Inverse Higgs phenomena as duals of holonomic constraints


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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.

Publication Date
2022
Online Publication Date
2022-02-07
Acceptance Date
2022-01-18
Keywords
inverse Higgs constraints, goldstone bosons, jet manifolds, differential geometry, holonomic constraints, non-holonomic constraints
Journal Title
Journal of Physics A: Mathematical and Theoretical
Journal ISSN
1751-8113
1751-8121
Volume Title
55
Publisher
IOP Publishing
Sponsorship
Science and Technology Facilities Council (ST/P000681/1)
Science and Technology Facilities Council (ST/S505316/1)
STFC (ST/T000694/1)