Heterotic Black Holes Research Articles

The first law and Wald entropy formula of heterotic stringy black holes at first order in \u03b1\u2032

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. We carefully take into account all the symmetries of the theory and, 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.In the proof we use momentum maps to define covariant variations and Lie derivatives and restricted generalized zeroth laws which state the closedness of certain differential forms on the bifurcation sphere and imply the constancy of the associated potentials on it.We study the relation between our entropy formula and other formulae that have been used in the literature.

The small black hole illusion

Small black holes in string theory are characterized by a classically singular horizon with vanishing Bekenstein-Hawking entropy. It has been argued that higher- curvature corrections resolve the horizon and that the associated Wald entropy is in agreement with the microscopic degeneracy. In this note we study the heterotic two-charge small black hole and question this result, which we claim is caused by a misidentification of the fundamental constituents of the system studied when higher-curvature interactions are present. On the one hand, we show that quadratic curvature corrections do not solve the singular horizon of small black holes. On the other, we argue that the resolution of the heterotic small black hole reported in the literature involves the introduction of solitonic 5-branes, whose asymptotic charge vanishes due to a screening effect induced by the higher- curvature interactions, and a Kaluza-Klein monopole, whose charge remains unscreened.

Exact charges from heterotic black holes

We derive exact relations to all orders in the α′ expansion for the charges of a bound system of heterotic strings, solitonic 5-branes and, optionally, a Kaluza-Klein monopole. The expressions, which differ from those of the zeroth-order supergravity approximation, coincide with the values obtained when only the corrections of quadratic order in curvature are included. Our computation relies on the consistency of string theory as a quantum theory of gravity; the relations follow from the matching of the Wald entropy with the microscopic degeneracy. In the heterotic frame, the higher-curvature terms behave as delocalized sources that introduce a shift between near-horizon and asymptotic charges. On the other hand, when described in terms of lower-dimensional effective fields, the solution carries constant charges over space which coincide with those of the asymptotic heterotic fields. In addition, we describe why the Gauss-Bonnet term, which only captures a subset of the relevant corrections of quadratic order in curvature, in some cases succeeds to reproduce the correct value for the Wald entropy, while fails in others.

Five-dimensional heterotic black holes and its dual IR-CFT

We analyze the possible dynamical emergence of IR conformal field theory describing the low- energy excitations of near-extremal black holes in five-dimensional compactification of heterotic strings. We find that, by tuning the mass and charges in such a way that the extremal black holes have a classically vanishing horizon area, the near-horizon develops an AdS3 throat and when we combine the low-energy limit with vanishing Newton coupling constant, the system has a dual conformal field theory description. We compare our results with c-extremization and the Kerr/CFT predictions.

Heterotic black horizons

We show that the supersymmetric near horizon geometry of heterotic black holes is either an AdS 3 fibration over a 7-dimensional manifold which admits a G 2 structure compatible with a connection with skew-symmetric torsion, or it is a product $ {\mathbb{R}^{1,1}} \times {\mathcal{S}^8} $ , where $ {\mathcal{S}^8} $ is a holonomy Spin(7) manifold, preserving 2 and 1 supersymmetries respectively. Moreover, we demonstrate that the AdS 3 class of heterotic horizons can preserve 4, 6 and 8 supersymmetries provided that the geometry of the base space is further restricted. Similarly $ {\mathbb{R}^{1,1}} \times {\mathcal{S}^8} $ horizons with extended supersymmetry are products of $ {\mathbb{R}^{1,1}} $ with special holonomy manifolds. We have also found that the heterotic horizons with 8 supersymmetries are locally isometric to AdS 3 × S 3 × T 4, AdS 3 × S 3 × K 3 or $ {\mathbb{R}^{1,1}} \times {T^4} \times {K_3} $ , where the radii of AdS 3 and S 3 are equal and the dilaton is constant.

