Perturbation theory near N = 2 superconformal fixed points in two-dimensional field theory

Nuclear Physics B348 (1991) 525-542 North-Holland

PERTURBATION THEORY NEAR N = 2 SUPERCONFORMAL FIXED POINTS IN TWO-DIMENSIONAL FIELD THEORY W.A. LEAFHERRMANN Jefferson Physical Laboratory, Harvard University, Cambridge, MA 0213A USA

Received 9 July 1990 The neighborhoods of N = 2 superconformal fixed points corresponding to the discrete series of minimal models are examined using perturbative renormalization group analysis. The renormalization group flow generated by the least relevant fields is explicitly calculated, and it is shown that no nontrivial fixed points exist in the perturbative regime.

1. Introduction Recently much progress has been made in the study of the relevant perturbations of two-dimensional conformal field theories [1-6]. This work is motivated by their role in describing the behavior of two-dimensional statistical systems, and of string theory, away from criticality. Zamolodchikov [1] has shown that given an exact solution corresponding to a fixed point, the neighborhood around that fixed point can be explored perturbatively using deformations by fields of dimension close to two. These fields are nearly marginal, and those with dimension slightly less than two are least relevant . In particular, calculations may be performed when some small number of the least relevant fields form a closed subalgebra in the space of nearly marginal fields . The possible deformations by other relevant fields may then be consistently neglected. This analysis has been performed in the cases of the N = 0 [1, 2] and N = 1 [3,4] discrete series of unitary minimal models, MP and SMP , and has been extended to the series of general SUM coset models SUP(2) X SUl(2)/SUP +r(2) [7], all in the large-p limit. In these examples, the least relevant field with dimension closest to two forms a closed subalgebra by itself, and perturbation by this field generates the following renormalization group flow pattern: MP -> Mp -1 SM p -+ SMp -2 , SUP(2)

X

SU,(2)/SUP +1(2) -> SUP -1 (2)

X SUI(2)/SUP

* Bitnet address : leaf@ huhepl.hepnet, Chuhepl .bitnet, or Chuhepl.harvard .edu 0550-3213/91/$03 .50 ©1991 - Elsevier Science Publishers B.V. (North-Holland)

2 .

526

W.A. Leaf-He

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rronfomwlfeedpoints

In this paper we continue this analysis and consider perturbations by the least relevant fields in the N = 2 superconformal minimal models. In sect. 2 we first review conformal perturbation theory, and show how the Callan-Symanzik equation can be used to calculate the anomalous dimensions of fields, as well as the -functions of the theory, once the perturbed correlation functions of the theory have been computed in the neighborhood of a fixed point. In sect. 3 we apply these methods to the N = 2 superconformal minimal models, and calculate the 8-functions corresponding to the least relevant deformations of these theories . We conclude in sect. 4 with a brief discussion of N = 2 Landau-Ginzburg effective lagrangians and the relation of our results to the invariance of the Witten index, Tr( - 1) F, in the context of perturbation theory. 2. Perturbation theory near fixed points Following Zamolodchikov [1], we first discuss perturbation theory near conformal fixed points. The correlation functions in the theory are defined by the functional integral A,(xi) . . . AN(XN)%

= .f [ _21(A]e _s[161Ai(xi) . . . AN(XN)')

where .0 is some set of "fundamental" fields in the theory, S[O(x)] is the euclidean action, and the Ai(xi) are some local fields in the theory. Here the usual normalization factor of Z-1 has been included in the definition of S[O]. We shall assume that S may be expressed as the integral of a local lagrangian density Y(x):

S=f d2xy(x) .

(2.2)

Under an infinitesimal coordinate transformation x" -+x'" =x," +EI'(x),

(2 .3)

the change in the action is related to the energy-momentum tensor T,,,, by

as= -

f d2x 27r 1

ÔN.Ev

TA v(x)

(2.4)

In particular, for an infinitesimal scale transformation x', ` -*x'~, =x 1' + üx',' ,

(2.5)

W.A. Leaf-Herrmann / Scrper+conf

points

327

we have 8S= -

A V

fd2x

(2 .h)

so that -2,P(x) -+ .7'(x)

=-12-9(x)

+A 2xp ax,a + 1 ~(x ) - A ,

x),

(2.7)

where 0(x) = â P`,x). We shall further assume that the correlation functions (2.1) depend on some set of n "coupling constants" g` = (g l, . . ., g n ). Then we have the relation

a CI

g

, ( A,(xl ) . . . AN(XN))

