Five‐dimensional black holes in heterotic string theory

Abstract

We review recent results on near-horizon static black hole solutions and entropy in R 2 -corrected N = 2 SUGRA in D = 5, focusing on actions connected to heterotic string compactified on K3×S 1 . Comparison with α ′ -perturbative results, results obtained by using simple Gauss-Bonnet R 2 -correction, OSV conjecture and microscopic stringy description (for small black holes) shows that situation in D = 5 is, in a sense, even more interesting then in D = 4.Copyright line will be provided by the publisher 1 5-dimensional black holes in higher derivative N = 2 SUGRA Bosonic part of the Lagrangian for the N = 2 supergravity action in five dimensions is given bywhere A 2 = A α i ab A i ab α and v 2 = v ab v ab . Also,A bosonic field content of the theory is the following. We have Weyl multiplet which contains the fünfbein e a µ , the two-form auxiliary field v ab , and the scalar auxiliary field D. There are n V vector multiplets enumerated by I = 1, . . . , n V , each containing the one-form gauge field A I (with the two-form field strength F I = dA I ), and the scalar M I . Scalar fields A i α , which are belonging to the hypermultiplet, can be gauge fixed and the convenient choice is given by A 2 = −2, ∂ a A α i = 0. Lagrangian (1) can be obtained from 11-dimensional SUGRA by compactifying on six-dimensional Calabi-Yau spaces. Then M I have interpretation as moduli (volumes of (1, 1)-cycles), and c IJK as intersection numbers. Condition N = 1 is a condition of real special geometry.

α′-exact entropies for BPS and non-BPS extremal dyonic black holes in heterotic string theory from ten-dimensional supersymmetry

“…Five-dimensional N = 2 off-shell supersymmetric action with R 2 terms, obtained in [6] by supersymmetrizing gravitational Chern-Simons term, again gives black hole entropy [8,9,10] which exactly agrees with CFT statistical entropy in BPS case (1.4), and fails already at first order in the non-BPS case [10,11]. 4 As for the action with pure Gauss-Bonnet correction, now it gives agreement in BPS case only at α ′1 -order, and again completely fails in the non-BPS case.…”

“…"GB" entropy differs starting from α ′2 -order [11]. Again, it is interesting to compare our nearhorizon solution "susy10" with "susy5", given in Eq.…”

“…−1 (see [11] for notations), relation valid in the two-derivative approximation. But, in this approximation, from the on-shell condition for prepotential N ≡ M 1 M 2 M 3 = 1 (real special geometry) follows that one could use also…”

Heterotic black holes

“…Comparison of (6.46) with the corresponding entropy obtained from full heterotic effective action given in (5.60), shows complete disagreement (starting already at first order in α ). 21 Note that relations in (6.43) satisfy real geometry condition N = M 1 M 2 M 3 = 1, which we showed to be violated by higher-derivative terms. This means that interpretations of fields in effective action as geometric moduli of compactification manifold are receiving corrections anf one has to be careful in such identifications.…”

“…This means that interpretations of fields in effective action as geometric moduli of compactification manifold are receiving corrections anf one has to be careful in such identifications. 22 It is interesting that choice T = (M 3 /M 2 ) 1/2 (which gives different result when higher-derivative corrections are includeed) leads to u T = n/w, which means that this could be a correct identification with heterotic string compactification modulus (radius of S 1 ) [21].…”