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The prepotential of N=2 SU(2)×SU(2) supersymmetric Yang–Mills theory with bifundamental matter
21 October 1999
Physics Letters B 465 Ž1999. 155–162
The prepotential of N s 2 SUž 2/ = SUž 2/ supersymmetric Yang–Mills theory with bifundamental matter Ulrike Feichtinger
1
Theory DiÕision, CERN, GeneÕa, Switzerland Received 25 August 1999; accepted 2 September 1999 Editor: L. Alvarez-Gaume´
Abstract We study the non-perturbative, instanton-corrected effective action of the N s 2 SUŽ2. = SUŽ2. supersymmetric Yang–Mills theory with a massless hypermultiplet in the bifundamental representation. Starting from the appropriate hyperelliptic curve, we determine the periods and the exact holomorphic prepotential in a certain weak coupling expansion. We discuss the dependence of the solution on the parameter q s L22rL12 and several other interesting properties. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Many field theoretical results have been obtained by considering brane configurations in string theory and M-theory. In particular M-theory five-branes provide a very fruitful approach to N s 2 w1,2x and N s 1 w3–5x supersymmetric field theories. In this note we study the case of product gauge group SUŽ2. = SUŽ2. with a hypermultiplet in the bifundamental representation. Seiberg–Witten curves of theories with product gauge groups have been obtained from the M-theory w2x and the geometric engineering approach w6x. The precise relation between the moduli of the curve and the moduli of the field theory has been described in w7x. Yet another account of supersymmetric field theories with product gauge groups of the form G s Ł i SUŽ Ni . has been obtained in the framework of 1
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Calogero–Moser systems, using certain limits of SUŽ N . supersymmetric Yang–Mills theory with matter in the adjoint representation w8x. In this approach the dynamical scales of the different gauge group factors are all equal to the scale of the underlying SUŽ N . group. One-instanton predictions for the prepotential of N s 2 supersymmetric field theories with gauge group SUŽ N1 . = SUŽ N2 . and massless matter in the bifundamental representation have been made by using a perturbation expansion of the non-hyperelliptic curve around its hyperelliptic approximation w9x. In the present paper we study the instanton expansion for the case of a massless hypermultiplet in the bifundamental representation of SUŽ2. = SUŽ2.. Specifically we will compute the periods as solutions of the Picard–Fuchs equations and obtain in this way the holomorphic prepotential F that governs the low-energy effective action of the theory. The solution reproduces the existing results in the appropriate
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 4 6 - 1
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limits and has interesting properties in the general region of moduli space. 2. The setup By considering the appropriate brane configuration in M-theory w2x one finds that the defining polynomial for the Seiberg–Witten curve of N s 2 supersymmetric Yang–Mills theory with a gauge group of the form Ł in SUŽ k i . is given by P Ž x ,t . st nq 1 q p k 1Ž x . t n q p k 2Ž x . t ny 1 q . . . qpk nŽ x . t q c s 0 .
Ž 2.1 .
The p k iŽ x . are polynomials of order k i in x, and c is a constant that depends on the dynamical scales L i of the different gauge group factors. In this expression the variables x and t correspond to the 6 10 combinations x 4 q ix 5 and eyŽ x qi x .r R in the notation of w2x. For the case of SUŽ2. = SUŽ2. with a massless hypermultiplet in the bifundamental representation, the explicit expressions for the polynomials p k iŽ x . in Ž2.1. have been derived by using symmetries and classical limits w7x. The curve for the massive case is given by P Ž x ,t . s L12 t 3 q t 2 x 2 y u q
Fig. 1. The brane configuration that gives rise to a four-dimensional field theory with gauge group SUŽ2.= SUŽ2. and a hypermultiplet in the bifundamental representation. Vertical lines represent five-branes, horizontal lines are four-branes.
SUŽ2. can be obtained by moving the two four-branes of one gauge group factor to x s `, so that the resulting configuration consists only of two fourbranes stretched between two five-branes. To obtain the curve of this theory we have to take the limit Õ ™ `, m ™ ` while keeping L12 Ž m 2 y Õ . s L40 fixed Žsee Fig. 1.. In the case of gauge group SUŽ2. = SUŽ2. the polynomials p 1Ž x . and p 2 Ž x . are quadratic in x; we can therefore rewrite Ž2.2. in hyperelliptic form by iy redefining 2 x ™ :
'2 t Ž1q t . 2
y s tŽ 1 q t . Ž2
L12
t 3 q t 2 Ž L12 y 2 u .
qt Ž L22 y 2 Õ . q 2 L22 . .
Ž 2.3 .
2
The discriminant D of this curve consists of two factors, D s D1 D2 , with
L22
D1 s 4 Ž u y Õ q 12 L12 y 12 L22 . ,
L12
2
2
q t Ž x q m. y Õ q
2
q L22s0 .
D2 s 8
L16 L22 q 359
L14 L24 q 8
L12 L62 y 48
Ž 2.4 . L14 L22
Ž 2.2 .
q 148 L12 L42 u q 96 L12 L22 u 2 y 4 L42 u 2
Here u and Õ are the moduli of the two gauge group factors, L i are the corresponding dynamical scales, and m is the bare mass of the hypermultiplet. Note that the dimensionless parameter q s L 22rL12 cannot be eliminated from the curve by rescaling of the moduli, unless q s 1. Exchanging the two gauge group factors, i.e. u l Õ, L1 l L 2 , t l 1t , and x l yŽ x q m., leaves the curve Ž2.2. invariant. Pulling the rightmost five-brane to x 6 s ` gives the brane configuration of a theory with gauge group SUŽ2. and two fundamental flavours with masses m1 and m 2 . To make this transformation manifest in the curve one has to perform the limit L2 ™ 0, Õ ™ 14 Ž m1 y m 2 . 2 and m ™ 21 Ž m1 q m 2 .. The theory with pure gauge group
y 64 L22 u 3 q 148 L14 L22 Õ y 48 L12 L24 Õ
u
y 304 L12 L22 u Õ q 16 L22 u 2 Õ y 4 L14 Õ 2 q 96 L12 L22 Õ 2 q 16 L12 u Õ 2 y 16 u 2 Õ 2 y 64 L12 Õ 3 .
Ž 2.5 .
As expected, D1 and D2 are symmetric under the exchange of the two gauge group factors. In general the vanishing of discriminant factors describes points in moduli space where extra massless states appear. Following the analysis of w7x, the vanishing of D1
2
In the following we set the bare mass m of the hypermultiplet to zero.