N

_

A

F (A I(x j ) . . . Bi,.A .(xa) . . . AN(xn))

a=1 -

f d2y( A,(xi) . . . AN(XN)0i(y))

7

(2.8)

where Bi. a is a linear operator acting on A .(x a ) given by

and we have introduced the spinless fields Oi(x) defined by

a

Ol(x ; g) = agl

(2.10)

g) -

From reparametrization invariance we have the following Ward identity for an infinitesimal scale transformation, where we have used (2.6): N

a

Ai(xi) . . 2 a ax~` +Da Aa(xa) . . . AN(xN) a=1 a

= -1 V

f d2y(A I(x l ) . . . AN(XN) 19(y))

9

(2 .11)

where D is the linear operator describing the variation of the field A(x) under the scale transformation (2.5), A(0) --+A'(0) =A(0) + ADA(0) .

(2 .12)

TWA. Leaf-Herrmann / Superconfonnal fixedpoints

528

Assuming that the theory is renormalizable, we may express the trace of the energy-momentum tensor in the form O(x) = 7r

n

F=i ß" Oi(x)1

(2.13)

where the coefficients ßi are 13-functions, and are related to the scale-dependent coupling constants g i(A) by the renormalization group equations dg' ß`( g) = da '

(2 .14)

Combining eqs. (2.8) and (2.11) we arrive at the Callan-Symanzik equation

a=1

"t a a ~ +Ta '7-21 xâ ax~a i=1 ß a g

A1(x1) . . .

AN(XN)% = 0,

(2 .15)

where T is the anomalous dimension operator defined by T(g) = D(g) + E ß` (g)Bi i=1

(2.16)

Eq. (2.15) expresses the action of the renormalization group on the correlation functions of the theory; the effect of a scale transformation on the fields may be re-expressed as an appropriate change in the coupling constants g', specified by (2.14). From the definitions (2.10), (2.13) and (2.16), and using the scaling behavior of Y in (2.7), the following relationship must hold: n

Foi

=1

n

yi j (g) (pi ~,

;=1

8i

.i -

aßi O; . ag i

(2.17)

This equation will be used to relate the O(g1) coefficients in a perturbative expansion of the anomalous dimension matrix yij(g) to the O(g") coefficients in the expansion of the ß-functions. We shall use the Callan-Symanzik equation (2 .15) to perturbatively analyze the renormalization group flow of conformal theories deformed by relevant fields . In general, the lagrangian density Y(x) describing a conformal field theory is not known. However, for certain classes of conformal field theories, known as minimal models, the constraints imposed by symmetries and the requirement of unitarity are sufficient to completely determine all the correlation functions of the theory [8]. We can consider perturbing a conformal theory by some set of fields 0i, then, using eq. (2.8), it is possible to determine the correlation functions of the deformed

W.A. Leaf-Hemmann / Supereonfomalfixed points

529

theory as a perturbative expansion in the associated coupling constants g`, where at the fixed point g` = 0. In particular, we shall consider perturbations by least relevant primary fields. Virasoro primary fields #a(Z, z) satisfy the conditions A

A

LnOa = L. çb a = 0, A

LOO. = ., .O d

n > 0,

(2.18)

_A

LOçba

(2.19)

where L,,O(z, z) = , dw (w -z)n+1T(w)O(z, z), 27ri

(2.20)

A _ _ dw _ Ln«Z-Z) - ~2Tll Op

(2.21)

_ _ z)n+1T .(w)do(Z  Z)

Here we have introduced the usual complex coordinates z = x 1 + ix 2, 2 =X 1 - ix 2, with T(z) = TZZ = -!(TI, - T22 - î 2T12), and similarily for TM. We shall normalize the primary fields so that (4fi'r(Z'y)4fi'S(0'0)) =Saßz- 2A.2 -2j,,

Furthermore, we shall restrict our analysis to spinless fields, such that Then the dimension of such fields is 24 a. Nearly marginal fields satisfy da

= 1- E a ,

1E.1 - E <<

1.

