Related papers: Quantum Liouville theory versus quantized Teichm\"…
The configuration space of the reduced Hamiltonian formulation of quantum gravity has been shown, for non-Ricci flat metrics, to be a higher-dimensional analogue of the Teichm\"{u}ller space of conformal structures on a Riemann surface. In…
We describe a self-consistent canonical quantization of Liouville theory in terms of canonical free fields. In order to keep the non-linear Liouville dynamics, we use the solution of the Liouville equation as a canonical transformation.…
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal…
Liouville theory describes the dynamics of surfaces with constant negative curvature and can be used to study the Weil-Petersson geometry of the moduli space of Riemann surfaces. This leads to an efficient algorithm to compute the…
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of…
The problem of definition of zero modes for quantum Liouville model is discussed and corresponding Hilbert space representation is constructed.
In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…
The notion of quantum embedding is considered for two classes of examples: quantum coadjoint orbits in Lie coalgebras and quantum symplectic leaves in spaces with non-Lie permutation relations. A method for constructing irreducible…
We have studied the quantum Liouville theory on the Riemann sphere with n>3 punctures. While considering the theory on the Riemann surfaces with n=4 punctures, the quantum theory near an arbitrary but fixed puncture can be obtained via…
A conformal map from a Riemann surface to a Euclidean space of dimension greater than or equal to three is explained by using the Clifford algebra, in a similar fashion to quaternionic holomorphic geometry of surfaces in the Euclidean…
We will propose a derivation of the correspondence between certain gauge theories with N=2 supersymmetry and conformal field theory discovered by Alday, Gaiotto and Tachikawa in the spirit of Seiberg-Witten theory. Based on certain results…
We give a recursive method to compute the classical conformal blocks in Liouville field theory. The values of the expansion coefficients are given by an algebraic scheme which works to all orders. The algebraic expression of the intervening…
We give an irreducible decomposition of the so-called local representations (see arXiv:0707.2151) of the quantum Teichm\"uller space $\mathcal{T}_q(\Sigma)$ where $\Sigma$ is a punctured surface of genus $g>0$ and $q$ is a primitive $N$-th…
This note is an attempt to extend "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. We introduce certain Hecke-type operators on vector bundles on an algebraic surface. The crucial observation is that the algebra…
We extend the Wigner-Weyl-Moyal phase-space formulation of quantum mechanics to general curved configuration spaces. The underlying phase space is based on the chosen coordinates of the manifold and their canonically conjugate momenta. The…
Deterministic dynamical models are discussed which can be described in quantum mechanical terms. -- In particular, a local quantum field theory is presented which is a supersymmetric classical model. The Hilbert space approach of Koopman…
We propose the graph description of Teichm\"uller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe…
Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether…
The paper presents an analog of the old result by the author and V. Voevodsky, according to which a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can…
Quantum spaces with $\frak{su}(2)$ noncommutativity can be modelled by using a family of $SO(3)$-equivariant differential $^*$-representations. The quantization maps are determined from the combination of the Wigner theorem for $SU(2)$ with…