The Cano-Ruipérez 4-dimensional string effective action is the simplest consistent truncation of the first-order in $\textit{α}$ heterotic string effective action compactified on T . This theory contains, on top of the metric, a dilaton and an axion that couple to the Gauss-Bonnet term and to the Pontrjagin density which suggests a very strong relation between the physical and geometrical properties of its solutions. In this paper we study the thermodynamics of the string black-hole solutions of this theory using Wald’s formalism. We construct the on-shell closed generalized Komar, dilaton and axion 2-form charges, and we used them to find the Smarr formula (that we test on the Schwarzschild and Kerr solutions) and to prove no-(primary)-hair theorems for the scalar. We find that $\textit{α}$ plays the role of a thermodynamical variable with an associated chemical potential. We also notice the occurrence of non-standard gravitational charges in the Wald entropy and scalar charges.

Spain

Tomás Ortín

Germany

Matteo Zatti

On the thermodynamics of the black holes of the Cano-Ruipérez 4-dimensional string effective action

We find solutions of the heterotic string effective action describing the first-order α  ′  corrections to two-charge black holes at finite temperature. Making explicit use of these solutions, we compute the corrections to the thermodynamic quantities: temperature, chemical potentials, mass, charges and entropy. We check that the first law of black hole mechanics is satisfied and that the thermodynamics agrees with the one extracted from the Euclidean on-shell action. Finally, we show that our results are in agreement with the corrections for the thermodynamics recently predicted by Chen, Maldacena and Witten.

Italy

Stefano Massai

Alejandro Ruipérez

Revisiting α ′ corrections to heterotic two-charge black holes

We compute the first-order α ′ corrections to a family of 4-dimensional, 4-charge, non-extremal black hole solutions of Heterotic Supergravity in the case with 3 independent charges. The solutions are fully analytic, reproduce the extremal limit previously found in the literature and, applying T-duality, they transform as expected. If we reduce to the case with a single independent charge we obtain the corrections to four embeddings of the Reissner-Nordström black hole in string theory. We completely characterize the black hole thermodynamics computing the Hawking temperature, Wald entropy, mass, gauge charges and their dual thermodynamic potentials. We verify that all these quantities are related by the first law of extended black hole mechanics and the Smarr formula once we include a potential associated to the dimensionful parameter α ′ and the scalar charges. We found that the latter are not identified with the poles at infinity of the scalar fields because they receive α ′ corrections.

α ′ corrections to 4-dimensional non-extremal stringy black holes

Conventional wisdom dictates that ℤ N  factors in the integral cohomology group H  p (X  n , ℤ) of a compact manifold X  n  cannot be computed via smooth p-forms. We revisit this lore in light of the dimensional reduction of string theory on X  n , endowed with a G-structure metric that leads to a supersymmetric EFT. If massive p-form eigenmodes of the Laplacian enter the EFT, then torsion cycles coupling to them will have a non-trivial smeared delta form, that is an EFT long-wavelength description of p-form currents of the (n − p)-cycles of X  n . We conjecture that, whenever torsion cycles are calibrated, their linking number can be computed via their smeared delta forms. From the EFT viewpoint, a torsion factor in cohomology corresponds to a ℤ N  gauge symmetry realised by a Stückelberg-like action, and calibrated torsion cycles to BPS objects that source the massive fields involved in it.

Gonzalo Casas

Fernando Marchesano

Torsion in cohomology and dimensional reduction

We study the thermodynamics of the 4-dimensional electrically charged black-hole solutions of the simplest 5-dimensional Kaluza-Klein theory using Wald’s formalism. We show how the electric work term present in the 4-dimensional first law of black-hole thermodynamics arises in the purely gravitational 5-dimensional framework. In particular, we find an interesting geometric interpretation of the 4-dimensional electrostatic potential similar to the angular velocity in rotating black holes. Furthermore, we show how the momentum map equation arises from demanding compatibility between the timelike Killing vector of the black-hole solution and the spatial Killing vector of the 5-dimensional background.

Carmen Gómez-Fayrén

Patrick Meessen

Wald entropy in Kaluza-Klein black holes

We revisit the first law of black hole thermodynamics in 4-dimensional theories containing scalar and Abelian vector fields coupled to gravity using Wald’s formalism and a new definition of scalar charge as an integral over a 2-surface which satisfies a Gauss law in the background of stationary black-hole spacetimes. We focus on ungauged supergravity-inspired theories with symmetric sigma models whose symmetries generate electric-magnetic dualities leaving invariant their equations of motion. Our manifestly duality-invariant form of the first law is compatible with the one obtained by of Gibbons, Kallosh and Kol. We also obtain the general expression for the scalar charges of a stationary black hole in terms of the other physical parameters of the solution and the position of the horizon, generalizing the expression obtained by Pacilio for dilaton black holes.

Chile

Romina Ballesteros

On scalar charges and black hole thermodynamics

We compute the first-order α′ corrections of the non-extremal Strominger-Vafa black hole and its non-supersymmetric counterparts in the framework of the Bergshoeff-de Roo formulation of the heterotic superstring effective action. The solution passes several tests: its extremal limit is the one found in an earlier publication and the effect of a T duality transformation on it is another solution of the same form with T dual charges. We compute the Hawking temperature and Wald entropy showing that they are related by the first law and Smarr formula. On the other hand, these two contain additional terms in which the dimensionful parameter α′ plays the role of thermodynamical variable.

Belgium

Pablo Cano

Non-extremal, α′-corrected black holes in 5-dimensional heterotic superstring theory

We consider massive type IIA orientifold compactifications of the form AdS4 × X 6, where X 6 admits a Calabi-Yau metric and is threaded by background fluxes. From a 4d viewpoint, fluxes generate a potential whose vacua have been classified, including one   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 and three perturbatively stable   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 branches. We reproduce this result from a 10d viewpoint, by solving the type IIA equations at the same level of detail as previously done for the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 branch. All solutions exhibit localised sources and parametric scale separation. We then analyse the non-perturbative stability of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 branches. We consider new 4d membranes, obtained from wrapping D8-branes on X 6 or D6-branes on its divisors, threaded by non-diluted worldvolume fluxes. Using them we show that all branches are compatible with the Weak Gravity Conjecture for membranes. In fact, most vacua satisfy the sharpened conjecture that predicts superextremal membranes in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0 settings, except for a subset whose non-perturbative stability remains an open problem.

Joan Quirant

New instabilities for non-supersymmetric AdS4 orientifold vacua

We study extremal 5- and 4-dimensional black hole solutions of the Heterotic Superstring effective action at first order in α′ with, respectively, 3 and 4 charges of arbitrary signs. For a particular choice of the relative signs of these charges the solutions are supersymmetric, but other signs give rise to non-supersymmetric extremal black holes. Although at zeroth order all these solutions are formally identical, we show that the α′ corrections are drastically different depending on how we break supersymmetry. We provide fully analytic   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$ (α′) solutions and we compute their charges, masses and entropies and check that they are invariant under α′-corrected T-duality transformations. The masses of some of these black holes are corrected in a complicated way, but we show that the shift is always negative, in agreement with the Weak Gravity Conjecture. Our formula for the corrected entropy of the four-dimensional black holes generalizes previous results obtained from the entropy function method, as we consider a more general way in which supersymmetry can be broken.