- Email: [email protected]
N = 4 SYM on ∑ × S2 and its topological reduction
ELSEVIER
Nuclear Physics B 551 (1999) 467--489 www.elsevier.nl/locate/npe
N = 4 S Y M on
× S 2 and its t o p o l o g i c a l reduction A. Imaanpur 1
Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, SA 5005, Australia Received 31 March 1998; revised 8 December 1998; accepted 29 March 1999
Abstract We consider the twisted N = 4 SYM on ~: × S2. In the limit that S2 shrinks to zero size the four-dimensional theory reduces to a two-dimensional SYM theory. We compute the correlation function:~ of a set of BRST cohomology classes in the reduced theory perturbed by mass. (~) 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Topological field theory; Duality
1. Intt~)duction Topoilogical field theories [ 1] have proven to be a useful tool in the investigation o f the non-perturbative characteristics o f supersymmetric gauge theories. There is an interplay between certain supersymmetric gauge theories and their corresponding topological versions: one can use topological results on smooth manifolds to learn about the underlying physical theory; conversely, one may use the physical arguments to gain new insight iinto the topological structure o f the manifold on which fields are defined [2]. As an example o f the first - i.e. using the results on the mathematical side to learn about physics - consider the N = 4 S Y M theory. This theory has been conjectured to have an exact S L ( 2 , Z ) duality [3]. Since this duality relates the weak and strong coupling behaviour o f the theory, to test the conjecture one needs quantities such as the partitior~ function to be computed non-perturbatively. This is a formidable task and one actually does not know how to proceed in this direction. This is where topological field l E-mail: [email protected] 0550-3213/99/$ - see frontmatter (~) 1999 Published by Elsevier Science B.V. All rights reserved. PII S0550-3213 (99)00221-7
468
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
theory comes to provide an alternative approach to the problem. Instead of the physical theory, one considers the corresponding topological field theory obtained by a procedure called twisting. The basic characteristics of the theory, such as SL(2, Z) invariance, remain intact under twisting, so one hopes to see the realization of this symmetry in the twisted model. In [4] it has been shown, using the known facts about the structure of the moduli space of instantons and the associated Euler characteristic, the partition function of N = 4 twisted theory on some specific manifolds can be computed. So in this way it has become possible to make some exact and non-perturbative statements about the theory and its self-duality properties. In this article, we will study the N = 2 reduction of the above theory obtained by mass perturbation for the hypermultiplet. This theory is still believed to be S-dual [ 5 ]. We will compute the correlation functions of a set of specific operators using a method of calculation similar to that of [6]. Twisted N = 2 and N = 4 SYM theories on product manifolds 2? x C, where 2 and C are both Riemann surfaces, have been considered in [7]. There it was shown that, in the limit C shrinks to zero, the four-dimensional theory generically reduces to an effective two-dimensional sigma model. However, when C is a Riemann sphere - as is the case of interest in the present paper - things are a bit different. The dimension of the self-dual harmonic 2-forms, b +, is one in this case. Hence the connection is reducible. It follows then that the path integral may get contribution from the so called u-plane [8,9]. Moreover, when b + = 1, there is a wall in the space of one parameter metrics. On crossing this wall the partition function may change its value. Here we will compute the path integral in a chamber where S2 shrinks to zero. We consider SO(3) bundles such that the restricted bundle over S2 is trivial. Bundles which restrict non-trivially on S2 give zero contribution to the path integral. This is so because in the limit that S2 shrinks to zero size, the path integral localizes on the moduli space of fiat connections in the S2 direction. However, it can be shown that for a fiat bundle over S 2, transition functions are trivial and the bundle must be trivial. Therefore non-trivial SO(3) bundles on S2 do not admit flat connections. The organization of this paper is as follows. In Section 2 we consider the twisted N = 4 Lagrangian on 2 × S 2. In the limit where S 2 shrinks it is shown how the four-dimensional theory reduces to an effective two-dimensional theory. The fixed point equations imply, in the case of a non-trivial SO(3) bundle over ,~, that the partition function of this reduced theory is in fact the Euler characteristic of the moduli space of fiat connections on 2L A mass perturbation makes the path integral calculation more tractable - particularly for the limiting two-dimensional theory. In Section 3, we show how this comes about. Perturbing by the mass allows most fields to be integrated out, and reduces the path integral to a finite-dimensional integral which can be easily performed. In Section 4 we discuss the result. Although we have not yet managed to give an explicit check of S-duality, we have isolated the problems involved and hope to return to this in later work.
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
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2. T w i s t e d N = 4 o n 2; X S 2 a n d its r e d u c t i o n
The key point in twisting [ 1] is to redefine the global space-time symmetry such that at least one component of the supercharge becomes scalar under the new defined spacetime symmetry. This procedure crucially depends on the existence of a suitable global R-symmetry. N = 4 SYM theory in four dimensions has a global SU(4) symmetry such that the supercharges transform under the 4 of this symmetry. First one needs to see how this representation transforms under the space-time symmetry group SU(2)L x SU(2)R. There are [4] three possibilities for the decomposition which give rise to singlets under the twisting: (i) (2,1) @ ( 1 , 2 ) ; (ii) (1,2) @ (1,2); (iii) (1,2) @ (1,1) ~ (1,1). As in [4], we will consider the case (ii) where, after twisting, two components of the supercharges turn out to be singlets and therefore square to zero. The scalar fields of the physical theory, which transform under the 6 of SU(4), now transform under the new rotation group, SU(2)L x SU(2)~, as 3(1, 1) @ (1, 3), three singlets and one self-dual 2-form. Having determined how the new fields transform under the new symmetry group, what remains is to rewrite the Lagrangian in terms of these new fields on fiat R 4. This Lagrangian can then be defined on an arbitrary smooth four manifold while preserving those two BRST like symmetries. Let us start our discussion with the twisted N = 4 Lagrangian 2 in four dimensions [~,10],
£=: ~2tr rl - D~*AD~'~b + ½[-I~([-I~ - 2v/2DuC + 4x/~DVBp~)
+ ~1H
/xv
+ (Hm,- 2F~,,-4i[B/.q,,B~I -4i[Bm,,C])
+4¢/~D~ x/~v + 4 ~(/~Dv~ ~v + 5(/~D~ ( - ~b/~D/%7 +iv'2~m't~m,, A]
-
iv~xm'[ Xz,,, qb] + i2v~tpZU [Xzu, C]
+i4v/2~ uv [Xup, Bvt'] - i v ~ x u v [(, Bm' ] - iv/2~u,, [77, Buv ] +i4v'2Ou IX,.. Bin'] - i v Y 2 , [.~'". ~b] i +iv/20t, [ ¢ u, A] - i2v/20~ [,~ u, C] + ~ - - ~ ( [ ( , 2t]
(1) As mentioned, the action is invariant under two BRST transformations. However, for us it is enough to consider one of them, which reads [ 10]
2The Lagrangian that we use is actually different from the one constructedin [ 10] by a BRST exact term -¼a(~[4,, al).
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
470
6( = 4i[ C, (b ] ,
aAu = - 2 ~ u ,
= v%, at/ = 2i[a, ~b],
a ~ = o,
= &,
a,.
8 ~ . = 2i[ B ~ , ~ ] , ac =
1
6/-1u = 2x/2i[,~,,~b],
,
~X/,,. = H.v,
6Hz, v = 2v"2i[ X/,v, dp].
In this article we choose ~b and a to be two independent real scalars. This will render the Lagrangian hermitian and allow us to treat 4~ and a independently. The generators of the SU(2) group are chosen to be hermitian T a = ~2tr a with tr (TaT b) = 6 ab. The theory enjoys an exact U(1) ghost symmetry under which ~j,, ~uv, ( have charge 1, Xt,~, r/,/~u charge - 1 , while ~b and a have charges 2 and - 2 , respectively. All other fields have ghost number zero. Take the underlying manifold to be 2 x S 2. Let us denote the indices on ,,~ by i, j , . . . and those on S 2 by a, b . . . . . We define 1
1
F. =
x.
1
=
1
Iftij =
Bi./ = - - t i j b ,
~ "~IV,~I 6`JO'
2V'~l
(2)
and the same for indices on S 2,
Bah = 2 + 2 6abb',
1 , Xab ----"~226abX,
~ab = 2 ~26abl~lt.
(3)
Here gl and g2 denote the determinant of the metric on ,Y and S 2 respectively. The fields Hu~, B~,v, Xu~ and ~t,v are all self-dual. Note that 2x/~ 1
2-"--~tij abBab = "~'-~tij£: 1
= 4 gg~2g2
tab bt
1
6ijb' = 2v/~ 16ijb',
(4)
where we have used that 6abtab = 6abta'b'gaa, gbb, = 2g2 a n d gaa'gbb'ta, b, = 6 ab. Also we chose 612 1 and so e12 = g2; thus, for example, we have Bab l---L-tabbt Hence we = = 2v,~2 • conclude that
b=b',
X = X ',
~ =~'.
In Ref. [7] it was shown that upon shrinking the metric on ~, one gets an effective two-dimensional sigma model governing the maps from S 2 to .A4, where .A4 is the moduli space of solutions to Hitchin's equations. Although the twisted theory is supposed to be topological, since the space of self-dual harmonic forms in this case is one-dimensional
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
471
one may not get the same effective theory if one instead shrinks S2. In that case we will see tha~! the effective theory which emerges is a two-dimensional twisted SYM theory, as conjectured in [7]. Thereto, we now scale the metric on S z by a factor of e. Notice that the definitions (2) and (3) are consistent with this scaling, since both sides of the self-duality constraints scale with the same power of e. After integrating out the auxiliary fields, the bosonic part of the Lagrangian reads EB = ~2tr {-D~aD~qb - (DuC - 2D~Bru) 2
1 (F~+ + 2i[Bt, o, B ~ ] + 2i[Bt~r, C] )2}
(5)
2
where 17+ = I ( F A- *F) and • is the Hodge duality operation. Thus we can write
- - ~:
'Iv~F~t+r F "'+'-'/v'~Ft,,FUU ~'/v ~ -
(*F)gu Fgr"
The last term is the instanton number and is metric independent. Using this, and the fact that Bur is self-dual, we write the last term in (5) as
_!~" _ 2iF~Zr([Buo, Bpu] + [Bur, C ] ) 4 " / z r - ~,~r + 2 ( [ B u a ' B ~ ] + [Bur ' C ] ) 2 -
Zl ( *F)urF~r
_- -~] {FijFiJ + 8iFiJ([Bij, C] + [Bia, Bajl) - 8 ( [Bij, C] + [Bia, B~] )2 + (,F)ijFiJ
+ FaOFa° + 8iFaO ( [Bah, C] + [ B~i, Bib] ) --8( [Bah, C] + [Bai, Bib] )2 + (,F)aoFaO}
- ( F a + + 2i[Baj, BJi] + 2i[Bab, B~.] + 2i[Bai, C]) 2 = -¼(Fij + 4i[Bij, C] )2 _ ¼(Fao + 4i[Bal, Bib] )2 -- 2iF ij [ Bia, B~ ] - 2iF ab [ Bat,, C ] + 4 [ B ij, C ] [ Bia, Baj ] +4[ Bab, C] [ Bai, Bib] -- ¼( *V)ijF ij - ¼( *Y)at, F ab) - ( F + + 2i[Baj,BJi] + 2i[Bab, B b.] + 2i[Bai, C]) 2. In the last equality we noted that for a self-dual antisymmetric tensor we have S/~ = S~6. In particular
[Bab, C] 2 = [Bij, C] 2, tr ( [ Bai, Bib] [ B aj, Bj b ] ) = tr ( [ Bia, B~ ] [B ib, Bbj ] ). After scaling the metric, then, the Lagrangian splits to three parts: E=£1
+ Eo + E - t ,
where/2n scales as e n. Specifically,
A. hnaanpur/Nuclear Physics B 551 (1999) 467-489
472
E !_ -- DiADiO - DiCDiC - DibDib + --~ll~ 2 iJDibDjC - ~1 ( f + 2i[b,C] )2 LI = -~tr 4 i 2 i ~ + V/_~-------(e JOiDjx + --~1 e J Y(iDj~b + Y(iDi( - ~biDirl + i v ~ ¢ [$, A] - i4v'2X[X, 0] + i4x/2~ [X, C] - 2iv'2x[(, b] 2x/~ i - - i v ~ [ r I, b] + i--~ll el~i[f(j, b] - iv/2f(i[f( i, 0] + i v ~ i [ ~ i, A]
i , i -i2x/~¢i[~(i,c] + ~--~sr[sr ~.] -- - ~ ' [ r / , C ]
(6)
+ 2 [ ¢ , b] [a,b] + 2 1 0 , C ] [ A , C ] / ' 1 L o = - ~ t r { - D a ) t D a O - ( DaC + "-j2eabDbb- 2DiBia) z
+4( DaBai) ( DiC - 2DJBj i) - ( F + + 2i[ Baj, BJi] +2i[ Bab, Bbi] + 2i[ Bai, C] )2 _ 1 ( ,F)ijFiJ _ .~(
ab
--2i( F 0 + 2i[ Bij, C ] ) [ B ia, BaJ ] - 2i( Fab + 2i[ Bai, Bib] ) [ B ab, C ] +4~[iDalX ia + 4)([iDa]~ ia + f(aDa( - ~baDarl + 4~aDbX ab + 4 f(aDb~t ab + 2iv/'2~ai[ ~tai, A ] -- 2ix/~xai[ xai, 0 l + i4v/2~ai[xai, C ] + i4v/2~ab[xai, Bb i]
+i4v"2~ia[xib, Bab] +
i4v/2g/ai[xab, Bi b ] + i 4 v ~ a i [ X a j, Bi j ]
+i4v'2~ ia [Xij, Bail + i 4 v ~ tij [Xia, Bj a ] - 2iV~Xai[(, B ai ] - 2iv/2(tai[rl, B ai] +i4x/2Oa[f(b, B °b ] + i4V~¢a[f(i, B ai ] + i4~v/2¢i[)(a, B ia ] - iv/2f(a[)( a, 0] +iv/'2@a [ O a , h ]
--
i2v~/'a [,~"a, CI + 410, Bail [,,~, Bail ~, J
(7)
and ~--1 ~- ee'--strl {_4(DaBai) 2 - l(Fab +4i[Bai, Bib])2 } .