(2.22)

as = da . (2.23)

Under these conditions we expect that the renormalization group might possess nontrivial behavior in the weakly coupled regime, g - E . The strategy for analyzing the neighborhood of a fixed point is as follows. At the fixed point, all the conformal weights of the fields are known, and by our choice of normalization (2.22) the anomalous dimension matrix yj ' is diagonal, so by integrating (2.17) we may immediately compute the O(g) terms in the 8-functions, which by definition vanish at the point g'= 0. Using the derivative of the correlation functions with respect to the couplings, given by eq. (2.8), we may calculate the O(g) terms in the two-point functions of the perturbed theory. By applying the Callan-Symanzik equation (2.15) to these two-point functions, the O(g) terms in the anomalous dimension matrix y,' may be computed, since both the coordinate dependence and the (3-functions are now known to O(g). Again integrating (2.17) with respect to the coupling constants g', the B-functions can then be determined to O(g 2). Proceeding in this fashion, the 8-functions may be calculated perturbatively order by order in g'.

W.A . Leaf-Herrmann / Superconformal fixedpoints

530

From eq. (2.8) we have

(X)Oï (0) ) + ~Pi

lag k

(2 .24)

- ,f d'y( 0°(x)0°(0)0'(y» ,

where (Bj.O,)° = (Bj.Oj)jg=o and 0°(x) = Oi(x; g = 0). From a geometric viewpoint, we may take the coupling constants g` as coordinates in the infinite-dimensional space of two-dimensional field theories. Zamolodchikov [1] has proposed the two-point function (2 .25)

Gj;(g) =( Oi(x)O;(0)) IX2= 1

as a natural candidate for a metric on this space. In this language, the arbitrariness in choosing a coordinate system in the space of field theories corresponds to the freedom of field redefinition within the theory. Hence the choice of a particular coordinate system is equivalent to the choice of specific field normalization conditions. For the case in which the field d5° are nearly marginal primary fields, Zamolodchikov has shown that by choosing the couplings g ` such that (2 .26)

( (Pi (x)O;(0)i 1x2=1= Gj;(g) = Sij + o(g 2 ) ~ we may write a

a g

k \

pi( x ) oj( O )l

Ig=O =

(

x 2) -"i-Ai

j

Cijk{Iik [(x

2 ) ,;_,

' -

(x2)1

_a,

+ (i

H J)},

(2.27) where Iik ' = ~ . +

7r $k

_ ~.

(1 + O(E3)) ,

(2.28)

and the Ciik are the operator product coefficients. It then follows from the Callan-Symanzik equation (2.15) that yi ' = vi s i ' + 7TC, k i9k +

O(g2) ,

(2 .29)

W.A. Leaf-Herrmann / Scperconfornml fired points

531

and therefore, from eq. (2.17), i

jk

(2.30)

(g

where there is no summation implied in the first term on the right. In the case of a single coupling constant, if C * 0, there exists another fixed point at g = 2e/zrC. For completeness, the details of this calculation are discussed in appendix A. If these 13-functions vanish for some nontrivial values of the couplings, then there are other fixed points in a neighborhood of the initial fixed point.

3. Application to N= 2 superconformal minimal models We now turn to the case of the N = 2 superconformal minimal models. N= 2 superconformal field theories have, in addition to the conserved current T(z) which generates analytic coordinate transformations, two anticommuting conserved currents, G+(z) and G- (z), which generate supersymmetry transformations, as well as a U(1) current J(z). In our normalization conventions the currents G±(z) carry U(1) charges ± 2. The N = 2 superconformal unitary minimal models [9-11] are classified by the value of their Virasoro central charge cp = 3 - 6/(p + 2), p = 1, 2, . . ., oo . The space of fields which form a closed algebra with respect to the operator product expansion can be divided into two sectors: the Neveu-Schwarz (NS) sector and the Ramond (R) sector. The primary fields in each sector are labeled by two integers l and m, where 1= 0, 1, . . . , p and m = - l, - l + 2, . . ., l. In the NS sector the Virasoro primary fields may be written as superfields (3 Nm(Z) = ~Vm(Z) +O+Y'm(Z) +0 ~m(Z) +0 0+ "`n(Z)'

.1)

where Z = (z, 0+, 0 -) are the holomorphic coordinates of superspace, and only the holomorphic structure of the superfield has been shown. In two dimensions the superfields will also depend on the anti-holomorphic coordinates Z = ( z, 8+, V).). The conformal weight 4;,1 and U(1) charge qm of the lowest component of the superfield, cpm, are given by QI

