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53
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 202 (20 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.
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 38 (19 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.
Product representation of dyon partition function in chl models
"... Preprint typeset in JHEP style HYPER VERSION hepth/0602254 ..."
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Cited by 31 (16 self)
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Preprint typeset in JHEP style HYPER VERSION hepth/0602254
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 24 (2 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 17 (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 16 (4 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 14 (6 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
Direct Integration of the Topological String
, 2007
"... We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non–holomorphic dependence of the amplitudes, and relies on the interplay between non–holomorphicity and modular ..."
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Cited by 14 (3 self)
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We present a new method to solve the holomorphic anomaly equations governing the free energies of type B topological strings. The method is based on direct integration with respect to the non–holomorphic dependence of the amplitudes, and relies on the interplay between non–holomorphicity and modularity properties of the topological string amplitudes. We develop a formalism valid for any Calabi–Yau manifold and we study in detail two examples, providing closed expressions for the amplitudes at low genus, as well as a discussion of the boundary conditions that fix the holomorphic ambiguity. The first example is the noncompact Calabi–Yau underlying Seiberg–Witten theory and its gravitational corrections. The second example is the Enriques Calabi–Yau, which we solve in full generality up to genus six. We discuss various aspects of this model: we obtain a new method to generate holomorphic automorphic forms on the Enriques moduli space, we write down a new product formula for the fiber amplitudes at all genus, and we analyze in detail the field theory limit. This allows us to uncover the modularity properties of SU(2), N = 2 super Yang–Mills theory with four massless hypermultiplets.
K3 SURFACES, LORENTZIAN KAC–MOODY ALGEBRAS AND MIRROR SYMMETRY
"... Abstract. We consider the variant of the Mirror Symmetry Conjecture for K3surfaces which relates “geometry ” of curves on a general member of a family of K3surfaces with “algebraic functions ” on the moduli of the mirror family. Lorentzian Kac–Moody algebras are involved in this construction. We giv ..."
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Cited by 13 (11 self)
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Abstract. We consider the variant of the Mirror Symmetry Conjecture for K3surfaces which relates “geometry ” of curves on a general member of a family of K3surfaces with “algebraic functions ” on the moduli of the mirror family. Lorentzian Kac–Moody algebras are involved in this construction. We give several examples when this conjecture is valid. 0.
Derived categories of coherent sheaves
 Proceedings of the 2006 ICM
, 2006
"... Abstract. We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of nonsingular CalabiYau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. ..."
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Cited by 13 (0 self)
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Abstract. We discuss derived categories of coherent sheaves on algebraic varieties. We focus on the case of nonsingular CalabiYau varieties and consider two unsolved problems: proving that birational varieties have equivalent derived categories, and computing the group of derived autoequivalences. We also introduce the space of stability conditions on a triangulated category and explain its relevance to these two problems. 1.