Leon A. Takhtajan

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We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasiFuchsian deformation(More)
Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case(More)
We study the Hilbert manifold structure on T0(1) — the connected component of the identity of the Hilbert manifold T (1). We characterize points on T0(1) in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators B1 and B4, associated with the points in T0(1) via conformal welding, are Hilbert-Schmidt. We define a “universal Liouville(More)
We continue the study of quantum Liouville theory through Polyakov’s functional integral [1, 2], started in [3]. We derive the perturbation expansion for Schwinger’s generating functional for connected multi-point correlation functions involving stress-energy tensor, give the “dynamical” proof of the Virasoro symmetry of the theory and compute the value of(More)
Inspired by Polyakov’s original formulation [1, 2] of quantum Liouville theory through functional integral, we analyze perturbation expansion around a classical solution. We show the validity of conformal Ward identities for puncture operators and prove that their conformal dimension is given by the classical expression. We also prove that total quantum(More)
For a family of compact Riemann surfaces Xt of genus g > 1, parameterized by the Schottky space Sg, we define a natural basis of H(Xt, ω n Xt ) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F (n) on Sg which generalizes the classical product(More)
We show equivalence between the standard weak coupling regime c > 25 of the the two-dimensional quantum gravity and regime h < 1/2 of the original geometric approach of Polyakov [1, 2], developed in [3, 4, 5]. 1 In this letter I shall demonstrate the equivalence of two approaches to the twodimensional quantum gravity. The first approach, called geometric,(More)