fulltext

Topological gauge theory associated with Vafa-Witten theory

YU9 JIOHTA

Research Institute for Mathematical Sciences, Kyoto University - Sakyoku, Kyoto 606, Japan

(ricevuto il 29 Agosto 1997; approvato il 7 Novembre 1997)

Summary. — A topological gauge theory associated with Vafa-Witten theory for

N 44 supersymmetric Yang-Mills theory is presented. The gauge structure is

studied by considering commutators of the infinitesimal gauge transformation and is found to be first-stage reducible with on-shell reducibility. The quantization is obtained by using Batalin-Vilkovisky algorithm. By a suitable gauge fixing, the resulting quantum action is found to be consistent with the one obtained in Mathai-Quillen formula.

PACS: 11.15 – Gauge field theories. PACS 12.60.Jv – Supersymmetric models.

PACS 02.40 – Geometry, differential geometry, and topology.

1. - Introduction

It is a classical statement that electromagnetic duality is observed in Maxwell theory with magnetic monopoles. Electromagnetic duality for gauge theories was conjectured by Montonen and Olive. In this case, the gauge group G is exchanged with dual gauge group G× whose weight lattice is dual of G.

Recently, Seiberg and Witten [1, 2] proposed a version of Montonen-Olive duality for N 42 supersymmetric SU(2) Yang-Mills theories and found that their quantum moduli spaces were controlled by certain elliptic curves of genus one. However, N 44 theory is believed to verify the Montonen-Olive duality conjecture. In fact, Vafa and Witten [3] studied this duality conjecture by using a topologically twisted N 44 theory and evaluated the topological partition functions on various non-trivial four-manifolds. They showed that the twisted _{N 44 theory supported the Montonen-Olive} conjecture.

This paper provides the topological field theory for this N 44 theory in view of BRST quantization of a classical Gaussian action. In sect. 2, we discuss the gauge structure of the gauge algebra. The starting point is the action including random Gaussian auxiliary fields. By considering various commutators of the infinitesimal

gauge transformation, we find that the gauge algebra is first-stage reducible and we obtain various structure functions. Consequently, Batalin-Vilkovisky algorithm [4-8] is found to be the best choice for quantization (this algorithm was used for several topological field theories [9, 10]). The gauge structure functions can be also obtained by solving a master equation. In sect. 3, we construct the topological action. We obtain the BRST law in sect. 4. We discuss the topological observables in sect. 4. In sect. 5, we construct an off-shell action. Section 6 discusses the topological observables. Section 7 is a summary. The reader will find that our construction is consistent with ones recently constructed in [11] (see also [12]) or in Mathai-Quillen formalism [13-15]. For notations, see appendix A.

2. – Gauge structure

We define the starting action by

Sc4 1 2



X d4 x tr

(

_NsN2_{1 NkN}2

)

, (2.1) where

.

/

´

NsN2fsmnsmn4

g

Hmn2 Fmn12 1 2 [C , B 1 mn] 2 grs 4 [B 1 mr, Bns1]

h

2 , NkN2fkmkm4 (HAm2 DnBnm12 DmC)2 (2.2) and Dm4 ¯m2 iAm. (2.3)

In (2.2), Hmn and HAm are auxiliary fields, Bmn is a self-dual 2-form, C is a scalar and all

fields are Lie algebra valued. For simplicity, we will omit the sign «1» for Bmn

throughout the paper. Note that all fields belong to the adjoint representation of G, so Dmactually acts as “adjoint”.

The ghost number assortment for the fields is given by

.

/

´

Ama 01 Bmna 01 Ca 01 Hmna 01 H Aa m 01_, (2.4)

where the signature of the superscript means the Grassmann parity.

1327 transformation given by

.

`

`

/

`

`

´

dAm4 Dmu 1em, dBmn4 i[u , Bmn] 1BAmn, dC 4i[u, C]1CA, dHmn4 i[u , Hmn] 1Pmnrs1

k

D[res]1 1 2

(

[CA, Bmn] 1 [C, B A mn]

)

1 1gab 4 ( [B Ara_{, B}sb ] 1 [Bra_{, B}Asb_{] )}

l

_, d HAm4 i[u , HAm] 1D n B A nm1 DmCA2i[e n , Bnm] 2i[em, C] , (2.5) where A[iBj]4 AiBj2 AjBi (2.6)

means antisymmetrization. In the above expressions, ua

R is Yang-Mills gauge transformation parameter, ea

m the topological shift parameter on the space of Ama, and

B Aa

mnand CAaare the topological shift parameters on the space of Bmna and Ca, respectively.

