Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Down the rabbit hole with theories of class S

Shlomo S. Razamat, Brian Willett

TL;DR

The paper addresses how to extract physical and mathematical structure of 4d class S theories by dimensionally reducing to 3d N=4 theories, where partition functions and mirror symmetry become powerful computational tools.It develops explicit 3d constructions based on T[SU(2)] building blocks and star-shaped quivers, deriving partition functions, holomorphic blocks, and dualities that reproduce and illuminate 4d results, including surface defects and S-duality relations.A central theme is the mapping between 4d difference operators and 3d line operators, with T[SU(2)] blocks shown to be eigenfunctions of these operators, and the Hall-Littlewood/Schur limits providing clean ties between 4d Higgs/ Coulomb data and 3d moduli-space indices.The work culminates in a cohesive dictionary linking 4d Hall-Littlewood indices to 3d Higgs/Coulomb indices, and in connecting Macdonald polynomial structures to q-integral representations, thereby forging a comprehensive 4d–3d correspondence for class S theories.

Abstract

We review some of the properties of 3d N=4 theories obtained by dimensionally reducing theories of class S. We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.

Down the rabbit hole with theories of class S

TL;DR

The paper addresses how to extract physical and mathematical structure of 4d class S theories by dimensionally reducing to 3d N=4 theories, where partition functions and mirror symmetry become powerful computational tools.It develops explicit 3d constructions based on T[SU(2)] building blocks and star-shaped quivers, deriving partition functions, holomorphic blocks, and dualities that reproduce and illuminate 4d results, including surface defects and S-duality relations.A central theme is the mapping between 4d difference operators and 3d line operators, with T[SU(2)] blocks shown to be eigenfunctions of these operators, and the Hall-Littlewood/Schur limits providing clean ties between 4d Higgs/ Coulomb data and 3d moduli-space indices.The work culminates in a cohesive dictionary linking 4d Hall-Littlewood indices to 3d Higgs/Coulomb indices, and in connecting Macdonald polynomial structures to q-integral representations, thereby forging a comprehensive 4d–3d correspondence for class S theories.

Abstract

We review some of the properties of 3d N=4 theories obtained by dimensionally reducing theories of class S. We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.

Paper Structure

This paper contains 12 sections, 133 equations.

Table of Contents

  1. Introduction
  2. ${\cal N}=2$ class ${\cal S}$ theories in $4d$
  3. Brief review of ${\cal N}=4$ $3d$ generalities
  4. ${\cal N}=4$ $3d$ partition functions
  5. Useful limits of the $3d$ partition functions
  6. Examples
  7. Down the rabbit hole
  8. Partition Functions of $T(SU(2))$
  9. Class ${\frak s}$ theories and their star-shaped quiver mirrors
  10. Line operators, difference operators, and eigenfunctions
  11. Poles
  12. $\dots$ and back again: $4d$ shards of $3d$ mirrors