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99
Vertex algebras and algebraic curves
 Mathematical Surveys and Monographs 88 (2001), Amer. Math.Soc. MR1849359 (2003f:17036
"... Vertex operators appeared in the early days of string theory as local operators describing propagation of string states. Mathematical analogues of these operators were discovered in representation theory of affine KacMoody algebras in the works of Lepowsky–Wilson [LW] and I. Frenkel–Kac [FK]. In or ..."
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Cited by 93 (9 self)
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Vertex operators appeared in the early days of string theory as local operators describing propagation of string states. Mathematical analogues of these operators were discovered in representation theory of affine KacMoody algebras in the works of Lepowsky–Wilson [LW] and I. Frenkel–Kac [FK]. In order to formalize the emerging structure and motivated
Modular invariance of trace functions in orbifold theory, qalg/9703016. [DLi
 C. Dong and
, 1996
"... theory ..."
Arithmetic properties of mirror map and quantum coupling, hepth/9411234. 30
 lectures on Theta I , Progress in Math
"... Abstract: We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1parameter deformations of CalabiYau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 su ..."
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Cited by 48 (3 self)
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Abstract: We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1parameter deformations of CalabiYau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of the Jfunction. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function field Q(J). This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the ConwayNorton conjecture (proved by Borcherds using FrenkelLepowskyMeurman’s Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the GriessFischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced mod p. Under the mirror hypothesis and an integrality assumption, we derive mod p congruences for the Fourier coefficients. For the quintics, we deduce (at least for 5 ̸ d) that the degree d instanton numbers nd are divisible by 53 – a fact first conjectured by Clemens.
Tensor products of modules for a vertex operator algebras and vertex tensor categories
 in: Lie Theory and Geometry, in honor of Bertram Kostant
, 1994
"... In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announ ..."
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Cited by 44 (5 self)
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In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1].
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.
Conformal Field Theory and Elliptic Cohomology
"... The purpose of the present paper is to address an old question (posed by Segal [37]) to find a geometric construction of elliptic cohomology. This question has recently become much more pressing due to the work of Mike Hopkins and Haynes Miller [19], who constructed exactly the “right”, or universal ..."
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Cited by 37 (9 self)
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The purpose of the present paper is to address an old question (posed by Segal [37]) to find a geometric construction of elliptic cohomology. This question has recently become much more pressing due to the work of Mike Hopkins and Haynes Miller [19], who constructed exactly the “right”, or universal, elliptic cohomology,
Representation Theory of the Vertex Algebra W1
 Transform. Groups
, 1996
"... In our paper [KR] we began a systematic study of representations of the universal central extension ̂ D of the Lie algebra of differential operators on the circle. This study was continued in the paper [FKRW] in the framework of vertex algebra theory. It was shown that the associated to ̂ D simple v ..."
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Cited by 36 (2 self)
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In our paper [KR] we began a systematic study of representations of the universal central extension ̂ D of the Lie algebra of differential operators on the circle. This study was continued in the paper [FKRW] in the framework of vertex algebra theory. It was shown that the associated to ̂ D simple vertex algebra W1+∞,N with positive integral central charge N is isomorphic to the classical vertex algebra W(glN), which led to a classification of modules over W1+∞,N. In the present paper we study the remaining nontrivial case, that of a negative central charge −N. The basic tool is the decomposition of N pairs of free charged bosons with respect to glN and the commuting with glN Lie algebra of infinite matrices ̂ gl.
Modular data: the algebraic combinatorics of conformal field theory, preprint math.QA/0103044
"... This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It is essentially selfcontained, apart from some of the background motivation (Section I) and examples (Section III) which are inc ..."
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Cited by 32 (5 self)
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This paper is primarily intended as an introduction for mathematicians to some of the rich algebraic combinatorics arising in for instance conformal field theory (CFT). It is essentially selfcontained, apart from some of the background motivation (Section I) and examples (Section III) which are included to give the reader a sense of the context.
Vertex operator algebras and operads
, 1993
"... Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, nary operations for all n greater than or equal to 0, not just binary products. In this paper, a reformulation o ..."
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Cited by 27 (4 self)
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Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, nary operations for all n greater than or equal to 0, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (twodimensional) “complex ” geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (onedimensional) “real ” geometric objects. In effect, the standard analogy between pointparticle theory and string theory is being shown to manifest itself at a more fundamental mathematical level. 1