=

4(p+2)

-

m2 4(p+2) ,

m

(3.2)

The primary fields in the R sector are labeled Rm(z) and have conformal weights

W.A. Leaf-Herrmann / Superconformal fixed points

532

and U(1) charges given by

din=

1(1 +2) (m ±1) 2 1 + 4(p+2) ! 4(p+2)

g,

r

q,n=

2m + p 4(p+2)

(3.3)

In general, to explore the neighborhood of a fixed point we must include in our analysis perturbations by all the nearly marginal fields, since under the action of the renormalization group there can be significant mixing among fields whose anomalous dimensions differ by O(g ). However, if a subspace of these nearly marginal fields forms a closed subalgebra, then we may consistently restrict our analysis to this subspace. The fusion rules and structure constants of the operator algebra for the N = 2 superconformal minimal models have been calculated [12], and in particular the subspace generated by the superconformal families [ Nô] and [N,p] forms a closed subalgebra. From eqs. (2.23) and (3.2) it is clear that in the subalgebra generated_by [Nô ] and [NP,,] the only nearly marginal fields in the large-p limit are qi+p and 4 pP with weights (3.4)

P(Z)

The descendent fields `YpP and ~+P vanish because Np are chiral superfields, satisfying D !NP= 0, where D ± = a/a0 ++ 0 ±a1az. In passing we note that the N = 2 superconformal algebra implies that all NS superfields N+ ~(Z) are chiral superfields. A deformation by qi+p and ~pP preserves supersymmetry, since the perturbation may be written in the form of an integral of a superpotential, W(Z, Z) = N+ (Z, Z ), over the superspace coordinates 0 + and 8 +. Let us define the following linear combinations of these fields:

P(Z)

P

0l(Z, Z) =

~2(Z, Z)

1

(4APP)-'[qV+P(z,2) +

~ p P(z, Z)]

,

(3 .5)

_ 72 (4ap) - 1 [O+P(Z, Z) - ,~~`PP-P(Z, Z)] .

(3 .6)

We denote the perturbed lagrangian density by Y(P)(x ; g), where Y(P)(x ; 0) describes the N= 2 superconformal fixed point corresponding to the minimal

W.A. Leaf-Herrmann / Superconformal fired points

533

model with central charge cp = 3 - 6/(p + 2). Then a ~~p)( Z, Z ; g), Oj(z, Z ; g) = 'agi

-Oj (z, z ;0) = ei(Z, î),

(3.7)

where the coupling constants are g` _ (g 1, g2) . In the case of N = 2 superconformal minimal models, all the three-point functions involving only the fields 0 1 and 402 vanish due to the conservation of the U(1) charge. Therefore the Cijk in (2.27) vanish, as does the O(g) term in the expansion of , so we may consistently set (Bkoi(X ; g)) ~g=o

=0.

(3.8)

From eq. (2.30) we see that pi

= Eigi + o(g3) ,

(3 .9)

and if the coefficients of the O(g3) terms are of order unity, then we may expect that another fixed point would exist in the perturbative regime, g - ~E_ . We shall now check this hypothesis . The O(g 2) term in the expansion of yields a2

ag ag k ~

~~(X)~;(o))

A \ /Ai( X)(BIBk~;)O(0) ~g- O _ <(BlBk~i)o(X) ;(0) +

+ f d2 wd 2y( O;(X)O;(0)4k(Y)41(w)) ,

(3 .10)

where we have used (3.8). The integral on the right-hand side may be expressed, via the definitions (3.5) and (3.6), in terms of the integrals of the four-point P(x1)_P function < p p( X 2 )1# pp( X 3)4' pp( X 4 )> . The calculation of this four-point function and the integrals involved is discussed in appendix B. The result, in the limit of large p, is

f d2 wd 2 y(O;(X)O;(0)MY)0r(w) ) - 47r2

p

(X2)-2c1-2s

)

[ Srjskl+SijS;l +( -1 )s'' 3 ilSikI

+O(1/p2) . (3 .11)

If we interpret G;;(g), defined in eq. (2.25), as a candidate for a metric on the space of coupling constants, then we may choose normal coordinates at g = 0 on

W.A.