(8)
Now, in sending e to zero path integral localizes around the solutions of the following equations:
Fab + 4i[ Bai , B i b ] = O, DaBai =0.
(9)
In Appendix A we show that these equations imply
Fab = Bai = 0,
(10)
and from Fab = 0 it follows that the instanton number vanishes. A flat connection on the sphere can be written globally as
Aa = g-13ag
473
A. Imaanpur/Nuclear Physics B 551 (1999) 467--489
for some gauge group element g. Therefore, the connection A is A
:=
Aidx i + (g-laag)dxa.
We gauge transform A such that it lies in 27 direction A .--* g A g -1 + gdg -1 = g ( A i d x i ) g -1 + g(cOig-l)dx i = A~dx i . A a = 0 and Bai -- 0 , ~ 0 greatly simplifies. However, because of the zero-modes of the operator da, one still has to keep the order e terms in El. We expand all fields in terms of eigenfunctions of da and denote the zero-modes by a superscript 0. Effectively we do the following substitution: Setting
~ ( z , ~; w, a,) ~ ¢~°(z, Z) + ¢~(z, ~; w, a,), where ~ ( z , Z; w, ~i,) on the r.h.s, stands for the non-zero-modes. The kinetic part of £0 then reads 1
~o kin = ~-~tr {--0a.,~coa~ -- ( O a f "F Eab~bb) 2 -- ( tgaAi) 2 "F 4~J[iValX ia +4,~[iVa] ~ia _.}_~(a~a¢ _ ~ja~a~] -'F 4~Ja~b/tvab -1- 4~(aVb~ab}.
( 1 1)
Since )(ai and ~ai are self-dual and since there are no holomorphic 1-forms on the sphere (see Appendix A), £o kin is non-degenerate. Thus in doing the integral over non-zero-modes, one may drop the terms which are order of 6. Keeping terms of order one, the integral over 71, ~', X, ~, ~hi and /~i results in a set of delta functions imposing the following constraints: ~7a X ai -~ O,
~ atff ai .~ O,
V a ~ a ----0,
eabVa~b -~ O,
Va~( a = O,
eabVa~(b = 0.
(12)
AS mentioned, these equations have no non-trivial solutions on the sphere. Setting these fields to zero, £0 reduces to £o = Ltre2
(--Oa,,~a(l~_
(o~aC) 2 _
(0ab) 2 _ (~ami) 2 } ,
where fields are all non-zero-modes. Using the equation of motion for A i
we
obtain
dta'Ai + terms proportional to e = 0
0. The same happens for ~b, b and C fields. So in the limit e ~ 0 all non-zero-modes can be set to zero and one is left with a copy o f / : l in which fields now depend only on coordinates on X. From now on we call this reduced Lagrangian £ and drop the superscript 0 on zero-modes. The reduced Lagrangian, £, which now describes a two-dimensional TFT, can be obtained by the BRST variation of V, where as
Ai is a non-zero-mode this equation implies that, up to e order, Ai
=
A. hnaanpur/Nuclear Physics B 551 (1999) 467.-.489
474
V --ezL
2v~ j "~ q-x(2n-2f-4i[b'C]) ftr{½ yc'(,,9 , -2v/2DiCq-'-~l~jiDb)
,
2 v ~ a(2D~0i + 2 v ' 2 i [ ~ , b ] + v ~ i [ ( , C l )
},
(13)
and the BRST transformations of the two-dimensional fields are (6 -= {Q . . . . }) ab = V'2~,
ac = -~2 ~,
a¢,~ = - v~ioicb, aYci = iFI~,
8~ = - 2 [ b , ~ b ] ,
8( = - 4 [ C , ff]
6 X = ill,
a[--Ii = 2v'2i[f(i, ~],
a H = 2x/-2i[x, ~bl,
aa =v%a~ =o, an = -2[a,~],
aAi
= -2~//i,
The fixed points around which path integral localizes are those configurations that are BRST invariant. Thus, setting 8 X = H = 0 and 8f(i = / : / / = 0 and using the equation of motion for H and Hi we find the fixed point equations
s= f + 2i[b,C] = 0 , 1
k= DiC + - - a i j D J b
(14)
= 0.
v%
Squaring these equations implies that 0=
f
=f
tr (7'lsl 2+lkl 2) tr
{½1fl2+ 21[b,C]12+ 2if[b,C] + IDiCI2+IDibl2+ 2--~-,ilDiCDib~
v'gl
J"
Using the definition of f in (2), we can see that the third and the last term cancel against each other. Therefore this integral is zero if and only if f=O,
[b,C] = 0 ,
DiC =Dib = 0.
(15)
Requiring that there are no reducible connections (as is the case for fiat non-trivial SO(3) bundles) it follows that the only solutions are C = b = 0. Therefore, following Ref. [4], it can be seen that in this case the partition function is nothing but the Euler characteristic of the moduli space of flat connections over ~.
3. Perturbing by mass term The theory discussed so far does not have a mass gap [ 11 ]. To make the calculations more feasible we perturb the theory such that it has a mass gap. 3 This enables us to 3 A similar perturbation has been considered in [4,12] for N = 4 SYM theory in four dimensions.
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
475
integra~e out most fields and reduce the path integral to a finite-dimensional one. The reduced two-dimensional theory has a U(1) ghost number symmetry coming directly from the non-anomalous U(1) symmetry of the underlying four-dimensional N = 4 SYM theory. Because of supersymmetry, the measure for non-zero-modes is invariant under the U(1) action. The ghost and the antighost zero-modes, on the other hand, obey the same equations of motion such that there are equal number of ghost and antighost zero-modes. This renders the measure to be invariant under the ghost symmetry of the action. Therefore the ghost symmetry is anomaly free. As the measure is invariant under this symmetry, the correlation function of any operator that has a ghost charge is zero. Therefore, this symmetry allows us to perturb the Lagrangian, by adding gauge invariant terms with non-zero ghost number, without changing the partition function. Thus, for example, since the mass term for the hypermultiplet (as we will see presently) consists of a term with negative ghost number and a term which is BRST exact, one expects that the partition function is invariant under perturb:ing the Lagrangian by a mass term for the hypermultiplet. One can even go further to argue that an additional mass term for the chiral multiplet q~ (which contains ~b and A) still leaves the partition function invariant [4,12]. In the following we are interested in the correlation functions of a set of BRST cohomology classes of the form i(e)=
1_ f t r ( 47reJ
i -~bF
1 ) e / 2 + ~ p A ~p + 3--f~2 tr~b.
Part of this factor with an extra BRST exact term provides the mass for the chiral multiplet ~/, [2]. The remaining part may have a non-vanishing expectation value in the mass-deformed theory. This, in particular, implies that, in contrast to the partition function, the correlation functions of I ( e ) (in the perturbed theory by mass for the hypermultiplet) may depend on the mass parameter. The next problem is to give a mass to X, ~7, A, ~ and (. This can be achieved by adding V' and W' to V, where
Vt =-e--22 / d/x tr {XA},
(16)
V"=
(17)
f
tr {(,C - ½(b}.
To give a mass term to the bosonic fields b, C and the fermionic o n e ,,~i we change the BRST tlransformation rules for Hi, ~ and ( to the following ones: ~
-
~3m~Oi= 2x/~i[ Y(i, ok] + -'--~meoY(J, x/gl 6n~ = - 2 [ b , ~b] + imC, 6m( = - 4 [ C , ~b] - 2imb.
(18)
A. Imaanpur/Nuclear Physics B 551 (1999) 467--489
476
Even though the metric is explicitly introduced via the above first BRST transformation rule, note that the extra term is still invariant under metric rescaling (eij ~" gl ). Thus, in the following we will consider the theory defined by the deformed action 1 .-. 11ll~t S = I ( e ) + itm (V + tV' + ~rnv )
= l(e) +
l/
{
dtz tr DiADi(b + DiCDiC + DibDib - ---~--oeJDibDjC VOl 2 i"
z + ½ ( f + 2i[b, C] + tA) 2 + 2ix/2tXTq - ½[m[2C 2 - lira2 2bz i~ +-~(¢~ -
m --E ~' "
. • , j X~tX~J + 2ifndp[b,C] - 2im,~[b,C]
4i r 2i i" -k--~l e '~lliDjx q- --~ll e Jf(iDj~lI -4- i2iDi ( - i~liDiTI - x / 2 ( t [~, al + 4 x / 2 X [ X , ~bl - 4x/2~ [X, C1 + 2 v ~ x [ sr, bl
+v~[n,
2v~ ~j
b] - ~ , e
~ [ ~ j , b] + v ~ [ ~
1
,
q-2vl2¢i[ f(i, c ] - ~ ( [ ~ 2 [ ~ ~ b]
1
~,~] - v~¢~[¢ ~, a] ,
( A] -4- " - ~ ( [ rI C]
[k,b] - 2[~b,C] [A,C] }.
( 19)
Notice that although the new BRST charge does not square to a gauge transformation (because of those new terms proportional to m), Lagrangian remains BRST invariant. This can be understood if we notice that 8.2, acting on fields generates (up to a gauge transformation) a U(1) action. Let 6r - iv~m 6 m2 a n d f l = _ b + i C , ~b = _ d T t + 6 ( , t h e n U(1) group acts as 6rfl = --ifl,
6T~b = --i0,
1 ~" v~1%xJ'
~r2i . . . .
t~rBi
1
v~l
= --eijF~
j.
Thus the fields fl, 0, Xz and /4z all have charge - 1 , with their complex conjugate having charge +1. All other fields have zero charge under this U(1) group. The fact that S is invariant under 6m then follows since V,V' and V" all have zero U(1) charge. Before continuing the analysis, it is important to understand the relation between the perturbed and unperturbed theories. Since the perturbing terms proportional to t and th are BRST exact, one may expect that correlation functions are going to be independent of these two parameters, but actually this is not true in general: adding 6mV' and 6mV" to the Lagrangian may result in some new set of fixed points flowing in from infinity and deforming the original moduli space of solutions [6] such that the path integral gets contributions from these new fixed points. The theory will be independent of t and ~ if in varying these parameters the Lagrangian remains non-degenerate and the perturbation does not introduce new components to the moduli space of fixed points.
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
477
We first discuss the situation for t = 0 with arbitrary m and r~. The perturbed Lagrangian (with t = 0) can also be derived upon reducing the N = 4 theory broken to N ---- 2 by the mass term. Had we started with N = 2 theory with one massive hyperrrmltiplet in the adjoint representation of the gauge group in four dimensions, we would have ended up with the same above perturbed Lagrangian after reduction. The fixed point equations are those of (14) together with (setting t~m~ = 6 , , ( = ~.,,7 = ~m¢i = O)
[~l,~b] = ½mfl,
[A,~b] = 0 , Dic~=O.
