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Instanton Recursion Relations for the Effective Prepotential in N=2 Super Yang-Mills


UCLA/99/22 June, 1999

arXiv:hep-th/9906193v2 30 Aug 1999

Instanton Recursion Relations for the E?ective Prepotential in N =2 Super Yang-Mills
1

Gordon Chan2 , Eric D’Hoker3

Department of Physics and Astronomy University of California Los Angeles, CA 90095, USA

Abstract Linear recursion relations for the instanton corrections to the e?ective prepotential of N =2 supersymmetric gauge theories with an arbitrary number of hypermultiplets in the fundamental representation of an arbitrary classical gauge group are derived. The construction proceeds from the Seiberg-Witten solutions and the renormalization group type equations for the prepotential. Successive iterations of these recursion relations allow us to simply obtain instanton corrections to arbitrarily high order, which we exhibit explicitly up to 6-th order. For gauge groups SU (2) and SU (3), our results agree with previous ones.

1 Research supported in part by the National Science Foundation under grants PHY9819686 and PHY-95-31023 2 E-mail: morgoth@physics.ucla.edu 3 E-mail: dhoker@physics.ucla.edu

1. Introduction
Over the past few years there has been great progress in understanding non-perturbative dynamics of N =2 SUSY gauge theories starting with the SU(2) case [1], further generalizations to other gauge groups [2] and additions of matter hypermultiplets [3]. Non-perturbative corrections in weak coupling correspond to instanton e?ects [4] which were evaluated using ?eld theory techniques to one instanton [4][5][6][7] and two instanton [8] orders. Some of the previous instanton calculations using the Seiberg-Witten ansatz were performed by solving the Picard-Fuchs equations for the period integrals corresponding to the quantum moduli parameters representing the set of vacuum expectation values of the Higgs ?elds [9][17][18]. Other previous calculations involved solving the period integrals directly [10], and were found to be in agreement with the Picard-Fuchs and ?eld theory results. In an intriguing paper [11], a recursion relation for the instanton corrections to the e?ective prepotential F was found for the pure SU (2) case which led us to seek a generalization of this result for any gauge group and number of matter hypermultiplets. In a related development, the Seiberg-Witten equations were viewed analogously to the Whitham hierarchy equations [12] and the WDVV equations [13]. Nonlinear recursion relations for the instanton corrections involving Jacobi θ-functions (which themselves involve τij as in [16]) were derived starting from the Whitham hierarchy equations [14]. The beta function of the prepotential F , ?rst observed in [15] and later proved in [16], is the starting point of our discussion and provides a very direct way at calculating instanton corrections to the prepotential F without having to perform complicated hyperelliptic integrals and immediately obtaining rational expressions without θ-functions. The Seiberg-Witten solutions for classical gauge groups SU (N ), SO (N ), and Sp(N ) and the renormalization group like equation for the prepotential F , led us to the discovery of a general recursion relation expressing the n ? th order instanton correction to the prepotential F in terms of the (n ? 1)th, · · · , 1st order instanton corrections for super Yang-Mills theory with matter hypermultiplets in the fundamental representation of classical gauge groups G. We start o? by reviewing the Seiberg-Witten solution for N =2 super Yang-Mills theory with hypermultiplets in the fundamental representation of any arbitrary classical gauge group. The renormalization group type equation 1

for the prepotential F is discussed next and it is shown how it can be used to determine the instanton corrections to the prepotential to arbitrary order in an e?cient manner. Recursion relations for the instanton corrections are then derived and shown to reproduce previous results.