534

this space,

®f--He

n / SuperconfWmalf"ed points

that t

(0» =_~ = GQj(g) = âij - ~ Rtkjtg kg + o(g3) '

(3 .12)

where RTkjl is the Riemann curvature tensor evaluated at the point g = 0. In the case of two coupling constants there is only one nonvanishing independent component, 81212, and those components related to it by symmetry. Using the normalization condition (3.12) we can calculate this component by taking a linear combination of the relations (3.10) to eliminate the wave-function renormalization terms, and we find that Rl?,? = F~°~~~ - 2( F?,11 + F11  ) = ®,

where Fjjk,

=

,

2 d wd"'y \ 'Ag(x)4j(0)40k(Y)40`(u')>

(3.13)

I,2=1 .

(3.14)

The fact that R,2,2 vanishes is a consequence of the U(1) invariance, since all the correlation functions involving (Bk45)° vanish by (3.8). Therefore may write, after solving for the wave-function renormalization terms, 40gt i) gk

~

(X) j(0))

Ig -o

=

47r 2 P

[(x2)-

X 1 3ij431A-

+

20- 2

_ (x2)-20-F)]

ôik ail + ( - 1 ) a 'j(5i1Sjk] + O(1/P 2 )

(3.15)

Integrating the above expression for < O,(x)4bj(0)> and applying the CallanSymanzik equation (2.15) to the two-point function, we find that the O(g2) term in the expansion of the anomalous dimension matrix vanishes in the large-p limit: Hence

Yi J =,à ` a;' + O(E292) + O(g3) .

(3.16)

p' = Eg` + O(E2g ; ) + O(ga),

3 .17

and to this order in perturbation theory we conclude that there are no nontrivial fixed points. 4. Discussion and conclusions

We shall first consider the results of this calculation from the point of view of Landau-Ginzburg effective lagrangians. It has been proposed [4-6] that the N = 2

W.A. Leaf-Herrmann /

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Super+confornud find points

superconformal minimal model with central charge cp = 3 - 6/(p + 2) fied with the multicritical behavior described by the action Sp

= f d2z d40 K(O, ;F) + (gf d2

z d20

op+2

+

h.c. ,

identi-

(4.1)

where 0 is a complex chiral superfield, satisfying D ' o = D' o = ®. Here we restrict our discussion to the A-series in the ADE classification of modular invariant conformal theories . The first term, which is integrated over all o¬ superspace, is known as a D-term, and includes the kinetic terms of the act" ential The second term, known as an F-term, is the integral of the supe 45p+2. be a hob rphic W(4p) = In general the superpotential W is constrained to function of the fields . Non-renormalization theorems for N = 2 theories in two ization dimensions [131 imply that the superpotential is invariant under reno group flow, if we assume that these non-renormalization theorems hold nonperturbatively. Consider the infrared limit of the action_Sp. Vafa and Warner [5] have conjectured that there exists some D-term K (0, 0) such that the action SP is conformally invariant. Assuming the invariance of the superpotential under this flow, it then follows that the fixed points of N= 2 theories can be classified according to their respective superpotentials . At the fixed point, we may use the invariance of the F-term under the action of the renormalization group to determine the scaling dimensions of the chiral fields. For instance, consider the action (4.1). Under a rescaling of the two-dimensional metric, gp.v -~ 'k -2 g Lv , we have f d2z d2 0 ~,A -'f d2z d28. The F-term is invariant if /(p 2)-0, and we accompany this transformation with the field redefinition 45 --* 1l' + this allows us to identify the chiral field 0 with the chiral primary N1(Z, Z), having conformal weights (112(p + 2),1/2(p + 2)), in the N= 2 superconformal minimal model with central charge cp. This association may be extended by identifying the fields 01, I = 1, 2, . . ., p, with the conformal primary fields N;. We expect OP" to be identified with a descendent of N1 by the equations of motion determined by the action. One might ask whether the above conclusion concerning nearby fixed points is a direct consequence of the putative non-renormalization theorems in N = 2 supersymmetry. In the language of Landau-Ginzburg effective lagrangians the calculation dealt with in this paper is equivalent to considering the superpotential W(0) = gop+2

+

,7 , g 45p

(4.2)

where we view the g' term as a perturbation of the N= 2 minimal model with central charge cp. Here g is the value of the fixed point coupling in that theory. From the scaling arguments discussed above, we can easily determine the lowest