As mentioned above, since the “1” sign for Bmn is suppressed, the self-dual projector

should be actually inserted into appropriate positions.

Equation (2.5) involves a degeneracy. In particular for the identification

u 4L , em4 2DmL, BAmn4 2i[L , Bmn] , CA4 2i[L , C] ,

(2.7)

the above gauge algebra closes on-shell; the equations of motion of both Hmnand HAmare

required to close (2.5), i.e.,

.

`

`

/

`

`

´

dAm4 0 , dBmn4 0 , dC 40 , dHmn4 i

k

L, Hmn2 Fmn12 1 2 [C , Bmn] 2 grs 4 [Bmr, Bns]

l

N

on-shell4 0 , d HAm4 i[L , HAm2 DnBnm2 DmC] Non-shell4 0 . (2.8)

Accordingly, (2.5) is a first-stage reducible gauge algebra with on-shell reducibility and therefore we should take Batalin-Vilkovisky quantization procedure. Several examples of first-stage reducible gauge theory are known, e.g., topological Yang-Mills gauge theory [9, 10], Seiberg-Witten theory and their associated lower-dimensional cases.

Let us further study the gauge structure of our gauge algebra. In general, for any fields fi_{, the local symmetry can be written as}

dfi

4 Raiea,

(2.9)

where the indices refer to the labels of the fields with a local parameter ea_{. In the case}

of dfi

higher-stage reducible cases, see ref. [10]). For the action Sc, we have

dSc4 Sc , idfi4 Sc , iRaiea,

(2.10)

where the index after comma means the right derivative by the indexed field, e.g., Sc , j4 ¯rSc/¯fj.

As is shown in (2.7) and (2.8), we can find zero-eigenvectors satisfying

Ri

aZaa4 0 .

(2.11)

However, for a reducible case, we have more generally

RaiZaa4 Sc , jBaji,

(2.12)

when equations of motion are used. For the case at hand, Ri

aand Zaaare found to be

.

`

`

`

`

`

`

/

`

`

`

`

`

`

´

RAma ub 4 ¯mdab1 facbAmc, RAma eb n 4 dabdmn, RBmna ub 4 facbBmnc , RBmna B Ab rs4 dabdmrdns, RuCba4 f_acbCc, R_CACba_{4 dab}, RHmna ub 4 facbHmnc , RHmna eb a 4 P 1 mnrs(¯[rds] adab1 facbAc[rds] a) , RHmna B Ab rs4 i 2 facbC c_d mrdns1 i gab 4 ( fabcB c nbdmrdas1 facbBmac dnrdbs) , BHmna CAb ₄ i 2fabcB c mn, RHAam ub4 facbHAcm, RHAam BAbnr4 ¯ n_d mrdab1 facbAncdmr, RHAam C Ab _{4 ¯m}d_{ab1 facb}A_mc, RHAam eb n 4 fabc(B c nm1 dmnCc) (2.13) and Zua Lb4 d_ab, Ze a m Lb4 2¯mdab2 facbA c m, Z B Aa mn Lb 4 fabcB c mn, Z C Aa Lb4 fabcCc. (2.14)

1329 On the other hand, Bji

a can be easily read from (2.8) and in fact for Hmn and HAm we

obtain BHmnaHrsb a C2a4 2 fabc 2 f c_d mrdns, BH Aa mHAbn a C2a4 2 fabc 2 f c_d mn, (2.15)

while for Am, Bmnand C we do not obtain the Bajias is expected from (2.8).

Next, let us consider commutators of the gauge transformation d1 and d2 with

parameters u1, u2, e1m, e2m etc., where 1 or 2 refers to the label of d1 or d2 (do not

confuse with the space-time or Lie algebra indices). In this case, we have [d1, d2] fi4 RgiTabg ea1eb2,

(2.16) where Tg

abis the structure function of the gauge algebra and we have omitted a trivial

gauge transformation term which is often denoted as Eji

ab (for details, see Gomis et

al. [7]).

For instance, for HAmwe find

[d1, d2] HAm4

[

HAm, [u2, u1]

]

1

[

Bnm, [u1, en2]

]

1

[

Bnm, [en1, u2]

]

1

[

C , [u1, e2m]

]

1

(2.17)

1

[

C , [e1

m, u2]

]

2 iDn[u1, BA2nm] 2iDm[u1, CA2] 1iDm[u2, CA1] 1iDn[u2, BA1nm]

or equivalently, [d1, d2] HAm4 R H Aa m ub fbdeud1ue21 R H Aa m eb n fbde(u d 1ene21 end1 ue2) 1 (2.18) 1RH Aa m CAb f_bde(ud₁CAe₂1 CAd₁ue₂) 1R H Aa m B Ab nrfbde(u e 1BAe2 nr1 BAd1 nrue2) . Thus we get

.

