Nicolai mapping and stochastic identities in supersymmetric fields theories

PHYSICS REPORTS (Review Section of Physics Letters) 137, No. 1(1986) 55—62. North-Holland, Amsterdam

Nicolai Mapping and Stochastic Identities in Supersymmetric Field Theories V. DE ALFARO Istituto di Fisica Teorica, Università di Torino, Italy Istituto Nazionale di Fisica Nucleare, sez. di Torino

S. FUBINI CERN, Geneva, Switzerland

G. FURLAN Istituto di Fisica Teorica, Universitd di Trieste, Italy Istituto Nazionale di Fisica Nucleare, sez. di Trieste International Centre for Theoretical Physics, Trieste

G. VENEZIANO CERN, Geneva, Switzerland

Among the interesting features of Supersymmetry (SUSY) theories, a very interesting property, first suggested by Nicolai [1], is the fact that, after integration over the fermionic fields in a functional approach, the resulting functional integral becomes a Gaussian form in a new set of bosonic variables. The connection between the old fields and the new variables is the so-called Nicolai map, and of course strong interest is raised when this map can be proved to be local (or almost local) in space-time; this was known to happen in SUSY Potential Theory and in 2-dimensional cases. Conversely, Parisi and Sourlas [2] have shown that in some cases a supersymmetry naturally arises from considering stochastic equations in real space. The point of view which we have taken in becoming interested in the field is the investigation of certain ‘stochastic identities’ (SI) involving bosonic Green’s functions, which can be obtained as a consequence of the Nicolai procedure when the latter can be proved to exist. Our point of view is to derive SI directly from SUSY, independently of whether a local Nicolai mapping is possible. The existence and form of such identities is of valuable help in the understanding of the connection between SUSY and the stochastic approach. We have been able to show the existence of the local Nicolai mapping for the N = 1 gauge SUSY theory in 4 dimensions, provided one adopts the light cone gauge, thus offering the first example of the mapping for a theory in physical space-time. At this meeting I shall only sketch the problem and give an idea of the methods employed. Those who are interested in the matter will find a more exhaustive treatment in a forthcoming paper. Let us first illustrate the well-known case of potential SUSY. We start from the Euclidean form of the SUSY action for a system of commuting x.(s) and anticommuting c/1(s) fields (i = 1,. , N; .

.

ds=idt): (1)

IIB~IF

0370-1573/86 /$3.50

©

Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

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New avenues in quantum theory and general relativity

where

1B-~Jds[x~+(aIw)]

IF =

J

ds {~+ c/1~a~~W}.

(2)

(3)

The superpotential W is our input. The action (1) is invariant under the SUSY transformations ~ix,= —ç (4)

=

—~

(5) =

1,—

aW.

In order to obtain the Nicolai result let us introduce

N,=~1+a1W.

(6)

The bosonic action is equivalent to 1B.~fdS1~i

(7)

and the fermionic action (3) is

‘F

=

J

ds1/i~11d/ds + a~a1W)1/~.

(8)

Now we consider the generating functional (9)

~

The fermionic integrals can be performed: Z =

f fl

dx~e’~Det(~~1 d/ds + ~

(10)

V. de Alfaro et al., Nicolai mapping and stochastic identities in supersymmetric field theories

57

Now, from (6), ~N,(s

1)/&x1(s2)

=

(~d/ds + ~,81W)8(s~ s2). —

(11)

Hence Det(~N,/3x1)=

J

dc/, d1/~e~’~’

(12)

and we are left with the effective bosonic functional Z8=Jfld~exp(_~JdsN~)

(13)

which is Gaussian in the variables N,. Thus it is very easy to have the vacuum expectation values in terms of these new variables: (N6(s1)” ~N,(s~))=

~ r,s

ôrs6(Sr_Ss)(

fl

N~(sp)).

(14)

pr,s

We may call these identities the ‘stochastic identities’ (SI). Thus SUSY Quantum Mechanics becomes elementary when expressed in terms of the Nicolai variables N,. The only complication in solving the bosonic sector arises from our interest in vacuum expectation values of products of the original fields x to which we attach a physical meaning. One then needs to solve for the x’s in terms of the N’s, i.e. to invert the functional equation (6). This approach has been named ‘infradiagrammatic’ by Cecotti and Girardello [3]to whose paper we refer for details. From this treatment emerges the importance of the Euclidean form of the functional approach, since the functional equation (6) has real coefficients. Now we start again from the action and want to establish directly the stochastic identities via a general method, without recurring to the Nicolai map. This method will be of great help for the more complicated case of field theory. Let us go back to the action (1). In order to have the correct SUSY algebra one introduces a set of auxiliary fields .Z~: I=Jds{~2+c/~_~z2+c/aaw~+zaw}.