(20)
If ~b is not identically zero then, being covariantly constant, it never vanishes and, in particular, can be diagonalized globally such that the bundle E splits as a sum of line bundles'. [ 13]. Moreover, if fl 4= 0, the first equation in (20) fixes ~b (up to a sign) m(1
01)
~=-g- O-
(21)
with/3 as
Now Eqs. (14) become f + 21~12 = 0, b B = (og - i A g ) ~ = 0
(notice f = ~fo'3, 1~ where f here is the U ( 1 ) curvature). Note that ~b = lmo-3 corresponds to a point, in the classical moduli space of vacua, where a component of the hypermultiplet becomes massless. 4 The relevant fixed points are then determined by the above equations. Clearly one can then argue that the path integral over massless modes computes the Euler characteristic of the moduli space of U ( 1 ) fiat connections. However, to evaluate the contribution of this singular point to the path integral, one still has to do the integral over the massive modes. This is not an easy task, but there is a special case where this point (~b = 41-mo-3) does not make any contribution. This occurs upon restricting to the non-trivial SO(3) bundles. As discussed above, a non-zero ~b breaks the gauge group down to U ( I ) . In particular, SO(3) bundles split as E=: L @ O ~ L - j ,
(23)
where L is the U ( 1 ) line bundle and O is a trivial line bundle. In this case, w2(E), which measures the non-triviality of the bundle E, turns out to be the mod two reduction of Cl(L), the first Chern class of L [4]. Thus if f = 0, as is required by Eqs. (14), 4 AS Eq. (20) fixes ~b up to a sign, there are indeed two such singular points in the classical moduli space of vacua.
A. l m a a n p u r / N u c l e a r Physics B 551 (1999) 467-489
478
w2(E) has to be zero - implying that fiat non-trivial SO(3) bundles do not admit reducible connections. Therefore, in this case, the point ~b = lmo'3 does not conl~ibute to the path integral. Let us now discuss the case that t =~ 0. The fixed point equations (14) turn into the following equations (~ =- b + iC with ezz = i`/~lgze):
f + [B,B] + ta=0, /5/~ = 0,
D/~ = 0.
(24)
The vanishing argument now fails; f = / ? = 0 (and a = 0) are not the only solutions, as there are new fixed points with f =~ 0 contributing to the partition function. Since the connection is not bounded to be flat any more, a set of U(1) connections, in all classes of U(1) bundles, appear in the moduli space of solutions. Moreover, the point ~b = lmo'3 may contribute to the path integral even for non-trivial bundles. In the following we single out this point from our discussion and treat it independently.
3.1. Integrating A, rI and X Perturbing by V' now allows us to integrate out the fields a, r/ and X. Using the equations of motion for a and r / w e get t2A= D2~b -
t ( f + 2 i [ b , C ] ) +2im[b,C]
+v~[~,t] and
1 x-- 2v t
1
v/2[@i, I//i]
+
,
+ _---~_[~" (1 + 2 [ b , [ ¢ , b l ] + 2 [ C , [ ¢ , C 1 1
2,/2
( _Di~i_ F iv/~[b,~]
i
+ .~[c,
sr]
(25)
}.
Putting these back into the Lagrangian yields +
1
f
tr(DiCDiC+ DibDi b - - ; le iJ D, bDjC+ "-2i ~ r 11 eij~,¥il.)jlll
2v~ ij r , 1 +if(iDi# - ---~11e I,bil.,~j, o] -llm[2C2
'{
+t
--
4- 2V~i[~(
llml2b2 + -iFn ~(0- -
i, C] 4- V~(i[)( i, ~]
m_.__m__e..-i.--j 2x/-~,,X X + 2iFn4)[b, C]
(
( f + 2i[b,C]) D2d~+2im[b,C] + v~[Oi, Oi] + v~[~,~l
,
+~--~[d',(l + 2[b, [4~,bl] +2[C,[4~,C11 +~--~i ( - D i q / + i v ~ [ b , ~ ] +
i--~[C,#])
)
A. lmaanpur/Nuclear Physics B 551 (1999) 467-489
x (--~lEDkOt-i4"¢~[C,~] -4 kl -- 2 i v ~ [ sr, b])+-~-~ D2d~+2im[b,C] +~[(,(1)
479
}
+ v ~ ( [ 0 i , 0 i] + [ ~ , ~ ]
+ 2 [ b , [4,, bll + 2 [ C , [4,,C11
(
,
+v~ -DiOi + iv~([b,~] + ~ [ C , ( I )
Terms proportional to
1/t are indeed BRST
)
trivial, and can be written
i
i
"-~.~3m{(f +2i[b,C])(-Di~bi+iv~[b,~t]+-~[C,(])}. Terms proportional to
lit 2 are
also combining into
i ~-~t3m { (-DiOi-+ iv/'2[b,~] +--~i + v ' 2 [ 0 t , 0~1 + v ~ [ ¢ , ~ l
[C,(])
+ ~ -1- ~ [ ( , (1 +
(DZcb+2im[b,C]
2[b,
[4~,bll + 2 [ C , [4,,C11
)1
In the effective Lagrangian (26), the kinetic terms are non-degenerate for all values of t and since those terms proportional to t are still in a BRST exact form, the path integral does noel depend on t.
3.2. Large-t limit and the integration over b, C, 5, ~ As argued above, for non-trivial SO(3) bundles the point ~ -- lmo-3 does not contribute. For t 4= 0, because of tile supersymmetry, even after integrating out .t, rI and X the singularJtty still persists at t r ~ 2 = ~m 2. As we have chosen ~ to be a real scalar field, reality c,f the action requires that m to be a real parameter. However, to regulate the contribution of the points in the neighbourhood of tr~b2 = ira2, we allow m to have a small imaginary part. If there is going to be any singularity when ~ approaches m, it has to show up in the final result when we take the limit Im m ---* 0. This can be thought of as a kind of regularization by analytic continuation. Now let us consider the large limit of t. Since the kinetic terms remain non-degenerate we can actually take t ~ e~. Using the auxiliary field/1i, in this limit we are left with the action
, ~ i ( ffli - 2x/~OiC + ~2V~ e j i O Y b ~" .] ~- V_---~-o" 2i e , I ij~ Xil")Jl~t ~ r S = -~1 f dtx tr l - ~n x/gl
A. lmaanpur/NuclearPhysicsB 551 (1999)467--489
480
+i](iDi ~
irh
-- ½1ml2C 2 -
~
m
~i
~
'
½1ml2b2 + -~(~b - - ~ l l eijX X ~ + 2irhO[b, C]
2~ ,~.
}
V,~'---~I~: l[Ti[)(j,b] q'- 2X/2~i[]( i, C] + v~](i[2 i, ~b] + I(e).
£ can still be written as a sum of BRST exact term
i6m
{1{( e2
, + ~fn(~bC 1 l ~(i f-li - 2v/2DiC + ---~l eji o j o) - ½(b)
and l(e). The integral over C gives a factor of (det
l f ~
l-i-
m
2" +~l-~(Diffli -- 2[,~i, iml -
(~eImllz ) ) - 1 / 2 ~.~,
2i
}}
and leaves
i.
~bi ] )2 _ 2 rh [b, ~b] 2 + 2iv/'2 [b, ~ ] (Di[-I i m rn
~
-
2[~'i, ~bi] )
v/2 ejiftioJ b _ ½1m[2b2 + --~(~p ifn ~ - ---~l 2x/~ e q ~Pitxj, ~ b] }_ + I(e). X/~l Next we would like to integrate out b, ( and ~. It is easy to integrate out ( and ~ using their equations of motion. In the evaluation of determinants, which appear in doing the integral over b and finally over Xi, we always assume that ¢ is a constant field. This can be justified finally when the integral over the gauge fields constrains ¢ to be constant. The equation of motion for b yields
= [--~I2K
--~leJDjHi+
--~leJ[X,,~. ~bj] - 2~i[(Dif-Ii-2t2oi'i])¢']m
' (27)
where we have defined (A and B are Lie algebra indices)
KAB :~ (1-- -~tr ~p2)-I (~AB -- -~¢A~pB) . Replacing b in the action, we obtain
,f d~ tr
s=I(~) + ~
-~HH~+ v'~2i[~,.¢1
-- ~Eij()(i)(J-- ~[2Dlf(IDi,J) + [-~(Diffli- 2[)(i,~Ii])2 }
1 ijDjHi+ - -Wo2 + T ~1 /\|-~evgl voe × \(--l---~"eklDlflkx/gl
iJ[xi,- ¢1] - ~[m (Di[-1i -212i,~i]),4,].)a K AB
+ -~gl Ekl[~(k'~'ll]- 2~im[(DlI-ll-
2['~/'01])'~]
'
A. hnaanpur/Nuclear Physics B 551 (1999) 467--489
481
and a factor of 1
-1
where (ad~b)aB = --faBcqbc and 120 indicates the space of zero-forms. The following equations are easily derived:
ata { ( Dl)( I ) ( Di ~-Ii - 2 [,i'/, ¢i] ) } i( DiI--Ii - 2[,~i, ~/Ii ] )2 __ 2x/~i( Dif(i) [ Dtf(t, qb] + ~2---~_meiJDif(jDlf(l, x/gl
and
I" l
ijo , x , -+
~[Dif(i'~] }
ieiJ
,
,
x/gl2i i=
2 [(Di[_ii _ 2[2i,~bil),¢b]" m 82m{--~gle'TDjfci+~[Dif(i'dP]}=iv~mDlf(t
4iV~[[Dlf(lf'b]'qb]'m
Using these, the action can be written as
if
S = l ( e ) + -~
dl.t tr
l ~i (_~nHi_4_V~f(i[fgi, qb]__ ~
z i e2~rnl2 /
dl.t 8m{tr((Dlf(l)(Diff-I i-
2[/~i,0i]))
2~
/ eq ~ 2i )A + ~--~1DjXi -Jr---m[Di)(i' qb] K AB × ~--~l(DlHk -- 2[,~k, ~/1) + --[(DIHlm - 2['~/'~k/])' ~b]
"
(28)
Note that the integration over b, C, ( and ~ has not destroyed the manifest BRST exactnes:g of the action, in particular, the variation of S with respect to fit is still a BRST commutator.
3.3• Large-Tn limit and the final reduction We note the partition function is formally independent of fit (since the variation of the partition function with respect to fit gives an BRST exact expression) and is really independent of fit if in varying fit the Lagrangian remains non-degenerate with a good behaviour at infinity in field space. The mass term for Xq, the term Oi/~f, and the form of the cohomology classes that we have added by hand, guarantee that this is actually
A. lmaanpur/NuclearPhysicsB 551 (1999)467--489
482
the case. Having this freedom in the value of th, we simply set rh = c~. This leaves us with the action
1 { - s Hl-ia-a-Hi S=I(e)+-jfd~.
~(ia
1 (---'~l~lmlfij~aB--2ifABCqbCgij))(JB I,
and the partition function reads
2 e ~im12) t , - -(d---det - - - tS -(T2~:~) -m z ~b)=) ) 1/2]~, Z[e,m] = f ~)(ai,~li, qb, I-li,~(i) ( (det ~_L ~-Z 8-~-(ad e -s , / O0®E The explicit appearance of m on the l.h.s, reminds us that, although independent of rh, Z does depend on m. This is so because m was introduced through the BRST transformation laws. This is reminiscent of holomorphicity of N = 1 theories in four dimensions. Doing the integral over )~i gives a similar determinant, but this time over the space of 1-forms. Putting all pieces together one gets ([detm (1- ~ad [detm (1
~ad
~b)]
)
~b)] a~®e :P®E /
xexp [ ( -i1 ~
tr ( ~v~ i- - ~ b F + ~-1¢ A ¢ )
3}-'"2e n i trq~2) ] .
(29)
Notice that, as expected, ~ cancels out between the fermionic and bosonic determinants. The integral over ~i provides a symplectic measure for the gauge fields Ai [6]. Performing the path integral over ~b and Ai is now straightforward. In Appendix B, using the Faddeev-Popov gauge fixing technique, it has been shown that the integral over the gauge fields constrains ~b to be constant and hence the path integral calculation reduces to a finite-dimensional integral over constant ~b [ 15 ]. Explicitly, for S0(3) gauge group we have
× exp (_iv/~4)(2~ 1)
~E:+' ) .~~
(3o)
4. Discussion
We have reduced the calculation of the correlation functions in the mass-deformed theory to a finite-dimensional integral in (30). We can now perform the sum over n which results in a delta function restricting ~b to obey the following equation:
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
(ix/2~b~ = ( m -~- 22v"-2~b'~ 2 --'~)
exp \ ~ j
483
(31)
Therefore, Zig, m] = m 3(g-l) V ~th2-2g 4,, x exp
-t
4¢r
1-
~b~
32rr2j ,
7 (32)
where ,bs is a solution to Eq. (31). A similar result for the correlation functions of a topological field theory corresponding to the Hitchin equations has been derived in [ 14]. To this one still has to add the contribution of the point ~b = ¼too'3. However, note that from the discussion we had in Section 3, for non-trivial SO(3) bundles, this point contributes only if we perturb to t # 0.5 Thus if we are interested in the limit of t = 0, we can just ignore the contribution of this point. In conclusion we note two observations. Firstly, the result is m-dependent as might be expected from the discussion in Section 3. Note in particular that the expression (30) has the right behaviour when m ---+ oo; in this limit, the solutions of Eq. (31) are reduced to ~bs = 2V/'2"n'2/
and therefore Eq. (30) reduces to the expression for the corresponding correlation functions in the say pure N = 2 theory [6]. The extra factor, m 3(g-l), is left from the integration over the heavy fields in that limit. The power of m is in accord with the dimension of the moduli space of fiat connections which is dina A4 = 6g - 6. Any two zero-modes of Xi are absorbed by the corresponding mass term in the Lagrangian and gives a power of m. Secondly, we recall that, in general, S-duality relates the strong and weak couplings and swaps the gauge group with its dual group. However, as in the limit where S2 shrinks only instantons with k = 0 contribute to the path integral, unlike [4], the correlators in the effective theory do not depend on the modular parameter "r". Hence the action of S-duality is now simply to exchange the gauge group SU(2) with SO(3). Thus to derive that S-duality holds in this calculations, we must extend it for the SU(2) case; in particular, the contribution of the point ~b = lmo'3 must be taken into account. Amusingly, one can infer properties of this contribution by demanding S-duality. 5 As discussed in Ref. [6], the contribution of the original moduli space is invariant under perturbing to t~O.
