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Leon A. Takhtajan

Publications and source records attributed to Leon A. Takhtajan.

24 records · Page 2Linked to original sources
arXiv2004

Weil-Petersson metric on the universal Teichmuller space II. Kahler potential and period mapping

We study the Hilbert manifold structure on $T_{0}(1)$ -- the connected component of the identity of the Hilbert manifold T(1). We characterize points on $T_{0}(1)$ in terms of Bers and pre-Bers embeddings, and prove that the Grunsky operators $B_{1}$ and $B_{4}$, associated with the points in $T_{0}(1)$ via conformal welding, are Hilbert-Schmidt. We define a ``universal Liouville action'' -- a real-valued function $\SSS_{1}$ on $T_{0}(1)$, and prove that it is a Kähler potential of the Weil-Petersson metric on $T_{0}(1)$. We also prove that $\SSS_{1}$ is $-\tfrac{1}{12π}$ times the logarithm of the Fredholm determinant of associated quasi-circle, which generalizes classical results of Schiffer and Hawley. We define the universal period mapping $\hat{\cP}: T(1)\to\cB(\ell^{2})$ of T(1) into the Banach space of bounded operators on the Hilbert space $\ell^{2}$, prove that $\hat{\cP}$ is a holomorphic mapping of Banach manifolds, and show that $\hat{\cP}$ coincides with the period mapping introduced by Kurillov and Yuriev and Nag and Sullivan. We prove that the restriction of $\hat{\cP}$ to $T_{0}(1)$ is an inclusion of $T_{0}(1)$ into the Segal-Wilson universal Grassmannian, which is a holomorphic mapping of Hilbert manifolds. We also prove that the image of the topological group $S$ of symmetric homeomorphisms of $S^{1}$ under the mapping $\hat{\cP}$ consists of compact operators on $\ell^{2}$.

math.CV↗
arXiv2002

Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography

We rigorously define the Liouville action functional for finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that the classical action - the critical point of the Liouville action functional, considered as a function on the quasi-Fuchsian deformation space, is an antiderivative of a 1-form given by the difference of Fuchsian and quasi-Fuchsian projective connections. This result can be considered as global quasi-Fuchsian reciprocity which implies McMullen's quasi-Fuchsian reciprocity. We prove that the classical action is a Kahler potential of the Weil-Petersson metric. We also prove that Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3-manifold associated with a quasi-Fuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasi-Fuchsian groups and their free combinations.

math.CV↗
arXiv2001

Free bosons and tau-functions for compact Riemann surfaces and closed smooth Jordan curves I. Current correlation functions

We study families of quantum field theories of free bosons on a compact Riemann surface of genus g. For the case g > 0 these theories are parameterized by holomorphic line bundles of degree g-1, and for the case g=0 - by smooth closed Jordan curves on the complex plane. In both cases we define a notion of tau-function as a partition function of the theory and evaluate it explicitly. For the case g > 0 the tau-function is an analytic torsion [3], and for the case g=0 - the regularized energy of a certain natural pseudo-measure on the interior domain of a closed curve. For these cases we rigorously prove the Ward identities for the current correlation functions and determine them explicitly. For the case g > 0 these functions coincide with those obtained in [21,36] using bosonization. For the case g=0 the tau-function we have defined coincides with the tau-function introduced in [29,44,24] as a dispersionless limit of the Sato's tau-function for the two-dimensional Toda hierarchy. As a corollary of the Ward identities, we obtain recent results [44,24] on relations between conformal maps of exterior domains and tau-functions. For this case we also define a Hermitian metric on the space of all contours of given area. As another corollary of the Ward identities we prove that the introduced metric is Kahler and the logarithm of the tau-function is its Kahler potential.

math.QA↗
arXiv2000

Generating Functional in CFT on Riemann Surfaces II: Homological Aspects

We revisit and generalize our previous algebraic construction of the chiral effective action for Conformal Field Theory on higher genus Riemann surfaces. We show that the action functional can be obtained by evaluating a certain Deligne cohomology class over the fundamental class of the underlying topological surface. This Deligne class is constructed by applying a descent procedure with respect to a Čech resolution of any covering map of a Riemann surface. Detailed calculations are presented in the two cases of an ordinary Čech cover, and of the universal covering map, which was used in our previous approach. We also establish a dictionary that allows to use the same formalism for different covering morphisms. The Deligne cohomology class we obtain depends on a point in the Earle-Eells fibration over the Teichmüller space, and on a smooth coboundary for the Schwarzian cocycle associated to the base-point Riemann surface. From it, we obtain a variational characterization of Hubbard's universal family of projective structures, showing that the locus of critical points for the chiral action under fiberwise variation along the Earle-Eells fibration is naturally identified with the universal projective structure.

math.AT↗
arXiv1996

Generating Functional in CFT and Effective Action for Two-Dimensional Quantum Gravity on Higher Genus Riemann Surfaces

We formulate and solve the analog of the universal Conformal Ward Identity for the stress-energy tensor on a compact Riemann surface of genus $g>1$, and present a rigorous invariant formulation of the chiral sector in the induced two-dimensional gravity on higher genus Riemann surfaces. Our construction of the action functional uses various double complexes naturally associated with a Riemann surface, with computations that are quite similar to descent calculations in BRST cohomology theory. We also provide an interpretation for the action functional in terms of the geometry of different fiber spaces over the Teichmüller space of compact Riemann surfaces of genus $g>1$.

hep-th↗