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299
Integrable Structure of Conformal Field Theory II. Qoperator and DDV equation
, 1996
"... This paper is a direct continuation of [1] where we begun the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators Q \Sigma () which act in highest weight Virasoro module and commute for different values of the parameter . These operators appear ..."
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Cited by 92 (14 self)
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This paper is a direct continuation of [1] where we begun the study of the integrable structures in Conformal Field Theory. We show here how to construct the operators Q \Sigma () which act in highest weight Virasoro module and commute for different values of the parameter . These operators appear to be the CFT analogs of the Q  matrix of Baxter [2], in particular they satisfy famous Baxter's T \Gamma Q equation. We also show that under natural assumptions about analytic properties of the operators Q() as the functions of the Baxter's relation allows one to derive the nonlinear integral equations of Destride Vega (DDV) [3] for the eigenvalues of the Qoperators. We then use the DDV equation to obtain the asymptotic expansions of the Q  operators at large ; it is remarkable that unlike the expansions of the T operators of [1], the asymptotic series for Q() contains the "dual" nonlocal Integrals of Motion along with the local ones. We also discuss an intriguing relation between the ...
Categorical mirror symmetry: the elliptic curve
 Adv. Theor. Math. Phys
, 1998
"... We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M ..."
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Cited by 81 (10 self)
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We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M is an elliptic curve and ˜ M is its dual curve, exhibiting the dictionary in detail.
Separation of variables. New trends
, 1995
"... The review is based on the author’s papers since 1985 in which a new approach to the separation of variables (SoV) has being developed. It is argued that SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown ..."
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Cited by 54 (2 self)
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The review is based on the author’s papers since 1985 in which a new approach to the separation of variables (SoV) has being developed. It is argued that SoV, understood generally enough, could be the most universal tool to solve integrable models of the classical and quantum mechanics. It is shown that the standard construction of the actionangle variables from the poles of the BakerAkhiezer function can be interpreted as a variant of SoV, and moreover, for many particular models it has a direct quantum counterpart. The list of the models discussed includes XXX and XYZ magnets, Gaudin model, Nonlinear Schrödinger equation, SL(3)invariant magnetic chain. New results for the 3particle quantum CalogeroMoser system are reported. Contents
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
"... ..."
Quantum Geometry of Isolated Horizons and Black Hole Entropy
, 2000
"... Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a nonperturbative quantization of the sector of general relativity, coupled to matter, admitting nonrotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polym ..."
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Cited by 45 (3 self)
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Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a nonperturbative quantization of the sector of general relativity, coupled to matter, admitting nonrotating isolated horizons as inner boundaries. The emphasis is on the quantum geometry of the horizon. Polymer excitations of the bulk quantum geometry pierce the horizon endowing it with area. The intrinsic geometry of the horizon is then described by the quantum ChernSimons theory of a U(1) connection on a punctured 2sphere, the horizon. Subtle mathematical features of the quantum ChernSimons theory turn out to be important for the existence of a coherent quantum theory of the horizon geometry. Heuristically, the intrinsic geometry is flat everywhere except at the punctures. The distributional curvature of the U(1) connection at the punctures gives rise to quantized deficit angles which account for the overall curvature. For macroscopic black holes, the logarithm of the number of these horizon microstates is proportional to the area, irrespective of the values of (nongravitational) charges. Thus, the black hole entropy can be accounted for entirely by the quantum states of the horizon geometry. Our analysis is applicable to all nonrotating black holes, including the astrophysically interesting ones which are very far from extremality. Furthermore, cosmological horizons (to which statistical mechanical considerations are known to apply) are naturally incorporated. An effort has been made to make the paper selfcontained by including short reviews of the background material.
Applications of Arithmetical Geometry to Cryptographic Constructions
 Proceedings of the Fifth International Conference on Finite Fields and Applications
"... Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use me ..."
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Cited by 41 (1 self)
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Public key cryptosystems are very important tools for data transmission. Their performance and security depend on the underlying crypto primitives. In this paper we describe one such primitive: The Discrete Logarithm (DL) in cyclic groups of prime order (Section 1). To construct DLsystems we use methods from algebraic and arithmetic geometry and especially the theory of abelian varieties over finite fields. It is explained why Jacobian varieties of hyperelliptic curves of genus 4 are candidates for cryptographically "good" abelian varieties (Section 2). In the third section we describe the (constructive and destructive) role played by Galois theory: Local and global Galois representation theory is used to count points on abelian varieties over finite fields and we give some applications of Weil descent and Tate duality.
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,
Homological mirror symmetry with higher products
 in Proceedings of the Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, 247–259. AMS and International
, 2001
"... The homological mirror symmetry conjecture formulated by M. Kontsevich in [6] claims that derived categories of Fukaya’s symplectic A∞categogy F(M) of a CalabiYau manifold M and of coherent sheaves on a mirror dual CalabiYau manifold X are equivalent. In particular, this means that one can identif ..."
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Cited by 33 (5 self)
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The homological mirror symmetry conjecture formulated by M. Kontsevich in [6] claims that derived categories of Fukaya’s symplectic A∞categogy F(M) of a CalabiYau manifold M and of coherent sheaves on a mirror dual CalabiYau manifold X are equivalent. In particular, this means that one can identify the associative product on Extgroups between coherent sheaves on X with the corresponding product in the Floer cohomology of Lagrangians submanifolds in M (defined by Fukaya in [2]). The drawback of this conjecture is that one has an A∞category on the symplectic side of the story and the usual category on the complex side, so one has to make the usual category out of F(M). In this note we fix this problem by constructing an A∞category on the complex side and formulate a more general conjecture involving A∞categories on both sides. Let X be a compact complex manifold equipped with a hermitian metric. Inspired by Merkulov’s paper [7] we define an A∞category