2. The Seiberg-Witten Solution for Arbitrary Classical Gauge Group G
The Seiberg-Witten (SW) ansatz gives a prescription for determining the prepotential of the e?ective action for N =2 supersymmetric Yang-Mills gauge theories, as well as for determining the spectrum of BPS states. We consider, N =2 SUSY gauge theories with classical gauge groups SU (r + 1), SO (2r + 1), Sp(2r ) and SO (2r ), of rank r and number of colours Nc = r + 1, 2r + 1, 2r , and 2r respectively. We include Nf hypermultiplets in the fundamental representation of the gauge group, with bare masses mj , j = 1, · · · , Nf . We restrict to the asymptotically free theories; this limits the hypermultiplet contents Nf . (ie. Nf < 2Nc for SU (Nc )). The classical vacuum expectation value of the gauge scalar φ is parameterized by complex moduli a ?k , k = 1, · · · , r as follows SU (r + 1) φ = diag [? a1 , · · · , ? ar , a ?r+1 ] a ?1 + · · · + a ?r + a ?r+1 = 0 SO (2r + 1) φ = diag [A1, · · · , Ar , 0] Sp(2r ) φ = diag [? a1 , ?a ?1 , · · · , a ?r , ?a ?r ] (1) 0 a ?k SO (2r ) φ = diag [A1, · · · , Ar ] Ak = ?a ?k 0 For generic a ?k ’s, the gauge symmetry is broken to U (1)r and the dynamics is that of an Abelian Coulomb phase. The Wilson e?ective Lagrangian of the quantum theory to leading order in the low momentum expansion in the Abelian Coulomb phase is completely characterized by a complex analytic prepotential F (a). The SW ansatz for determining the full prepotential F is based on a choice of a ?bration of spectral curves over the space of vacua, and of a meromorphic 1-form dλ on each of these curves. The renormalized order parameters ak of the theory, their duals aD,k , and the prepotential F are

2

given by 2πiak = dλ, 2πiaD,k = dλ, aD,k = ?F ?ak (2)

Ak

Bk

with Ak , Bk a suitable set of homology cycles on the spectral curves [2]. For all N =2 supersymmetric gauge theories based on classical gauge groups with Nf hypermultiplets in the fundamental representation of the gauge group, the spectral curves and meromorphic 1-forms are y 2 = A2 (x) ? B (x) x AB ′ dλ = A′ ? dx y 2B where
r +1 Nf

(3)

SU (r + 1) SO (2r + 1) Sp(2r )1 SO (2r ) ? ≡ Λq with Λ
? ? ? ? ?

A(x) =
r a 2

(x ? a ?k )
k =1

?2 B (x) = Λ
Nf

(x + mj )
j =1

A(x) = x

(x ?
k =1

a ?2 k)

? 2 xb B (x) = Λ

(x2 ? m2 j)
j =1

(4)

SU (r + 1) SO (2r + 1) Sp(2r ) SO (2r )

q q q q

= r + 1 ? Nf /2 = 2r ? 1 ? Nf = 2r + 2 ? Nf = 2r ? 2 ? Nf

a=0 a=2 a=0

b=2 b=0 b=4

(5)

respectively. The spectral curves (4) for SO (2r + 1), Sp(2r ) and SO (2r ) can be obtained from the SU (2r ) spectral curve by a suitable restriction on the classical moduli a ?k ’s and masses [19]. For gauge theories with classical gauge groups and asymptotically free coupling obeying the constraint q > 0, general arguments based on the holomorphicity of F , perturbative non-renormalization theorems beyond 1-loop
For simplicity, we restrict attention here to the Sp(2r) case with at least two massless hypermultiplets. The cases with one or no massless hypermultiplets may be treated accordingly [19].
1

3

order, the nature of instanton corrections, and restrictions of U (1)R invariance, constrain F to have the form 2q F (a) = πi ?
i ∞ r

a2 i
i=1 Nf

i + 4π

α

(α · a)2 (α · a) log Λ2
2

+

Λ2mq (m) F (a) m=1 2mπi

(λi · a + mj )2 ? (λi · a + mj )2 log Λ2 j =1 (6)

?

where λi = ±ei for SO and Sp and λi = ei for SU in an orthonormal basis ei , and α are the roots of the gauge group G. (The SU (r ) solution requires ). an additional overall factor of 1 2 The terms on the right side are respectively the classical prepotential, the contribution of perturbative one-loop e?ects, and m-instanton processes contributions 2 . Λ is the dynamically generated scale of the theory.

