N=4 Superconformal Algebra and the Entropy of HyperKahler Manifolds

We study the elliptic genera of hyperKahler manifolds using the representation theory of N=4 superconformal algebra. We consider the decomposition of the elliptic genera in terms of N=4 irreducible characters, and derive the rate of increase of the multiplicities of half-BPS representations making use of Rademacher expansion. Exponential increase of the multiplicity suggests that we can associate the notion of an entropy to the geometry of hyperKahler manifolds. In the case of symmetric products of K3 surfaces our entropy agrees with the black hole entropy of D5-D1 system.Comment: 25 pages, 1 figur

How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?

Single centered supersymmetric black holes in four dimensions have spherically symmetric horizon and hence carry zero angular momentum. This leads to a specific sign of the helicity trace index associated with these black holes. Since the latter are given by the Fourier expansion coefficients of appropriate meromorphic modular forms of Sp(2,Z) or its subgroup, we are led to a specific prediction for the signs of a subset of these Fourier coefficients which represent contributions from single centered black holes only. We explicitly test these predictions for the modular forms which compute the index of quarter BPS black holes in heterotic string theory on T^6, as well as in Z_N CHL models for N=2,3,5,7.Comment: LaTeX file, 17 pages, 1 figur

Black Hole Entropy Function, Attractors and Precision Counting of Microstates

In these lecture notes we describe recent progress in our understanding of attractor mechanism and entropy of extremal black holes based on the entropy function formalism. We also describe precise computation of the microscopic degeneracy of a class of quarter BPS dyons in N=4 supersymmetric string theories, and compare the statistical entropy of these dyons, expanded in inverse powers of electric and magnetic charges, with a similar expansion of the corresponding black hole entropy. This comparison is extended to include the contribution to the entropy from multi-centered black holes as well.Comment: LaTeX file, 196 pages, based on lectures given at various schools; v2: added appendix E containing analysis of the multiple D5-brane system, expanded discussion on duality orbits, other minor changes, references added; v3: equations (5.6.20) and (5.6.21) corrected; v4: minor corrections to equations (C.19), (C.20

Jacobi forms and a certain space of modular forms.

Let M\sb2k-2(m) be the space of holomorphic modular forms of weight 2k-2 on Γ\sb 0(m) and let J\sbk,m be the space of Jacobi forms of weight k and index m in the sense of Eichler-Zagier [\it M. Eichler and \it D. Zagier, The theory of Jacobi forms (Prog. Math. 55)(Birkhäuser 1985; Zbl 0554.10018)]. The main point in the proof of the Saito-Kurokawa conjecture was the isomorphism between J\sbk,1 and M\sb2k-2(1) as modules over the Hecke algebra. \par In the impressive paper under review the authors deal with the general case for the index m. There exists a canonical subspace \frak M\sp- \sb2k-2(m) of M\sp-\sb2k-2(m), which can be described by properties of the Euler factors of the L-series attached to a modular form and which contains the space of newforms. Here ``-'' means that the L-series satisfies a functional equation under s\mapsto 2k-2-s with root number -1. The Main Theorem says that J\sbk,m and \frak M\sp- \sb2k-2(m) are isomorphic as modules over the Hecke algebra. \par In \S 1 the trace of the Hecke operator T(ℓ) on J\sbk,m with ℓ relatively prime to m is computed as an application of the general trace formula for Jacobi forms. Then the Eichler-Selberg trace formula is used in order to express \texttr(T(ℓ),J\sbk,m) as linear combinations of \texttr(T(ℓ),M\sb2k-2\spnew,-(m')), m'\vert m. In \S 3 the isomorphy is proved, where the proof moreover gives a collection of explicit lifting maps. In the Appendix the authors derive a formula for a certain class number involving Gauss sums associated to binary quadratic forms

A trace formula for Jacobi forms

The authors state and derive a completely explicit trace formula for double coset operators acting on spaces of Jacobi forms. As a side result some nice formulas concerning Gauss sums drop out. A specialization of this general trace formula is considered in the paper reviewed below (Zbl 0651.10020). Here it turns out that the space of Jacobi forms of weight k and index m is Hecke equivariantly isomorphic to a certain subspace of elliptic modular forms of weight 2k-2 and level m

Counting zeros in quaternion algebras using Jacobi forms

We use the theory of Jacobi forms to study the number of elements in a maximal order of a definite quaternion algebra over the field of rational numbers whose characteristic polynomial equals a given polynomial. A certain weighted average of such numbers equals (up to some trivial factors) the Hurwitz class number H(4n−r^2). As a consequence we obtain new proofs for Eichler’s trace formula and for formulas for the class and type number of definite quaternion algebras. As a secondary result we derive explicit formulas for Jacobi Eisenstein series of weight 2 on Γ_0(N) and for the action of Hecke operators on Jacobi theta series associated to maximal orders of definite quaternion algebras