Gravitational corrections in supersymmetric gauge theory and matrix models

Abstract

Gravitational corrections in N = 1 and N = 2 supersymmetric gauge theories are obtained from topological string amplitudes. We show how they are recovered in matrix model computations. This provides a test of the proposal by Dijkgraaf and Vafa beyond the planar limit. Both, matrix model and topological string theory, are used to check a conjecture of Nekrasov concerning these gravitational couplings in Seiberg-Witten theory. Our analysis is performed for those gauge theories which are related to the cubic matrix model, i.e. pure SU (2) SeibergWitten theory and N = 2 U (N ) SYM broken to N = 1 via a cubic superpotential. We outline the computation of the topological amplitudes for the local Calabi-Yau manifolds which are relevant for these two cases.

Quantum geometry of resurgent perturbative/nonperturbative relations

“…In the gauge theory formalism, this "quantum Matone relation" generalizes the classical Matone relation [87] with the inclusion of gravitational couplings [88][89][90][91][92], and it also has a natural interpretation in all-orders WKB [27,60,63]. At the classical level, → 0, the relation (1.5) is a simple consequence of the associated classical Picard-Fuchs equation, which characterizes the energy dependence of the classical action variables (see section 3.1.1 below).…”

“…It is remarkable that for a class of quantum problems, when the actions become non-trivial functions of , the only modification to the classical Wronskian condition, to all-orders in , is the extra derivative terms on the left-hand-side of (1.6). Ultimately this fact can be traced to the N = 2 supersymmetry of the related quantum field theories [42][43][44][88][89][90][91][92].…”

Chiral trace relations in Ω-deformed N = 2 $$ \mathcal{N}=2 $$ theories

“…This 4d Poincaré breaking deformation depends on two parameters ε 1 , ε 2 and is the main ingredient in the localization approach of [3,5,48] where it is needed to regularize the multi-instanton moduli space [4,6,8,[49][50][51][52][53]. 3 The Ω-deformed prepotential expanded around the undeformed limit is the generating function of higher genus amplitudes of the N = 2 topological string [18,[58][59][60][61][62][63] and satisfies a holomorphic anomaly equation [17,[64][65][66]. At finite ε 1 , ε 2 , the (deformed) partition function is also a fundamental object within the AGT correspondence [11].…”

“…Investigations in this direction have been discussed in [82,83]. Another option is to exploit the topological string in the spirit of [66,84,85]. Finally, one can exploit AGT correspondence.…”

Determinant formulas for matrix model free energy

“…The authors of [5] proposed a new anzatz for F 1 in the two-cut case (with absent double points) and made a perturbative check. Their formula in fact comes from the correspondence between the so called topological B-model on the local Calabi-Yau geometry II and the cubic matrix model.…”