Remarks on effective action and entanglement entropy of Maxwell field in generic gauge

TL;DR

This paper investigates how the effective action and entanglement entropy in Maxwell theory depend on gauge fixing, demonstrating gauge invariance of certain contact terms through heat kernel analysis and regulator comparisons.

Contribution

It provides a detailed analysis of gauge fixing dependence in Maxwell theory's effective action and entanglement entropy, establishing gauge invariance of the Kabat contact term.

Findings

Effective action and entanglement entropy are gauge fixing independent under certain regulator conditions.

The heat kernel coefficients for nonminimal operators can be expressed via minimal Laplacian operators.

The gauge invariance of the Kabat contact term is supported by regulator analysis.

Abstract

We analyze the dependence of the effective action and the entanglement entropy in the Maxwell theory on the gauge fixing parameter $a$ in $d$ dimensions. For a generic value of $a$ the corresponding vector operator is nonminimal. The operator can be diagonalized in terms of the transverse and longitudinal modes. Using this factorization we obtain an expression for the heat kernel coefficients of the nonminimal operator in terms of the coefficients of two minimal Beltrami-Laplace operators acting on 0- and 1-forms. This expression agrees with an earlier result by Gilkey et al. Working in a regularization scheme with the dimensionful UV regulators we introduce three different regulators: for transverse, longitudinal and ghost modes, respectively. We then show that the effective action and the entanglement entropy do not depend on the gauge fixing parameter $a$ provided the certain ($a$-dependent) relations are imposed on the regulators. Comparing the entanglement entropy with the black hole entropy expressed in terms of the induced Newton's constant we conclude that their difference, the so-called Kabat's contact term, does not depend on the gauge fixing parameter $a$. We consider this as an indication of gauge invariance of the contact term.