3. Renormalization Group Type Equations
In [16], a renormalization group type equation for the prepotential F was derived using the SW ansatz equations (2) q r 2 ?F = a ? Λ ?Λ πi k=1 k (7)

up to an additive term independent of ak and a ?k which is physically immaterial. (The SU (r ) case requires an additional factor of 1 ). 2 In [10] an e?cient algorithm was presented for calculating the renormalized order parameters ak and their duals aD,k in terms of the classical order ? is parameters a ?k to any order of perturbation theory in a regime where Λ small and the a ?k ’s are well-separated. The calculation of ak starts o? from equations (2) and (3) producing a ?nal result


ak =
m=0

? 2m ?(m) (? Λ a) k

(8)

2 The normalization of the instanton contributions in the present paper di?ers from that 1 1 for SU (Nc ) and 2mπi for SO(2r + 1), Sp(2r), and SO(2r). For of [10][19] by a factor 4mπi our purposes, it will be convenient to use the normalization of (6).

4

where we set ?k (? a) ≡ a ?k , and we have ?k (? a) = with SU (r + 1) SO (2r + 1) Sp(2r )1 SO (2r ) Sk (x, a) =
(m)

(0)

1 22m (m!)2

? ?a ?k

2m?1

Sk (? ak , ? a)m , m = 0

(9)

+ mj ) 2 l=k (x ? al ) Nf x2 j =1 (x2 ? m2 j) Sk (x, a) = 2 (x + ak )2 l=k (x2 ? a2 l) Nf ? 2 2 2 j =1 (x ? mj ) Sk (x, a) = 2 (x + ak )2 l=k (x2 ? a2 l) N f (x2 ? m2 x4 j =1 j) Sk (x, a) = 2 2 2 (x + ak ) l=k (x ? a2 l)

Nf j =1 (x

(10)

? de?ned as previously (5). and Λ Equations (7)(8)(9) (10) su?ces to determine the prepotential F in terms ? 2. of the renormalized order parameters ak order by order in powers of Λ

4. Recursion Relation for the Prepotential F
A very direct way of deriving the form of the instanton corrections to the prepotential F starts o? from the beta function on the right hand side of (7). Substituting the ansatz for the prepotential (6) into the beta function (7), one obtains r r ∞ 2 ? 2m F (m) (a) Λ (11) a ?2 = a + k k
k =1 k =1 m=1

Substituting (8) into (11), one obtains
r ∞ 2

0 = + Λ

? 2m ?(m) (? Λ a) k


r

?

(?k (? a))2
k =1

(0)

(12)

k =1 m=0 ∞ ? 2m m=1

F (m)
n=0

? 2n ?(n) (? Λ a) k

? 2 in the last term and replacing the a Expanding in powers of Λ ?k ’s with ak ’s,

5

the m-th order instanton correction to the prepotential F takes on the form
r

? F (m) (a) =
k =1

? ? ? ?

m

i,j =0 i+j =m

m?1

+
n=1

1 n!

?k (a)?k (a)? ?
n?1 r

(i)

(j )

? ? ? ?

β1 ,···,βn+1 =1 α1 ,···,αn =1 β1 +···+βn+1 =m

? ? (βn+1 ) (βi ) ?α (a) ? F (a) i i=1 j =1 ?aαj (13)

n

n

which is a linear recursion relation for F (m) (a) in terms of the lower order instanton corrections F (m?1) (a), . . . , F (1) (a). The intriguing part about the recursion relation (13) for F (n) (a) is that it is linear in F (n?1) (a), · · · , F (1) (a) and is valid for all classical gauge groups with the number of hypermultiplets in the fundamental representation constrained by q > 0. Previous recursion relations [11] were only valid for SU (2) with no hypermultiplets and were non-linear.

