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Ramanujan’s mock theta functions and their applications, Séminaire Bourbaki
"... One of the most romantic stories in the history of mathematics is that of the friendship between Hardy and Ramanujan. It began and ended with two famous letters. The first, sent by Ramanujan to Hardy in 1913, presents its author as a penniless clerk in a Madras shipping office who has made some disc ..."
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Cited by 34 (0 self)
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One of the most romantic stories in the history of mathematics is that of the friendship between Hardy and Ramanujan. It began and ended with two famous letters. The first, sent by Ramanujan to Hardy in 1913, presents its author as a penniless clerk in a Madras shipping office who has made some discoveries that “are termed by the
Automorphic properties of generating functions for generalized rank moments and Durfee symbols
, 2009
"... We define twoparameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the kmarked Durfee symbol. We prove that three specializations of the associated generating functions are socalled quasimock theta functions, while a fourth specialization gi ..."
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Cited by 16 (8 self)
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We define twoparameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the kmarked Durfee symbol. We prove that three specializations of the associated generating functions are socalled quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a twoparameter generalization of Andrews’ smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using qseries identities relating the relevant generating functions to known mock theta functions.
Fermionic expressions for the characters of c(p,1) logarithmic conformal field theories
 Nucl. Phys. B
"... We present fermionic quasiparticle sum representations consisting of a single fundamental fermionic form for all characters of the logarithmic conformal field theory models with central charge cp,1, p ≥ 2, and suggest a physical interpretation. We also show that it is possible to correctly extract ..."
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Cited by 9 (2 self)
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We present fermionic quasiparticle sum representations consisting of a single fundamental fermionic form for all characters of the logarithmic conformal field theory models with central charge cp,1, p ≥ 2, and suggest a physical interpretation. We also show that it is possible to correctly extract dilogarithm identities.
Higgs bundles, gauge theories and quantum groups
 Commun. Math. Phys
"... The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding twodimensional topological U(N) gauge theory reproduce quantum wave functions ..."
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Cited by 4 (1 self)
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The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding twodimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the Nparticle sector. This implies the full equivalence between the above gauge theory and the Nparticle subsector of the quantum theory of Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of degenerate double affine Hecke algebra. We propose similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra. The relation with the Nahm transform and the geometric Langlands correspondence is briefly discussed
SL(2, Z) – invariant spaces spanned by modular units
, 2005
"... Characters of rational vertex operator algebras (RVOAs) arising, e.g., in 2 dimensional conformal field theories often belong (after suitable normalization) to the (multiplicative) semigroup E + of modular units whose Fourier expansions are in 1 + q Z≥0[q], up to a fractional power of q. If, furth ..."
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Characters of rational vertex operator algebras (RVOAs) arising, e.g., in 2 dimensional conformal field theories often belong (after suitable normalization) to the (multiplicative) semigroup E + of modular units whose Fourier expansions are in 1 + q Z≥0[q], up to a fractional power of q. If, furthermore, all characters of a RVOA share this property then we have an example of what we call modular sets, i.e. finite subsets of E + whose elements (additively) span a vector space which is invariant under the usual action of SL(2, Z). The appearance of modular sets is always linked to the appearance of other interesting phenomena. The first nontrivial example is provided by the functions appearing in the two classical RogersRamanujan identities, and generalizations of these identities known from combinatorial theory yield further examples. The classification of modular sets and RVOAs seems to be related. This article is a first step towards the understanding of modular sets. We give an explicit description of the group of modular units generated by E +, we prove a certain finiteness result for modular sets contained in a natural semisubgroup E ∗ of E +, and we discuss consequences, in particular a method for effectively enumerating all modular sets in E∗.
Integrable deformations of CFTs and the discrete Hirota equations
, 905
"... One of the current major targets in physics is the understanding of integrable quantum field theories in two dimensions. In theories with diagonal scattering matrix there is a convincing conjecture for the spectrum of the Hamiltonian. Suppose that there are K species of particles with masses ma, a = ..."
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One of the current major targets in physics is the understanding of integrable quantum field theories in two dimensions. In theories with diagonal scattering matrix there is a convincing conjecture for the spectrum of the Hamiltonian. Suppose that there are K species of particles with masses ma, a = 1,...,K, and a
NUMBER THEORY IN PHYSICS
"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."
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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The
What the characters of irreducible subrepresentations of Jordan cells can tell us about LCFT
"... In this article, we review some aspects of logarithmic conformal field theories which can be inferred from the characters of irreducible submodules of indecomposable modules. We will mainly consider the W(2, 2p − 1, 2p − 1, 2p − 1) series of triplet algebras and a bit logarithmic extensions of the ..."
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In this article, we review some aspects of logarithmic conformal field theories which can be inferred from the characters of irreducible submodules of indecomposable modules. We will mainly consider the W(2, 2p − 1, 2p − 1, 2p − 1) series of triplet algebras and a bit logarithmic extensions of the minimal Virasoro models. Since in all known examples of logarithmic conformal field theories the vacuum representation of the maximally extended chiral symmetry algebra is an irreducible submodule of a larger, indecomposable module, its character provides a lot of nontrivial information about the theory such as a set of functions which spans the space of all torus amplitudes. Despite such characters being modular forms of inhomogeneous weight, they fit in the ADETclassification of fermionic sum representations. Thus, they show that logarithmic conformal field theories naturally have to be taken into account when attempting to classify rational conformal field theories.
MOTIVES: AN INTRODUCTORY SURVEY FOR PHYSICISTS
"... We survey certain accessible aspects of Grothendieck’s theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic th ..."
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We survey certain accessible aspects of Grothendieck’s theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic theory and theoretical physics.