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determines the subspace of the moduli space where the Coulomb branch meets the Higgs branch and the gauge group SUŽ2. = SUŽ2. is broken to the diagonal SUŽ2.. Therefore the extra massless states that occur at this point are two components of the bifundamental hypermultiplet. The vanishing of the second discriminant factor describes the singular locus where other massless states appear, in particular monopoles. The classical intersection point of the Coulomb branch and the Higgs branch at u s Õ gets shifted by non-perturbative effects if the two scales do not have equal values, i.e. if q / 1. This dependence is inferred by the terms L2i r2 in the curve Ž2.2. and could formally be avoided by choosing the defining polynomial to be
The second-order derivatives of F with respect to a i imply the integrability condition
P Ž x ,t . s L12 t 3 q t 2 Ž x 2 y u q L12 .
lA
qt Ž x 2 y Õ q L22 . q L22 s 0.
Ž 2.6 .
Then the two scales no longer explicitly appear in D1. In order to reduce Ž2.6. to the curve for SUŽ2. with two flavours we then have to perform a shift in the modulus u in addition to the above-stated transformation: u ™ u s u q 7 L12r8. This means that the origins of the two moduli spaces would simply be shifted by 7 L12r8, though classically the moduli still coincide. The polynomial Ž2.3. describes a Riemann surface S of genus 2. If we choose a symplectic homology basis of S , i.e. a i and bi Ž i s 1,2. with intersection pairing a i l b j s d i j , a i l a j s bi l b j s 0, we can define the period integrals in terms of a properly chosen meromorphic one-form l: ai s
Ha l ,
aD i s
i
Hb l .
Ž 2.7 .
i
The periods are functions of the moduli u and Õ, a i Ž u,Õ . is identified with the scalar component of the N s 1 chiral multiplet in the i-th gauge group factor, a D i Ž u,Õ . is its dual. Once the periods a i and a D i are known, the exact quantum-corrected prepotential F Ž a k . of the theory can be computed by integrating the relations aD i Ž ak . s
E F Ž ak . E ai
.
E aD i Ž ak . E aj
s
E aD j Ž ak . E ai
.
Ž 2.9 .
The meromorphic one-form l of the curve Ž2.2. is given by w1x
lA
x dt t
,
Ž 2.10 .
or, after the redefinition x™
iy
'2 t Ž 1 q t .
,
by y dt 2
t Ž1qt .
,
Ž 2.11 .
where the proportionality factor is determined by the requirements
El EÕ
dt s y
El and
Eu
t dt s y
.
Ž 2.12 .
It is rather tedious to perform the integrals Ž2.7. explicitly. However, the periods as functions of the moduli u and Õ satisfy a system of partial differential equations, the so-called Picard–Fuchs equations. This method to obtain the periods is relatively straightforward and will be the subject of the next sections.
3. The Picard–Fuchs equations Starting with the hyperelliptic form Ž2.3. of the curve 3 for SUŽ2. = SUŽ2., we will derive the system of partial differential equations for the periods Ha i l and Hb i l. On a Riemann surface S of genus 2 there are two holomorphic differentials and two meromorphic differentials with no residues Žabelian differentials of first and second kind. w10x. Therefore we can choose dxry, x dxry, x 2 dxry, x 3 dxry4 as basis of mero-
Ž 2.8 . 3
For convenience we replace t by x in the following.
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morphic forms on S . When considering derivatives of the meromorphic one-form l with respect to the moduli, only four of them will be linearly independent up to exact forms. The linear relations between these derivatives define the Picard–Fuchs equations. Derivatives of l with respect to u and Õ involve terms of the form f Ž x . dxry n with some polynomials f Ž x .. In order to express f Ž x . dxry n in terms of abelian differentials of first and second kind, we need a method to reduce high powers of x in the numerator and high powers of y in the denominator. For the latter we use the fact that the discriminant of a polynomial pŽ x . of order n can always be written in the form w11x
D s a Ž x . p Ž x . q b Ž x . pX Ž x . ,
Ž 3.1 .
where aŽ x ., resp. bŽ x ., is a polynomial of order Ž n y 2., resp. Ž n y 1., in x. With this formula it can easily be shown that the following relation holds up to exact forms:
With these formulae we can re-express the meromorphic one-form Ž2.11. in terms of abelian differentials of first and second kind
l s Ž 2 Õ y L22 .
y
q Ž 4 u q 2 Õ y 2 L12 y L 22 .
q 4 Ž u y 2 L12 .
x 2 dx y
y 6 L12
x 3 dx y
.
x dx y
Ž 3.4 .
When we consider derivatives of l with respect to u and Õ up to second order, we find that these derivatives satisfy two linear relations as only four out of six are linearly independent. These two relations give rise to the following Picard–Fuchs operators: L 1 s Ž 5 L12 q 2 u . Eu2 q 2 Ž L12 y L22 y 2 u q 2 Õ . Eu EÕ y Ž 5 L22 q 2 Õ . EÕ2 q Eu y EÕ ,
Ž 3.5 .
L 2 s Ž L14 q 14 L12 L22 y 4 u 2 y 4 L12 Õ . Eu2
fŽ x.
y Ž 14 L12 L22 q L24 y 4 L22 u y 4 Õ 2 . EÕ2
yn 2
s
dx
y 2 Ž 15 L12 L22 y 2 L22 u y 2 L12 Õ
d
1 a Ž x . f Ž x . q n y 2 dx Ž b Ž x . f Ž x . . . D y ny 2 Ž 3.2 .
Following w11x we derive a formula to reduce high powers of x that is valid up to addition of total derivatives: xn sy y
y
Ž 2 n y 7 . L22 x ny 4 Ž 2 n y 3 . L12 y Ž n y 3. Ž 2 n y 3.
Ž 2 n y 5. y Ž 2 n y 3. y
Ž n y 2. Ž 2 n y 3.
y4 u Õ . Eu EÕ q 1.
Ž 3.6 .
Note that the first Žsecond. operator is antisymmetric Žsymmetric. under the exchange of the two gauge group factors.
4. The periods and the prepotential
Ž L12 q L22 y 2 u y 2 Õ .
x ny 2
2 L12
y
We are interested in the prepotential of SUŽ2. = SUŽ2. supersymmetric Yang–Mills theory in the weak coupling regime, which is parametrized by large values of the moduli u and Õ. As usual, if we have two variables that simultaneously become large, we have to specify an appropriate coordinate patch in which the limit is well-defined. By inspection of the first discriminant factor Ž2.4., we choose the variables to be
Ž 3.3 .
z1 s
Ž 3 L22 y 2 Õ .
x ny 3
L12
y
Ž 3 L12 y 2 u .
x ny 1
L12
y
.
L12 u
and
z2 s
u y Õ q 12 Ž L12 y L22 .
L12
. Ž 4.1 .