-4 . Leaf-Herrmann / Superconfomial fixed points

order scale dependence of the renormalized coupling gß =90 A

gI :

i -ps(p+2)+o(g') a

(4 .3)

since by the non-renormalization theorems the only contribution comes from wave-function renormalization. We see that this behavior is in agreement with the first term in the perturbative expansion of the j9-function (3.17). Of course, g will also be renormalized by terms of O(g'), since at finite g' the theory is not scale invariant. On the basis of the scaling behavior of the Landau-Ginzburg effective lagrangian we expect the above relevant perturbation to carry the theory in the infrared limit to the theory with central charge cp - 2. Although we can deduce the fixed point to which the perturbation will cause the theory to flow, we cannot determine whether there is a nearby critical point in the p -4 00 limit without a scheme to perform perturbative computations in the Landau-Ginzburg theory near a fixed point, where the theory is clearly not a free theory. We can better understand our result from topological considerations, specifically involving the Witten index, Tr(-1)F. The value of Tr(-1) F in the N = 2 superconformal minimal model with central charge c p is equal to the number of Ramond ground states, so Tr( -1) F = p. Since Tr(-1) F is a topological invariant, one would not expect this index to change within the context of perturbative renormalization group flow. This behavior is also apparent in the flow pattern of the N = 1 superconformal minimal models, SM... [3,41. These minimal models are characterized by a Virasoro central charge c, = - 12/m(m + 2), where m = 2,3, . . ., oo. In these theories, Tr(-1) F = + -2,10 + (-1)"') and perturbation by the least relevant field generates the flow m ---> m - 2, thus preserving the value of the Witten index. The results of this calculation support the notion that Tr(-1) F is conserved under perturbative renormalization group flows which do not violate supersymmetry . I would like to acknowledge helpful conversations with Jon Bagger, Paul Fendley, Mark Goulian, Brian Greene, and Carl Schmidt. I would especially like to thank Cumrun Vafa for bringing this problem to may attention, and for several illuminating discussions. This work was supported in part by NSF grant PHY-8714654.

Appendix A Here we review the details in the calculation of the lowest order terms in the 13-functions, a result first established by Zamolodchikov [1] and discussed above in sect. 2.

IVA. Leaf-Hemmann / SWerconfommal

fixed ` s

537

Conformal invariance requires the three-point function of three (spinless) primary fields to be of the following form: Oi(Z1, Z1)40j(Z29 Z2)Y0k(Z37 2 3)/ = CijkIZ121

2a3

IZ231

2A,

IZ131

2A2 ,

(A .1)

where z 12 = z 1 - z 2 , etc., and A 1 = Ai - dj - ak, etc. The C,jk are the operator product expansion coefficients. Using the equation

f

r(a)

1)T(1 _a) T( 1 _ß) r «2 -a-

d2zIZI -2al1 _zI -2 ß=Or r(a +ß-

(A.2)

we can write the integral of the above three-point function over the point (z3 , '13) as follows:

f d2Z3C 0i(Z1 , 21)'Aj(Z2') Z2) 4I'k(Z39 Z3)l = IZ121u1-,` -,~-,~)Cr~k fij

(A.3)

~ F(2ak -1)l'(1 +Ai -aj - ak )I'(1 +aj -a i - ak) . r(ai+ak+ak-ai)r(2-2ak)

(AA)

where L_k _

Under certain conditions, e.g. if ®k =1 + ai -A,., this integral may have to be regulated in some fashion. In that case, the operators Bi and D, defined in (2.9) and (2.12), also depend on the method of regularization, but the linear combination of these operators defining the anomalous dimension operator T in (2.16) is regularization independent. For the case in which the primary fields are nearly marginal, i.e. ai = 1 - Ei where Ei << 1, we have iii

k=

2?TEk

-(1+0(,- 3 ))(Ei + Ek - Ej) (Ej + Ek - Ei)

(A .5)

We see that the third term on the right-hand side of (2.24) is given by eq. (A.3), assuming that no regularization is necessary. From the normalization conditions (2.22) we see that the "wave-function renormalization" terms on the right-hand side of eq. (2.24) must be of the form «Bk~i)0(x)0°(0) )(X (x2) -2A'

(A.6)

and be symmetric in i and k as well, from (2 .9) and (2.10). It should be noted that, for nearly marginal fields, any subtractions necessary to make (A.3) finite will have