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
484
Acknowledgements I am very grateful to Jim McCarthy for all his support and assistance throughout this work. I would also like to thank Nicholas Buchdahl for useful discussions.
Appendix A. The vanishing argument In this appendix we want to discuss the solutions to Eqs. (9):
k= Fab q-4i[Bai, Bib]= O, s = DaBai = O. Let us first analyze the second equation. After squaring we get tr ( D a Bai) 2 = _ f tr B ai (DaDbBbi) =
-
tBai~ -- Bai[Da, Db]Bbi) tr ~, Lib D a °Dbi -~"
= f tr ((OaBbi)(Ob nai) -4- eabnainb i - iB ai [Fab, n b ] )
= / tr ((DaBbi + DbBai - DbBai) (Db Bai) + 1 R B aiBai - iB ai [ Fab, Bbi ] )
=f = ((DbBai)2
-
1 2 -[- ½R Ba'Bai " -iBai[Fab, n b ] ) , ~(D[aBbli) (A.I)
where we used the fact that in two dimensions, the Ricci tensor takes a simple form, I Rab = ~gabR,
and
[ Da, Db ] B ci = RCdabBdi "4- i[ Fab, Bci], [ Da, Db ] B ai = Rab Bai 4- i[ Fab, Bai].
(A.2)
Since B~u is self-dual, we have Bw~ = B~z = O. Hence
(D[aBb]i) (D[aB b]i) = (Og, Bwz ) (Oa'B wz ) q- (OwBa,~) (OWB ~ )
= (DWBwz)(DwB wz ) + (D~B~e ~(D:vB ~ ) = (DaBai) (Db Bbi) .
(A.3)
Putting this back into (A.1) we get
3 / t r ( D a B a i ) 2 = f tr ((DbBai)2-} - I R BaiBai-iBa'[Fab, Bbi] ) .
(A.4)
A. hnaanpur/Nuclear Physics B 551 (1999) 467-489
485
Upon adding the squares of the sections k and s, we have
f tr(¼kZ+3s2)=f tr {¼(Fab)2--4[Bai,
Bib]2
+ 2iFao [ B ai, B i b] .q_ 2( DbBai ) 2 "-1- R BaiBai - 2iBai[ Fab, Bbi] }
=
f
tr {-~(Fab) -- 4[B,~i, Bib] 2 + 2(DbB~i) 2 + R BaiB,i}.
1
2
The right-hand side vanishes if and only if k = s = O. However, for sphere (R > O) all terms on the r.h.s, are positive definite so a solution to k = s = 0 has necessarily B ai = O. This leaves us with the equation
Fa,~ = O. This equation implies that the connection is locally a pure gauge Aa = u - l d a u for some S U ( 2 ) matrix u. However, as the transition functions for SU(2) bundles on the sphere are trivial, the connection can be written globally as a pure gauge and be gauged away. Moreover, one can argue that this can be done continuously all over 27. Thus we can set A~ = 0 ,everywhere. More rigorously, if {U~} is an open covering of 2 by contractible sets and {V/} is an open covering of S 2 by such sets, the sets U~ x V/give an open cover of ,~ × S 2 by contractible sets. On the intersection of two patches, the connection A now satisfies
A,~, = g~.~jABjgait~ j q.- g~.lBjdgail~j,
or
dgaiBj -'k ABjgaiflj - gaiBjAai = O.
Since the S 2 component of the curvature is zero we have that (A~),,i = u~ .l dau,~i. Putting this in the above equation yields
da( uaigaiBjuB; ) = O. Therefore g,,~i#j -- uotigaiBjUB; does not depend on the coordinates of S 2. This implies that the gaiflj'S are a set of locally constant transition functions equivalent to g,~iaj and for a fixed point on 27 define a map from S l to SU(2). This map is trivial so ~i~j belongs to the conjugacy class of the identity
gaic~j : gaig~j I = uaigcdajUa'-j I"
or
(g~iluai)gaiaj(g-~jlUotj) -1 = l.
Now consider (g~.lUcti)gctiflj(gfl;Uflj)--l.
This is a constant matrix in the S 2 direc-
tion. Si~:ce g,~i/~j = gaifligfliflj it is equal to (gailuai)gctifli(gflilufli) -l , and since g,~i/3j = g~i,~jgc~j~q it is equal to ( ~ l u , ~ j ) g , ~ j # j ( ~ l u # j ) - l . Thus it is in fact independent of the index i and therefore defines a matrix ~,~ depending only on x E Ua/3 and satisfying the cocycle condition. 6 Since the transition functions are independent of i, therefore (A.~)ai do not depend on the index i and Aa can be gauged away. 6 The proof of this part was provided by Nicholas Buchdahl.
486
A. hnaanpur/Nuclear Physics B 551 (1999) 467-489
It is now easy to see that the flatness condition, F~b = 0, necessarily requires the instanton number to be zero. The curvature locally takes the form F=dA+AAA.
Therefore locally we can write tr(FAF)=dtr(AAdA+2AAAAA), but since Aa = O, the instanton number reads k = 8~I 2f
trFAF=
I f 8~r2
XxS 2
d¢ t r ( A ~ . A d c A ~ ) ,
~xS 2
where the subindex C indicates differentiating with respect to the coordinates on S2. Note that the integrand is still a local one. However, we showed that the transition functions are independent of the local coordinates on S2. Therefore, for a fixed point on X, A~ is globally defined on S2. This means that the integral over S2 is a total divergence and gives zero for the instanton number. In summary, we have learned that if the bundle E admits a flat connection in S2 direction then it has to be trivial (for those bundles that are classified only by instanton number) and k, the instanton number, is zero.
Appendix B. Faddeev-Popov gauge fixing In this appendix we want to show how Eq. (30) is obtained starting from (29). To evaluate the path integral over gauge fields and ~b, following Ref. [ 15], we choose the so-called unitary gauge in which one rotates the Lie algebra valued field ~ba to the Caftan subalgebra by conjugation, i.e. we choose ~b+ = 0, where q~ = ~3q'3 -}- ~b+'/'+ q- ~b_7"_.
This gauge can always be achieved at least locally, but there might be some topological obstruction to impose it globally [15]. Implementing this gauge in the path integral requires to introduce the Faddeev-Popov ghosts c and antighosts g together with a bosonic auxiliary field b. These fields transform under a BRST operator 8 like ~q~-t- = q-ic+q~3,
~
= bi,
~ b 3 : 0,
•c4- = 0,
6b~ = O.
(B.1)
The Faddeev-Popov prescription consists of adding a BRST-trivial term iS(P_qb+ + g+fb_ ) = ib_
to the action in (29). It is now clear that the integration over b will impose the gauge condition; ~ba: = 0. We have tribE = q~3F3 = q~3(dA3 + (A A A)3) = ~b3(dA3 q- i v ~ A l A A2),
A. lmaanpur/Nuclear Physics B 551 (1999) 467-489
487
therefore, defining ~b _= ~b3, A _= A3 and F _= F3, the action in (29) turns into
S=~
ck d A - q b A 1 A A 2 +
8) / c~2
+
dtz(g-fbc+-P+Obc-).
Integration over Faddeev-Popov ghosts gives [det ~bz] ao(~), while over Al and A2 results in .2 a -I/2 det@ J n~(~O " Using the Hodge decomposition theorem we can express the product of these two determinants as [clet ~b2] u0(~0 2 1/2 • [det ~b ] H~(Xg) When E is a non-trivial SO(3) bundle we write the curvature of the reduced U ( 1 ) bundle as F = 27r(2n + 1)to + dA, where w is the volume form ( f z co = 1) and 2n+l
=~-~
is the first Chern class which characterizes the U ( 1 ) bundle. To gauge fix the residual U ( 1 ) symmetry A --* A + dc~, we again appeal to the Faddeev-Popov prescription. We demand that a selected slice be normal to the gauge orbit,
(d,~, A) = O, which implies that d t A = O. Imposing this gauge, the action is l 4~r2/(iv/'27r(2n+l)~b~o+8~b2+--~2(#bdA+bd'A+'d*dc) )•
The kinetic term for A vanishes for A a harmonic 1-form, i.e. when dA = 0 and d t A = O. Hence there is still a residual symmetry under
A--*A+y, b --* b + constant, c --~ c + constant,
A. hnaanpur/NuclearPhysicsB 551 (1999)467-489
488
where 3' is a harmonic 1-form. Integration over the zero-modes of b and c and over the harmonic 1-forms gives an unspecified constant factor that can be simply absorbed in the normalization. Therefore we need only be concerned about the non-zero-modes. Dropping the harmonic part of A, it can be written globally and uniquely as
A = da + *dfl, for some zero-forms a and/3. The action then looks like 4~"21
iv'27r(2n+l)~bw+
d~2+--~(obd,d/3+bd,da+ed,dc)
£ and the measure is
DA = DotD/3 det [dd t ] ao.
(B.2)
Note that ,2 = ( _ l ) p when acting on a p-form and d t = - • d*. The integral over b and ce results in a determinant, det [ddt]~0l, which cancels the Jacobian in (B.2). Also the integral over/3 gives a delta function
6(ddt~b) -- det [ddt]~ 18(~b).
(B.3)
Notice that since we are integrating over non-zero-modes the delta function on the righthand side is a delta function on non-constant ~b's. The determinant in Eq. (B.3) gets cancelled against the determinant coming from the ghosts. At the end we are left with a finite-dimensional integral over constant ~b fields, ([detm(1-~ad~b)])
Z[e'm]=n~czfdq~
[detm(1
x exp ( _ i v , ~ b ( 2 4 ; 1)
[det~b2]no
~_~)]a'®e
2 l/2 n°®el [det~b ]m
e~b2 "~ 3-~2j.
(K4)
Using the Riemann-Roch formula dimO I ® L - d i m O ° ® L= g - 1 - c l ( L ) and the definition of Euler characteristic of a Riemann surface X(~g) = 2 b ° - b l = 2 - 2 g , and the fact that q~ is now a constant, we can write the partition function as
Z[.,g#l] :#'n3(g-l)Z
/'d4>
.~z ~ ×exp which is Eq. (30).
( _ , v , ~ b ( 2 4 ~ 1)
(l_ e~b2 "~ 3--7-42) '
"-1 {m-
t m + 2v/2"4b) (B.5)
A. Imaanpur/Nuclear Physics B 551 (1999) 467--489
489
Refene~nces [1] [2] [3] [4] [5] [6] [71 [8] [9] [10] [11] [12] [13] [ 14] [15]
E. Witten, Comm. Math. Phys. 117 (1988) 353. E. Witten, J. Math. Phys. 35 (1994) 5101, hep-th/9403195. C. Montonen and D. Olive, Phys. Lett. B 72 (1977) 117. C. Vafa and E. Witten, Nucl. Phys. B 431 (1994) 3, hep-th/9408074. N. Seiberg and E. Witten, Nucl.Phys. B 431 (1994) 484, hep-th/9408099. E. Witten, J. Geom. Phys. 9 (1992) 303, hep-th/9204083. M. Bershadsky, A. Johansen, V. Sadov, and C. Vafa, Nucl. Phys. B 448 (1995) 166, hep-th/9501096. G. Moore and E. Witten, hep-th/9709193. A. Losev, N. Nekrasov and S. Shatashvili, hep-th/9711108. J.M.F. Labastida and C. Lozano, Nucl. Phys. B 502 (1997) 741, hep-th/9702106. E. Witten, Nucl. Phys. B 460 (1996) 335, hep-th/9510135. J.M.E Labastida and C. Lozano, hep-th/9711132. D. Freed and K. Uhlenbeck, Instantons and Four-Manifolds (Springer, Berlin, 1984). G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9712241. M. Blau and G. Thompson, hep-th/9310144.
