Spontaneous symmetry breaking: a view from derived geometry

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.