J+ =0,

J+ =(i/4rr)U-'(O+ U)+(1/g2)[D+, O _ ( U - O+ U ) ] - x / ~ Z _ X _ .

(20)

The first term in (20) is Witten's current [ 1 ]; the term in 1/g 2 originates in the gauge field action [7], and the last one, in the fermionic action. There is also an arbitrariness in the choice of the initial condition for U [eq. (5)], which gives rise to an invariance of the effective action (12) under [ 7 ]: U--,g2(x- ) U.

(21)

This also is an invariance of the quantum measure, and corresponds to the conserved current O+J_ =0,

J_ = ( i / 4 n ) U ( 0

U - ~ ) + ( i / g Z ) [ U O U - ' , O + ( U O U - 1 ) ] - x / 2 U z + z + U -~

(22)

J+ and J_ are the generators of two commuting Kac-Moody algebras. The transformation in (17) implies J+ = ~ - ~ J + I ] + (i/4n)O-~(0+O),

J'_ =J_ ,

(23)

J+.=J+

(24)

while the symmetry in (20) gives J'_=g2J_12-' +(i/4z0£2(0_12-'),

.

Both central extensions originate in the Wess-Zumino functional W[ U] (i.e., in the fermionic chiral anomaly) [1,7]. Going back to the non-hnear supersymmetry of eq. (1), it can be reinterpreted in terms of the degrees of freedom characterizing the effective theory. In doing so, we change U into Ue~, which mimics a chiral transformation in ~ [ see eqs. (i) and (8) ] and find the compensating transformation law for ZIn the first place, we choose ot=a_Z+,

0+a_ =0.

(25)

This reflects into the following variations in the different pieces of the effective action: 119

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6V=-itr(~d2x[O2_(U-tO+U)][D+,a_X+]),

(26)

6W=--~tr(fdZx[(O_U-')Ul[D+,a_X+ ] ) ,

(27)

~X=-x/~tr(fd2xz+x+[D+,a_X+]),

(28)

which can clearly be compensated through a simultaneous change in X+. Thus, the action (12) is invariant under

U~U exp( ot_x +),

(29)

X- ~ X - , Z+ -~X+ + ( a _ / x / ~ ) {(i/4rt) U-'( O_ U) + (i/g2)[O 2_( u-'a+ u)] +x/2x+z+ }. Associated to this symmetry, there is a conserved current:

O+G_ =0,

G_ =tr(x+{(-l/4n)U-'(O_ U)-(1/gE)[O2_(U-'O+ U)] +ix/~X+X+ }) .

(30)

Notice that this is a classical conservation law, since the fermionic integration measure is not invariant under the transformation (29), which is non-linear in X+. Nevertheless, it is easy to see that (29) transforms the fermionic current UZ+ U-' into the Kac-Moody bosonic current J_ in eq. (22), i.e., they are in the same supermultiplet with respect to this classical supersymmetry. However, the fermionic part of the effective action enjoys an additional symmetry, already pointed about in ref. [ 4 ]; X changes by the integral of a total derivative when

X+ ~ X + - a - X + X + •

(31)

The combined effect of this invariance, together with the supersymmetry in (29) corresponds to a quasilinear transformation of the fields, and has an associated quantum conserved current ~_ = t r ( z + { ( - 1/4n) U-t(O_ U) - (1/g2)[02_( U-'O+ U)] +i]x/2X+)~+ }) •

(32)

Another supersymmetric invariance of the action is obtained by choosing

a=a+Z_,

0 a+=0.

(33)

Similarly to the previous case, one finds that the effective action (12) remains invariant under the simultaneous changes

U-*Uexp(a+X_) , X---')~- + ( a+/x/~)tr{ (i/4n) U-'( O+ U) + (1/g2)[D+ , O_( U-' O+ U)]}, X+ --'X+ - a +

(34)

tr{x+ , )~_ } .

The corresponding classically conserved current is G+ = t r ( x _ {( - l/4r0 U-'(O+ U) + (i/g2)[D+, O_ ( U - ' 0 + U)] }).

(35)

Being a non-linear transformation, (34) is not an invariance of the quantum measure. But this time, the nonlinearity cannot be eliminated through a combination with a symmetry of the type (31 ). However, the fer120

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mionic current X- and the K a c - M o o d y current J+ are related by a combined transformation under (34) and X - - , X - -o~ +x I X _

(36)

,

thus constituting a super K a c - M o o d y current at a classical level, while X + remains unchanged. Finally, we study the behavior o f our effective theory in the strong-coupling limit (g2__. o9), where it can be brought to the form o f the supersymmetric W e s s - Z u m i n o model, the superconformal and super-Kac-Moodyinvariant theory discussed in ref. [ 4 ], through a redefinition of fields:

g=U, ~+=-ix//~U(U-~x_U), ~u_=-ix//~U(Uz+U-~),

(37)

in terms o f which the effective action reads

1 fd2xdZ o OG D G + 1 - ~ f d2x d20 dt G + ~ s = 1-g-~n

(IT)G+ysDG)

(38)

where

G=g+iO~+ ½0OF,

(39)

and F is the auxiliary field, which makes the supersymmetry linear in superspace (for conventions, see ref. [4]). This identification can only be made at a classical level, because the fermionic transformation in (37) (an axial transformation followed by a non-unitary one) implies the appearance o f a non-trivial jacobian. To summarize, we showed that the introduction of gauge field does not spoil the supersymmetry between two species o f fermions. By bosonizing one o f them, we were able to reinterpret it as a supersymmetry between bosons and fermions. In studying the invariances o f the resulting effective theory, we found two commuting K a c - M o o d y algebras. Moreover, we showed that in the strong-coupling limit, the model turns out to be classically equivalent to the supersymmetric Wess-Zumino model. That this equivalence is only true at a classical level reflects in the fact that the supersymmetries carrying our K a c - M o o d y currents into their supersymmetric fermionic partners are not q u a n t u m symmetries o f our effective theory. We thank our colleagues o f the Laboratorio de Fisica Te6rica for useful discussions. This work was supported by C O N I C E T , CIC Pcia. de Buenos Aires and SECYT, Argentina.

References [ 1] E. Witten, Commun. Math. Phys. 92 (1984) 455. [2] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [3] D. Nemeschansky and S. Yankielowicz, Phys. Rev. Lett. 54 (1985) 620; C. Lovelace, Phys. Lett. B 135 (1984) 75; S. Jain et al., Phys. Rev. D 32 (1985) 2713; A. Sen, Phys. Rev. Lett. 55 (1985) 1846. [4] P. Di Vecchia et al., Nucl. Phys. B 253 (1985) 701. [ 5 ] E. Abdalla and M.C.B. Abdalla, Phys. Lett. B 152 (1985) 59. [ 6 ] P. Windey, preprint LBL-20423. [7] H. Falomir and E.M. Santangelo, Phys. Rev. Lett. 56 (1986) 1659. [8] M. Henneaux and L. Mezincescu, Phys. Lett. B 152 (1985) 340. [9] I. Antoniadis et al., SLAC-PUB-3789 (1985). [ 10] R.E. Gamboa Saravi et al., Nucl. Phys. B 185 (1981) 239.

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