U. Feichtingerr Physics Letters B 465 (1999) 155–162
This choice is not symmetric under the exchange of the two gauge group factors, but of course there exists the analogous patch in the moduli space where the role ˆ of the two moduli is interchanged. Using Frobenius’ method, we find two series solutions and two logarithmic solutions with indices Žy 12 ,0. and Ž 12 ,1. in the coordinate patch of the moduli space parametrized by Ž4.1.: ` r2 w 1 Ž z 1 , z 2 . s zy1 1
Ý
a i j z 1i z 2j ,
Ž 4.2 .
bi j z 1i z 2j ,
Ž 4.3 .
i , js0 `
w 2 Ž z 1 , z 2 . s z 11r2 z 2
Ý
159
Therefore the one-loop contributions of the pure gauge couplings are given by
t 11 ; y4ln Ž a1 . q ln Ž a12 y a22 . , t 22 ; y4ln Ž a2 . q ln Ž a12 y a22 . .
Ž 4.7 .
In order to examine the asymptotic behaviour of the periods, we integrate the gauge couplings t i i in Ž4.7. to obtain a D i Ž a k . s Ht i i Ž a k . da i , introduce a new variable ´ s a1 y a 2 , and expand around the point Ž a1 , ´ . s Ž`,0.. This procedure gives the leading terms of the periods a D i : a D1 ; y2 a1 ln Ž a1 . y ´ ln Ž a1 . q ´ ln Ž ´ . q . . . ,
i , js0
a D 2 ; y2 a1 ln Ž a1 . q 3 ´ ln Ž a1 . y ´ ln Ž ´ . q . . . . Ž 4.8 .
w 3 Ž z 1 , z 2 . s w1 Ž z 1 , z 2 . ln Ž z 1 . ` r2 q zy1 1
Ý
c i j z 1i z 2j ,
Ž 4.4 .
i , js0
w4 Ž z 1 , z 2 . s w 2 Ž z 1 , z 2 . ln Ž z 13 z 22 . ` r2 q zy1 1
Ý
d i j z 1i z 2j .
Ž 4.5 .
y2 A a1 ln Ž a1 . q Ž A y 8 B . ´ ln Ž a1 .
i , js0
The first few coefficients of these series expansions are listed in Appendix A. In order to get the periods we have to consider linear combinations of the solutions that match the infrared asymptotic behaviour. To analyse these leading terms one can either evaluate the lowest order of the integrals Ž2.7. explicitly or one uses semi-classical relations of the gauge couplings and the integrability condition Ž2.9.. The gauge couplings t i j of the theory are related to the prepotential F by
ti j s
E 2F Ž a k . E ai E a j
s
E aD i E aj
.
To compare the solutions Ž4.4. and Ž4.5. with these expansions, we have to replace the moduli u and Õ by the corresponding classical expressions u s a12 and Õ s a 22 . We apply the same procedure as above to a linear combination of the logarithmic solutions A w 3 q B w4 and find, in leading order:
Ž 4.6 .
The one-loop terms of the pure couplings t i i consist of the logarithmic contributions of the massless spectrum. In each gauge group factor there is a gauge field in the adjoint representation; moreover there is a hypermultiplet in the bifundamental representation.
q 4 B ´ ln Ž ´ . q . . . .
Ž 4.9 .
Hence the periods a D i are reproduced by the following linear combinations of the solutions a D1
L1 aD 2
L1
s A1w 1 q B1w 2 q w 3 q 14 w4 , s A 2 w 1 q B2 w 2 q w 3 y 14 w4 ,
Ž 4.10 .
where the coefficients A1 , A 2 , B1 , B2 are still undetermined. We also find that the periods a i are given by a1
L1
s w 1 q 14 w 2 ,
a2
L1
s w 1 y 14 w 2 .
Ž 4.11 .
It can easily be checked that in the limits of SUŽ2. Nf s 0, resp. Nf s 2, stated in Section 2, a1 reduces to the corresponding periods known in the literature. By imposing the integrability condition Ž2.9., one finds that A 2 s A1 and B2 s yB1.
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The prepotential can be computed by integrating Eq. Ž2.8.. To this end we have to know the magnetic periods a D i as functions of the electric periods a k . By inverting the series expansions of a k in Ž4.11. we get the variables z i and in consequence the magnetic periods as functions of a k : a D1 Ž a1 ,a2 .
with the following expressions for the first few terms: Fclass s Ž a12 q a22 . Ž 12 q 14 A1 q B1 q 12 ln Ž 2 . . q a1 a 2 Ž 12 A1 y 2 B1 y ln Ž 2 . . ,
Ž 4.15 .
F1- loop s y2 a12 ln Ž a1 . y 2 a 22 ln Ž a2 .
1 2
s a1 Ž A1 q 2 B1 q ln Ž 2 . . 2
q a 2 Ž 21 A1 y 2 B1 y ln Ž 2 . . y 4 a1 ln Ž a1 .
q 12 Ž a1 q a2 . ln Ž a1 q a 2 .
q Ž a1 q a 2 . ln Ž a1 q a2 .
q 12 Ž a1 y a2 . ln Ž a1 y a 2 . ,
2
q Ž a1 y a 2 . ln Ž a1 y a2 .
L12
1 q
ž
2 a1
a 22 a12
Ž
32 2 a71 1
q8 q
yq
a13
F1- inst s
/
L12 4
Ž a12 y a22 .
ž
1 a12
a14 y 12
a12 a 22 q 15
a 24
. sy
2
a1 a62
Ž5
a12 y 3
a22
L14 4
q...
.
q
Ž 4.12 . y16 q
a D 2 Ž a1 ,a2 .
1 y
a12
q a 2 Ž 21 A1 q 2 B1 q ln Ž 2 . . y 4 a2 ln Ž a2 . q Ž a1 q a 2 . ln Ž a1 q a2 . y Ž a1 y a 2 . ln Ž a1 y a2 . y
2 a2 y
1 2
ž
L14
a2
32
a16
q q
2
1 a72
/
a22
,
Ž 4.17 .
a22
qq2
a 22 a12
Ž3
Ž 15
yq
a12 a22
a12 y 5
1
. q 8 q a3
a12 a22 q a24
.
q...
Ž 4.13 . Notice that exchanging a1 and a 2 transforms a D1 into a D 2 and vice versa. Integrating the expression Ž4.12., resp. Ž4.13., with respect to a1 , resp. a2 , yields the prepotential `
F s Fclass q F1 - loop q
Ý Fn - inst , ns1
384
qq 3
2
a14 y 12
L16
q3 q
/
a 22
F3- inst s y
a12
ž
a 24
Ž 4.14 .