Higher-order corrections to mass-charge relation of extremal black holes

We investigate the hypothesis that the higher-derivative corrections always make extremal non-supersymmetric black holes lighter than the classical bound and self-repulsive. This hypothesis was recently formulated in the context of the so-called swampland program. One of our examples involves an extremal heterotic black hole in four dimensions. We also calculate the effect of general four-derivative terms in Maxwell-Einstein theories in D dimensions. The results are consistent with the conjecture.

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.

Dual heterotic black holes in four and two dimensions

We consider a class of extremal and non-extremal four-dimensional black hole solutions occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional solutions can be connected to extremal and non-extremal two-dimensional heterotic black hole solutions through a change in the asymptotic behaviour of the harmonic functions associated with the Kaluza-Klein monopole and with the five-brane. This change in the asymptotic behaviour can be achieved by a sequence of S and T- S- T duality transformations in four dimensions. These transformations are implemented by performing a reduction on a two-torus with Lorentzian signature. We argue that the same mechanism can be applied to extremal and non-extremal black hole solutions in the FHSV model.

Charged heterotic black holes in four and two dimensions

We consider four-dimensional charged black holes occuring in toroidally compactified heterotic string theory, whose ten-dimensional interpretation involves a Kaluza-Klein monopole and a five-brane. We show that these four-dimensional black holes can be connected to two-dimensional charged heterotic black holes upon removal of the constants appearing in the harmonic functions associated with the Kaluza-Klein monopole and the five-brane.

E7(7) duality, BPS black-hole evolution and fixed scalars

We study the general equations determining BPS black holes by using a solvable Lie algebra representation for the homogeneous scalar manifold U/H of extended supergravity. In particular we focus on the N = 8 case and we perform a general group-theoretical analysis of the Killing spinor equation enforcing the BPS condition. Its solutions parametrize the U-duality orbits of BPS solutions that are characterized by having 40 of the 70 scalars fixed to constant values. These scalars belong to hypermultiplets in the N = 2 decomposition of the N = 8 theory. Indeed, it is shown that those decompositions of the solvable Lie algebra into appropriate subalgebras which are enforced by the existence of BPS black holes are the same that single out consistent truncations of the N = 8 theory to interacting theories with lower supersymmetry. As an exemplification of the method we consider the simplified case where the only non-zero fields are in the Cartan subalgebra H ⊃ Solv (U/H) and correspond to the radii of string toroidal compactification. Here we derive and solve the mixed system of first and second order non-linear differential equations obeyed by the metric and by the scalar fields. Doing so we retrieve the generating solutions of heterotic black holes with two charges. Finally, we show that the general N = 8 generating solution is based on the six-dimensional solvable subalgebra Solv[(SL(2,ℝ)/U(1))3].

Four-dimensional plane wave string solutions with coset CFT description

We present a number of D = 4 bosonic and heterotic string solutions with a covariantly constant null Killing vector which, like the solution of Nappi and Witten (NW), correspond to (gauged) WZW models and thus have a direct conformal field theory interpretation. A class of exact plane wave solutions (which includes the NW solution) is constructed by ‘boosting’ the twisted products of two D = 2 ‘cosmological’ or ‘black-hole’ cosets related to (G ⊗ G′)/(H ⊗ H′) (G, G′ = SL(2, R) or SU(2); H, H ′ = SO(1, 1) or SO(2)) gauged WZW models. We describe a general limiting procedure by which one can construct new solutions with a covariantly constant null Killing vector starting with known string backgrounds. By applying a non-abelian duality transformation to the NW model we find a D = 4 solution which admits a covariantly constant null Killing vector but is not a plane wave. Higher dimensional bosonic backgrounds with isometries can be interpreted as lower dimensional ones with extra gauge fields. Some of them are at the same time solutions of the heterotic string theory. In particular, the NW model represents also a D = 3 gravi-electromagnetic heterotic string plane wave. In addition to the (1, 1) supersymmetric embeddings of bosonic solutions we construct a number of non-trivial (1, 0) supersymmetric exact D = 4 heterotic string plane wave solutions some of which are related (by a boost and analytic continuation) to limiting cases of D = 4 heterotic black hole solutions.