Nuclear Physics B 551 (1999) 467--489 www.elsevier.nl/locate/npe
N = 4 S Y M on
× S 2 and its t o p o l o g i c a l reduction A. Imaanpur 1
Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, SA 5005, Australia Received 31 March 1998; revised 8 December 1998; accepted 29 March 1999
Abstract We consider the twisted N = 4 SYM on ~: × S2. In the limit that S2 shrinks to zero size the four-dimensional theory reduces to a two-dimensional SYM theory. We compute the correlation function:~ of a set of BRST cohomology classes in the reduced theory perturbed by mass. (~) 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Topological field theory; Duality
1. Intt~)duction Topoilogical field theories [ 1] have proven to be a useful tool in the investigation o f the non-perturbative characteristics o f supersymmetric gauge theories. There is an interplay between certain supersymmetric gauge theories and their corresponding topological versions: one can use topological results on smooth manifolds to learn about the underlying physical theory; conversely, one may use the physical arguments to gain new insight iinto the topological structure o f the manifold on which fields are defined [2]. As an example o f the first - i.e. using the results on the mathematical side to learn about physics - consider the N = 4 S Y M theory. This theory has been conjectured to have an exact S L ( 2 , Z ) duality [3]. Since this duality relates the weak and strong coupling behaviour o f the theory, to test the conjecture one needs quantities such as the partitior~ function to be computed non-perturbatively. This is a formidable task and one actually does not know how to proceed in this direction. This is where topological field l E-mail: [email protected] 0550-3213/99/$ - see frontmatter (~) 1999 Published by Elsevier Science B.V. All rights reserved. PII S0550-3213 (99)00221-7
468
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
theory comes to provide an alternative approach to the problem. Instead of the physical theory, one considers the corresponding topological field theory obtained by a procedure called twisting. The basic characteristics of the theory, such as SL(2, Z) invariance, remain intact under twisting, so one hopes to see the realization of this symmetry in the twisted model. In [4] it has been shown, using the known facts about the structure of the moduli space of instantons and the associated Euler characteristic, the partition function of N = 4 twisted theory on some specific manifolds can be computed. So in this way it has become possible to make some exact and non-perturbative statements about the theory and its self-duality properties. In this article, we will study the N = 2 reduction of the above theory obtained by mass perturbation for the hypermultiplet. This theory is still believed to be S-dual [ 5 ]. We will compute the correlation functions of a set of specific operators using a method of calculation similar to that of [6]. Twisted N = 2 and N = 4 SYM theories on product manifolds 2? x C, where 2 and C are both Riemann surfaces, have been considered in [7]. There it was shown that, in the limit C shrinks to zero, the four-dimensional theory generically reduces to an effective two-dimensional sigma model. However, when C is a Riemann sphere - as is the case of interest in the present paper - things are a bit different. The dimension of the self-dual harmonic 2-forms, b +, is one in this case. Hence the connection is reducible. It follows then that the path integral may get contribution from the so called u-plane [8,9]. Moreover, when b + = 1, there is a wall in the space of one parameter metrics. On crossing this wall the partition function may change its value. Here we will compute the path integral in a chamber where S2 shrinks to zero. We consider SO(3) bundles such that the restricted bundle over S2 is trivial. Bundles which restrict non-trivially on S2 give zero contribution to the path integral. This is so because in the limit that S2 shrinks to zero size, the path integral localizes on the moduli space of fiat connections in the S2 direction. However, it can be shown that for a fiat bundle over S 2, transition functions are trivial and the bundle must be trivial. Therefore non-trivial SO(3) bundles on S2 do not admit flat connections. The organization of this paper is as follows. In Section 2 we consider the twisted N = 4 Lagrangian on 2 × S 2. In the limit where S 2 shrinks it is shown how the four-dimensional theory reduces to an effective two-dimensional theory. The fixed point equations imply, in the case of a non-trivial SO(3) bundle over ,~, that the partition function of this reduced theory is in fact the Euler characteristic of the moduli space of fiat connections on 2L A mass perturbation makes the path integral calculation more tractable - particularly for the limiting two-dimensional theory. In Section 3, we show how this comes about. Perturbing by the mass allows most fields to be integrated out, and reduces the path integral to a finite-dimensional integral which can be easily performed. In Section 4 we discuss the result. Although we have not yet managed to give an explicit check of S-duality, we have isolated the problems involved and hope to return to this in later work.
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
469
2. T w i s t e d N = 4 o n 2; X S 2 a n d its r e d u c t i o n
The key point in twisting [ 1] is to redefine the global space-time symmetry such that at least one component of the supercharge becomes scalar under the new defined spacetime symmetry. This procedure crucially depends on the existence of a suitable global R-symmetry. N = 4 SYM theory in four dimensions has a global SU(4) symmetry such that the supercharges transform under the 4 of this symmetry. First one needs to see how this representation transforms under the space-time symmetry group SU(2)L x SU(2)R. There are [4] three possibilities for the decomposition which give rise to singlets under the twisting: (i) (2,1) @ ( 1 , 2 ) ; (ii) (1,2) @ (1,2); (iii) (1,2) @ (1,1) ~ (1,1). As in [4], we will consider the case (ii) where, after twisting, two components of the supercharges turn out to be singlets and therefore square to zero. The scalar fields of the physical theory, which transform under the 6 of SU(4), now transform under the new rotation group, SU(2)L x SU(2)~, as 3(1, 1) @ (1, 3), three singlets and one self-dual 2-form. Having determined how the new fields transform under the new symmetry group, what remains is to rewrite the Lagrangian in terms of these new fields on fiat R 4. This Lagrangian can then be defined on an arbitrary smooth four manifold while preserving those two BRST like symmetries. Let us start our discussion with the twisted N = 4 Lagrangian 2 in four dimensions [~,10],
£=: ~2tr rl - D~*AD~'~b + ½[-I~([-I~ - 2v/2DuC + 4x/~DVBp~)
+ ~1H
/xv
+ (Hm,- 2F~,,-4i[B/.q,,B~I -4i[Bm,,C])
+4¢/~D~ x/~v + 4 ~(/~Dv~ ~v + 5(/~D~ ( - ~b/~D/%7 +iv'2~m't~m,, A]
-
iv~xm'[ Xz,,, qb] + i2v~tpZU [Xzu, C]
+i4v/2~ uv [Xup, Bvt'] - i v ~ x u v [(, Bm' ] - iv/2~u,, [77, Buv ] +i4v'2Ou IX,.. Bin'] - i v Y 2 , [.~'". ~b] i +iv/20t, [ ¢ u, A] - i2v/20~ [,~ u, C] + ~ - - ~ ( [ ( , 2t]
(1) As mentioned, the action is invariant under two BRST transformations. However, for us it is enough to consider one of them, which reads [ 10]
2The Lagrangian that we use is actually different from the one constructedin [ 10] by a BRST exact term -¼a(~[4,, al).
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
470
6( = 4i[ C, (b ] ,
aAu = - 2 ~ u ,
= v%, at/ = 2i[a, ~b],
a ~ = o,
= &,
a,.
8 ~ . = 2i[ B ~ , ~ ] , ac =
1
6/-1u = 2x/2i[,~,,~b],
,
~X/,,. = H.v,
6Hz, v = 2v"2i[ X/,v, dp].
In this article we choose ~b and a to be two independent real scalars. This will render the Lagrangian hermitian and allow us to treat 4~ and a independently. The generators of the SU(2) group are chosen to be hermitian T a = ~2tr a with tr (TaT b) = 6 ab. The theory enjoys an exact U(1) ghost symmetry under which ~j,, ~uv, ( have charge 1, Xt,~, r/,/~u charge - 1 , while ~b and a have charges 2 and - 2 , respectively. All other fields have ghost number zero. Take the underlying manifold to be 2 x S 2. Let us denote the indices on ,,~ by i, j , . . . and those on S 2 by a, b . . . . . We define 1
1
F. =
x.
1
=
1
Iftij =
Bi./ = - - t i j b ,
~ "~IV,~I 6`JO'
2V'~l
(2)
and the same for indices on S 2,
Bah = 2 + 2 6abb',
1 , Xab ----"~226abX,
~ab = 2 ~26abl~lt.
(3)
Here gl and g2 denote the determinant of the metric on ,Y and S 2 respectively. The fields Hu~, B~,v, Xu~ and ~t,v are all self-dual. Note that 2x/~ 1
2-"--~tij abBab = "~'-~tij£: 1
= 4 gg~2g2
tab bt
1
6ijb' = 2v/~ 16ijb',
(4)
where we have used that 6abtab = 6abta'b'gaa, gbb, = 2g2 a n d gaa'gbb'ta, b, = 6 ab. Also we chose 612 1 and so e12 = g2; thus, for example, we have Bab l---L-tabbt Hence we = = 2v,~2 • conclude that
b=b',
X = X ',
~ =~'.
In Ref. [7] it was shown that upon shrinking the metric on ~, one gets an effective two-dimensional sigma model governing the maps from S 2 to .A4, where .A4 is the moduli space of solutions to Hitchin's equations. Although the twisted theory is supposed to be topological, since the space of self-dual harmonic forms in this case is one-dimensional
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
471
one may not get the same effective theory if one instead shrinks S2. In that case we will see tha~! the effective theory which emerges is a two-dimensional twisted SYM theory, as conjectured in [7]. Thereto, we now scale the metric on S z by a factor of e. Notice that the definitions (2) and (3) are consistent with this scaling, since both sides of the self-duality constraints scale with the same power of e. After integrating out the auxiliary fields, the bosonic part of the Lagrangian reads EB = ~2tr {-D~aD~qb - (DuC - 2D~Bru) 2
1 (F~+ + 2i[Bt, o, B ~ ] + 2i[Bt~r, C] )2}
(5)
2
where 17+ = I ( F A- *F) and • is the Hodge duality operation. Thus we can write
- - ~:
'Iv~F~t+r F "'+'-'/v'~Ft,,FUU ~'/v ~ -
(*F)gu Fgr"
The last term is the instanton number and is metric independent. Using this, and the fact that Bur is self-dual, we write the last term in (5) as
_!~" _ 2iF~Zr([Buo, Bpu] + [Bur, C ] ) 4 " / z r - ~,~r + 2 ( [ B u a ' B ~ ] + [Bur ' C ] ) 2 -
Zl ( *F)urF~r
_- -~] {FijFiJ + 8iFiJ([Bij, C] + [Bia, Bajl) - 8 ( [Bij, C] + [Bia, B~] )2 + (,F)ijFiJ
+ FaOFa° + 8iFaO ( [Bah, C] + [ B~i, Bib] ) --8( [Bah, C] + [Bai, Bib] )2 + (,F)aoFaO}
- ( F a + + 2i[Baj, BJi] + 2i[Bab, B~.] + 2i[Bai, C]) 2 = -¼(Fij + 4i[Bij, C] )2 _ ¼(Fao + 4i[Bal, Bib] )2 -- 2iF ij [ Bia, B~ ] - 2iF ab [ Bat,, C ] + 4 [ B ij, C ] [ Bia, Baj ] +4[ Bab, C] [ Bai, Bib] -- ¼( *V)ijF ij - ¼( *Y)at, F ab) - ( F + + 2i[Baj,BJi] + 2i[Bab, B b.] + 2i[Bai, C]) 2. In the last equality we noted that for a self-dual antisymmetric tensor we have S/~ = S~6. In particular
[Bab, C] 2 = [Bij, C] 2, tr ( [ Bai, Bib] [ B aj, Bj b ] ) = tr ( [ Bia, B~ ] [B ib, Bbj ] ). After scaling the metric, then, the Lagrangian splits to three parts: E=£1
+ Eo + E - t ,
where/2n scales as e n. Specifically,
A. hnaanpur/Nuclear Physics B 551 (1999) 467-489
472
E !_ -- DiADiO - DiCDiC - DibDib + --~ll~ 2 iJDibDjC - ~1 ( f + 2i[b,C] )2 LI = -~tr 4 i 2 i ~ + V/_~-------(e JOiDjx + --~1 e J Y(iDj~b + Y(iDi( - ~biDirl + i v ~ ¢ [$, A] - i4v'2X[X, 0] + i4x/2~ [X, C] - 2iv'2x[(, b] 2x/~ i - - i v ~ [ r I, b] + i--~ll el~i[f(j, b] - iv/2f(i[f( i, 0] + i v ~ i [ ~ i, A]
i , i -i2x/~¢i[~(i,c] + ~--~sr[sr ~.] -- - ~ ' [ r / , C ]
(6)
+ 2 [ ¢ , b] [a,b] + 2 1 0 , C ] [ A , C ] / ' 1 L o = - ~ t r { - D a ) t D a O - ( DaC + "-j2eabDbb- 2DiBia) z
+4( DaBai) ( DiC - 2DJBj i) - ( F + + 2i[ Baj, BJi] +2i[ Bab, Bbi] + 2i[ Bai, C] )2 _ 1 ( ,F)ijFiJ _ .~(
ab
--2i( F 0 + 2i[ Bij, C ] ) [ B ia, BaJ ] - 2i( Fab + 2i[ Bai, Bib] ) [ B ab, C ] +4~[iDalX ia + 4)([iDa]~ ia + f(aDa( - ~baDarl + 4~aDbX ab + 4 f(aDb~t ab + 2iv/'2~ai[ ~tai, A ] -- 2ix/~xai[ xai, 0 l + i4v/2~ai[xai, C ] + i4v/2~ab[xai, Bb i]
+i4v"2~ia[xib, Bab] +
i4v/2g/ai[xab, Bi b ] + i 4 v ~ a i [ X a j, Bi j ]
+i4v'2~ ia [Xij, Bail + i 4 v ~ tij [Xia, Bj a ] - 2iV~Xai[(, B ai ] - 2iv/2(tai[rl, B ai] +i4x/2Oa[f(b, B °b ] + i4V~¢a[f(i, B ai ] + i4~v/2¢i[)(a, B ia ] - iv/2f(a[)( a, 0] +iv/'2@a [ O a , h ]
--
i2v~/'a [,~"a, CI + 410, Bail [,,~, Bail ~, J
(7)
and ~--1 ~- ee'--strl {_4(DaBai) 2 - l(Fab +4i[Bai, Bib])2 } .
