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Black Hole Entropy Function, Attractors and Precision Counting of Microstates
, 2007
"... 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 strin ..."
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Cited by 326 (28 self)
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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 multicentered black holes as well.
On singular moduli
 J. reine angew. Math
, 1985
"... Introduction. “Singular moduli ” is the classical name for the values assumed by the modular invariant j(τ) (or by other modular functions) when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [ ..."
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Cited by 91 (1 self)
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Introduction. “Singular moduli ” is the classical name for the values assumed by the modular invariant j(τ) (or by other modular functions) when the argument is a quadratic irrationality. These values are algebraic numbers and have been studied intensively since the time of Kronecker and Weber. In [5], formulas for their norms, and for the norms of their differences, were obtained. Here we obtain instead a result for their traces, and a number of generalizations. The results are
Product representation of dyon partition function in chl models
"... Preprint typeset in JHEP style HYPER VERSION hepth/0602254 ..."
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Cited by 63 (24 self)
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Preprint typeset in JHEP style HYPER VERSION hepth/0602254
Automorphic forms and Lorentzian KacMoody algebras
 Part II,, Preprint RIMS 1122, Kyoto
, 1996
"... Abstract. Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl ..."
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Cited by 62 (25 self)
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Abstract. Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac–Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac–Moody algebras and formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II. 0.
Algebraic topology and modular forms
 Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), Higher Ed
, 2002
"... The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many ..."
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Cited by 53 (3 self)
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The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many
String Partition function and Infinite Products [arXiv:hepth/0002169
"... We continue to explore the conjectural expressions of the GromovWitten potentials for a class of elliptically and K3 fibered CalabiYau 3folds in the limit where the base P 1 of the K3 fibration becomes infinitely large. At least in this limit we argue that the string partition function ( = the ex ..."
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Cited by 40 (4 self)
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We continue to explore the conjectural expressions of the GromovWitten potentials for a class of elliptically and K3 fibered CalabiYau 3folds in the limit where the base P 1 of the K3 fibration becomes infinitely large. At least in this limit we argue that the string partition function ( = the exponential generating function of the GromovWitten potentials) can be expressed as an infinite product in which the Kähler moduli and the string coupling are treated somewhat on an equal footing. Technically speaking, we use the exponential lifting of a weight zero Jacobi form to reach the infinite product as in the celebrated work of Borcherds. However, the relevant Jacobi form is associated with a lattice of Lorentzian signature. A major part of this work is devoted to an attempt to interpret the infinite product or more precisely the Jacobi form in terms of the bound states of D2 and D0branes using a vortex description and its suitable generalization. 1
Moduli spaces of irreducible symplectic manifolds
, 2008
"... We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3 [2] manifolds with polarisation of degree 2d and split type is of general type if d ..."
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Cited by 30 (7 self)
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We study the moduli spaces of polarised irreducible symplectic manifolds. By a comparison with locally symmetric varieties of orthogonal type of dimension 20, we show that the moduli space of polarised deformation K3 [2] manifolds with polarisation of degree 2d and split type is of general type if d ≥ 12. MSC 2000: 14J15, 14J35, 32J27, 11E25, 11F55 0
The Kodaira dimension of the moduli of K3 surfaces
 Inventiones Math
"... The global Torelli theorem for projective K3 surfaces was first proved by PiatetskiiShapiro and Shafarevich 35 years ago, opening the way to treat moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasiprojective variety of dimension 19. For general d ver ..."
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Cited by 29 (11 self)
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The global Torelli theorem for projective K3 surfaces was first proved by PiatetskiiShapiro and Shafarevich 35 years ago, opening the way to treat moduli problems for K3 surfaces. The moduli space of polarised K3 surfaces of degree 2d is a quasiprojective variety of dimension 19. For general d very little has been known about the Kodaira dimension of these varieties. In this paper we present an almost complete solution to this problem. Our main result says that this moduli space is of general type for d> 61 and for d = 46, 50, 54, 58, 60. 0