W.A. Leaf-Herrmann / Superconformal fvred points

538

the same x-dependence as these wave-function renormalization terms. The choice in (2.26) then requires of coordinates

g`

(A .7)

where

This result leads directly to (2.27) and (2.28). To find the anomalous dimension matrix yt', we first note that integrating (2.27) gives

X {Iik'[(x2)~~-~' -

( x2) 1-ak J + (i H1)

+

o(g2 ) -

(A.9)

From the Callan-Symanzik equation (2.15) we have ~~ ~~~~~~ x)~i~0)~ + ~~r~x)~T~~~(0) ~~ (x==i

= 24i(Si; + C~jkgk[(Ei

+ Ek - E;)hk' -~

= 2[dl ~~; ~- 7TC~;kg k ]

-~ O(g2),

(1 H~)]

+ o(g2) (A.10)

where we have used (2.26). From this equation we may directly read off the components of the anomalous dimension matrix y~', and we find the result given in (2.29). Appendix B Here we discuss the calculation of the four-point function involving the least relevant superconformal descendents, ( ~+p( x l)`l~pp( x 2)~Ypp(x3)4'pp(x4)>

>

and the integrals leading to the result in (3.11). Using the parafermionic construction of the N = 2 superconformal theory [9,12], the lowest components of the

W.A. Leaf-Herrmann / Superconformal fixed points

539

chiral primary superfields NP p(z, 0 +, 0-), where D ±Nß p = 0, are given by

(p

pp(z)=Xpp(z)

.exp

±PP(z) i 2P P+2

(B.1)

and *PP (z) = lim (w -z)G (w)(PPP(z)

-_

2P p - 2p(z) X; -2(Z) :exp i p +2 2P(P + 2)

(B.2)

g p p(z) = lim (w - z)G +(w),ppp(z) -

2P

p

2p(z) X-p+2( Z)

:exp l

2P(P +2)

~'

(B.3)

where G ± (w) are the two conserved supercurrents, Xm(z) are the lowest dimensional fields of the Zp parafermion theory with dimensions 1(P - 1) r = + d 2P(P + 2)

1 2-m2

4p

,

1=0,1, . . .,P,

-1 _< m < 1,

(B.4)

and p(z) is a free scalar field. To calculate the required four-point function, we use the following relation between the parafermion fields X1 and the primary fields Om of SUM WZW theories [14]: pm(Z) =Xim(Z) :exp i ,

where j = 0,1/2, . . ., p/2 and -j < m Xp-2(Z1)Xpp+2(Z2)Xp-2(Z3)Xpp+2(

-

Z12Z14Z23Z34_ Z13Z24

)(p-2) 2 /2p

%2

<

m

(B .5)

p(z) :,

j. We then see that

Z 4)) 1( Z1)~pp/2+1( Z2)~p/2

1( Z3)~pp/2+1(

Z 4)) (B .6)

IVA. Leafdie

ann / S

reonformal fixedpoints

where z;j = z, - z,, and (

P/2

1 ( Z 1)

p/2+ 1(Z2 ) 2

_

( Z14Z32)

P! )

-p /2+ 1(Z4)%

p/2 1(Z3)

a P -1

Ô

1

(X, 2

p-1 t~X

ap -1

a

3

lX

-1 (~XP

p

1

[(X14X32) U(X ; Z )1

Ix;=O~ (B .7)

and z = z,2 zU/z, 4 z3,, x = X 1 ,X34 /X 14x3 , . The auxiliary variables xi are isospin coordinates denoting the components of SUM representations. The function U(X; z) may be written in the form U(x ;Z)=(X-z)PUP(z),

(B.8)

where UP(z) is the solution of the differential equation z(z-1)

d UP(z) =p(1-2z)U(z), P dz 2

(B .9)

which is just the statement of the SUM Ward identities for this particular representation . Hence U(x ; z) = (X -z) P [z(1 -z)]

p/2 .

(B .10)

Substituting this into (B.7) we find op/2 i( z

=

1 P

-p/2+1( Z2)Op/2? 1( Z 3)~ P p/2+1( Z4)}

(1 - z) p-2 [Z14Z32Z( 1

-z)]

-p12 [(p -

1)(1 +z 2 )

+

(1

-z)21 .