(8)
Now, in sending e to zero path integral localizes around the solutions of the following equations:
Fab + 4i[ Bai , B i b ] = O, DaBai =0.
(9)
In Appendix A we show that these equations imply
Fab = Bai = 0,
(10)
and from Fab = 0 it follows that the instanton number vanishes. A flat connection on the sphere can be written globally as
Aa = g-13ag
473
A. Imaanpur/Nuclear Physics B 551 (1999) 467--489
for some gauge group element g. Therefore, the connection A is A
:=
Aidx i + (g-laag)dxa.
We gauge transform A such that it lies in 27 direction A .--* g A g -1 + gdg -1 = g ( A i d x i ) g -1 + g(cOig-l)dx i = A~dx i . A a = 0 and Bai -- 0 , ~ 0 greatly simplifies. However, because of the zero-modes of the operator da, one still has to keep the order e terms in El. We expand all fields in terms of eigenfunctions of da and denote the zero-modes by a superscript 0. Effectively we do the following substitution: Setting
~ ( z , ~; w, a,) ~ ¢~°(z, Z) + ¢~(z, ~; w, a,), where ~ ( z , Z; w, ~i,) on the r.h.s, stands for the non-zero-modes. The kinetic part of £0 then reads 1
~o kin = ~-~tr {--0a.,~coa~ -- ( O a f "F Eab~bb) 2 -- ( tgaAi) 2 "F 4~J[iValX ia +4,~[iVa] ~ia _.}_~(a~a¢ _ ~ja~a~] -'F 4~Ja~b/tvab -1- 4~(aVb~ab}.
( 1 1)
Since )(ai and ~ai are self-dual and since there are no holomorphic 1-forms on the sphere (see Appendix A), £o kin is non-degenerate. Thus in doing the integral over non-zero-modes, one may drop the terms which are order of 6. Keeping terms of order one, the integral over 71, ~', X, ~, ~hi and /~i results in a set of delta functions imposing the following constraints: ~7a X ai -~ O,
~ atff ai .~ O,
V a ~ a ----0,
eabVa~b -~ O,
Va~( a = O,
eabVa~(b = 0.
(12)
AS mentioned, these equations have no non-trivial solutions on the sphere. Setting these fields to zero, £0 reduces to £o = Ltre2
(--Oa,,~a(l~_
(o~aC) 2 _
(0ab) 2 _ (~ami) 2 } ,
where fields are all non-zero-modes. Using the equation of motion for A i
we
obtain
dta'Ai + terms proportional to e = 0
0. The same happens for ~b, b and C fields. So in the limit e ~ 0 all non-zero-modes can be set to zero and one is left with a copy o f / : l in which fields now depend only on coordinates on X. From now on we call this reduced Lagrangian £ and drop the superscript 0 on zero-modes. The reduced Lagrangian, £, which now describes a two-dimensional TFT, can be obtained by the BRST variation of V, where as
Ai is a non-zero-mode this equation implies that, up to e order, Ai
=
A. hnaanpur/Nuclear Physics B 551 (1999) 467.-.489
474
V --ezL
2v~ j "~ q-x(2n-2f-4i[b'C]) ftr{½ yc'(,,9 , -2v/2DiCq-'-~l~jiDb)
,
2 v ~ a(2D~0i + 2 v ' 2 i [ ~ , b ] + v ~ i [ ( , C l )
},
(13)
and the BRST transformations of the two-dimensional fields are (6 -= {Q . . . . }) ab = V'2~,
ac = -~2 ~,
a¢,~ = - v~ioicb, aYci = iFI~,
8~ = - 2 [ b , ~ b ] ,
8( = - 4 [ C , ff]
6 X = ill,
a[--Ii = 2v'2i[f(i, ~],
a H = 2x/-2i[x, ~bl,
aa =v%a~ =o, an = -2[a,~],
aAi
= -2~//i,
The fixed points around which path integral localizes are those configurations that are BRST invariant. Thus, setting 8 X = H = 0 and 8f(i = / : / / = 0 and using the equation of motion for H and Hi we find the fixed point equations
s= f + 2i[b,C] = 0 , 1
k= DiC + - - a i j D J b
(14)
= 0.
v%
Squaring these equations implies that 0=
f
=f
tr (7'lsl 2+lkl 2) tr
{½1fl2+ 21[b,C]12+ 2if[b,C] + IDiCI2+IDibl2+ 2--~-,ilDiCDib~
v'gl
J"
Using the definition of f in (2), we can see that the third and the last term cancel against each other. Therefore this integral is zero if and only if f=O,
[b,C] = 0 ,
DiC =Dib = 0.
(15)
Requiring that there are no reducible connections (as is the case for fiat non-trivial SO(3) bundles) it follows that the only solutions are C = b = 0. Therefore, following Ref. [4], it can be seen that in this case the partition function is nothing but the Euler characteristic of the moduli space of flat connections over ~.
3. Perturbing by mass term The theory discussed so far does not have a mass gap [ 11 ]. To make the calculations more feasible we perturb the theory such that it has a mass gap. 3 This enables us to 3 A similar perturbation has been considered in [4,12] for N = 4 SYM theory in four dimensions.
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
475
integra~e out most fields and reduce the path integral to a finite-dimensional one. The reduced two-dimensional theory has a U(1) ghost number symmetry coming directly from the non-anomalous U(1) symmetry of the underlying four-dimensional N = 4 SYM theory. Because of supersymmetry, the measure for non-zero-modes is invariant under the U(1) action. The ghost and the antighost zero-modes, on the other hand, obey the same equations of motion such that there are equal number of ghost and antighost zero-modes. This renders the measure to be invariant under the ghost symmetry of the action. Therefore the ghost symmetry is anomaly free. As the measure is invariant under this symmetry, the correlation function of any operator that has a ghost charge is zero. Therefore, this symmetry allows us to perturb the Lagrangian, by adding gauge invariant terms with non-zero ghost number, without changing the partition function. Thus, for example, since the mass term for the hypermultiplet (as we will see presently) consists of a term with negative ghost number and a term which is BRST exact, one expects that the partition function is invariant under perturb:ing the Lagrangian by a mass term for the hypermultiplet. One can even go further to argue that an additional mass term for the chiral multiplet q~ (which contains ~b and A) still leaves the partition function invariant [4,12]. In the following we are interested in the correlation functions of a set of BRST cohomology classes of the form i(e)=
1_ f t r ( 47reJ
i -~bF
1 ) e / 2 + ~ p A ~p + 3--f~2 tr~b.
Part of this factor with an extra BRST exact term provides the mass for the chiral multiplet ~/, [2]. The remaining part may have a non-vanishing expectation value in the mass-deformed theory. This, in particular, implies that, in contrast to the partition function, the correlation functions of I ( e ) (in the perturbed theory by mass for the hypermultiplet) may depend on the mass parameter. The next problem is to give a mass to X, ~7, A, ~ and (. This can be achieved by adding V' and W' to V, where
Vt =-e--22 / d/x tr {XA},
(16)
V"=
(17)
f
tr {(,C - ½(b}.
To give a mass term to the bosonic fields b, C and the fermionic o n e ,,~i we change the BRST tlransformation rules for Hi, ~ and ( to the following ones: ~
-
~3m~Oi= 2x/~i[ Y(i, ok] + -'--~meoY(J, x/gl 6n~ = - 2 [ b , ~b] + imC, 6m( = - 4 [ C , ~b] - 2imb.
(18)
A. Imaanpur/Nuclear Physics B 551 (1999) 467--489
476
Even though the metric is explicitly introduced via the above first BRST transformation rule, note that the extra term is still invariant under metric rescaling (eij ~" gl ). Thus, in the following we will consider the theory defined by the deformed action 1 .-. 11ll~t S = I ( e ) + itm (V + tV' + ~rnv )
= l(e) +
l/
{
dtz tr DiADi(b + DiCDiC + DibDib - ---~--oeJDibDjC VOl 2 i"
z + ½ ( f + 2i[b, C] + tA) 2 + 2ix/2tXTq - ½[m[2C 2 - lira2 2bz i~ +-~(¢~ -
m --E ~' "
. • , j X~tX~J + 2ifndp[b,C] - 2im,~[b,C]
4i r 2i i" -k--~l e '~lliDjx q- --~ll e Jf(iDj~lI -4- i2iDi ( - i~liDiTI - x / 2 ( t [~, al + 4 x / 2 X [ X , ~bl - 4x/2~ [X, C1 + 2 v ~ x [ sr, bl
+v~[n,
2v~ ~j
b] - ~ , e
~ [ ~ j , b] + v ~ [ ~
1
,
q-2vl2¢i[ f(i, c ] - ~ ( [ ~ 2 [ ~ ~ b]
1
~,~] - v~¢~[¢ ~, a] ,
( A] -4- " - ~ ( [ rI C]
[k,b] - 2[~b,C] [A,C] }.
( 19)
Notice that although the new BRST charge does not square to a gauge transformation (because of those new terms proportional to m), Lagrangian remains BRST invariant. This can be understood if we notice that 8.2, acting on fields generates (up to a gauge transformation) a U(1) action. Let 6r - iv~m 6 m2 a n d f l = _ b + i C , ~b = _ d T t + 6 ( , t h e n U(1) group acts as 6rfl = --ifl,
6T~b = --i0,
1 ~" v~1%xJ'
~r2i . . . .
t~rBi
1
v~l
= --eijF~
j.
Thus the fields fl, 0, Xz and /4z all have charge - 1 , with their complex conjugate having charge +1. All other fields have zero charge under this U(1) group. The fact that S is invariant under 6m then follows since V,V' and V" all have zero U(1) charge. Before continuing the analysis, it is important to understand the relation between the perturbed and unperturbed theories. Since the perturbing terms proportional to t and th are BRST exact, one may expect that correlation functions are going to be independent of these two parameters, but actually this is not true in general: adding 6mV' and 6mV" to the Lagrangian may result in some new set of fixed points flowing in from infinity and deforming the original moduli space of solutions [6] such that the path integral gets contributions from these new fixed points. The theory will be independent of t and ~ if in varying these parameters the Lagrangian remains non-degenerate and the perturbation does not introduce new components to the moduli space of fixed points.
A. lmaanpur/Nuclear Physics B 551 (1999) 467--489
477
We first discuss the situation for t = 0 with arbitrary m and r~. The perturbed Lagrangian (with t = 0) can also be derived upon reducing the N = 4 theory broken to N ---- 2 by the mass term. Had we started with N = 2 theory with one massive hyperrrmltiplet in the adjoint representation of the gauge group in four dimensions, we would have ended up with the same above perturbed Lagrangian after reduction. The fixed point equations are those of (14) together with (setting t~m~ = 6 , , ( = ~.,,7 = ~m¢i = O)
[~l,~b] = ½mfl,
[A,~b] = 0 , Dic~=O.