ž
a14 a14 a82
ž
5 a12 5 a12
a18
/
a22 9
y
a 22
5 y
a12
1 y
a14
1
a 42
Ž a12 y a22 . 1
a22
Ž a12 y a22 .
128
1 a12
L14
s a1 Ž 12 A1 y 2 B1 y ln Ž 2 . .
1 L12
1
yq
F2- inst
L14 1 1
y
yq
a12 a22
Ž 4.16 .
/
ž
a 22
a12
1
a12
y
, 5
y
q3 q2
.
ž
/
9
5
a22 1 a 42
a22
/
Ž 4.18 .
/ ž
1 a12
5 y
a 22
/
Ž 4.19 .
As expected, all contributions to the prepotential are totally symmetric under the exchange of the two gauge group factors. The constants A1 and B1 are determined by the classical couplings. The one-loop contribution F1- loop is in agreement with the corresponding term of the prepotential reported in w8x and w9x. Up to an overall scaling factor of the two scales, F1- inst coincides with the one-instanton term in the prepotential of w9x. If the bare mass m of the hypermultiplet in the prepotential obtained in w8x is set to zero, the one-instanton term coincides with F1- inst
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Table 1 The non-vanishing coefficients of the series solutions i
j
ai j
bi j
0 1
0 0
1
1 y 34 Ž 1 q 23 q .
2
0
y 321 Ž 1 q 8 q .
3
0
1 128
4 1 2
0 1 1
5 y 2048 Ž 1 q 48 q q 96 q 2 . 1 y4 y 161 Ž 1 y 2 q .
1 2
21 128
3 2
1 4
y
Ž 1 q 43 q q 49 q 2 . 135 y 128 Ž 1 q 2 q q 34 q 2 q 278 q 3 . 2835 8 8 3 32 3 16 4 2048 Ž 1 q 3 q q 3 q q 27 q q 81 q . 27 32
Ž 1 q 24 q .
Ž1y
12 7
4 7
qy q
2
1 4
y 163 Ž 1 q 4 q . 45 y 128 Ž 1 y 163 q y 4 q 2 .
.
1 16
1 8
for q s 1, but the terms proportional to q in F2- inst are not present. This is a priori no contradiction, since sending the bare mass to zero is a singular limit. The instanton terms cannot be written as simple sums of contributions stemming from the two subgroups, but show a non-trivial mixing of a1 and a 2 , as it is expected in the presence of a hypermultiplet in the bifundamental representation. In particular the n-th instanton term is not just a sum of contributions proportional to L2i n , but all possible combinations L12 i L22 j with i q j s n appear in Fn - inst . Terms proportional to odd powers of the scales are forbidden as they would violate the anomaly-free discrete symmetry of the theory w12x. It is quite striking that nearly all terms in the instanton expansion of the prepotential are propor-
tional to Ž a12 y a22 . and therefore vanish for a1 s " a 2 . In either of the two cases two components of the hypermultiplet become massless and one expects to recover the point in moduli space where the Coulomb branch meets the Higgs branch and the gauge group is broken to the diagonal SUŽ2.. Continuity of the prepotential at the intersection point ensures that F collapses to the prepotential of pure SUŽ2.. Indeed, if one sets a1 s " a 2 s a in the prepotential Ž4.14., the non-vanishing terms reproduce the prepotential of SUŽ2. theory with the dynamical scale L2 s 4 L1 L 2 to all calculated orders. In w7x a different argumentation, namely considerations of the brane configuration and the curve, led to the insight that the same point in moduli space is determined by the equation u q 12 L12 s Õ q 12 L22 . At first sight it is quite astounding that this condition is
Table 2 The non-vanishing coefficients of the logarithmic solutions i
j
ci j
di j
0 1
0 0
1 y 14
1
2 3 4 1 2 3 2
0
y
0
5 384
0 1 1 1 2
y 1 40 37 160
y y
9 4
Ž1q 31 6144
Ž 1 y 89 q .
25 y 32 Ž 1 y 258 q y 258 q 2 .
1 32 48 5
Ž1q
768 31
288 360 2 32 q y 179 q y 179 q3 . Ž 1 q 179 1584 1008 2 48 3 4 y Ž 1 q 419 q y 419 q y 512 419 q y 419 q . 179 384
q. qq
816 31
q
2
.
2095 6144
2 15
Ž1q 193 1280 29 160
18 37
214 y 87 20 Ž 1 q 261 q .
q.
Ž1q
564 193
qq
128 193
q
2
.
141 20 23 15
289 2 Ž 1 q 117 94 q q 564 q .
U. Feichtingerr Physics Letters B 465 (1999) 155–162
162
equivalent to a12 s a22 , but it is clear that these two relations describe the same physical scenario.
Acknowledgements I would like to thank W. Lerche and P. Mayr, as well as P. Kaste and S. Stieberger for valuable discussions.
Appendix A. Coefficients of the solutions In Table 1 the non-vanishing coefficients of the series solutions Ž4.2. and Ž4.3. are listed up to fourth order in the variables
z1 s
L12 u
uyÕq and
z2 s
1
Ž L12 y L22 . 2 . L12
In Table 2 the corresponding coefficients for the logarithmic solutions Ž4.4. and Ž4.5. are listed.
References w1x A. Klemm, W. Lerche, P. Mayr, C. Vafa, N. Warner, Nucl. Phys. B 477 Ž1996. 746, hep-thr9604034. w2x E. Witten, Nucl. Phys. B 500 Ž1997. 3, hep-thr9703166. w3x E. Witten, Nucl. Phys. B 507 Ž1997. 658, hep-thr9706109. w4x K. Hori, H. Ooguri, Y. Oz, Adv. Theor. Math. Phys. 1 Ž1998. 1, hep-thr9706082. w5x A. Brandhuber, N. Itzhaki, V. Kaplunovsky, J. Sonnenschein, S. Yankielowicz, Phys. Lett. B 410 Ž1997. 27, hepthr9706127. w6x S. Katz, P. Mayr, C. Vafa, Adv. Theor. Math. Phys. 1 Ž1998. 53, hep-thr9706110. w7x J. Ehrlich, A. Naqvi, L. Randall, Phys. Rev. D 58 Ž1998. 046002, hep-thr9801108. w8x E. D’Hoker, D.H. Phong, Nucl. Phys. B 513 Ž1998. 405, hep-thr9709053. w9x I. Ennes, S. Naculich, H. Rhedin, H. Schnitzer, Nucl. Phys. B 536 Ž1998. 245, hep-thr9806144. w10x H.M. Farkas, I. Kra, Riemann Surfaces, Springer Verlag, Berlin, 1980. w11x A. Klemm, S. Theisen, M. Schmidt, Int. J. Mod. Phys. A 7 Ž1992. 6215. w12x N. Seiberg, E. Witten, Nucl. Phys. B 431 Ž1994. 484, hep-thr9408099.
