Thomas Mohaupt

Corrections to macroscopic supersymmetric black hole entropy

We determine the corrections to the entropy of extremal black holes arising from terms quadratic in the Riemann tensor in N=2, D=4 supergravity theories. We follow Wald’s proposal to modify the Bekenstein-Hawking area law. The new entropy formula, whose value only depends on the electric/magnetic charges, is expressed in terms of a single holomorphic function and is consistent with electric-magnetic duality. For string effective field theories arising from Calabi-Yau compactifications, our result for the entropy of a certain class of extremal black-hole solutions fully agrees with the counting of microstates performed some time ago by Maldacena, Strominger, Witten and by Vafa.

Stationary BPS solutions in N=2 supergravity with R**2 interactions

We analyze a broad class of stationary solutions with residual N = 1 supersymmetry of four-dimensional N = 2 supergravity theories with terms quadratic in the Weyl tensor. These terms are encoded in a holomorphic function, which determines the most relevant part of the action and which plays a central role in our analysis. The solutions include extremal black holes and rotating field configurations, and may have multiple centers. We prove that they are expressed in terms of harmonic functions associated with the electric and magnetic charges carried by the solutions by a proper generalization of the so-called stabilization equations. Electric/magnetic duality is manifest throughout the analysis. We also prove that spacetimes with unbroken supersymmetry are fully determined by electric and magnetic charges. This result establishes the so-called fixed-point behavior according to which the moduli fields must flow towards certain prescribed values on a fully supersymmetric horizon, but now in a more general context with higher-order curvature interactions. We briefly comment on the implications of our results for the metric on the moduli space of extremal black hole solutions.

Macroscopic entropy formulae and non-holomorphic corrections for supersymmetric black holes

In four-dimensional N = 2 compactifications of string theory or M-theory, modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions are crucial for finding agreement between the macroscopic entropy obtained from supergravity and subleading corrections to the microscopic entropy obtained via state counting. Here we compute the modifications to the area law for various classes of black holes, such as heterotic black holes, stemming from certain higher-derivative gravitational Wilsonian coupling functions. We consider the extension to heterotic N = 4 supersymmetric black holes and their type-II duals and we discuss its implications for the corresponding micro-state counting. In the effective field theory approach the Wilsonian coupling functions are known to receive non-holomorphic corrections. We discuss how to incorporate such corrections into macroscopic entropy formulae so as to render them invariant under duality transformations, and we give a concrete example thereof.

Deviations from the Area Law for Supersymmetric Black Holes

We review modifications of the Bekenstein-Hawking area law for black hole entropy in the presence of higher-derivative interactions. In four-dimensional N = 2 compactifications of string theory or M-theory these modifications are crucial for finding agreement between the macroscopic entropy obtained from supergravity and the microscopic entropy obtained by counting states in string or M-theory. Our discussion is based on the effective Wilsonian action, which in the context of N = 2 supersymmetric theories is defined in terms of holomorphic quantities. At the end we briefly indicate how to incorporate non-holomorphic corrections.

Classical and quantum N = 2 supersymmetric black holes

Abstract We use heterotic/type-H prepotentials to study quantum/classical black holes with half the N = 2, D = 4 supersymmetries unbroken. We show that, in the case of heterotic string compactifications, the perturbatively corrected entropy formula is given by the tree-level entropy formula with the tree-level coupling constant replaced by the perturbative coupling constant. In the case of type-II compactifications, we display a new entropy/area formula associated with axion-free black-hole solutions, which depends on the electric and magnetic charges as well as on certain topological data of Calabi-Yau three-folds, namely the intersection numbers, the second Chem class and the Euler number of the three-fold. We show that, for both heterotic and type-II theories, there is the possibility to relax the usual requirement of the non-vanishing of some of the charges and still have a finite entropy.

Area law corrections from state counting and supergravity

Modifications of the Bekenstein-Hawking area law for black holes are crucial in order to find agreement between the microscopic entropy based on state counting and the macroscopic entropy based on an effective field theory computation. We discuss this and related issues for the case of four-dimensional N = 2 supersymmetric black holes. We also briefly comment on the state counting for N = 4 and 8 black holes.

Higher-order black-hole solutions in N=2 supergravity and Calabi-Yau string backgrounds

Abstract Based on special geometry, we consider corrections to N=2 extremal black-hole solutions and their entropies originating from higher-order derivative terms in N=2 supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2n+2) vectors. We apply the formalism to N=2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature.

BPS spectra and non-perturbative gravitational couplings in N = 2, 4 supersymmetric string theories

Abstract We study the BPS spectrum in D = 4, N = 4 heterotic string compactifications, with some emphasis on intermediate N = 4 BPS states. These intermediate states, which can become short in N = 2 compactifications, are crucial for establishing an S ↔ T exchange symmetry in N = 2 compactifications. We discuss the implications of a possible S ↔ T exchange symmetry for the N = 2 BPS spectrum. Then we present the exact result for the one-loop corrections to gravitational couplings in one of the heterotic N = 2 models recently discussed by Harvey and Moore. We conjecture this model to have an S ↔ T exchange symmetry. This exchange symmetry. This exchange symmetry can then be used to evaluate non-perturbative corrections to gravitational couplings in some of the non-perturbative regions (chambers) in this particular model and also in other heterotic models.

MODULAR SYMMETRIES OF N = 2 BLACK HOLES

We discuss the transformation properties of classical extremal N = 2 black hole solutions in S-T-U like models under S and T duality. Using invariants of (subgroups of) the triality group, which is the symmetry group of the classical BPS mass formula, the transformation properties of the moduli on the event horizon and of the entropy under these transformations become manifest. We also comment on quantum corrections and we make a conjecture for the one-loop corrected entropy.

Black holes in supergravity and string theory

We give an elementary introduction to black holes in supergravity and string theory. The focus is on the role of BPS solutions in four- and higher-dimensional supergravity and in string theory. Basic ideas and techniques are explained in detail, including exercises with solutions.