Komar integral and Smarr formula for axion-dilaton black holes versus S duality

TL;DR

This paper develops a duality-invariant Komar integral for axion-dilaton black holes, deriving a Smarr relation that accounts for electric-magnetic duality and extends the understanding of black hole thermodynamics in these theories.

Contribution

It introduces a new, invariant Komar integral formalism for axion-dilaton gravity and derives a corresponding Smarr relation considering duality symmetries.

Findings

Komar integral is invariant under SL(2,R) duality.

Derived a generalized Smarr formula for axion-dilaton black holes.

Validated the formula with static black hole solutions.

Abstract

We construct the Komar integral for axion-dilaton gravity using Wald's formalism and momentum maps and we use it to derive a Smarr relation for stationary axion-dilaton black holes. While the Wald-Noether 2-form charge is not invariant under SL(2,R) electric-magnetic duality transformations because Wald's formalism does not account for magnetic charges and potentials, the Komar integral constructed with it turns out to be invariant and, in more general theories, it will be fully symplectic invariant. We check the Smarr formula obtained with the most general family of static axion-dilaton black holes.