Physics Letters B 465 Ž1999. 155–162
The prepotential of N s 2 SUž 2/ = SUž 2/ supersymmetric Yang–Mills theory with bifundamental matter Ulrike Feichtinger
1
Theory DiÕision, CERN, GeneÕa, Switzerland Received 25 August 1999; accepted 2 September 1999 Editor: L. Alvarez-Gaume´
Abstract We study the non-perturbative, instanton-corrected effective action of the N s 2 SUŽ2. = SUŽ2. supersymmetric Yang–Mills theory with a massless hypermultiplet in the bifundamental representation. Starting from the appropriate hyperelliptic curve, we determine the periods and the exact holomorphic prepotential in a certain weak coupling expansion. We discuss the dependence of the solution on the parameter q s L22rL12 and several other interesting properties. q 1999 Published by Elsevier Science B.V. All rights reserved.
1. Introduction Many field theoretical results have been obtained by considering brane configurations in string theory and M-theory. In particular M-theory five-branes provide a very fruitful approach to N s 2 w1,2x and N s 1 w3–5x supersymmetric field theories. In this note we study the case of product gauge group SUŽ2. = SUŽ2. with a hypermultiplet in the bifundamental representation. Seiberg–Witten curves of theories with product gauge groups have been obtained from the M-theory w2x and the geometric engineering approach w6x. The precise relation between the moduli of the curve and the moduli of the field theory has been described in w7x. Yet another account of supersymmetric field theories with product gauge groups of the form G s Ł i SUŽ Ni . has been obtained in the framework of 1
E-mail: [email protected]
Calogero–Moser systems, using certain limits of SUŽ N . supersymmetric Yang–Mills theory with matter in the adjoint representation w8x. In this approach the dynamical scales of the different gauge group factors are all equal to the scale of the underlying SUŽ N . group. One-instanton predictions for the prepotential of N s 2 supersymmetric field theories with gauge group SUŽ N1 . = SUŽ N2 . and massless matter in the bifundamental representation have been made by using a perturbation expansion of the non-hyperelliptic curve around its hyperelliptic approximation w9x. In the present paper we study the instanton expansion for the case of a massless hypermultiplet in the bifundamental representation of SUŽ2. = SUŽ2.. Specifically we will compute the periods as solutions of the Picard–Fuchs equations and obtain in this way the holomorphic prepotential F that governs the low-energy effective action of the theory. The solution reproduces the existing results in the appropriate
0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 4 6 - 1
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limits and has interesting properties in the general region of moduli space. 2. The setup By considering the appropriate brane configuration in M-theory w2x one finds that the defining polynomial for the Seiberg–Witten curve of N s 2 supersymmetric Yang–Mills theory with a gauge group of the form Ł in SUŽ k i . is given by P Ž x ,t . st nq 1 q p k 1Ž x . t n q p k 2Ž x . t ny 1 q . . . qpk nŽ x . t q c s 0 .
Ž 2.1 .
The p k iŽ x . are polynomials of order k i in x, and c is a constant that depends on the dynamical scales L i of the different gauge group factors. In this expression the variables x and t correspond to the 6 10 combinations x 4 q ix 5 and eyŽ x qi x .r R in the notation of w2x. For the case of SUŽ2. = SUŽ2. with a massless hypermultiplet in the bifundamental representation, the explicit expressions for the polynomials p k iŽ x . in Ž2.1. have been derived by using symmetries and classical limits w7x. The curve for the massive case is given by P Ž x ,t . s L12 t 3 q t 2 x 2 y u q
Fig. 1. The brane configuration that gives rise to a four-dimensional field theory with gauge group SUŽ2.= SUŽ2. and a hypermultiplet in the bifundamental representation. Vertical lines represent five-branes, horizontal lines are four-branes.
SUŽ2. can be obtained by moving the two four-branes of one gauge group factor to x s `, so that the resulting configuration consists only of two fourbranes stretched between two five-branes. To obtain the curve of this theory we have to take the limit Õ ™ `, m ™ ` while keeping L12 Ž m 2 y Õ . s L40 fixed Žsee Fig. 1.. In the case of gauge group SUŽ2. = SUŽ2. the polynomials p 1Ž x . and p 2 Ž x . are quadratic in x; we can therefore rewrite Ž2.2. in hyperelliptic form by iy redefining 2 x ™ :
'2 t Ž1q t . 2
y s tŽ 1 q t . Ž2
L12
t 3 q t 2 Ž L12 y 2 u .
qt Ž L22 y 2 Õ . q 2 L22 . .
Ž 2.3 .
2
The discriminant D of this curve consists of two factors, D s D1 D2 , with
L22
D1 s 4 Ž u y Õ q 12 L12 y 12 L22 . ,
L12
2
2
q t Ž x q m. y Õ q
2
q L22s0 .
D2 s 8
L16 L22 q 359
L14 L24 q 8
L12 L62 y 48
Ž 2.4 . L14 L22
Ž 2.2 .
q 148 L12 L42 u q 96 L12 L22 u 2 y 4 L42 u 2
Here u and Õ are the moduli of the two gauge group factors, L i are the corresponding dynamical scales, and m is the bare mass of the hypermultiplet. Note that the dimensionless parameter q s L 22rL12 cannot be eliminated from the curve by rescaling of the moduli, unless q s 1. Exchanging the two gauge group factors, i.e. u l Õ, L1 l L 2 , t l 1t , and x l yŽ x q m., leaves the curve Ž2.2. invariant. Pulling the rightmost five-brane to x 6 s ` gives the brane configuration of a theory with gauge group SUŽ2. and two fundamental flavours with masses m1 and m 2 . To make this transformation manifest in the curve one has to perform the limit L2 ™ 0, Õ ™ 14 Ž m1 y m 2 . 2 and m ™ 21 Ž m1 q m 2 .. The theory with pure gauge group
y 64 L22 u 3 q 148 L14 L22 Õ y 48 L12 L24 Õ
u
y 304 L12 L22 u Õ q 16 L22 u 2 Õ y 4 L14 Õ 2 q 96 L12 L22 Õ 2 q 16 L12 u Õ 2 y 16 u 2 Õ 2 y 64 L12 Õ 3 .
Ž 2.5 .
As expected, D1 and D2 are symmetric under the exchange of the two gauge group factors. In general the vanishing of discriminant factors describes points in moduli space where extra massless states appear. Following the analysis of w7x, the vanishing of D1
2
In the following we set the bare mass m of the hypermultiplet to zero.