/

´

Tua gbC b 1C g 14 fabc 2 c b_cc_{, T}CAa gbC b 1 C g 14 2facbcbjc, T B Aa mn gb C b 1C g 14 2facbcbcAcmn, Team gbC b 1C g 14 2facbcbccm. (2.19)

Taking the degeneracy of the gauge algebra, e.g., eq. (2.7), into account, we can rewrite (2.16) as [d1, d2] fi4 RgiZagAbbaeb1Lb2. (2.20) Thus we find [d1, d2] HAm4

[

HAm2 DnBnm2 DmC , [f2, u1]

]

. (2.21) Accordingly, we obtain ALa abC b 1C a 24 fabcfbcc. (2.22)

We can also consider the Jacobi identity of the gauge algebra, but we do not get any new (higher-order) structure functions. Thus the study of the gauge structure is completed at this stage.

In summary, we have obtained the structure functions (2.15), (2.19) and (2.22) from the commutators of the gauge algebra.

3. – Topological action

The various gauge structure functions obtained in the previous section can also be easily found as solution to a master equation in Batalin-Vilkovisky algorithm which involves the introduction of anti-fields as a helpful device. In this section, we take the Batalin-Vilkovisky algorithm and construct the topological action by solving the master equation.

For this purpose, we assign the ghost for the local parameters such that

ua K ca, ea mK cam, CAaK ja, BAamnK cAamn (3.1) and La K fa, (3.2) where cAa

mn is self-dual. The ghosts in (3.1) are first generations, whereas fais a second

generation ghost. Their Grassmann parity and ghost number are given by

.

/

´

ca 12 ca m 12 ja 12 cAa mn 12 fa 21_. (3.3)

The minimal set called Fminof fields consists of (2.4) and (3.3).

Field-anti-field formalism requires the introduction of the set F*min of the antifields

carrying opposite statistics to Fmin. The content of F*minis given by

.

/

´

A* a m 212 B* a mn 212 C* a 212 H* a mn 212 H A* a m 221 c* a 221 c* a m 221 j* a 221 c A* a mn 221 f* a 232. (3.4)

The Batalin-Vilkovisky algorithm works for solving the master equation ¯rS ¯FA ¯lS ¯F*A 2 ¯rS ¯F*A ¯lS ¯FA 4 0 , (3.5)

where r (l) denotes right (left) derivative, for an like bosonic object S. This action-like object plays two roles. The one is a generating function of BRST transformation and the other is a generating function of quantum action. However, in order to see this role, suitable boundary conditions such as

.

`

/

`

´

S(Fmin, 0 ) 4Sc, ¯l ¯f *i ¯rS ¯Ca 1

N

F*min4 0 4 Rai, ¯l ¯C *1 a ¯rS ¯C *2 b

N

F*min4 04 Z a b (3.6) should be imposed.

1331 The solution to the master equation is given by the expansion

S 4Sc1 F*i RaiC1a1 C *1 a(ZbaC2b1 T a bgC g 1C b 1) 1C *2 gA g baC a 1C b 21 F*i F*j BajiC2a1 R , (3.7) where Ca

1(C2a) denotes generally the first (second) generation ghost and only relevant

terms in our case are shown. For simplicity, we have omitted the integration over X in the above expansion. We use FAmin4 (fi, C1a, C

b

2), where fidenote generally the fields.

In (3.7), the expansion coefficients Ri

a, Zba, Tbga, etc. are determined from (3.5), but for

the case at hand they coincide with those obtained in the previous section. Therefore studying the gauge structure and solving the master equation are equivalent in this sense.

