Spontaneous symmetry breaking: A view from derived geometry
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 whose elements we call the fields; typically consists of sections of a fibre bundle over a (super)manifold. There is also a function on the space 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 acts by symmetries. Since the space of all solutions is preserved by the action of , each solution lives on some -orbit, which has the form with . Note that the variational derivative 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 in the space of all solutions, it is natural to ask how the Lie algebra 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 of solutions up to gauge equivalence. If one examines the BV theory at a solution , its linearization around corresponds to a free theory. Let 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 ) 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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