Holomorphic Blocks in Three Dimensions

TL;DR

This paper introduces holomorphic blocks as fundamental components of 3D N=2 gauge theory partition functions, revealing their properties, calculation methods, and dual interpretations in various physical contexts.

Contribution

It proposes a new technique for calculating holomorphic blocks and explores their properties and dualities, connecting gauge theories with Chern-Simons and topological string theories.

Findings

Holomorphic blocks decompose sphere partition functions and indices.

Blocks correspond to massive vacua and BPS states.

New effective method inspired by supersymmetric quantum mechanics.

Abstract

We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.