# The first law and Wald entropy formula of heterotic stringy black holes at first order in α

- Elgood, Zachary

- Tomás Ortín Miguel Director

Defence university: Universidad Autónoma de Madrid

Fecha de defensa: 29 September 2022

- José Luis Fernández Barbón Chair
- Pablo Bueno Gómez Secretary
- Bert Janssen Committee member
- José Manuel Izquierdo Rodríguez Committee member
- Luis Javier Garay Elizondo Committee member

Type: Thesis

## Abstract

Black-hole thermodynamics is probably one of the most active fields of research in The- oretical Physics. It interconnects seemingly disparate areas of Physics such as Gravity, Quantum Field Theory, and Information Theory, providing deep insights in all of them. While initially valid only for General Relativity, Wald and collaborators developed a new approach to demonstrate the first law of black hole mechanics in general diffeomorphism- invariant theories, beyond General Relativity. As a by-product, this approach lead to the identification of an expression that plays the role of entropy (Wald entropy) in the first law in theories beyond General Relativity. However, the first laws and the entropy formulas derived in the literature with this formalism (the Iyer-Wald prescription) present severe shortcomings in certain string theories, such as missing work terms in the first laws and lack of gauge invariance of the entropy formula. This prevents a fair comparison with the microscopic entropy computed using other techniques (AdS/CFT correspondence etc.). The main goal of this thesis is to identify the roots of these problems and fix them. As we will see, the root of these problems is the inadequate treatment of the fields that exhibit some kind of gauge freedom. These are, as a matter of fact, all fields except for scalars and the metric (if one does not use the vielbein formalism). This thesis is divided into two parts. The first section will involve compactifying the heterotic string action on S1 , allowing us to compute re-derive the Buscher rules and prove T duality. We will then use the Iyer-Wald formula in the dimensionally reduced action to derive an entropy formula that can be applied to black-hole solutions which can be obtained by a single non-trivial compactification on a circle and discuss its invariance under the α0 -corrected T duality transformations. Specifically, we shall apply it to the Strominger-Vafa extremal black hole. We will demonstrate that in addition to the lack of gauge invariance, there exists an ambiguity in applying the formula, as applying it to d = 10 and d = 5 yields two different results that differ by a factor of 2. As previously mentioned, Iyer-Wald formula cannot be applied unambiguously in the case of the heterotic string case, as one of the main assumptions was that all fields behaved as tensors. However, all fields apart from the metric and scalars possess gauge freedoms, and their transformations under diffeomorphisms are always coupled to gauge transformations. This serves as motivation for the second section of the thesis, where we determine the first law of black hole thermodynamics in a gauge-invariant way, introducing gauge-covariant transformations under diffeomorphisms (gauge covariant Lie derivatives). The construction of these transformations involves the definition of “momentum maps” associated to field strengths and the vectors that generate their symmetries. These objects play the role of generalized thermodynamical potentials in the first law and satisfy the restricted generalized zeroth laws. After testing our ideas on the d-dimensional Reissner-Nordström-Tangherlini black hole in the context of the Einstein-Maxwell theory, we turn our focus to the heterotic string case. Initially, we examine the case of the heterotic string theory up to zeroth order α0 compactified on a torus. This theory is interesting because of the black-hole solutions it admits, and because of the Abelian Chern-Simons terms present in the Kalb- Ramond 3-form field strength. The presence of those terms induces the so-called Nicolai- Townsend gauge transformations of the Kalb-Ramond 2-form. These terms and gauge transformations, appear in the 10-dimensional theory at first order in α' in a much more complicated way (non-Abelian, gravitational) and this model can be used as a toy model to test our ideas. We show how to deal with all these gauge symmetries deriving the first law in terms of manifestly gauge-invariant quantities. Explicitly, we will demonstrate this in the case of a non-extremal, charged, black ring solution of pure N = 1, d = 5 supergravity embedded in the Heterotic Superstring effective field theory. In the final chapter, we arrive at our main result, based on the work of the previous chapters. We derive the first law of black hole mechanics in the context of the Heterotic Superstring effective action to first order in α' using Wald’s formalism, taking into account all the symmetries of the theory. This requires additional care due to the presence of the non-Abelian Lorentz and Yang-Mills Chern-Simon terms found in the Kalb-Ramond field strength. As a result, we obtain a manifestly gauge- and Lorentz-invariant entropy formula in which all the terms can be computed explicitly. An entropy formula with these properties allows unambiguous calculations of macroscopic black-hole entropies to first order in α' that can be reliably used in a comparison with the microscopic ones. Such a formula was still lacking in the literature