Spontaneous symmetry breaking: A view from derived geometry

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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 feed the Goldstone bosons to a hungry, massless gauge boson). 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 ξ limit.

Section snippets

An overview of the problem and our perspective

We start by sketching the basic situation mathematically; in the following sections we will unpack the quintessential example of symmetry breaking in both physical and mathematical styles.

Recall the broad outlines of classical field theory. We begin with a space F whose elements we call the fields; typically F consists of sections of a fibre bundle over a (super)manifold. There is also a function S on the space F that we call the action functional, and according to the principle of least

Breaking a global symmetry

Let us consider a field theory on which a group G acts by symmetries. Since the space of all solutions is preserved by the action of G, each solution ϕ0 lives on some G-orbit, which has the form GH with H=Stab(ϕ0). Note that the variational derivative δS of the action must vanish in the orbit directions, but not in the directions transverse to the orbit.

When we examine the formal neighbourhood of ϕ0 in the space of all solutions, it is natural to ask how the Lie algebra h=Lie(H) acts on this

Breaking a gauge symmetry

We now turn to the more interesting situation: breaking of gauge symmetries. We will spend most of our time analysing two classic examples, but before delving into them, we outline the basic setup.

In brief, consider a theory with gauge symmetries, which has a global derived stack SolG of solutions up to gauge equivalence. If one examines the BV theory at a solution x, its linearization around x corresponds to a free theory. Let Tx denote the cochain complex encoding this free theory. In

Towards quantization of Yang–Mills–Higgs theories

It is beyond the scope of this paper to construct the perturbative quantization of the BV theory for the spontaneously broken gauge symmetry. For those familiar with the textbook story, however, we rapidly sketch how the ’t Hooft (aka Rξ) gauges appear in our articulation of the theories. With those in hand, it is possible to deploy the usual power-counting arguments to verify renormalizability à la ’t Hooft–Veltman.

Acknowledgements

We benefited from discussions with a number of people, notably Ingmar Saberi, Brian Williams, and Philsang Yoo. We are also grateful to Eugene Rabinovich for his helpful comments on an earlier draft. We thank in particular David Carchedi, who gave us extensive feedback on the draft and has discussed further refinements of this approach, and John Huerta, whose careful reading and thoughtful suggestions caught many issues and improved the paper substantially. The National Science Foundation,

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