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determines the subspace of the moduli space where the Coulomb branch meets the Higgs branch and the gauge group SUŽ2. = SUŽ2. is broken to the diagonal SUŽ2.. Therefore the extra massless states that occur at this point are two components of the bifundamental hypermultiplet. The vanishing of the second discriminant factor describes the singular locus where other massless states appear, in particular monopoles. The classical intersection point of the Coulomb branch and the Higgs branch at u s Õ gets shifted by non-perturbative effects if the two scales do not have equal values, i.e. if q / 1. This dependence is inferred by the terms L2i r2 in the curve Ž2.2. and could formally be avoided by choosing the defining polynomial to be
The second-order derivatives of F with respect to a i imply the integrability condition
P Ž x ,t . s L12 t 3 q t 2 Ž x 2 y u q L12 .
lA
qt Ž x 2 y Õ q L22 . q L22 s 0.
Ž 2.6 .
Then the two scales no longer explicitly appear in D1. In order to reduce Ž2.6. to the curve for SUŽ2. with two flavours we then have to perform a shift in the modulus u in addition to the above-stated transformation: u ™ u s u q 7 L12r8. This means that the origins of the two moduli spaces would simply be shifted by 7 L12r8, though classically the moduli still coincide. The polynomial Ž2.3. describes a Riemann surface S of genus 2. If we choose a symplectic homology basis of S , i.e. a i and bi Ž i s 1,2. with intersection pairing a i l b j s d i j , a i l a j s bi l b j s 0, we can define the period integrals in terms of a properly chosen meromorphic one-form l: ai s
Ha l ,
aD i s
i
Hb l .
Ž 2.7 .
i
The periods are functions of the moduli u and Õ, a i Ž u,Õ . is identified with the scalar component of the N s 1 chiral multiplet in the i-th gauge group factor, a D i Ž u,Õ . is its dual. Once the periods a i and a D i are known, the exact quantum-corrected prepotential F Ž a k . of the theory can be computed by integrating the relations aD i Ž ak . s
E F Ž ak . E ai
.
E aD i Ž ak . E aj
s
E aD j Ž ak . E ai
.
Ž 2.9 .
The meromorphic one-form l of the curve Ž2.2. is given by w1x
lA
x dt t
,
Ž 2.10 .
or, after the redefinition x™
iy
'2 t Ž 1 q t .
,
by y dt 2
t Ž1qt .
,
Ž 2.11 .
where the proportionality factor is determined by the requirements
El EÕ
dt s y
El and
Eu
t dt s y
.
Ž 2.12 .
It is rather tedious to perform the integrals Ž2.7. explicitly. However, the periods as functions of the moduli u and Õ satisfy a system of partial differential equations, the so-called Picard–Fuchs equations. This method to obtain the periods is relatively straightforward and will be the subject of the next sections.
3. The Picard–Fuchs equations Starting with the hyperelliptic form Ž2.3. of the curve 3 for SUŽ2. = SUŽ2., we will derive the system of partial differential equations for the periods Ha i l and Hb i l. On a Riemann surface S of genus 2 there are two holomorphic differentials and two meromorphic differentials with no residues Žabelian differentials of first and second kind. w10x. Therefore we can choose dxry, x dxry, x 2 dxry, x 3 dxry4 as basis of mero-
Ž 2.8 . 3
For convenience we replace t by x in the following.
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morphic forms on S . When considering derivatives of the meromorphic one-form l with respect to the moduli, only four of them will be linearly independent up to exact forms. The linear relations between these derivatives define the Picard–Fuchs equations. Derivatives of l with respect to u and Õ involve terms of the form f Ž x . dxry n with some polynomials f Ž x .. In order to express f Ž x . dxry n in terms of abelian differentials of first and second kind, we need a method to reduce high powers of x in the numerator and high powers of y in the denominator. For the latter we use the fact that the discriminant of a polynomial pŽ x . of order n can always be written in the form w11x
D s a Ž x . p Ž x . q b Ž x . pX Ž x . ,
Ž 3.1 .
where aŽ x ., resp. bŽ x ., is a polynomial of order Ž n y 2., resp. Ž n y 1., in x. With this formula it can easily be shown that the following relation holds up to exact forms:
With these formulae we can re-express the meromorphic one-form Ž2.11. in terms of abelian differentials of first and second kind
l s Ž 2 Õ y L22 .
y
q Ž 4 u q 2 Õ y 2 L12 y L 22 .
q 4 Ž u y 2 L12 .
x 2 dx y
y 6 L12
x 3 dx y
.
x dx y
Ž 3.4 .
When we consider derivatives of l with respect to u and Õ up to second order, we find that these derivatives satisfy two linear relations as only four out of six are linearly independent. These two relations give rise to the following Picard–Fuchs operators: L 1 s Ž 5 L12 q 2 u . Eu2 q 2 Ž L12 y L22 y 2 u q 2 Õ . Eu EÕ y Ž 5 L22 q 2 Õ . EÕ2 q Eu y EÕ ,
Ž 3.5 .
L 2 s Ž L14 q 14 L12 L22 y 4 u 2 y 4 L12 Õ . Eu2
fŽ x.
y Ž 14 L12 L22 q L24 y 4 L22 u y 4 Õ 2 . EÕ2
yn 2
s
dx
y 2 Ž 15 L12 L22 y 2 L22 u y 2 L12 Õ
d
1 a Ž x . f Ž x . q n y 2 dx Ž b Ž x . f Ž x . . . D y ny 2 Ž 3.2 .
Following w11x we derive a formula to reduce high powers of x that is valid up to addition of total derivatives: xn sy y
y
Ž 2 n y 7 . L22 x ny 4 Ž 2 n y 3 . L12 y Ž n y 3. Ž 2 n y 3.
Ž 2 n y 5. y Ž 2 n y 3. y
Ž n y 2. Ž 2 n y 3.
y4 u Õ . Eu EÕ q 1.
Ž 3.6 .
Note that the first Žsecond. operator is antisymmetric Žsymmetric. under the exchange of the two gauge group factors.
4. The periods and the prepotential
Ž L12 q L22 y 2 u y 2 Õ .
x ny 2
2 L12
y
We are interested in the prepotential of SUŽ2. = SUŽ2. supersymmetric Yang–Mills theory in the weak coupling regime, which is parametrized by large values of the moduli u and Õ. As usual, if we have two variables that simultaneously become large, we have to specify an appropriate coordinate patch in which the limit is well-defined. By inspection of the first discriminant factor Ž2.4., we choose the variables to be
Ž 3.3 .
z1 s
Ž 3 L22 y 2 Õ .
x ny 3
L12
y
Ž 3 L12 y 2 u .
x ny 1
L12
y
.
L12 u
and
z2 s
u y Õ q 12 Ž L12 y L22 .
L12
. Ž 4.1 .
