3d expansions of 5d instanton partition functions

Abstract

We propose a set of novel expansions of Nekrasov's instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on C 2 q,t −1 × S 1 , we show that the instanton partition function admits expansions in terms of partition functions of unitary gauge theories living on the 3d subspaces C q × S 1 , C t −1 × S 1 and their intersection along S 1 . These new expansions are natural from the BPS/CFT viewpoint, as they can be matched with W q,t correlators involving an arbitrary number of screening charges of two kinds. Our constructions generalize and interpolate existing results in the literature.

Bootstrapping the S5 partition function

“…While our Higgs branch result formally follows from the residue calculation of the matrix integral, we do not have a rigorous proof of its equivalence with the Coulomb branch expression due to some assumption on the contributing poles. However, we support our proposal by showing that it has all the expected properties, both from the the gauge theory and 5d AGT perspectives: i) the partition function has poles at special values of the mass parameters due to the pinching mechanism of the integration contour, and taking the residues can be interpreted as RG flows to defect theories; our proposal indeed assumes the form of a discrete sum whose summands capture the partition functions of codimension 2 and 4 defects which are consistent with the Higgs phase from the brane picture; ii) our proposal looks like the compact space version of a similar result in the 5d Ω-background [58], which can be proven combinatorially; iii) the defect partition functions can manifestly be constructed through the q-Virasoro modular triple [59], in agreement with [35,36]; quite interestingly, it turns out that the free boson correlators form a natural basis for solutions to q-deformed Ward identities or q-conformal blocks, and a similar approach has been recently used to study W 3 conformal blocks of Toda 2d CFT [60].…”

“…In practice, the Coulomb branch parameters are set to special points where the instanton series truncates to a vortex partition function, and it turns out that the generating function of the defect theory also satisfies analogous constraints [78]. This observation has led us to the study of codimension 2 (and 4) expansions of the instanton series [58], finding that the full Ωbackground partition function can be naturally assembled from lower dimensional pieces.…”

On refined Chern–Simons and refined ABJ matrix models

“…The partition function of ABJ theory can be expressed as a matrix integral over two sets of independent variables {x i } N i=1 and {y a } M a=1 which can be interpreted as the eigenvalues of an Hermitean super-matrix in the Lie super-algebra of U (N |M). The usual Vandermonde determinant in the measure is first substituted by the Cauchy determinant and then q, t-deformed to the function (see [40][41][42])…”

Intersecting defects and supergroup gauge theory

“…We are thus left with a 3d-1d defect coupled to a free bulk, and the non-trivial vortex-like part of the partition function coincides with the specialization of the instanton partition function. Compact and non-compact space versions of our setup were already studied in several papers [32][33][34][35]. The deformation parameters q and t −1 are related to the usual 4d conventions as follows…”