Abstract
Abstract We study the non-perturbative properties of $ \mathcal{N}=2 $ super conformal field theories in four dimensions using localization techniques. In particular we consider SU(2) gauge theories, deformed by a generic ϵ-background, with four fundamental flavors or with one adjoint hypermultiplet. In both cases we explicitly compute the first few instanton corrections to the partition function and the prepotential using Nekrasov’s approach. These results allow us to reconstruct exact expressions involving quasi-modular functions of the bare gauge coupling constant and to show that the prepotential terms satisfy a modular anomaly equation that takes the form of a recursion relation with an explicitly ϵ-dependent term. We then investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory.
Highlights
Coupling receives perturbative corrections at 1-loop and non-perturbative corrections due to instantons, and the corresponding effective action follows from a prepotential F that is a holomorphic function of the vacuum expectation value a of the adjoint vector multiplet, of the flavor masses, if any, and of the dynamically generated scale in asymptotically free theories or of the bare gauge coupling constant τ0 in conformal models
We investigate the implications of this recursion relation on the modular properties of the effective theory and find that with a suitable redefinition of the prepotential and of the effective coupling it is possible, at least up to the third order in the deformation parameters, to cast the S-duality relations in the same form as they appear in the Seiberg-Witten solution of the undeformed theory
As shown in [4] for the N = 2∗ theory and more recently in [5] for the Nf = 4 theory, by organizing the effective prepotential F as a series in inverse powers of a and by exploiting a recursion relation hidden in the SW curve, it is possible to write the various terms of F as exact functions of the bare coupling
Summary
We can express the instanton partition functions Zk of the Nf = 4 theory as sums of terms related to pairs of Young tableaux. The a-independent terms in (3.3), instead, are not invariant under the SO(8) flavor group. That a-independent terms in the partition functions, and in the prepotential, are irrelevant for the gauge theory dynamics, and can be neglected. The second line represents the contribution of the gauge vector multiplet and the last line that of the fundamental hypermultiplets. From this expression, by selecting the appropriate Young tableaux, one can obtain the various terms of the instanton partition function and their dependence on the ǫl parameters
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