Boundary charges and integral identities for solitons in $(d+1)$-dimensional field theories

TL;DR

This paper develops a new set of integral identities for solitons in various field theories, introducing boundary charges and generalizing known identities like Pohozaev's, applicable to topological defects and textures in multiple dimensions.

Contribution

It introduces a 3-parameter family of integral identities with boundary charges for spherically symmetric solitons, expanding the analytical tools beyond Derrick's theorem and virial relations.

Findings

Derived five boundary charges related to soliton profile functions

Generalized the Pohozaev identity for a broad class of theories

Connected topological winding numbers to integral identities in vortex solutions

Abstract

We establish a 3-parameter family of integral identities to be used on a class of theories possessing solitons with spherical symmetry in $d$ spatial dimensions. The construction provides five boundary charges that are related to certain integrals of the profile functions of the solitons in question. The framework is quite generic and we give examples of both topological defects (like vortices and monopoles) and topological textures (like Skyrmions) in 2 and 3 dimensions. The class of theories considered here is based on a kinetic term and three functionals often encountered in reduced Lagrangians for solitons. One particularly interesting case provides a generalization of the well-known Pohozaev identity. Our construction, however, is fundamentally different from scaling arguments behind Derrick's theorem and virial relations. For BPS vortices, we find interestingly an infinity of integrals simply related to the topological winding number.