5. Instanton Expansion of the Prepotential F
? 2 , the ?rst six instanton corrections (13) to Order by order in powers of Λ the prepotential F are
r (0) (1)

? F (1) (a) =
k =1 r

2?k (a)?k (a) 2?k (a)?k (a) + (?k (a))2 +
k =1 r k =1 (0) (2) (1) r

(14) ?k (a)
(1)

?F (2) (a) = ?F (3) (a) =
k =1 r

? F (1) (a) ?ak

(15)

2?k (a)?k (a) + 2?k (a)?k (a) ?k (a)
k =1 (1)

(0)

(3)

(1)

(2)

+

? F (1) (a) ? F (2) (a) (2) +?k (a) ?ak ?ak (16)

1 r ? 2 F (1) (a) (1) (1) + ? (a)?m (a) 2! k,m=1 k ?ak ?am

6

r

?F (4) (a) =
k =1 r

2?k (a)?k (a) + 2?k (a)?k (a) + (?k (a))2 ?k (a)
k =1 (1)

(0)

(4)

(1)

(3)

(2)

+

? F (3) (a) ? F (2) (a) ? F (1) (a) (2) (3) +?k (a) +?k (a) ?ak ?ak ?ak

1 r ? 2 F (1) (a) ? 2 F (2) (a) (1) (1) (2) (1) + + 2?k (a)?m (a) ? (a)?m (a) 2! k,m=1 k ?ak ?am ?ak ?am + ?F (5) (a) =
k =1 r r ? 3 F (1) (a) 1 (1) (1) ?k (a)?(1) ( a )? ( a ) m n 3! k,m,n=1 ?ak ?am ?an r

(17)
(2) (3)

2?k (a)?k (a) + 2?k (a)?k (a)+ 2?k (a)?k (a) ?k (a)
k =1 (1)

(0)

(5)

(1)

(4)

+ +

? F (3) (a) ? F (2) (a) ? F (4) (a) (2) (3) + ?k (a) + ?k (a) ?ak ?ak ?ak (a) 1 r ? 2 F (3) (a) (1) + ?k (a)?(1) m (a) 2! k,m=1 ?ak ?am

?F (4) ?k (a)
(1)

(1)

?ak

+ 2?k (a)?(3) m (a)

? 2 F (1) (a) ?ak ?am 2 (1) ? F (a) ? 2 F (2) (a) (2) (1) (2) + ?k (a)?(2) ( a ) + 2? ( a )? ( a ) m m k ?ak ?am ?ak ?am

r 1 ? 3 F (2) (a) (1) (1) (1) + ? (a)?m (a)?n (a) 3! k,m,n=1 k ?ak ?am ?an (1) + 3?k (a)?(1) m (a)?n (a) (2)

? 3 F (1) (a) ?ak ?am ?an (18)

+ ?F (6) (a) =

r 1 ? 4 F (2) (a) (1) (1) (1) ?k (a)?(1) ( a )? ( a )? ( a ) m n l 4! k,m,n,l=1 ?ak ?am ?an ?al r k =1

2?k (a)?k (a) + 2?k (a)?k (a) + 2?k (a)?k (a)
(3) r k =1

(0)

(6)

(1)

(5)

(2)

(4)

+ (?k (a))2 + + ?k (a)
(3) (3)

?k (a)

(1)

? F (5) (a) ? F (4) (a) (2) + ?k (a) ?ak ?ak

? F (2) (a) ? F (1) (a) ? F (a) (4) (5) + ?k (a) + ?k (a) ?ak ?ak ?ak 7

+

? 2 F (1) (a) ? 2 F (4) (a) 1 r (1) (1) (4) + 2? ( a )? ( a ) ?k (a)?(1) ( a ) m m k 2! k,m=1 ?ak ?am ?ak ?am
(2)

+ ?k (a)?(2) m (a)

? 2 F (2) (a) ? 2 F (3) (a) (1) + 2?k (a)?(2) ( a ) m ?ak ?am ?ak ?am 2 (2) ? F (a) ? 2 F (1) (a) (1) (2) (3) + 2?k (a)?(3) ( a ) + 2? ( a )? ( a ) m m k ?ak ?am ?ak ?am
r ? 3 F (3) (a) 1 (1) (1) ?k (a)?(1) ( a )? ( a ) m n 3! k,m,n=1 ?ak ?am ?an (1)

+

(3) + 3?k (a)?(1) m (a)?n (a)