(20)
If ~b is not identically zero then, being covariantly constant, it never vanishes and, in particular, can be diagonalized globally such that the bundle E splits as a sum of line bundles'. [ 13]. Moreover, if fl 4= 0, the first equation in (20) fixes ~b (up to a sign) m(1
01)
~=-g- O-
(21)
with/3 as
Now Eqs. (14) become f + 21~12 = 0, b B = (og - i A g ) ~ = 0
(notice f = ~fo'3, 1~ where f here is the U ( 1 ) curvature). Note that ~b = lmo-3 corresponds to a point, in the classical moduli space of vacua, where a component of the hypermultiplet becomes massless. 4 The relevant fixed points are then determined by the above equations. Clearly one can then argue that the path integral over massless modes computes the Euler characteristic of the moduli space of U ( 1 ) fiat connections. However, to evaluate the contribution of this singular point to the path integral, one still has to do the integral over the massive modes. This is not an easy task, but there is a special case where this point (~b = 41-mo-3) does not make any contribution. This occurs upon restricting to the non-trivial SO(3) bundles. As discussed above, a non-zero ~b breaks the gauge group down to U ( I ) . In particular, SO(3) bundles split as E=: L @ O ~ L - j ,
(23)
where L is the U ( 1 ) line bundle and O is a trivial line bundle. In this case, w2(E), which measures the non-triviality of the bundle E, turns out to be the mod two reduction of Cl(L), the first Chern class of L [4]. Thus if f = 0, as is required by Eqs. (14), 4 AS Eq. (20) fixes ~b up to a sign, there are indeed two such singular points in the classical moduli space of vacua.
A. l m a a n p u r / N u c l e a r Physics B 551 (1999) 467-489
478
w2(E) has to be zero - implying that fiat non-trivial SO(3) bundles do not admit reducible connections. Therefore, in this case, the point ~b = lmo'3 does not conl~ibute to the path integral. Let us now discuss the case that t =~ 0. The fixed point equations (14) turn into the following equations (~ =- b + iC with ezz = i`/~lgze):
f + [B,B] + ta=0, /5/~ = 0,
D/~ = 0.
(24)
The vanishing argument now fails; f = / ? = 0 (and a = 0) are not the only solutions, as there are new fixed points with f =~ 0 contributing to the partition function. Since the connection is not bounded to be flat any more, a set of U(1) connections, in all classes of U(1) bundles, appear in the moduli space of solutions. Moreover, the point ~b = lmo'3 may contribute to the path integral even for non-trivial bundles. In the following we single out this point from our discussion and treat it independently.
3.1. Integrating A, rI and X Perturbing by V' now allows us to integrate out the fields a, r/ and X. Using the equations of motion for a and r / w e get t2A= D2~b -
t ( f + 2 i [ b , C ] ) +2im[b,C]
+v~[~,t] and
1 x-- 2v t
1
v/2[@i, I//i]
+
,
+ _---~_[~" (1 + 2 [ b , [ ¢ , b l ] + 2 [ C , [ ¢ , C 1 1
2,/2
( _Di~i_ F iv/~[b,~]
i
+ .~[c,
sr]
(25)
}.
Putting these back into the Lagrangian yields +
1
f
tr(DiCDiC+ DibDi b - - ; le iJ D, bDjC+ "-2i ~ r 11 eij~,¥il.)jlll
2v~ ij r , 1 +if(iDi# - ---~11e I,bil.,~j, o] -llm[2C2
'{
+t
--
4- 2V~i[~(
llml2b2 + -iFn ~(0- -
i, C] 4- V~(i[)( i, ~]
m_.__m__e..-i.--j 2x/-~,,X X + 2iFn4)[b, C]
(
( f + 2i[b,C]) D2d~+2im[b,C] + v~[Oi, Oi] + v~[~,~l
,
+~--~[d',(l + 2[b, [4~,bl] +2[C,[4~,C11 +~--~i ( - D i q / + i v ~ [ b , ~ ] +
i--~[C,#])
)
A. lmaanpur/Nuclear Physics B 551 (1999) 467-489
x (--~lEDkOt-i4"¢~[C,~] -4 kl -- 2 i v ~ [ sr, b])+-~-~ D2d~+2im[b,C] +~[(,(1)
479
}
+ v ~ ( [ 0 i , 0 i] + [ ~ , ~ ]
+ 2 [ b , [4,, bll + 2 [ C , [4,,C11
(
,
+v~ -DiOi + iv~([b,~] + ~ [ C , ( I )
Terms proportional to
1/t are indeed BRST
)
trivial, and can be written
i
i
"-~.~3m{(f +2i[b,C])(-Di~bi+iv~[b,~t]+-~[C,(])}. Terms proportional to
lit 2 are
also combining into
i ~-~t3m { (-DiOi-+ iv/'2[b,~] +--~i + v ' 2 [ 0 t , 0~1 + v ~ [ ¢ , ~ l
[C,(])
+ ~ -1- ~ [ ( , (1 +
(DZcb+2im[b,C]
2[b,
[4~,bll + 2 [ C , [4,,C11
)1
In the effective Lagrangian (26), the kinetic terms are non-degenerate for all values of t and since those terms proportional to t are still in a BRST exact form, the path integral does noel depend on t.
3.2. Large-t limit and the integration over b, C, 5, ~ As argued above, for non-trivial SO(3) bundles the point ~ -- lmo-3 does not contribute. For t 4= 0, because of tile supersymmetry, even after integrating out .t, rI and X the singularJtty still persists at t r ~ 2 = ~m 2. As we have chosen ~ to be a real scalar field, reality c,f the action requires that m to be a real parameter. However, to regulate the contribution of the points in the neighbourhood of tr~b2 = ira2, we allow m to have a small imaginary part. If there is going to be any singularity when ~ approaches m, it has to show up in the final result when we take the limit Im m ---* 0. This can be thought of as a kind of regularization by analytic continuation. Now let us consider the large limit of t. Since the kinetic terms remain non-degenerate we can actually take t ~ e~. Using the auxiliary field/1i, in this limit we are left with the action
, ~ i ( ffli - 2x/~OiC + ~2V~ e j i O Y b ~" .] ~- V_---~-o" 2i e , I ij~ Xil")Jl~t ~ r S = -~1 f dtx tr l - ~n x/gl
A. lmaanpur/NuclearPhysicsB 551 (1999)467--489
480
+i](iDi ~
irh
-- ½1ml2C 2 -
~
m
~i
~
'
½1ml2b2 + -~(~b - - ~ l l eijX X ~ + 2irhO[b, C]
2~ ,~.
}
V,~'---~I~: l[Ti[)(j,b] q'- 2X/2~i[]( i, C] + v~](i[2 i, ~b] + I(e).
£ can still be written as a sum of BRST exact term
i6m
{1{( e2
, + ~fn(~bC 1 l ~(i f-li - 2v/2DiC + ---~l eji o j o) - ½(b)
and l(e). The integral over C gives a factor of (det
l f ~
l-i-
m
2" +~l-~(Diffli -- 2[,~i, iml -
(~eImllz ) ) - 1 / 2 ~.~,
2i
}}
and leaves
i.
~bi ] )2 _ 2 rh [b, ~b] 2 + 2iv/'2 [b, ~ ] (Di[-I i m rn
~
-
2[~'i, ~bi] )
v/2 ejiftioJ b _ ½1m[2b2 + --~(~p ifn ~ - ---~l 2x/~ e q ~Pitxj, ~ b] }_ + I(e). X/~l Next we would like to integrate out b, ( and ~. It is easy to integrate out ( and ~ using their equations of motion. In the evaluation of determinants, which appear in doing the integral over b and finally over Xi, we always assume that ¢ is a constant field. This can be justified finally when the integral over the gauge fields constrains ¢ to be constant. The equation of motion for b yields
= [--~I2K
--~leJDjHi+
--~leJ[X,,~. ~bj] - 2~i[(Dif-Ii-2t2oi'i])¢']m
' (27)
where we have defined (A and B are Lie algebra indices)
KAB :~ (1-- -~tr ~p2)-I (~AB -- -~¢A~pB) . Replacing b in the action, we obtain
,f d~ tr
s=I(~) + ~
-~HH~+ v'~2i[~,.¢1
-- ~Eij()(i)(J-- ~[2Dlf(IDi,J) + [-~(Diffli- 2[)(i,~Ii])2 }
1 ijDjHi+ - -Wo2 + T ~1 /\|-~evgl voe × \(--l---~"eklDlflkx/gl
iJ[xi,- ¢1] - ~[m (Di[-1i -212i,~i]),4,].)a K AB
+ -~gl Ekl[~(k'~'ll]- 2~im[(DlI-ll-
2['~/'01])'~]
'
A. hnaanpur/Nuclear Physics B 551 (1999) 467--489
481
and a factor of 1
-1
where (ad~b)aB = --faBcqbc and 120 indicates the space of zero-forms. The following equations are easily derived:
ata { ( Dl)( I ) ( Di ~-Ii - 2 [,i'/, ¢i] ) } i( DiI--Ii - 2[,~i, ~/Ii ] )2 __ 2x/~i( Dif(i) [ Dtf(t, qb] + ~2---~_meiJDif(jDlf(l, x/gl
and
I" l
ijo , x , -+
~[Dif(i'~] }
ieiJ
,
,
x/gl2i i=
2 [(Di[_ii _ 2[2i,~bil),¢b]" m 82m{--~gle'TDjfci+~[Dif(i'dP]}=iv~mDlf(t
4iV~[[Dlf(lf'b]'qb]'m
Using these, the action can be written as
if
S = l ( e ) + -~
dl.t tr
l ~i (_~nHi_4_V~f(i[fgi, qb]__ ~
z i e2~rnl2 /
dl.t 8m{tr((Dlf(l)(Diff-I i-
2[/~i,0i]))
2~
/ eq ~ 2i )A + ~--~1DjXi -Jr---m[Di)(i' qb] K AB × ~--~l(DlHk -- 2[,~k, ~/1) + --[(DIHlm - 2['~/'~k/])' ~b]
"
(28)
Note that the integration over b, C, ( and ~ has not destroyed the manifest BRST exactnes:g of the action, in particular, the variation of S with respect to fit is still a BRST commutator.
3.3• Large-Tn limit and the final reduction We note the partition function is formally independent of fit (since the variation of the partition function with respect to fit gives an BRST exact expression) and is really independent of fit if in varying fit the Lagrangian remains non-degenerate with a good behaviour at infinity in field space. The mass term for Xq, the term Oi/~f, and the form of the cohomology classes that we have added by hand, guarantee that this is actually
A. lmaanpur/NuclearPhysicsB 551 (1999)467--489
482
the case. Having this freedom in the value of th, we simply set rh = c~. This leaves us with the action
1 { - s Hl-ia-a-Hi S=I(e)+-jfd~.
~(ia
1 (---'~l~lmlfij~aB--2ifABCqbCgij))(JB I,
and the partition function reads
2 e ~im12) t , - -(d---det - - - tS -(T2~:~) -m z ~b)=) ) 1/2]~, Z[e,m] = f ~)(ai,~li, qb, I-li,~(i) ( (det ~_L ~-Z 8-~-(ad e -s , / O0®E The explicit appearance of m on the l.h.s, reminds us that, although independent of rh, Z does depend on m. This is so because m was introduced through the BRST transformation laws. This is reminiscent of holomorphicity of N = 1 theories in four dimensions. Doing the integral over )~i gives a similar determinant, but this time over the space of 1-forms. Putting all pieces together one gets ([detm (1- ~ad [detm (1
~ad
~b)]
)
~b)] a~®e :P®E /
xexp [ ( -i1 ~
tr ( ~v~ i- - ~ b F + ~-1¢ A ¢ )
3}-'"2e n i trq~2) ] .
(29)
Notice that, as expected, ~ cancels out between the fermionic and bosonic determinants. The integral over ~i provides a symplectic measure for the gauge fields Ai [6]. Performing the path integral over ~b and Ai is now straightforward. In Appendix B, using the Faddeev-Popov gauge fixing technique, it has been shown that the integral over the gauge fields constrains ~b to be constant and hence the path integral calculation reduces to a finite-dimensional integral over constant ~b [ 15 ]. Explicitly, for S0(3) gauge group we have
× exp (_iv/~4)(2~ 1)
~E:+' ) .~~
(3o)
4. Discussion
We have reduced the calculation of the correlation functions in the mass-deformed theory to a finite-dimensional integral in (30). We can now perform the sum over n which results in a delta function restricting ~b to obey the following equation:
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
(ix/2~b~ = ( m -~- 22v"-2~b'~ 2 --'~)
exp \ ~ j
483
(31)
Therefore, Zig, m] = m 3(g-l) V ~th2-2g 4,, x exp
-t
4¢r
1-
~b~
32rr2j ,
7 (32)
where ,bs is a solution to Eq. (31). A similar result for the correlation functions of a topological field theory corresponding to the Hitchin equations has been derived in [ 14]. To this one still has to add the contribution of the point ~b = ¼too'3. However, note that from the discussion we had in Section 3, for non-trivial SO(3) bundles, this point contributes only if we perturb to t # 0.5 Thus if we are interested in the limit of t = 0, we can just ignore the contribution of this point. In conclusion we note two observations. Firstly, the result is m-dependent as might be expected from the discussion in Section 3. Note in particular that the expression (30) has the right behaviour when m ---+ oo; in this limit, the solutions of Eq. (31) are reduced to ~bs = 2V/'2"n'2/
and therefore Eq. (30) reduces to the expression for the corresponding correlation functions in the say pure N = 2 theory [6]. The extra factor, m 3(g-l), is left from the integration over the heavy fields in that limit. The power of m is in accord with the dimension of the moduli space of fiat connections which is dina A4 = 6g - 6. Any two zero-modes of Xi are absorbed by the corresponding mass term in the Lagrangian and gives a power of m. Secondly, we recall that, in general, S-duality relates the strong and weak couplings and swaps the gauge group with its dual group. However, as in the limit where S2 shrinks only instantons with k = 0 contribute to the path integral, unlike [4], the correlators in the effective theory do not depend on the modular parameter "r". Hence the action of S-duality is now simply to exchange the gauge group SU(2) with SO(3). Thus to derive that S-duality holds in this calculations, we must extend it for the SU(2) case; in particular, the contribution of the point ~b = lmo'3 must be taken into account. Amusingly, one can infer properties of this contribution by demanding S-duality. 5 As discussed in Ref. [6], the contribution of the original moduli space is invariant under perturbing to t~O.