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This choice is not symmetric under the exchange of the two gauge group factors, but of course there exists the analogous patch in the moduli space where the role ˆ of the two moduli is interchanged. Using Frobenius’ method, we find two series solutions and two logarithmic solutions with indices Žy 12 ,0. and Ž 12 ,1. in the coordinate patch of the moduli space parametrized by Ž4.1.: ` r2 w 1 Ž z 1 , z 2 . s zy1 1
Ý
a i j z 1i z 2j ,
Ž 4.2 .
bi j z 1i z 2j ,
Ž 4.3 .
i , js0 `
w 2 Ž z 1 , z 2 . s z 11r2 z 2
Ý
159
Therefore the one-loop contributions of the pure gauge couplings are given by
t 11 ; y4ln Ž a1 . q ln Ž a12 y a22 . , t 22 ; y4ln Ž a2 . q ln Ž a12 y a22 . .
Ž 4.7 .
In order to examine the asymptotic behaviour of the periods, we integrate the gauge couplings t i i in Ž4.7. to obtain a D i Ž a k . s Ht i i Ž a k . da i , introduce a new variable ´ s a1 y a 2 , and expand around the point Ž a1 , ´ . s Ž`,0.. This procedure gives the leading terms of the periods a D i : a D1 ; y2 a1 ln Ž a1 . y ´ ln Ž a1 . q ´ ln Ž ´ . q . . . ,
i , js0
a D 2 ; y2 a1 ln Ž a1 . q 3 ´ ln Ž a1 . y ´ ln Ž ´ . q . . . . Ž 4.8 .
w 3 Ž z 1 , z 2 . s w1 Ž z 1 , z 2 . ln Ž z 1 . ` r2 q zy1 1
Ý
c i j z 1i z 2j ,
Ž 4.4 .
i , js0
w4 Ž z 1 , z 2 . s w 2 Ž z 1 , z 2 . ln Ž z 13 z 22 . ` r2 q zy1 1
Ý
d i j z 1i z 2j .
Ž 4.5 .
y2 A a1 ln Ž a1 . q Ž A y 8 B . ´ ln Ž a1 .
i , js0
The first few coefficients of these series expansions are listed in Appendix A. In order to get the periods we have to consider linear combinations of the solutions that match the infrared asymptotic behaviour. To analyse these leading terms one can either evaluate the lowest order of the integrals Ž2.7. explicitly or one uses semi-classical relations of the gauge couplings and the integrability condition Ž2.9.. The gauge couplings t i j of the theory are related to the prepotential F by
ti j s
E 2F Ž a k . E ai E a j
s
E aD i E aj
.
To compare the solutions Ž4.4. and Ž4.5. with these expansions, we have to replace the moduli u and Õ by the corresponding classical expressions u s a12 and Õ s a 22 . We apply the same procedure as above to a linear combination of the logarithmic solutions A w 3 q B w4 and find, in leading order:
Ž 4.6 .
The one-loop terms of the pure couplings t i i consist of the logarithmic contributions of the massless spectrum. In each gauge group factor there is a gauge field in the adjoint representation; moreover there is a hypermultiplet in the bifundamental representation.
q 4 B ´ ln Ž ´ . q . . . .
Ž 4.9 .
Hence the periods a D i are reproduced by the following linear combinations of the solutions a D1
L1 aD 2
L1
s A1w 1 q B1w 2 q w 3 q 14 w4 , s A 2 w 1 q B2 w 2 q w 3 y 14 w4 ,
Ž 4.10 .
where the coefficients A1 , A 2 , B1 , B2 are still undetermined. We also find that the periods a i are given by a1
L1
s w 1 q 14 w 2 ,
a2
L1
s w 1 y 14 w 2 .
Ž 4.11 .
It can easily be checked that in the limits of SUŽ2. Nf s 0, resp. Nf s 2, stated in Section 2, a1 reduces to the corresponding periods known in the literature. By imposing the integrability condition Ž2.9., one finds that A 2 s A1 and B2 s yB1.
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The prepotential can be computed by integrating Eq. Ž2.8.. To this end we have to know the magnetic periods a D i as functions of the electric periods a k . By inverting the series expansions of a k in Ž4.11. we get the variables z i and in consequence the magnetic periods as functions of a k : a D1 Ž a1 ,a2 .
with the following expressions for the first few terms: Fclass s Ž a12 q a22 . Ž 12 q 14 A1 q B1 q 12 ln Ž 2 . . q a1 a 2 Ž 12 A1 y 2 B1 y ln Ž 2 . . ,
Ž 4.15 .
F1- loop s y2 a12 ln Ž a1 . y 2 a 22 ln Ž a2 .
1 2
s a1 Ž A1 q 2 B1 q ln Ž 2 . . 2
q a 2 Ž 21 A1 y 2 B1 y ln Ž 2 . . y 4 a1 ln Ž a1 .
q 12 Ž a1 q a2 . ln Ž a1 q a 2 .
q Ž a1 q a 2 . ln Ž a1 q a2 .
q 12 Ž a1 y a2 . ln Ž a1 y a 2 . ,
2
q Ž a1 y a 2 . ln Ž a1 y a2 .
L12
1 q
ž
2 a1
a 22 a12
Ž
32 2 a71 1
q8 q
yq
a13
F1- inst s
/
L12 4
Ž a12 y a22 .
ž
1 a12
a14 y 12
a12 a 22 q 15
a 24
. sy
2
a1 a62
Ž5
a12 y 3
a22
L14 4
q...
.
q
Ž 4.12 . y16 q
a D 2 Ž a1 ,a2 .
1 y
a12
q a 2 Ž 21 A1 q 2 B1 q ln Ž 2 . . y 4 a2 ln Ž a2 . q Ž a1 q a 2 . ln Ž a1 q a2 . y Ž a1 y a 2 . ln Ž a1 y a2 . y
2 a2 y
1 2
ž
L14
a2
32
a16
q q
2
1 a72
/
a22
,
Ž 4.17 .
a22
qq2
a 22 a12
Ž3
Ž 15
yq
a12 a22
a12 y 5
1
. q 8 q a3
a12 a22 q a24
.
q...
Ž 4.13 . Notice that exchanging a1 and a 2 transforms a D1 into a D 2 and vice versa. Integrating the expression Ž4.12., resp. Ž4.13., with respect to a1 , resp. a2 , yields the prepotential `
F s Fclass q F1 - loop q
Ý Fn - inst , ns1
384
qq 3
2
a14 y 12
L16
q3 q
/
a 22
F3- inst s y
a12
ž
a 24
Ž 4.14 .
ž
a14 a14 a82
ž
5 a12 5 a12
a18
/
a22 9
y
a 22
5 y
a12
1 y
a14
1
a 42
Ž a12 y a22 . 1
a22
Ž a12 y a22 .