? 3 F (1) (a) ?ak ?am ?an ? 3 F (2) (a) (1) (2) ( a )? ( a ) + 3?k (a)?(1) m n ?ak ?am ?an ? 3 F (1) (a) (1) (2) + 3?k (a)?(2) ( a )? ( a ) m n ?ak ?am ?an
r ? 4 F (2) (a) 1 (1) (1) (1) ?k (a)?(1) ( a )? ( a )? ( a ) m n l 4! k,m,n,l=1 ?ak ?am ?an ?al (2) (1)

+

(1) + 4?k (a)?(1) m (a)?n (a)?l (a)

? 4 F (1) (a) ?ak ?am ?an ?al

+

r 1 ? 5 F (1) (a) (1) (1) (1) (1) ?k (a)?(1) ( a )? ( a )? ( a )? ( a ) m n o l 5! k,m,n,l,o=1 ?ak ?am ?an ?al ?ao

(19) A closer examination of the recursion relation (13) for the prepotential F reveals that there is always a term of the form
r

2
k =1

?k (a)?k (a) = 2
k =1

(0)

(n)

r

ak ?k (a)

(n)

(20)

When performing explicit calculations for special cases of Nc and Nf , it is useful to rewrite terms of the form (20) so that there are no ak ’s sitting out in front. Using the de?nition (10) of Sk (x, a) and performing contour integrals in the complex plane by residue methods as in [10], it can be shown that
r

2
k =1

(n) ak ?k (a)

(2n ? 1) = ? 2n?1 2 (n!)2 8

r k =1

? ?ak

2n?2

Sk (ak , a)n

(21)

up to an ak independent term that is physically immaterial for q > 0.

6. Comparison with Previous Results
In order to make explicit comparisons with results in the literature, the instanton corrections have to be rewritten in terms of symmetric polynomials in the ak ’s as follows. For SU (2), the existing results in the literature have the instanton expressions expressed in terms of a1 = 2a a2 = ?2a

(22)

Solving the recursion relation (13) for the pure SU (2) case, the explicit form for the n-th order instanton correction to the prepotential F was determined to be 1 (a) = (2a)4n?2
n j =1

F

(n)

4n ? 3 j?1

(?1)j ?1 j

n

bn1 · · · bnj
n1 ,···,nj =1 n1 +···+nj =n

(23) where bn = (2n ? 3)!! (n!)2

(24)

which agrees with previous results [9][10] 3 . Explicit evaluations for Nf = 0, 1, 2, 3 were performed, with Nf = 3 summarized here. F (1) = F (2) = F (3)
3 2

1 22 a2 1 28 a6

a2 (m1 + m2 + m3 ) + m1 m2 m3

2 2 2 2 2 2 2 2 2 2 2 2 a6 + a4 (m2 1 + m2 + m3 ) ? a (m1 m2 + m1 m3 + m2 m3 ) + 5m1 m2 m3 m1 m2 m3 2 2 2 2 2 2 2 2 2 = ?3a6 + 5a4 (m2 1 + m2 + m3 ) ? 7a (m1 m2 + m1 m3 + m2 m3 ) 211 a10

Our results agree exactly with those of [9] to eight instantons with the replacement 2 Λ →Λ 2 .

9

F (4)

