α′²-corrections to extremal dyonic black holes in heterotic string theory

Abstract

We calculate α ′2 -corrections to the entropy of the 5-dimensional 3-charge and the 4-dimensional 4-charge large extremal black holes using the low energy effective action of the heterotic string theory. In the 4-dimensional case, our results are in agreement with the microscopic statistical entropy both for the BPS and the non-BPS black holes. In the more interesting 5-dimensional case, where the direct microscopic stringy description is still unknown, our results for the BPS black holes are in agreement with the results obtained from the action supplemented with R 2 -correction obtained by supersymmetric completion of the gravitational Chern-Simons term. This agreement does not extend to the non-BPS black holes, for which we propose a different expression for the entropy. We show that the new expression is supported by certain α ′3 -order calculations, and by the arguments based on the AdS/CFT correspondence. Contents 1. Introduction: Motivation and results 1 2. D = 6 heterotic effective action 4 3. Entropy function and its expansion 7 4. 3-charge black holes in D = 5 9 5. 4-charge black holes in D = 4 13 A. Identification of charges 14 B. Near-horizon solutions 15 B.1 D = 5 3-charge extremal black holes 16 B.2 D = 4 4-charge extremal black holes 16C. On contributions from α ′2 and higher order terms in the action 18 the limit of small string coupling constant g s (free string limit), which for nw ≫ 1 is given by 1 [7,8,9] S (BP S) stat = 2π nw(N W + 4) , n > 0 .( 1.1) For n < 0 the corresponding states are non-BPS with the statistical entropy given byNote that (1.1) and (1.2) are exact in α ′ . Now, when one increases g s , it has been argued that at some point these states become black holes. While in this regime string theory becomes highly nonperturbative, it is expected that one can use low energy effective action (at least for large black holes). Indeed, in the lowest order in α ′ , the solutions which describe extremal black holes with the two electric (n and w) and the two magnetic charges (N and W ) were explicitly constructed [10]. The near-horizon effective string coupling constant is proportional to 1/|nw|, which means that one can neglect string loops for nw ≫ 1. Also, the expansion in α ′ is equivalent to the expansion in 1/|N W |. The Bekenstein-Hawking entropy is S bh(0) = 2π |nwN W |, in agreement with (1.1) and (1.2). The α ′ -corrections to the entropies were calculated in [11], with the results again in agreement with (1.1) and (1.2). Surprising results were obtained when the following two types of R 2 -corrections in the effective action were taken: (i) the supersymmetrized gravitational Chern-Simons term [12], (ii) the Gauss-Bonnet term [13]. Both of these actions give the black hole entropy in the BPS case in the exact agreement with the statistical entropy (1.1), while they do not reproduce (1.2) in the non-BPS case. These results are surprising because the full effective action contains an infinite number of additional terms, for which there is no obvious reason to produce a canceling c...

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

“…Again, a mystery is why R 2 -truncated actions which are incomplete already at first order, are giving agreement (exactly or perturbatively) with statistical entropy (only) for BPS black holes. That there is no generic problem with gravity description of non-BPS black holes is shown by explicit perturbative calculations (up to α ′2 -order) of black hole entropies which are in agreement with statistical entropies (1.2)-(1.5) [15,16]. Motivated by such puzzles, we committed ourselves to calculate near-horizon solutions and entropies of above mentioned extremal black holes directly from ten-dimensional heterotic effective action (and without using AdS/CFT conjecture).…”

Heterotic black holes

“…As sphere does not have boundaries, the inclusion of boundary total-derivative terms (which "covariantize" CS term) cannot change the result. This is exactly equal to the result conjectured in [26] (on the basis of α 3 -order perturbative results).…”

“…• Finally, we note that results from this section agree with perturbative calculations up to α 2 -order obtained in [25,26] by using the 4-derivative effective action derived in [29]. This is expected, as it was shown in [28] that this action is equivalent up to α 1 with the action used in this section.…”

“…This is expected, as it was shown in [28] that this action is equivalent up to α 1 with the action used in this section. We note that this action by itself is not working beyond α 2 order, which means that in the field redefinition scheme used in [29] one has to take into account also terms with more than four derivatives to obtain α -exact entropies [26].…”

On Six-Dimensional Fundamental Superstrings as Holograms

“…Since total central charge is already c = 26 we cannot add any more degrees of freedom. The resulting sigma model is then bosonic SL(2, R) 4 × ×U (1) 4 × U(1) 16 CFT. Now we consider the field content of the SCFT.…”