The Cosmological Constant and Volume-Preserving Diffeomorphism Invariants

TL;DR

This paper explores the quantum field theories of $(D-1)$-form fields on $D$-dimensional manifolds, revealing how observables relate to spacetime volume, intersection numbers, and scalar couplings, with implications for thermodynamics.

Contribution

It introduces a detailed analysis of observables in $(D-1)$-form field theories, linking them to geometric invariants and scalar potentials, and examines their thermodynamic behavior.

Findings

Vacuum expectation value of $T_{ab}$ depends on spacetime volume.

Correlation functions yield intersection numbers on the manifold.

Scalar couplings induce potentials after integrating out the form fields.

Abstract

Observables in the quantum field theories of $(D-1)$-form fields, $\ca$, on $D$-dimensional, compact and orientable manifolds, $M$, are computed. Computations of the vacuum value of $T_{ab}$ find it to be the metric times a function of the volume of spacetime, $\Omega(M)$. Part of this function of $\Omega$ is a finite zero-mode contribution. The correlation functions of another set of operators give intersection numbers on $M$. Furthermore, a similar computation for products of Wilson area operators results in a function of the volumes of the intersections of the submanifolds the operators are defined on. In addition, scalar field couplings are introduced and potentials are induced after integrating out the $\ca$ field. Lastly, the thermodynamics of the pure theories is found to be analogous to the zero-point motion of a scalar particle. The coupling of a Gaussian scalar field to the $\ca$ field is found to manifest itself on the free energy at high temperatures and/or small volumes.