Functional determinants, index theorems, and exact quantum black hole entropy

Abstract

Abstract:The exact quantum entropy of BPS black holes can be evaluated using localization in supergravity. An important ingredient in this program, that has been lacking so far, is the one-loop effect arising from the quadratic fluctuations of the exact deformation (the QV operator). We compute the fluctuation determinant for vector multiplets and hyper multiplets around Q-invariant off-shell configurations in four-dimensional N = 2 supergravity with AdS 2 × S 2 boundary conditions, using the Atiyah-Bott fixed-point index theorem and a subsequent zeta function regularization. Our results extend the large-charge on-shell entropy computations in the literature to a regime of finite charges. Based on our results, we present an exact formula for the quantum entropy of BPS black holes in N = 2 supergravity. We explain cancellations concerning 1 8 -BPS black holes in N = 8 supergravity that were observed in arXiv:1111.1161. We also make comments about the interpretation of a logarithmic term in the topological string partition function in the low energy supergravity theory.

Single-centered black hole microstate degeneracies from instantons in supergravity

“…At low energies the effective description of the theory is given by N = 4 supergravity coupled to 28 N = 4 gauge field multiplets specified by the compactification. The quarter-BPS black holes carry electric and magnetic charges (Q i e , Q i m ) (i = 1, · · · , 28), under these gauge fields, where i is a vector index under the T-duality group SO (6,22), and (Q e , Q m ) transform as a doublet under the S-duality group SL(2, Z). The U-duality group of the theory is SL(2, Z) × SO (6,22) One-fourth BPS dyonic states in the theory are completely labelled by the three continuous T-duality invariants: 14) and, in addition, some discrete charge invariants [40].…”

“…The quarter-BPS black holes carry electric and magnetic charges (Q i e , Q i m ) (i = 1, · · · , 28), under these gauge fields, where i is a vector index under the T-duality group SO (6,22), and (Q e , Q m ) transform as a doublet under the S-duality group SL(2, Z). The U-duality group of the theory is SL(2, Z) × SO (6,22) One-fourth BPS dyonic states in the theory are completely labelled by the three continuous T-duality invariants: 14) and, in addition, some discrete charge invariants [40]. As for the N = 8 example we write the compactification manifold as K3 × S 1 × S 1 , and we can choose a duality frame in which the black hole consists of the D1-D5-P system wrapping K3 × S 1 with Q 1 D1-branes, Q 5 D5-branes, n units of momentum on S 1 , one unit of KK-monopole charge and ℓ units of momentum on S 1 .…”

“…Indeed one needs to compute the one-loop determinant of the off-shell fluctuations of the non-BPS modes, and compute the induced measure from the supergravity field space. The former was computed explicitly in [6,58] for matter multiplets in N = 2 supergravity. Further it was argued that the symmetries fix the form of the graviton multiplet determinant up to a constant that can be matched to an on-shell computation [59].…”

“…The first time 6 we see this interference 4 We note that there is a textual error in the appendix of [55]. In the first paragraph of the appendix, it says that the coefficients c F m (n, ℓ) of the mock Jacobi forms are presented for the first four values of m, while what is really presented is d(n, ℓ, m) = (−1) ℓ c F m (n, ℓ) to emphasize the positivity of those numbers.…”

“…The exact computation of such functional integrals has been made possible due to the powerful technique of localization [2] applied to supergravity [3][4][5]. Although some important hurdles remain to be crossed in this program, we can -with some explicitly-stated assumptions -begin to write formulas for the perturbatively exact quantum entropy of a supersymmetric black hole in string theories with 8 supercharges in four dimensions [6]. We can thus compare the microscopic and macroscopic entropy formulas at a very precise level, pushing forward the earlier ideas of [7][8][9][10][11][12][13][14][15].…”