(15)

Put Z~=8.W+F,: I = Jds {~2+ ~

2+ c/8aWt~+~(aW)2}.

~F

Since F, appears quadratically only, the following identities are an immediate consequence:

(16)

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New avenues in quantum theory and general relativity

(F,(s))= 0 ~F,(s~)F~(s2))= —ô~8(s~

—

(17)

S2)

(F,(s1) A(s2)) = 0 where A(s) is any field not containing F’s. Now let us introduce the complete SUSY variation (18) {~ii} = 0,

{~I,~i}= —2 d/ds

=

2i d/dz’

(19)

whose action on the fields is

(20) JZ,-çii,. 2= 0.

For our procedure, we focus on generators S in the SUSY algebra such that ~i Now consider the bosonic variable 1~=~ic/,.

(21)

Then L11 = 0. Consider now the SUSY identity valid if SUSY is not spontaneously broken: i~,,)=0

~

(22)

which gives

~

1,,)=0.

(23)

The usual stochastic identities are recovered from eq. (23) by writing

1

as its on-shell expression plus

a purely Gaussian field F~:

i~=I~+F,.

(24)

Inserting (24) into (23) and exploiting eqs. (17) for F we obtain the SI:

=

(_1)~~/2 ~

8,,,,5(s)

.

(25)

V. de Alfaro et a!., Nicolai mapping and stochastic identities in supersym metric field theories

59

This is the general procedure. In quantum mechanics we have only the choices

S=zi

(26)

S=~.

(27)

The corresponding stochastic variables are the Nicolai map variables: either

P~=N,=x,+a,W

(28)

I~=I~=x,—a,w.

(29)

or

Note that N, and N, are linear combinations of even and odd quantities under time reversal. In this way SI containing N, are transformed into SI containing N,. This aspect is characteristic of SI also in field theories, where terms with different Lorentz character may appear in a stochastic variable, an interesting point on which we have no time to dwell here. The SI thus established are the right number to guess (if we had not known) that the N, realize the local Nicolai map. Many field models have been discussed in the literature [4] and we choose here to deal briefly with the only known case in four dimensions where the existence of a local Nicolai map has been established [5].This is the N = 1 supersymmetric Yang—Mills theory, whose action is I=

J

F±F4— ~Aa(O~D)a&,} d4x {~(F~+

where A, A are two-component spinor fields and o-” the quantities

(30)

=

(1, iu). In (30) we have used for the boson field

~

(31)

The index a is colour and will be neglected. The action (30) is equivalent, up to surface terms, to the usual Yang—Mills action, and suggests that the F, are the stochastic variables of the Nicolai map in the present case. This is indeed so. Let us start from the generating functional of the theory in the light-cone gauge:

Z =

J fl

where AL =

dA,, dA, dA,

ô(AL)

exp(—I) Detf~—LLAL}

(32)

A 3 — iA4,

and AL

=

0 is a light cone gauge. Now the Faddeev—Popov determinant is 4(x

D = Det{~—zLAL}= Det aL5

—

y).

(33)

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New avenues in quantum theory and general relativity

Further F=

J fl

dA, dA, exp(—IF) = Det(uD 84(x

—

y)).

(34)

Now the fundamental observation is that the product DF is the functional Jacobian from the A’s (in the light cone gauge) to the F’s: DF = Det(6A,/~Fk)

(i = 1,2,3;

k=

+, —,

3).

(35)

To see this, we must compute the functional determinant Det(~A,/~Ek). Let us introduce ‘polar’ components for both A,. and D,,: A~=A 1±iA2 AR=Ai+iA4,

(36)

AL=A3—iA4.

We can express the relationship between F’s and A’s as

iF~= [D±,DL] iF_

=

[DR, D_]

(37)

2iF3= [DR,DL1—[D+,D~] and in differential form idF÷= D+dAL—DLdA± idF=DRdA—DdAR

(38)

2idF3= DRdAL—DLdAR—D±dA+DdA~. It is a very important observation that in a light cone gauge, e.g. AL = 0, eq. (38) takes the form

/

idF÷\ i dF \2idF3/

7—aL 0 0 DR D_ —D~

) ( =

\

\

0 —D

—aL/

/dA±\

(

~ dA

(39)

.

\dA3J

Thus the matrix ~F,/~A1factors into a 1 x 1 and a 2 x 2 matrix. The relationship (35) is then rather straightforward.

Thus, from (32) we have Z

=

J fl

dF, expf

—~J

4x (F~+ F+F)}

(40)

d

which proves that in the gauge

AL

=

0 the three components

E + H

are indeed the stochastic variables.

