Sameer Murthy

Counting all dyons in \mathcal{N} = 4 string theory

For dyons in heterotic string theory compactified on a six-torus, with electric charge vector Q and magnetic charge vector P, the positive integer I ≡ gcd(Q ∧ P) is an invariant of the U-duality group. We propose the microscopic theory for computing the spectrum of all dyons for all values of I, generalizing earlier results that exist only for the simplest case of I = 1. Our derivation uses a combination of arguments from duality, 4d-5d lift, and a careful analysis of fermionic zero modes. The resulting degeneracy agrees with the black hole degeneracy for large charges and with the degeneracy of field-theory dyons for small charges. It naturally satisfies several physical requirements including integrality and duality invariance. As a byproduct, we also derive the microscopic (0, 4) superconformal field theory relevant for computing the spectrum of five-dimensional Strominger-Vafa black holes in ALE backgrounds and count the resulting degeneracies.

Quantum black holes, localization, and the topological string

We use localization to evaluate the functional integral of string field theory on AdS2 × S2 background corresponding to the near horizon geometry of supersymmetric black holes in 4d compactifications with

Supersymmetric index from black hole entropy

\mathcal{N} = 2

All solutions of the localization equations for \( \mathcal{N}=2 \) quantum black hole entropy

supersymmetry. In particular, for a theory containing nv + 1 vector multiplets, we show that the functional integral localizes exactly onto an ordinary integral over a finite-dimensional submanifold in the field space labeling a continuous family of instanton solutions in which auxiliary fields in the vector multiplets are excited with nontrivial dependence on AdS2 coordinates. These localizing solutions are universal in that they follow from the off-shell supersymmetry transformations and do not depend on the choice of the action. They are parametrized by nv + 1 real parameters {CI; I = 0,…, nv} that correspond to the values of the auxiliary fields at the center of AdS2. In the Type-IIA frame, assuming D-terms evaluate to zero on the solutions for reasons of supersymmetry, the classical part of the integrand equals the absolute square of the partition function of the topological string as conjectured by Ooguri, Strominger, and Vafa; however evaluated at the off-shell values of scalar fields at the center of AdS2. In addition, there are contributions from one-loop determinants, brane-instantons, and nonperturbative orbifolds that are in principle computable. These results thus provide a concrete method to compute exact quantum entropy of these black holes including all perturbative and nonperturbative corrections and can be used to establish a precise relation between the quantum degeneracies of black holes and the topological string.

Localization & exact holography

For BPS black holes with at least four unbroken supercharges, we describe how the macroscopic entropy can be used to compute an appropriate index, which can be then compared with the same index computed in the microscopic description. We obtain exact results incorporating all higher order quantum corrections in the limit when only one of the charges, representing momentum along an internal direction, approaches infinity keeping all other charges fixed at arbitrary finite values. In this limit, we find that the microscopic index is controlled by certain anomaly coefficients whereas the macroscopic index is controlled by the coefficients of certain Chern-Simons terms in the effective action. The equality between the macroscopic and the microscopic index then follows as a consequence of anomaly inflow. In contrast, the absolute degeneracy does not have any such simple expression in terms of the anomaly coefficients or coefficients of Chern-Simons terms. We apply our analysis to several examples of spinning black holes in five dimensions and non-spinning black holes in four dimensions to compute the index exactly in the limit when only one of the charges becomes large, and find perfect agreement with the result of exact microscopic counting. Our analysis resolves a puzzle involving M5-branes wrapped on a 5-cycle in K3 × T3.

Moonshine in fivebrane spacetimes

A bstractWe find the most general bosonic solution to the localization equations describing the contributions to the quantum entropy of supersymmetric black holes in four dimensional

Elliptic genera of ALE and ALF manifolds from gauged linear sigma models

\mathcal{N}=2

Corrections to the statistical entropy of five dimensional black holes

supergravity coupled to nv vector multiplets. This requires the analysis of the BPS equations of the corresponding off-shell supergravity (including fluctuations of the auxiliary fields) with AdS2 × S2 attractor boundary conditions. Our work completes and extends the results of arXiv:1012.0265 that were obtained for the vector multiplet sector, to include the fluctuations of all the fields of the off-shell supergravity. We find that, when the auxiliary SU(2) gauge field strength vanishes, the most general supersymmetric configuration preserving four supercharges is labelled by nv + 1 real parameters corresponding to the excitations of the conformal mode of the graviton and the scalars of the nv vector multiplets. In the general case, the localization manifold is labelled by an additional SU(2) triplet of one-forms and a scalar function.

F-Theory, Spinning Black Holes and Multi-string Branches

A bstractWe consider the AdS2/CFT1 holographic correspondence near the horizon of big four-dimensional black holes preserving four supersymmetries in toroidally compactified Type-II string theory. The boundary partition function of CFT1 is given by the known quantum degeneracies of these black holes. The bulk partition function is given by a functional integral over string fields in AdS2. Using recent results on localization we reduce the infinite-dimensional functional integral to a finite number of ordinary integrals over a space of localizing instantons. Under reasonable assumptions about the relevant terms in the effective action, these integrals can be evaluated exactly to obtain a bulk partition function. It precisely reproduces all terms in the exact Rademacher expansion of the boundary partition function as nontrivial functions of charges except for the Kloosterman sum which can in principle follow from an analysis of phases in the background of orbifolded instantons. Our results can be regarded as a step towards proving ‘exact holography’ in that the bulk and boundary partition functions computed independently agree for finite charges. Since the bulk partition function defines the quantum entropy of the black hole, our results enable the evaluation of perturbative as well as nonperturbative quantum corrections to the Bekenstein-Hawking-Wald entropy of these black holes.

Nonperturbative black hole entropy and Kloosterman sums

A bstractWe consider type II superstring theory on K3 × S1 × ℝ1,4 and study pertur-bative BPS states in the near-horizon background of two Neveu-Schwarz fivebranes whose world-volume wraps the K3 × S1 factor. These states are counted by the spacetime helicity supertrace χ2(τ) which we evaluate. We find a simple expression for χ2(τ) in terms of the completion of the mock modular form H(2)(τ) that has appeared recently in studies of the decomposition of the elliptic genus of K3 surfaces into characters of the N = 4 supercon-formal algebra and which manifests a moonshine connection to the Mathieu group M24.