General Actions of Extended Objects and Volume-Preserving Diffeomorphism

TL;DR

This paper explores actions for extended objects with various metric symmetries, demonstrating classical equivalences and the significance of volume-preserving diffeomorphisms, extending known string theories to higher dimensions.

Contribution

It establishes the classical equivalence of different metric-dependent actions and highlights the role of volume-preserving diffeomorphisms in string and extended object theories.

Findings

All nontrivial actions with both worldsheet and induced metrics are classically equivalent.

VPD symmetry is as restrictive as full diffeomorphism symmetry for classical actions.

Generalized Schild and Nambu-Goto actions are equivalent for higher-dimensional objects.

Abstract

We consider actions that are general functions of the worldsheet/worldvolume metric and the induced metric for extended objects embedded in spacetime as Riemannian manifolds, areal-metric manifolds, and volume-metric manifolds. For strings on a Riemannian spacetime, we consider general actions respecting volume-preserving diffeomorphisms (VPD), general diffeomorphisms, and diffeomorphisms with Weyl symmetry, respectively. Well-known Schild, Nambu-Goto, and Polyakov actions are included as special cases. We reach two main conclusions: (1) When actions are functions of both the worldsheet metric and induced metrics, all nontrivial self-consistent actions are classically equivalent. (2) As a physical constraint on the classical action, VPD symmetry is as strong as the full diffeomorphism symmetry. The discussion is then extended to strings in spacetime manifolds equipped with the areal or volume metrics. Then, we further consider higher-dimensional extended objects in spacetime defined with areal or volume metrics, and show the equivalence between the generalized Schild actions and the generalized Nambu-Goto action. We prove a general theorem on VPD that explains this equivalence. Incidentally, while only the areal metric is needed to define the string worldsheet action, we show that the Polyakov action with an areal-metric perturbation cannot describe critical strings without other interaction terms.