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General actions of extended objects and volume-preserving diffeomorphism
Phys. Rev. D 113, 126008 – Published 5 June, 2026
DOI: https://doi.org/10.1103/hg29-jvx5
Abstract
We consider actions that are general functions of the world sheet/world volume metric and the induced metric for extended objects embedded in spacetime as Riemannian manifolds, area-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, respectivelyn. 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 area or volume metrics. Then, we further consider higher-dimensional extended objects in spacetime defined with area 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 area metric is needed to define the string world sheet action, we show that the Polyakov action with an area-metric perturbation cannot describe critical strings without other interaction terms.
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