Phases of $\mathcal{N}=4$ SYM, S-duality and Nilpotent Cones

TL;DR

This paper explores the structure of vacua in four-dimensional $ abla$=4 SYM theory, using stratified spaces and Lie algebra sheets to analyze S-duality and boundary phenomena relevant to geometric Langlands programs.

Contribution

It introduces a stratified space framework for vacua in $ abla$=4 SYM, connecting Lie algebra sheets, S-duality, and boundary symmetry breaking to geometric Langlands theory.

Findings

Describes the space of vacua as a stratified space with different effective theories on each stratum.

Links the theory of sheets in Lie algebras to specific strata of $ abla$=4 SYM.

Identifies the Local and Global Nilpotent Cones as arising from boundary symmetry breaking.

Abstract

In this note, I describe the space of vacua $\mathcal{V}$ of four dimensional $\mathcal{N}=4$ SYM on $\mathbb{R}^4$ with gauge group a compact simple Lie Group $G$ as a stratified space. On each stratum, the low energy effective field theory is different. This language allows one to make precise the idea of moving in the space of vacua $\mathcal{V}$. A particular subset of the strata of $\mathcal{N}=4$ SYM can be efficiently described using the theory of sheets in a Lie algebra. For these strata, I study the conjectural action of S-duality. I also indicate some benefits of using such a language for the study of the available space of vacua $\overline{\mathcal{V}}$ on the boundary of GL twisted $\mathcal{N}=4$ SYM on a half-space $\mathbb{R}^3 \times \mathbb{R}^+$. As an application of boundary symmetry breaking, I indicate how a) the Local Nilpotent Cone arises as part of the available symmetry breaking choices on the boundary of the four dimensional theory and b) the Global Nilpotent Cone arises in the theory reduced down to two dimensions on a Riemann Surface $C$. These geometries play a critical role in the Local and Global Geometric Langlands Program(s).