2 2 + 9m2 1 m2 m3 1 2 2 8 4 4 4 = 20 14 a12 ? 6a10 (m2 1 + m2 + m3 ) + a [5(m1 + m2 + m3 ) 2 a 2 2 2 2 2 6 2 2 2 + 100(m2 1 m2 + m1 m3 + m2 m3 )] + a [1176m1 m2 m3 2 2 4 4 2 2 4 4 2 2 4 ? 126(m4 1 m2 + m1 m2 + m1 m3 + m1 m3 + m2 m3 + m2 m3 )] 4 4 4 4 4 2 2 2 2 2 2 + a4 [153(m4 1 m2 + m1 m3 + m2 m3 ) + 1332m1 m2 m3 (m1 + m2 + m3 )] 2 2 2 2 2 2 2 2 4 4 4 ? 1430a2 m2 1 m2 m3 (m1 m2 + m1 m3 + m2 m3 ) +1469m1 m2 m3 m1 m2 m3 2 2 = 35a12 ? 210a10 (m2 1 + m2 + m3 ) 23 18 2 a 4 2 2 2 2 3 2 2 + a8 [207(m4 1 + m2 + m3 ) + 1260(m1 m2 + m1 m3 + m2 m3 )] 2 2 4 4 2 2 4 4 2 2 4 ? 1210a6 (m4 1 m2 + m1 m2 + m1 m3 + m1 m3 + m2 m3 + m2 m3 ) 4 4 4 4 4 2 2 2 2 2 2 + a4 [1131(m4 1 m2 + m1 m3 + m2 m3 ) + 5960m1 m2 m3 (m1 + m2 + m3 )] 2 2 2 2 2 2 2 2 4 4 4 ? 5250a2 m2 1 m2 m3 (m1 m2 + m1 m3 + m2 m3 ) +4471m1 m2 m3 1 2 2 14 2 2 2 2 2 2 = 29 22 5a16 (m2 1 + m2 + m3 ) ? a [210(m1 m2 + m1 m3 + m2 m3 ) 2 a 4 4 12 6 6 6 2 2 2 + 14(m4 1 + m2 + m3 )] + a [9(m1 + m2 + m3 ) + 6507m1 m2 m3 2 2 4 4 2 2 4 4 2 2 4 + 801(m4 1 m2 + m1 m2 + m1 m3 + m1 m3 + m2 m3 + m2 m3 )] 2 2 6 6 2 2 6 6 2 2 6 ? a10 [660(m6 1 m2 + m1 m2 + m1 m3 + m1 m3 + m2 m3 + m2 m3 ) 4 4 4 4 4 2 2 2 2 2 2 + 330(m4 1 m2 + m1 m3 + m2 m3 ) + 24420m1 m2 m3 (m1 + m2 + m3 )] 4 4 6 6 4 4 6 6 4 4 6 + a8 [2769(m6 1 m2 + m1 m2 + m1 m3 + m1 m3 + m2 m3 + m2 m3 ) 2 2 4 4 4 2 2 2 2 2 2 + m2 1 m2 m3 (19851(m1 + m2 + m3 ) + 87945(m1 m2 + m1 m3 + m2 m3 ))] 4 4 2 2 2 2 2 2 ? a6 [295050m4 1 m2 m3 + 2310(m1 m2 + m1 m3 + m2 m3 ) 2 2 4 2 2 4 4 2 2 4 4 2 2 4 + 69510m2 1 m2 m3 (m1 m2 + m1 m2 + m1 m3 + m1 m3 + m2 m3 + m2 m3 )] 2 2 4 4 4 4 4 4 2 2 2 2 2 2 + a4 m2 1 m2 m3 [53839(m1 m2 + m1 m3 + m2 m3 ) + 224485m1 m2 m3 (m1 + m2 + m3 )] 4 4 2 2 2 2 2 2 6 6 6 ? 166896a2 m4 1 m2 m3 (m1 m2 + m1 m3 + m2 m3 ) + 121191m1 m2 m3

F (5)

F (6)

A check of the hypermultiplet decoupling limits of the Nf = 3 instanton corrections, by letting Λ3 m3 = Λ2 2 and sending m3 → ∞, reproduces the Nf = 2 results. A further decoupling of a second hypermultiplet, by letting Λ2 m2 = Λ2 1 and sending mc → ∞, reproduces the Nf = 1 results. Comparison with results in the literature [17][10][14] show an agreement to four instantons up to a rede?nition of the a ?k ’s as discussed in [10]. 10

For SU (3), the existing results in the literature have the instanton corrections expressed in terms of the invariant SU (3) symmetric polynomials u, v and the discriminant ? u = ?a1 a2 ? a1 a3 ? a2 a3 v = a1 a2 a3 ? = 4u3 ? 27v 3 and the p-th symmetric mass polynomials tp (m) =
j1 <···<jp

(25)

mj1 · · · mjp

(26)