After lengthy algebraic calculations using Fminand F*min, the minimal solution to the

master equation is found to be

S 4Sc1



X d4_{x tr DS ,} (3.8) where DS 4A *m (Dmc 1cm) 1B *mn(i[c , Bmn] 1cAmn) 1C *(i[c, C]1j)1 (3.9)

1 HA*m(i[c , HAm) 1DncAnm1 Dmj 2i[cn, Bnm] 2i[cm, C] ) 1

1H* mnP1 mnrs

k

i[c , Hrs] 1D[rcs]1 1 2( [j , B rs ] 1 [C, cArs ] ) 1 1gab 4 ( [c Ara_{, B}sb ] 1 [Bra_{, c}Asb_{] )}

l

2 i 2 ]H *mn, H * mn ( f 2 i 2]HA*m, H A* m ( f 2 2i cA*mn( [f , Bmn] 1 ]c, cAmn() 2 ij * ( [f , C] 1 ]c , j() 1 1c *

g

f 2 i 2 ]c , c(

h

2 c *m(D m f 1i]cm, c()2if*[f, c] . We augment Fminby adding the new fields

.

/

´

xa mn 212 da mn 01 dAa m 01 x Aa m 212 ea 01 ra 212 ha 212 f Aa 221 (3.10)

and the corresponding antifields

.

/

´

x* a mn 01 f A* a 12 r* a 01_, (3.11)

Then we look for the solution

S 84S(Fmin, F*min) 1



X

tr (x *mndmn1 xA* mdAm1 r * e 1 fA* h) d4x .

(3.12)

The gauge condition which leads to the action obtained by Mathai-Quillen formalism is given by

.

`

/

`

´

Hmn4 0 , H A m4 0 , ¯mAm4 0 , 2Dmc m 1 [cAmn, B mn ] 40 . (3.13)

Thus the gauge fermion (ghost number 21 with odd Grassmann parity) is given by C 4xmn_H

mn1 xAmHAm2 Am¯mr 1fA(Dmcm2 [cAmn, Bmn] ) ,

(3.14)

where we have performed a «partial integration».

In order to obtain the quantum action, we must eliminate the antifields by the restriction

F_{* 4} ¯rC ¯F . (3.15)

In this way, we obtain the topological action

Sq4 Sc1



X d4 x tr D SA , (3.16) where D SA_{4 i f}A_]cm, Dmc 1cm( 2 fA]cAmn, i[c , Bmn] 1cAmn( 1 (3.17)

1 xAm(i[c , HAm] 1DncAnm1 Dmj 2i[cn, Bnm] 2i[cm, C] ) 1

1xmnP1 mnrs

k

i[c , Hrs] 1D[rcs]1 1 2( [j , B rs ] 1 [C, cArs] ) 1 1gab 4 ( [c Ara_{, B}sb ] 1 [Bra_{, c}Asb_{] )}

l

2 i 2]xmn, x mn ( f 2 i 2]x A_{m, x}Am ( f 2 2i fA

[

Bmn, [f , Bmn] 1 ]c, cAmn(

]

1 r¯m(Dmc 1cm) 2 2 fA Dm(Dmf 1i]cm, c()1Hmndmn1 HAmdAm1 Am¯me 1 1(2Dmcm1 [cAmn, Bmn] ) h .

1333 As is easy to check, the topological action obtained in Mathai-Quillen formalism is given by SMQ4 ScNc 4Hmn4 HAm4 01



X d4_{x tr D S}A MQ, (3.18) where D SAMQ4 i fA]cm, cm( 2 fA]cAmn, cAmn( 2 i fA

[

Bmn, [f , Bmn]

]

1 r¯mcm1 (3.19) 1 xAm(DncAnm1 Dmj 2i[cn, Bnm] 2i[cm, C] ) 1 1xmnP1 mnrs

k

D[rcs]1 1 2(j , B rs ] 1 [C, cArs ] ) 1 1gab 4 ( [c Ara_{, B}sb ] 1 [Bra_{, c}Asb_{] )}

l

2 i 2]xmn, x mn ( f 2 i 2]x A_{m, x}Am ( f 2 2 fA DmDmf 1Am¯me 1 (2Dmcm1 [cAmn, Bmn] ) h .

The reader can easily check that Sq is consistent with that of the topological

Vafa-Witten theory in the Mathai-Quillen formalism, except for the appearance of Faddeev-Popov ghost.

4. – BRST transformation

The BRST transformation variation dBfor any field F can be obtained by

dBF 4e ¯rS 8 ¯F*

N

F* 4¯rC ¯F , (4.1)

where e is a constant parameter with odd Grassmann parity. Thus we get

.

