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Register a new userInstantons on ALE spaces and orbifold partitions
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We consider N=4 theories on ALE spaces of $A_{k-1}$ type. As is well known, their partition functions coincide with $A_{k-1}$ affine characters. We show that these partition functions are equal to the generating functions of some peculiar classes of partitions which we introduce under the name orbifold partitions. These orbifold partitions turn out to be related to the generalized Frobenius partitions introduced by G. E. Andrews some years ago. We relate the orbifold partitions to the blended partitions and interpret explicitly in terms of a free fermion system.
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We study SYM gauge theories living on ALE spaces. Using localization formulae we compute the prepotential (and its gravitational corrections) for SU(N) supersymmetric ${cal N}=2, 2^*$ gauge theories on ALE spaces of the $A_n$ type. Furthermore we derive the Poincar{e} polynomial describing the homologies of the corresponding moduli spaces of self-dual gauge connections. From these results we extract the ${cal N}=4$ partition function which is a modular form in agreement with the expectations of $SL(2,Z)$ duality.
We generalize Nakajima-Yoshioka blowup equations to arbitrary gauge group with hypermultiplets in arbitrary representations. Using our blowup equations, we compute the instanton partition functions for 4d N=2 and 5d N=1 gauge theories for arbitrary gauge theory with a large class of matter representations, without knowing explicit construction of the instanton moduli space. Our examples include exceptional gauge theories with fundamentals, SO(N) gauge theories with spinors, and SU(6) gauge theories with rank-3 antisymmetric hypers. Remarkably, the instanton partition function is completely determined by the perturbative part.
We derive the classical type IIB supergravity solution describing fractional D3-branes transverse to a C^2/Gamma orbifold singularity, for Gamma any Kleinian ADE subgroup. This solution fully describes the N=2 gauge theory with appropriate gauge groups and matter living on the branes, up to non-perturbative instanton contributions.
We compute the prepotential for gauge theories descending from ${cal N}=4$ SYM via quiver projections and mass deformations. This accounts for gauge theories with product gauge groups and bifundamental matter. The case of massive orientifold gauge theories with gauge group SO/Sp is also described. In the case with no gravitational corrections the results are shown to be in agreement with Seiberg-Witten analysis and previous results in the literature.
We investigate the classical geometry corresponding to a collection of fractional D3 branes in the orbifold limit of an ALE space. We discuss its interpretation in terms of the world-volume gauge theory on the branes, which is in general a non conformal N=2 Yang-Mills theory with matter. The twisted fields reproduce the perturbative behaviour of the gauge theory. We regulate the IR singularities for both twisted and untwisted fields by means of an enhancon mechanism qualitatively consistent with the gauge theory expectations. The five-form flux decreases logarithmically towards the IR with a coefficient dictated by the gauge theory beta-functions.
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