N=2 gauge theories, instanton moduli spaces and geometric representation theory

Abstract

We survey some of the AGT relations between N = 2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces. We treat the cases of gauge theories on both flat space and ALE spaces in some detail, and with emphasis on the implications arising from embedding them into supersymmetric theories in six dimensions. Along the way we construct new toric noncommutative ALE spaces using the general theory of complex algebraic deformations of toric varieties, and indicate how to generalise the construction of instanton moduli spaces. We also compute the equivariant partition functions of topologically twisted six-dimensional Yang-Mills theory with maximal supersymmetry in a general Ω-background, and use the construction to obtain novel reductions to theories in four dimensions.

Elliptic non-Abelian Donaldson-Thomas invariants of ℂ3

“…We underline that in the QM case the higher-rank result does not factorise in Abelian contributions, due to the presence of non-trivial twisted sectors under the orbifold. We instead confirm that the factorisation holds in the matrix model limit, as conjectured in [26] and verified in [44,53]. The relevant formula for the matrix model case is (3.23), that we checked with our techniques up to 8 th order in the instanton expansion.…”

“…By a further reduction on a second circle, we obtain a SUSY matrix integral with 4 supercharges. This case, known as rational, has been studied in [44]. It can be obtained from the trigonometric case in the limit β → 0, where β is the radius of the circle used to compute the Witten index in the path integral formulation.…”

“…Notice that now the function h has an index α that clarifies which plane partition in the coloured set it refers to. These expressions are the elliptic version of similar equations in [44], where the rational case was analysed. The dimensional reduction of these formulas to the trigonometric case is the following:…”

A Combinatorial Study on Quiver Varieties

“…For recent advances, see, e.g., [6,53] and references therein. In [53, Section 2], results of this paper are nicely recast into the framework of instanton counting on quotient stacks [C 2 /Z l ].…”

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

“…The equivariant partition function, therefore, depends on three equivariant parameters q 1,2,3 = e 1,2,3 . The equivariant integrals over the Q-closed field configurations localize on the fixed points, which are labelled by plane partitions, so that the instanton partition function is given by the sum over fixed points each taken with its equivariant index 4 [88,[130][131][132]]…”