`

`

`

`

`

`

`

`

/

`

`

`

`

`

`

`

`

´

dBAm4 2e(Dmc 1cm) , dBBmn4 2e(i[c , Bmn] 1cAmn) , dBC 42e(i[c, C]1j) , dBHmn4 2ePmnrs1

k

i[c , Hrs] 1D[rcs]1 1 2( [j , B rs ] 1 [C, cArs ] ) 1 1 gab 4 ( [c Ama , Bnb_{] 1 [B}ma, cAnb] )

l

_{1 ie[x}mn, f] ,

dBHAm4 2e

[

i[c , HAm] 1DncAnm1 Dmj 2i[cn, Bnm] 2i[cm, C] 2i[xAmn, f]

]

,

dBcAmn4 2ie( [f , Bmn] 1 ]c, cAmn() , dBj 42ie( [f, C]1 ]c, j() , dBc 4e

g

f 2 1 2]c , c(

h

, dBcm4 2e(Dmf 1i]cm, c() , dBf 4ie[f, c] , dBfA4 eh , dBr 4ee , dBxmn4 edmn, dBxAm4 e dAm, dBe 4dBh 4dBdmn4 dBdAm4 0 . (4.2) 5. – Off-shell action

Let us rewrite our topological action Sq to the off-shell form by the integration of

Hmnand HAm. The method is outlined in [10].

First, pick up the terms

1 2 (NkN

2

1 NsN2) 1ixAm[c , HAm] 1ixmn[c , Hmn] 1Hmndmn1 HAmdAm

(5.1)

in Sq, where the first term is the integrand of Scand we have used the self-duality of

Hmn. Integrating Hmnand HAm leads (5.1) to

21 2[ (Y1 mn) 2 1 (Y2 m)2] 1X mn 1 Y1 mn1 X m 2Y2 m, (5.2)

1335 where we have defined

.

`

/

`

´

X1 mn4 Fmn11 1 2[C , Bmn] 1 grs 4 [Bmr, Bns] , X2 m4 DnBnm1 DmC , Y1 mn4 i]xmn, c(1dmn, Y2 m4 i]xAm, c(1dAm. (5.3)

Then Sqcan be expressed by

Sq4 dB



X

C A d4_{x ,}

(5.4)

where CA is the off-shell gauge fermion given by C

A_{4 x}mn_(X

1 mn2 aY1 mn) 1xAm(X2 m2 bY2 m) 2Am¯mr 1fA(Dmcm2 [cAmn, Bmn] ) .

(5.5)

We have introduced arbitrary gauge fixing parameters a and b. The choice a 4b41/2 leads to the off-shell topological action.

The BRST transformation rule for (5.3) is given by

.

`

`

`

/

`

`

`

´

dBX1 mn4 2e

k

i

k

c , Fmn11 1 2[C , Bmn] 1 grs 4 [Bmr, Bns]

l

1 P 1 mnrsD[rcs]1 11 2( [j , Bmn] 1 [C, c A mn] ) 1 grs 4 ( [c A mr, Bns] 1 [Bmr, cAns] )

l

,

dBX2 m4 2e(Dmj 1DncAnm1 i[c , DmC 1DnBnm] 2i[cm, C] 2i[cn, Bnm] ) ,

dBY1 mn4 ie

g

[dmn, c] 1 [xmn, f] 2 1 2]c , c(

h

, dBY2 m4 ie

g

[dAm, c] 1 [xAm, f] 2 1 2]c , c(

h

. (5.6) 6. – Observables

The observables associated with the Chern classes can be easily found from the geometric sector. Let us define

. / ´ A 4A1c , F4F1c2f , (6.1)

where A and F are interpreted as a generalized gauge connection and its curvature 2-form (we have used the differential form notations to express the above quantities, but the meaning of the notations would be obvious).

The action of the BRST variation dBand the exterior derivative d is defined by

.

/

´

(d 1dB) A 2 i 2[ A , A] 4 F , (d 1dB) F2i[ A, F] 40 . (6.2)

By the ghost expansion of (6.2) we can easily recover the BRST transformation law in the previous section for the geometric sector.

Thus we find that the (i , 2 n 2i)-form Wn , i

.

`

`

`

`

/

`

`

`

`

´

Wn , 04 tr

g

fn n!

h

, Wn , 14 tr

g

fn 21 (n 21)!c

h

, Wn , 24 tr

g

fn 22 2(n 22)!c R c 2 fn 21 (n 21)!F

h

, Wn , 34 tr

g

fn 23 6(n 23)!c R c R c 2 cn 22 (n 22)!F R c

h

, Wn , 44 tr

g

fn 24 24(n 24)! c R c R c R c 2 2 f n 23 2(n 23)!F R c R c 1 fn 22 2(n 22)!F R F

h

. (6.3)

These obey the relation

dWn , i4 dBWn , i 11.

