Multisymplectic Constraint Analysis of Scalar Field Theories, Chern-Simons Gravity, and Bosonic String Theory

TL;DR

This paper advances the multisymplectic geometric framework for various field theories, analyzing their constraint structures and symmetries, including scalar fields, string theory, and Chern-Simons gravity, highlighting singular and regular cases.

Contribution

It develops the multisymplectic constraint analysis for diverse field theories, revealing their singular or regular nature and how symmetries are affected by constraints.

Findings

Scalar and Chern-Simons Lagrangians are singular in De Donder--Weyl formalism.

Nambu-Goto Lagrangian for string theory is regular.

Symmetries are influenced by the constraint structure in multisymplectic phase space.

Abstract

The (pre)multisymplectic geometry of the De Donder--Weyl formalism for field theories is further developed for a variety of field theories including a scalar field theory from the canonical Klein-Gordon action, the electric and magnetic Carrollian scalar field theories, bosonic string theory from the Nambu-Goto action, and $2+1$ gravity as a Chern-Simons theory. The Lagrangians for the scalar field theories and for $2+1$ Chern-Simons gravity are found to be singular in the De Donder--Weyl sense while the Nambu-Goto Lagrangian is found to be regular. Furthermore, the constraint structure of the premultisymplectic phase spaces of singular field theories is explained and applied to these theories. Finally, it is studied how symmetries are developed on the multisymplectic phase spaces in the presence of constraints.