Spontaneous symmetry breaking: a view from derived geometry

TL;DR

This paper explores symmetry breaking in field theories using derived geometry and the BV formalism, providing elegant formulations and reformulations of gauge-fixing conditions in the context of the Higgs mechanism.

Contribution

It introduces a derived geometric perspective on symmetry breaking in field theories, offering new reformulations of gauge-fixing conditions within the BV formalism.

Findings

Derived geometry captures symmetry breaking elegantly.

Reformulation of 't Hooft's gauge-fixing conditions in BV formalism.

Enhanced understanding of the Higgs mechanism through geometric language.

Abstract

We examine symmetry breaking in field theory within the framework of derived geometry, as applied to field theory via the Batalin-Vilkovisky formalism. Our emphasis is on the standard examples of Ginzburg-Landau and Yang-Mills-Higgs theories and is primarily interpretive. The rich, sophisticated language of derived geometry captures the physical story elegantly, allowing for sharp formulations of slogans (e.g., for the Higgs mechanism, that the unstable ghosts eat the Goldstone modes). Rewriting these results in the BV formalism provides, as one nice payoff, a reformulation of 't Hooft's family of gauge-fixing conditions for spontaneously broken gauge theory that behaves well in the $\xi \to \infty$ limit.