(6.4)

7. – Summary

In this paper, we have constructed the topological field theory associated with Vafa-Witten theory by the Batalin-Vilkovisky (BV) algorithm. In sect. 2, we have discussed the gauge structure of the gauge algebra of Vafa-Witten theory and determined various structure functions. In this section, we have found that the gauge algebra is first-stage reducible. Following to this result, we have constructed the topological action by the Batalin-Vilkovisky quantization algorithm. We have also checked the structure functions obtained in sect. 2 coincide with those of this algorithm. In sect. 4, we have obtained the BRST transformation law. In the present paper, anti-BRST transformation and related off-shell action are not included, but these would be obtained by using a method similar to Birmingham et al.’s [16]. On the other hand, the quantum action obtained in this paper is consistent with that of Mathai-Quillen formalism.

Before ending the paper, let us comment on the N 44 theory. As is well known, the N 44 theory can be twisted by three twistings, one of which is often called half-twisting as was discussed by Marcus [17]. It would be interesting to ask whether

1337 BV BRST quantization can lead to those twisted models, but such reports are not found anywhere. In this sense, the uniqueness of the off-shell action may depend on the model used. Nevertheless, it is interesting to note that one of actions of these three N 44 theories can be found by a standard BV quantization often used in N42 case, as is shown in this paper.

* * *

This work was completed while the author was a member of the Department of Mathematics of Hiroshima University.

AP P E N D I X

Notation

Let X be a compact orientable Riemannian four-manifold with local coordinates denoted by xm, where the index runs from 0 to 3, gmn be the metric of X (the volume

factor denoted by g 4det g), P be a principal bundle with a structure group G and Ambe

a connection 1-form on P (actually, Am is not a differential form but a component,

nevertheless we do not often distinguish the terminologies). The space-time indices often denoted by Greek letters are raised and lowered by gmn and repeated indices are

assumed to be summed over.

The Lie algebra g of G is defined by

[Ta_{, T}b

] 4ifabcTc,

(A.1)

where the Roman indices run from 1 to dim g and fabcis a structure constant of g. The

normalization for generators is

tr TaTb_{4 d}ab.

(A.2)

We define the curvature 2-form (gauge field strength) by Fmn4 ¯mAn2 ¯nAm2 i[Am, An] .

(A.3)

We often use the abbreviation of the Lie algebra index, e.g., Am4 AmaTa.

The self-dual part of Fmnis given by

F1

mn4 Pmnrs1 Frs,

(A.4)

where P1

mnrsis the self-dual projector defined by

P1 mnrs4 1 2

g

dmrdns1 kg 2 emnrs

h

. (A.5)

Here emnrsis a totally anti-symmetric tensor with e01234 11.

For simplicity’s sake, we assume that X is endowed with Euclidean signature and we omitkg throughout the paper.

R E F E R E N C E S

[1] SEIBERGN. and WITTENE., Nucl. Phys. B, 426 (1994) 19. [2] SEIBERGN. and WITTENE., Nucl. Phys. B, 431 (1994) 484. [3] VAFAC. and WITTENE., Nucl. Phys. B, 431 (1994) 3.

[4] BATALINI. A. and VILKOVISKYG. A., Phys. Lett. B, 102 (1981) 27. [5] BATALINI. A. and VILKOVISKYG. A., Phys. Rev. D, 28 (1983) 2567. [6] BATALINI. A. and VILKOVISKYG. A., J. Math. Phys., 26 (1985) 172. [7] GOMISJ., PARISJ. and SAMUELS., Phys. Rep., 259 (1995) 1. [8] HODGESP. J. and MOHAMMEDIN., Phys. Lett. B, 388 (1996) 761. [9] LABASTIDAJ. M. F. and PERNICIM., Phys. Lett. B, 212 (1988) 56.

[10] BIRMINGHAMD., BLAUM., RAKOWSKIM. and THOMPSONG., Phys. Rep., 209 (1991) 129. [11] WANGP., Phys. Lett. B, 378 (1996) 147.

[12] YAMRONY. P., Phys. Lett. B, 213 (1988) 325. [13] MATHAIV. and QUILLEND., Topology, 25 (1986) 85.

[14] ATIYAHM. F. and JEFFREYL., J. Geom. Phys., 7 (1990) 119. [15] BLAUM., J. Geom. Phys., 11 (1993) 95.

[16] BIRMINGHAMD., RAKOWSKIM. and THOMPSONG., Nucl. Phys. B, 315 (1989) 577. [17] MARCUSN., Nucl. Phys. B, 452 (1995) 331.