A. Imaanpur/Nuclear Physics B 551 (1999) 467-489
484
Acknowledgements I am very grateful to Jim McCarthy for all his support and assistance throughout this work. I would also like to thank Nicholas Buchdahl for useful discussions.
Appendix A. The vanishing argument In this appendix we want to discuss the solutions to Eqs. (9):
k= Fab q-4i[Bai, Bib]= O, s = DaBai = O. Let us first analyze the second equation. After squaring we get tr ( D a Bai) 2 = _ f tr B ai (DaDbBbi) =
-
tBai~ -- Bai[Da, Db]Bbi) tr ~, Lib D a °Dbi -~"
= f tr ((OaBbi)(Ob nai) -4- eabnainb i - iB ai [Fab, n b ] )
= / tr ((DaBbi + DbBai - DbBai) (Db Bai) + 1 R B aiBai - iB ai [ Fab, Bbi ] )
=f = ((DbBai)2
-
1 2 -[- ½R Ba'Bai " -iBai[Fab, n b ] ) , ~(D[aBbli) (A.I)
where we used the fact that in two dimensions, the Ricci tensor takes a simple form, I Rab = ~gabR,
and
[ Da, Db ] B ci = RCdabBdi "4- i[ Fab, Bci], [ Da, Db ] B ai = Rab Bai 4- i[ Fab, Bai].
(A.2)
Since B~u is self-dual, we have Bw~ = B~z = O. Hence
(D[aBb]i) (D[aB b]i) = (Og, Bwz ) (Oa'B wz ) q- (OwBa,~) (OWB ~ )
= (DWBwz)(DwB wz ) + (D~B~e ~(D:vB ~ ) = (DaBai) (Db Bbi) .
(A.3)
Putting this back into (A.1) we get
3 / t r ( D a B a i ) 2 = f tr ((DbBai)2-} - I R BaiBai-iBa'[Fab, Bbi] ) .
(A.4)
A. hnaanpur/Nuclear Physics B 551 (1999) 467-489
485
Upon adding the squares of the sections k and s, we have
f tr(¼kZ+3s2)=f tr {¼(Fab)2--4[Bai,
Bib]2
+ 2iFao [ B ai, B i b] .q_ 2( DbBai ) 2 "-1- R BaiBai - 2iBai[ Fab, Bbi] }
=
f
tr {-~(Fab) -- 4[B,~i, Bib] 2 + 2(DbB~i) 2 + R BaiB,i}.
1
2
The right-hand side vanishes if and only if k = s = O. However, for sphere (R > O) all terms on the r.h.s, are positive definite so a solution to k = s = 0 has necessarily B ai = O. This leaves us with the equation
Fa,~ = O. This equation implies that the connection is locally a pure gauge Aa = u - l d a u for some S U ( 2 ) matrix u. However, as the transition functions for SU(2) bundles on the sphere are trivial, the connection can be written globally as a pure gauge and be gauged away. Moreover, one can argue that this can be done continuously all over 27. Thus we can set A~ = 0 ,everywhere. More rigorously, if {U~} is an open covering of 2 by contractible sets and {V/} is an open covering of S 2 by such sets, the sets U~ x V/give an open cover of ,~ × S 2 by contractible sets. On the intersection of two patches, the connection A now satisfies
A,~, = g~.~jABjgait~ j q.- g~.lBjdgail~j,
or
dgaiBj -'k ABjgaiflj - gaiBjAai = O.
Since the S 2 component of the curvature is zero we have that (A~),,i = u~ .l dau,~i. Putting this in the above equation yields
da( uaigaiBjuB; ) = O. Therefore g,,~i#j -- uotigaiBjUB; does not depend on the coordinates of S 2. This implies that the gaiflj'S are a set of locally constant transition functions equivalent to g,~iaj and for a fixed point on 27 define a map from S l to SU(2). This map is trivial so ~i~j belongs to the conjugacy class of the identity
gaic~j : gaig~j I = uaigcdajUa'-j I"
or
(g~iluai)gaiaj(g-~jlUotj) -1 = l.
Now consider (g~.lUcti)gctiflj(gfl;Uflj)--l.
This is a constant matrix in the S 2 direc-
tion. Si~:ce g,~i/~j = gaifligfliflj it is equal to (gailuai)gctifli(gflilufli) -l , and since g,~i/3j = g~i,~jgc~j~q it is equal to ( ~ l u , ~ j ) g , ~ j # j ( ~ l u # j ) - l . Thus it is in fact independent of the index i and therefore defines a matrix ~,~ depending only on x E Ua/3 and satisfying the cocycle condition. 6 Since the transition functions are independent of i, therefore (A.~)ai do not depend on the index i and Aa can be gauged away. 6 The proof of this part was provided by Nicholas Buchdahl.
486
A. hnaanpur/Nuclear Physics B 551 (1999) 467-489
It is now easy to see that the flatness condition, F~b = 0, necessarily requires the instanton number to be zero. The curvature locally takes the form F=dA+AAA.
Therefore locally we can write tr(FAF)=dtr(AAdA+2AAAAA), but since Aa = O, the instanton number reads k = 8~I 2f
trFAF=
I f 8~r2
XxS 2
d¢ t r ( A ~ . A d c A ~ ) ,
~xS 2
where the subindex C indicates differentiating with respect to the coordinates on S2. Note that the integrand is still a local one. However, we showed that the transition functions are independent of the local coordinates on S2. Therefore, for a fixed point on X, A~ is globally defined on S2. This means that the integral over S2 is a total divergence and gives zero for the instanton number. In summary, we have learned that if the bundle E admits a flat connection in S2 direction then it has to be trivial (for those bundles that are classified only by instanton number) and k, the instanton number, is zero.
Appendix B. Faddeev-Popov gauge fixing In this appendix we want to show how Eq. (30) is obtained starting from (29). To evaluate the path integral over gauge fields and ~b, following Ref. [ 15], we choose the so-called unitary gauge in which one rotates the Lie algebra valued field ~ba to the Caftan subalgebra by conjugation, i.e. we choose ~b+ = 0, where q~ = ~3q'3 -}- ~b+'/'+ q- ~b_7"_.
This gauge can always be achieved at least locally, but there might be some topological obstruction to impose it globally [15]. Implementing this gauge in the path integral requires to introduce the Faddeev-Popov ghosts c and antighosts g together with a bosonic auxiliary field b. These fields transform under a BRST operator 8 like ~q~-t- = q-ic+q~3,
~
= bi,
~ b 3 : 0,
•c4- = 0,
6b~ = O.
(B.1)
The Faddeev-Popov prescription consists of adding a BRST-trivial term iS(P_qb+ + g+fb_ ) = ib_
to the action in (29). It is now clear that the integration over b will impose the gauge condition; ~ba: = 0. We have tribE = q~3F3 = q~3(dA3 + (A A A)3) = ~b3(dA3 q- i v ~ A l A A2),
A. lmaanpur/Nuclear Physics B 551 (1999) 467-489
487
therefore, defining ~b _= ~b3, A _= A3 and F _= F3, the action in (29) turns into
S=~
ck d A - q b A 1 A A 2 +
8) / c~2
+
dtz(g-fbc+-P+Obc-).
Integration over Faddeev-Popov ghosts gives [det ~bz] ao(~), while over Al and A2 results in .2 a -I/2 det@ J n~(~O " Using the Hodge decomposition theorem we can express the product of these two determinants as [clet ~b2] u0(~0 2 1/2 • [det ~b ] H~(Xg) When E is a non-trivial SO(3) bundle we write the curvature of the reduced U ( 1 ) bundle as F = 27r(2n + 1)to + dA, where w is the volume form ( f z co = 1) and 2n+l
=~-~
is the first Chern class which characterizes the U ( 1 ) bundle. To gauge fix the residual U ( 1 ) symmetry A --* A + dc~, we again appeal to the Faddeev-Popov prescription. We demand that a selected slice be normal to the gauge orbit,
(d,~, A) = O, which implies that d t A = O. Imposing this gauge, the action is l 4~r2/(iv/'27r(2n+l)~b~o+8~b2+--~2(#bdA+bd'A+'d*dc) )•
The kinetic term for A vanishes for A a harmonic 1-form, i.e. when dA = 0 and d t A = O. Hence there is still a residual symmetry under
A--*A+y, b --* b + constant, c --~ c + constant,
A. hnaanpur/NuclearPhysicsB 551 (1999)467-489
488
where 3' is a harmonic 1-form. Integration over the zero-modes of b and c and over the harmonic 1-forms gives an unspecified constant factor that can be simply absorbed in the normalization. Therefore we need only be concerned about the non-zero-modes. Dropping the harmonic part of A, it can be written globally and uniquely as
A = da + *dfl, for some zero-forms a and/3. The action then looks like 4~"21
iv'27r(2n+l)~bw+
d~2+--~(obd,d/3+bd,da+ed,dc)
£ and the measure is
DA = DotD/3 det [dd t ] ao.
(B.2)
Note that ,2 = ( _ l ) p when acting on a p-form and d t = - • d*. The integral over b and ce results in a determinant, det [ddt]~0l, which cancels the Jacobian in (B.2). Also the integral over/3 gives a delta function
6(ddt~b) -- det [ddt]~ 18(~b).
(B.3)
Notice that since we are integrating over non-zero-modes the delta function on the righthand side is a delta function on non-constant ~b's. The determinant in Eq. (B.3) gets cancelled against the determinant coming from the ghosts. At the end we are left with a finite-dimensional integral over constant ~b fields, ([detm(1-~ad~b)])
Z[e'm]=n~czfdq~
[detm(1
x exp ( _ i v , ~ b ( 2 4 ; 1)
[det~b2]no
~_~)]a'®e
2 l/2 n°®el [det~b ]m
e~b2 "~ 3-~2j.
(K4)
Using the Riemann-Roch formula dimO I ® L - d i m O ° ® L= g - 1 - c l ( L ) and the definition of Euler characteristic of a Riemann surface X(~g) = 2 b ° - b l = 2 - 2 g , and the fact that q~ is now a constant, we can write the partition function as
Z[.,g#l] :#'n3(g-l)Z
/'d4>
.~z ~ ×exp which is Eq. (30).
( _ , v , ~ b ( 2 4 ~ 1)
(l_ e~b2 "~ 3--7-42) '
"-1 {m-
t m + 2v/2"4b) (B.5)
A. Imaanpur/Nuclear Physics B 551 (1999) 467--489
489
Refene~nces [1] [2] [3] [4] [5] [6] [71 [8] [9] [10] [11] [12] [13] [ 14] [15]
E. Witten, Comm. Math. Phys. 117 (1988) 353. E. Witten, J. Math. Phys. 35 (1994) 5101, hep-th/9403195. C. Montonen and D. Olive, Phys. Lett. B 72 (1977) 117. C. Vafa and E. Witten, Nucl. Phys. B 431 (1994) 3, hep-th/9408074. N. Seiberg and E. Witten, Nucl.Phys. B 431 (1994) 484, hep-th/9408099. E. Witten, J. Geom. Phys. 9 (1992) 303, hep-th/9204083. M. Bershadsky, A. Johansen, V. Sadov, and C. Vafa, Nucl. Phys. B 448 (1995) 166, hep-th/9501096. G. Moore and E. Witten, hep-th/9709193. A. Losev, N. Nekrasov and S. Shatashvili, hep-th/9711108. J.M.F. Labastida and C. Lozano, Nucl. Phys. B 502 (1997) 741, hep-th/9702106. E. Witten, Nucl. Phys. B 460 (1996) 335, hep-th/9510135. J.M.E Labastida and C. Lozano, hep-th/9711132. D. Freed and K. Uhlenbeck, Instantons and Four-Manifolds (Springer, Berlin, 1984). G. Moore, N. Nekrasov and S. Shatashvili, hep-th/9712241. M. Blau and G. Thompson, hep-th/9310144.