V. de Alfaro et aL, Nicolai mapping and stochastic identities in supersymmetric field theories

61

In the previous treatment there is a delicate point since the gauge AL = 0 prevents the reality of all the components in Euclidean space. The interested reader may find more about this in a paper by the same authors [4]. Now we shall discuss the identification of the stochastic variables and shall obtain the SI using the general procedure already outlined in the case of potential theory. First, when considering the action (30) we need to fix the gauge. Now, we require that after fixing the gauge we can still retain a set of residual supersymmetry transformations, because we may then apply the same procedure as in quantum mechanics. This is the key observation. Now, under a SUSY transformation we have A) =

i~(nr)A

(41)

where n is a fixed vector and ~ is the SUSY parameter. Equation (41) tells us that we may have zi(nA) = 0 and ~ non vanishing. We need to have i.e. n2 = 0.

det(nu) = 0,

(42)

Thus n is a light-like vector. If we choose ~ constrained as

)

~

(43)

C

then there is a residual SUSY leaving AL invariant: =0

Z1 5AR =

‘i5A~2ieA2

—2ieA1

‘iSALO.

(44)

This result enables us to obtain directly the SI as in the case of potential theory. Consider the variation ~ applied to A: ~i5A1

—e(F3+iF)= sT1

A5A2= —eF÷=sT2.

(45)

For this operation z1~= 0. Thus F3 + iF and F+ are stochastic variables. It takes more effort to find that also 3LF is a stochastic variable. As a consequence the expected stochastic identities hold, e.g. (F~F+)=0 (F3F3) = t5~(x y) —

(46)

3LF lead to corresponding identities It can be also one that finds the stochastic relationsofinvolving involving F shown itself. Thus the exact amount SI to conclude that, in the light cone gauge, the

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New avenues in quantum theory and general relativity

components of E + H realize the Nicolai map discussed earlier. We shall not give all the annoying details in this report. Thus we have sketched this simple powerful method that enables one to construct the SI in SUSY. The method can of course be used to investigate SI in the various models of SUSY, and the reader is referred to ref. [4] where all the detail of the investigated cases can be found. In conclusion, one of the most striking features of certain theories endowed with SUSY is that they admit a local Nicolai mapping. When expressed in the form of SI, this local mapping leads to potentially important exact constraints on certain bosonic correlation functions. Our method of establishing directly the SI tries to provide a systematic approach to the matter. In particular, two main results have emerged from our approach: (1) We have been able to give a first example of local Nicolai mapping in a 4-dimensional field theory. Interestingly enough, that theory turns out to be the N = 1 super Yang—Mills theory in the light cone gauge. (2) Even when a local Nicolai mapping does not exist, we can write a nontrivial set of SI. The presence of SI appears to us to be an important feature of SUSY field theories, possibly representing the starting point of interesting developments such as: (a) tests of SUSY theories based on exact identities not involving directly the fermionic degrees of freedom; (b) a new understanding of SUSY theories at the perturbative level, based on the inversion of generalized identities of the type (6) between the original fields and the stochastic variables (the so-called ‘infradiagrammatic’ expansion, see ref. [3]). Supersymmetric regularization of the infradiagrams could be an easier task since they do not entail fermion lines and do not exhibit divergences worse than logarithmic. Work is in progress in these directions.

References [1] H. Nicolai, Phys. Lett. 89B (1980) 341; NucI. Phys. B 176 (1980) 419. [2] G. Parisi and N. Sourlas, Phys. Rev. Lett. 43 (1979) 244; NucI. Phys. B 206 (1982) 321. [3] 5. Cecotti and L. Girardello, Ann. Phys. (NY) 145 (1983) 81. [4] V. de Alfaro, S. Fubini, G. Furlan and G. Veneziano, NucI. Phys. B 225 (1985) 1. [5]V. de Alfaro, S. Fubini, G. Furlan and G. Veneziano, Phys. Lett. 142B (1984) 399.

Note added in proof In the last months the matter has undergone a noticeable development that we are glad to report briefly. On one side, the reduction to Gaussian form of the functional generator Z has been extended to general N = 1 coupled to supersymmetric matter, and to N = 1 in 6 dimensions [a]. On the other side, the SI for N = 1 gauge theory have been confirmed in perturbation theory by Floreanini, Leroy, Micheli and Rossi [b]. [a] V. de Alfaro, S. Fubini and G. Furlan, Phys. Lett. 163B (1985) 176. [b] R. Floreanini, J.P. Leroy, 1. Micheli and G.C. Rossi, Phys. Lett. 158B (1985) 47.