(B .11)

Including the scalar field contributions we find ( APp(Z1) pp( Z 2)4Pp(Z3W p(Z4) ) 1 ( 2p `' +2 (1 =P p

Z)-21(p+2)(Z12Z34)-2(p+1 )/(p+2)[(p

-

1)(1

+Z 2 ) +

(1

_Z)2I .

(B .12) For the result in (3.11) we require the integrals of (B.12), multiplied by its antiholomorphic part, over the pairs of coordinates x3, x4, and x2, x4 . For in-

W.A . Leaf-Herrrmann / Superconformal daredpoints

stance, the first of these integrals is given by I = f d2x3 d2x4( * +p(x1)*pp(x2) *+p(x3)*pp(x4)> P-2

( 2P

P

+ 2 )f d2Z3 d2Z 4 IZ12Z341-4('-f'11-ZI-4

X ((P - 1)(1 +z2) + (1-z)2I2 =P-2

2P 1-a(l-2F) d2z IZI -«1- E'll -ZI -4e 1z 12 f (p+2 )

I2

X J(p - 1)(1 +z 2 ) + (1-z) 2 J(z, z),

(B.13)

where J(z' 2) = f d2z3 I Z31 -2(1-2F)Il - z31-2(1-2E)I1- ZZ31-aF . To evaluate this integral we use the general result for Iz I < 1: 112b lu - Z1 2c = s(a + b + c)s(b) K (z) d2 u (u1 2a lu f s(a +c ( ' (B.12)

where s(a) = sin(rra), and K1(z)

00

= f du ua(u - 1) b(u -z)` 1

r(-a-b-c-1)r(b+1)

r(-a -c)

K2( Z)

(B.16a)

Z

= f du ua(1 - u) b (z-u)` 0

= Z1+a+~ r(a + 1) r(c + 1) F(-b,a+1 ;a+c+2 ;z), r(a+c+2)

(B .16b)

W-l . Leaf-Herr'nann / Superconfonnal foxed points

542

where F(a, b; c; `) is the hypergeometric function. Performing the integration over z 3 and 23 we find in the large-p limit J(z)

= 7r[p

+ 2(l - lnll - zI) + O(l/p2)a

(B.17)

.

The final integrations over z and z may be performed using the well-known

equation

f d2

Z IZI-2«Il

77

_ ZI-2ßZn(l _

Z)m

r( a +ß -1)r(1 + n -a) T(l +m - .3)

r(a)r(ß)r(2+n+m--a-P)

'

(B.18)

and the representation lnli - zi = lien Ill - zl"? - ï)/,q .

(B.19)

,~ -0

The final result is ( 2P )41 zi21_ P+ 2

4(i-2e)[4/P

+ o(1/P2)]

.

(B .20)

Similarly we find

1

d2 .X3d2X4(' +p(xl)`l1+p(x2)wpp(x3)~pp(xa))

=fJ(1/P 2 ) .

This pair of results lead directly to that given in (3.11). [1] A.B. Zamolodchikov,

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

References

Sov. J. Nucl. Phys. 46 (1987) 1090 A. Ludwig and J.L. Cardy, Nucl. Phys. B285 [FS19] (1987) 687 R.G. Pogosyan, Sov . J. Nucl. Phys. 48 (1988) 763 D.A. Kastor, E.J. Martinec and S.H. Shenker, Nucl. Phys. B316 (1989) 590 C. Vafa and N. Warner, Phys. Lett. B218 (1989) 51 E. Martinec, Phys. Lett. B217 (1989) 431 C. Crnkovic, G.M. Sotkov and M. Stanishkov, Phys. Lett. B226 (1989) 297 A.A. Belavin, A.M. Polyakov and A.B . Zamolodchikov, Nucl. Phys. B241 (1984) 333 A.B. Zamolodchikov and V.A. Fateev, Sov. Phys. JETP 63 (1986) 913 W. Boucher, D. Friedan and A. Kent, Phys . Lett . B172 (1986) 316 P. Di Vecchia, J.L. Petersen, M. Yu and H.B. Zheng, Phys. Lett. B174 (1986) 280 G. Mussardo, G. Sotkov and M. Stanishkov, Int. J. Mod . Phys. A4 (1989) 1135 P.S. Howe and P.C. West, Phys. Lett. B223 (1989) 377 A.B. Zamolodchikov and V.A. Fateev, Sov. J. Nucl. Phys. 43 (1986) 657

(8 .21)