Inverse Higgs phenomena as duals of holonomic constraints

TL;DR

This paper introduces a geometric framework for understanding inverse Higgs phenomena, revealing a duality with holonomic constraints that simplifies analysis and broadens the scope of applicable systems.

Contribution

It formulates inverse Higgs phenomena using differential geometry, identifying coholonomic constraints as duals to holonomic constraints, and clarifies their role in physical systems with Goldstone bosons.

Findings

Coholonomic constraints are dual to holonomic constraints in a categorical sense.

Systems with coholonomic constraints are equivalent to unconstrained systems, simplifying their study.

The formalism clarifies assumptions and extends applicability to more general dynamical systems.

Abstract

The inverse Higgs phenomenon, which plays an important r\^ole in physical systems with Goldstone bosons (such as the phonons in a crystal) involves nonholonomic mechanical constraints. By formulating field theories with symmetries and constraints in a general way using the language of differential geometry, we show that many examples of constraints in inverse Higgs phenomena fall into a special class, which we call coholonomic constraints, that are dual (in the sense of category theory) to holonomic constraints. Just as for holonomic constraints, systems with coholonomic constraints are equivalent to unconstrained systems (whose degrees of freedom are known as essential Goldstone bosons), making it easier to study their consistency and dynamics. The remaining examples of inverse Higgs phenomena in the literature require the dual of a slight generalisation of a holonomic constraint, which we call (co)meronomic. Our formalism simplifies and clarifies the many ad hoc assumptions and constructions present in the literature. In particular, it identifies which are necessary and which are merely convenient. It also opens the way to studying much more general dynamical examples, including systems which have no well-defined notion of a target space.