128
1 a12
L14
s a1 Ž 12 A1 y 2 B1 y ln Ž 2 . .
1 L12
1
yq
F2- inst
L14 1 1
y
yq
a12 a22
Ž 4.16 .
/
ž
a 22
a12
1
a12
y
, 5
y
q3 q2
.
ž
/
9
5
a22 1 a 42
a22
/
Ž 4.18 .
/ ž
1 a12
5 y
a 22
/
Ž 4.19 .
As expected, all contributions to the prepotential are totally symmetric under the exchange of the two gauge group factors. The constants A1 and B1 are determined by the classical couplings. The one-loop contribution F1- loop is in agreement with the corresponding term of the prepotential reported in w8x and w9x. Up to an overall scaling factor of the two scales, F1- inst coincides with the one-instanton term in the prepotential of w9x. If the bare mass m of the hypermultiplet in the prepotential obtained in w8x is set to zero, the one-instanton term coincides with F1- inst
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Table 1 The non-vanishing coefficients of the series solutions i
j
ai j
bi j
0 1
0 0
1
1 y 34 Ž 1 q 23 q .
2
0
y 321 Ž 1 q 8 q .
3
0
1 128
4 1 2
0 1 1
5 y 2048 Ž 1 q 48 q q 96 q 2 . 1 y4 y 161 Ž 1 y 2 q .
1 2
21 128
3 2
1 4
y
Ž 1 q 43 q q 49 q 2 . 135 y 128 Ž 1 q 2 q q 34 q 2 q 278 q 3 . 2835 8 8 3 32 3 16 4 2048 Ž 1 q 3 q q 3 q q 27 q q 81 q . 27 32
Ž 1 q 24 q .
Ž1y
12 7
4 7
qy q
2
1 4
y 163 Ž 1 q 4 q . 45 y 128 Ž 1 y 163 q y 4 q 2 .
.
1 16
1 8
for q s 1, but the terms proportional to q in F2- inst are not present. This is a priori no contradiction, since sending the bare mass to zero is a singular limit. The instanton terms cannot be written as simple sums of contributions stemming from the two subgroups, but show a non-trivial mixing of a1 and a 2 , as it is expected in the presence of a hypermultiplet in the bifundamental representation. In particular the n-th instanton term is not just a sum of contributions proportional to L2i n , but all possible combinations L12 i L22 j with i q j s n appear in Fn - inst . Terms proportional to odd powers of the scales are forbidden as they would violate the anomaly-free discrete symmetry of the theory w12x. It is quite striking that nearly all terms in the instanton expansion of the prepotential are propor-
tional to Ž a12 y a22 . and therefore vanish for a1 s " a 2 . In either of the two cases two components of the hypermultiplet become massless and one expects to recover the point in moduli space where the Coulomb branch meets the Higgs branch and the gauge group is broken to the diagonal SUŽ2.. Continuity of the prepotential at the intersection point ensures that F collapses to the prepotential of pure SUŽ2.. Indeed, if one sets a1 s " a 2 s a in the prepotential Ž4.14., the non-vanishing terms reproduce the prepotential of SUŽ2. theory with the dynamical scale L2 s 4 L1 L 2 to all calculated orders. In w7x a different argumentation, namely considerations of the brane configuration and the curve, led to the insight that the same point in moduli space is determined by the equation u q 12 L12 s Õ q 12 L22 . At first sight it is quite astounding that this condition is
Table 2 The non-vanishing coefficients of the logarithmic solutions i
j
ci j
di j
0 1
0 0
1 y 14
1
2 3 4 1 2 3 2
0
y
0
5 384
0 1 1 1 2
y 1 40 37 160
y y
9 4
Ž1q 31 6144
Ž 1 y 89 q .
25 y 32 Ž 1 y 258 q y 258 q 2 .
1 32 48 5
Ž1q
768 31
288 360 2 32 q y 179 q y 179 q3 . Ž 1 q 179 1584 1008 2 48 3 4 y Ž 1 q 419 q y 419 q y 512 419 q y 419 q . 179 384
q. qq
816 31
q
2
.
2095 6144
2 15
Ž1q 193 1280 29 160
18 37
214 y 87 20 Ž 1 q 261 q .
q.
Ž1q
564 193
128 193
q
2
.
141 20 23 15
289 2 Ž 1 q 117 94 q q 564 q .
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162
equivalent to a12 s a22 , but it is clear that these two relations describe the same physical scenario.
Acknowledgements I would like to thank W. Lerche and P. Mayr, as well as P. Kaste and S. Stieberger for valuable discussions.
Appendix A. Coefficients of the solutions In Table 1 the non-vanishing coefficients of the series solutions Ž4.2. and Ž4.3. are listed up to fourth order in the variables
z1 s
L12 u
uyÕq and
z2 s
1
Ž L12 y L22 . 2 . L12
In Table 2 the corresponding coefficients for the logarithmic solutions Ž4.4. and Ž4.5. are listed.
References w1x A. Klemm, W. Lerche, P. Mayr, C. Vafa, N. Warner, Nucl. Phys. B 477 Ž1996. 746, hep-thr9604034. w2x E. Witten, Nucl. Phys. B 500 Ž1997. 3, hep-thr9703166. w3x E. Witten, Nucl. Phys. B 507 Ž1997. 658, hep-thr9706109. w4x K. Hori, H. Ooguri, Y. Oz, Adv. Theor. Math. Phys. 1 Ž1998. 1, hep-thr9706082. w5x A. Brandhuber, N. Itzhaki, V. Kaplunovsky, J. Sonnenschein, S. Yankielowicz, Phys. Lett. B 410 Ž1997. 27, hepthr9706127. w6x S. Katz, P. Mayr, C. Vafa, Adv. Theor. Math. Phys. 1 Ž1998. 53, hep-thr9706110. w7x J. Ehrlich, A. Naqvi, L. Randall, Phys. Rev. D 58 Ž1998. 046002, hep-thr9801108. w8x E. D’Hoker, D.H. Phong, Nucl. Phys. B 513 Ž1998. 405, hep-thr9709053. w9x I. Ennes, S. Naculich, H. Rhedin, H. Schnitzer, Nucl. Phys. B 536 Ž1998. 245, hep-thr9806144. w10x H.M. Farkas, I. Kra, Riemann Surfaces, Springer Verlag, Berlin, 1980. w11x A. Klemm, S. Theisen, M. Schmidt, Int. J. Mod. Phys. A 7 Ž1992. 6215. w12x N. Seiberg, E. Witten, Nucl. Phys. B 431 Ž1994. 484, hep-thr9408099.