Explicit evaluations for Nf = 0, 1, 2, 3, 4, 5 were performed, and are summarized here for Nf = 0. F (1) = Λ6 F (2) = F (3) = F (4) = 3u ? Λ12 u 10935v 2 153 + 2 16 ?3 ? 3Λ18 u 4782969v 4 161109v 2 385 + + 3 16 2?5 2?4 ? 24 6 Λ u 1707362095023v 91216001799v 4 1254600981v 2 + + 4096 ?7 ?6 ?5 3048885 + ?4 5Λ30 u 3788227372819653v 8 277223767370307v 6 + 4096 10?9 10?8 6447389599341v 4 50110037721v 2 7400133 + + + 10?7 10?6 ?5 10 36 2319087386959542567v 8 3Λ u 24952152189682606959v + 65535 2?11 2?10 38185135433846901v 6 525166021552761v 4 + + ?9 ?8 2 5323867298775v 5295230391 + + 7 2? 2?6 11

F (5) =

F (6) =

A check of successive hypermultiplet decoupling limits of the Nf = 5 instanton corrections reproduces all of the Nf < 5 cases accordingly. Comparison with results in the literature [9][10][18][14] show an agreement to three instantons up to a rede?nition of the a ?k ’s as discussed in [10].

7. Summary
The recursion relations discovered in this paper improve considerably the ability to evaluate explicitly the non-perturbative instanton corrections to N =2 super Yang-Mills theories. Possible extensions to other problems like the strongly coupled N =2 SUSY SU (Nc ) Seiberg-Witten problem [20][21] were also investigated [22].

Acknowledgments
We are grateful to D.H. Phong for several helpful discussions and collaboration at the very early stage of this work. G.C. would like to thank NSERC for ?nancial support.

Appendix
A. Classical Moduli in Terms of Quantum Moduli
Another way of evaluating the beta function (7) of the prepotential F involves inverting (8) to get


a ?k ≡ ak +
m=1

? 2m β (m) (a) Λ k

(27)

where the βk (a)’s are functions of the renormalized order parameters ak . A very direct way of deriving the form of the βk (a)’s involves starting o? with (27) and substituting in equation (8) to get


0=
m=1

(m) ? 2m ?i Λ (? a) +

∞ m=1

? 2m βi(m) Λ

∞ m=0

? 2m ?(m) (? Λ a) k

(28)

12

? 2 in the second term and replacing the a Expanding in powers of Λ ?k ’s with ak ’s, one obtains ? βk (a) = ?k (a)
m?1 (m) (m)

+
n=1

1 n!

n?1

r

n

β1 ,···,βn+1 =1 α1 ,···,αn =1 i=1 β1 +···+βn+1 =m

(βi ) ?α (a) ? i

?

? ? (βn+1 ) βk (a) j =1 ?aαj (29)

n

?

? 2 , the ?rst few βk (a)’s are Order by order in powers of Λ ?βk (a) = ?k (a) ?βk (a) = ?k (a) +
l=1 (3) ?βk (a) (2) (2) r (1) (1)

?l (a)
r

(1)

?βk (a) ?al
(1) ?βk (a) ? (2) +?l (a) ?al

(1)

=

(3) ?k (a)

+

1 r ? 2 βk (a) (1) + ?l (a)?(1) m (a) 2! l,m=1 ?al ?am
(4) ?βk (a)

(2) ?β (a) ??(1) (a) k l ?al l=1 (1)

?

?

= +

(4) ?k (a)

r

+
l=1

?

(3) ?l (a)

(1) ?βk (a) ?

??(1) (a)
l

?βk (a) ?β (a) (2) +?l (a) k ?al ?al
r

(3)

(2)

?al

?

+ 2?l (a)?(2) m (a) +

(1)

r 1 ? 3 βk (a) (1) (1) ?l (a)?(1) ( a )? ( a ) m n 3! l,m,n=1 ?al ?am ?an

(1) ? 2 βk (a) ?

1 ? 2 βk (a) ??(1) (a)?(1) (a) + l m 2! l,m=1 ?al ?am
?

?

(2)

?al ?am

(1)

Substituting (29) into (11) reproduces the instanton corrections to the pre? 2. potential (13) order by order in Λ

13

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