Strong Coupling Expansion and Seiberg–witten–whitham Equations

We study the Seiberg-Witten-Whitham equations in the strong coupling regime of the N = 2 super Yang-Mills theory in the vicinity of the maximal singularities. In the case of SU(2) the Seiberg-Witten-Whitham equations fix completely the strong coupling expansion. For higher rank SU(N) they provide a set of non-trivial constraints on the form of this expansion. As an example, we study the off-diagonal couplings at the maximal point for which we propose an ansatz that fulfills all the equations. [email protected] [email protected] The low energy dynamics of N = 2 supersymmetric gauge theories is governed by a single holomorphic function F known as the effective prepotential. A self-consistent proposal for this function has been done a few years ago by Seiberg and Witten [1] in the SU(2) case and generalized to SU(N) by many authors [2]. The Seiberg-Witten solution for the effective theory of N = 2 super Yang-Mills can be embedded into the Whitham hierarchy associated to the periodic Toda lattice [3]. The link between both constructions is summarized in the statement that the prepotential of the N = 2 Yang-Mills theory corresponds to the logarithm of the Toda’s quasiclassical tau function. On the Whitham hierarchy side, there is a family of new variables entering into the prepotential known as slow times. They can be promoted to spurion superfields that softly break N = 2 supersymmetry down to N = 0 with higher Casimir perturbations [4], something that might drive the theory to a vacuum placed in the neighborhood of the so-called Argyres-Douglas singularities [5]. Aside from this physical motivation and its mathematical interest, the Seiberg-Witten-Whitham (SWW) correspondence also provides us with new techniques to explore the structure of the effective action of N = 2 theories. In Ref.[6], for example, the second derivatives of the prepotential with respect to the Whitham times were computed and shown to be given in terms of Riemann Theta functions associated to the root lattice of the gauge group. When evaluated in the SeibergWitten theory, these formulae were shown to provide a powerful tool to compute all instanton corrections to the effective prepotential of SU(N) N = 2 super Yang–Mills theory from the one-loop contribution through a recursive relation [4]. It is the aim of this letter to apply the Seiberg-Witten-Whitham equations to the strong coupling regime of the N = 2 super Yang-Mills theory with gauge group SU(N). We will start by considering N = 2 SU(2) super Yang-Mills theory and show that, starting from the Whitham hierarchy side, the exact effective prepotential near the monopole singularity is obtained in a remarkably simple way up to arbitrary order in the dual variable. We will then consider the generic SU(N) case and show that the SWW equations do not give a closed recursive procedure to perform the same computation. Nevertheless, as we will see, these equations can be used to study the couplings between different magnetic U(1) factors at the maximal singularities of the moduli space. Let us start with a brief description of our general framework. The low-energy dynamics of SU(N) N = 2 super Yang-Mills theory is described in terms of the hyperelliptic curve [2] y = P (λ, uk)− 4Λ , (1) where P (λ, uk) = λ N − ∑k=2 ukλ is the characteristic polynomial of SU(N) and 1 uk, k = 2, ..., N are the Casimirs of the gauge group. This curve can be identified with the spectral curve of the N site periodic Toda lattice and, moreover, the prepotential of the effective theory is essentially the logarithm of the corresponding quasiclassical tau function, hence depending on the slow times Tn, 1 ≤ n ≤ N − 1, of the corresponding Whitham hierarchy [3]. The derivatives of the prepotential with respect to the Whitham slow times have been obtained in Ref.[6]. When restricted to the submanifold Tn>1 = 0, where the Seiberg-Witten solution lives (we may identify T1 with Λ in this submanifold [4]), the first derivatives are simply ∂F ∂ log Λ = β 2πi H2 ∂F ∂Tn = β 2πin Hn+1 , (2) while the second order derivatives with respect to the Whitham slow times result in [4] ∂2F ∂(log Λ)2 = − β 2 2πi ∂H2 ∂ai ∂H2 ∂aj 1 iπ ∂τij log ΘE(0|τ) , ∂2F ∂ log Λ ∂Tn = − β 2 2πin ∂H2 ∂ai ∂Hn+1 ∂aj 1 iπ ∂τij log ΘE(0|τ) , (3) ∂2F ∂Tm∂Tn = − β 2πi ( Hm+1,n+1 + β mn ∂Hm+1 ∂ai ∂Hn+1 ∂aj 1 iπ ∂τij logΘE(0|τ) ) , with m,n ≥ 2 and β = 2N . The functions Hm,n are certain homogeneous combinations of the Casimirs uk, given by Hm+1,n+1 = N mn res∞ ( P(λ)dP n/N + (λ) ) and Hm+1 ≡ Hm+1,2 = um+1 +O(um) . Here (