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Register a new userInfinite-dimensional Teichm{u}ller spaces
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Authors
Firat Yac{s}ar
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In this paper, the Teichm{u}ller spaces of surfaces appear from two points of views: the conformal category and the hyperbolic category. In contrast to the case of surfaces of topologically finite type, the Teichm{u}ller spaces associated to surfaces of topologically infinite type depend on the choice of a base structure. In the setting of surfaces of infinite type, the Teichm{u}ller spaces can be endowed with different distance functions such as the length-spectrum distance, the bi-Lipschitz distance, the Fenchel-Nielsen distance, the Teichm{u}ller distance and there are other distance functions. Unlike the case of surfaces of topologically finite type, these distance functions are not equivalent. We introduce the finitely supported Teichm{u}ller space T f s H 0 associated to a base hyperbolic structure H 0 on a surface $Sigma$, provide its characterization by Fenchel-Nielsen coordinates and study its relation to the other Teichm{u}ller spaces. This paper also involves a study of the Teichm{u}ller space T 0 ls H 0 of asymptotically isometric hyperbolic structures and its Fenchel-Nielsen parameterization. We show that T f s H 0 is dense in T 0 ls H 0 , where both spaces are considered to be subspaces of the length-spectrum Teichm{u}ller space T ls H 0. Another result we present here is that asymptotically length-spectrum bounded Teichm{u}ller space A T ls H 0 is contractible. We also prove that if the base surface admits short curves then the orbit of every finitely supported hyperbolic surface is non-discrete under the action of the finitely supported mapping class group MCG f s $Sigma$ .
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We investigate the Teichm{u}ller parameters for a Euclidean multiple BTZ black hole spacetime. To induce a complex structure in the asymptotic boundary of such a spacetime, we consider the limit in which two black holes are at a large distance from each other. In this limit, we can approximately determine the period matrix (i.e., the Teichm{u}ller parameters) for the spacetime boundary by using a pinching parameter. The Teichm{u}ller parameters are essential in describing the partition function for the boundary conformal field theory (CFT). We provide an interpretation of the partition function for the genus two extremal boundary CFT proposed by Gaiotto and Yin that it is relevant to double BTZ black hole spacetime.
An almost Fuchsian 3-manifold is a quasi-Fuchsian manifold which contains an incompressible closed minimal surface with principal curvatures in the range of $(-1,1)$. Such a 3-manifold $M$ admits a foliation of parallel surfaces, whose locus in Teichm{u}ller space is represented as a path $gamma$, we show that $gamma$ joins the conformal structures of the two components of the conformal boundary of $M$. Moreover, we obtain an upper bound for the Teichm{u}ller distance between any two points on $gamma$, in particular, the Teichm{u}ller distance between the two components of the conformal boundary of $M$, in terms of the principal curvatures of the minimal surface in $M$. We also establish a new potential for the Weil-Petersson metric on Teichm{u}ller space.
Let $X_{0}$ be a complete hyperbolic surface of infinite type with geodesic boundary which admits a countable pair of pants decomposition. As an application of the Basmajian identity for complete bordered hyperbolic surfaces of infinite type with limit sets of 1-dimensional measure zero, we define an asymmetric metric (which is called arc metric) on the quasiconformal Teichmuller space $mathcal{T}(X_{0})$ provided that $X_{0}$ satisfies a geometric condition. Furthermore, we construct several examples of hyperbolic surfaces of infinite type satisfying the geometric condition and discuss the relation between the Shigas condition and the geometric condition.
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.
The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(rho_beta)$, for a thermal density operator $rho_beta$ at inverse temperature $beta$. More generally, $GAP(rho)$ is a probability measure on the unit sphere in Hilbert space for any density operator $rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(rho)$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorovs theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(rho)$ depends continuously on $rho$ in the sense that convergence of $rho$ in the trace norm implies weak convergence of $GAP(rho)$.
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