Hom-quantum groups I: quasi-triangular Hom-bialgebras

We introduce a Hom-type generalization of quantum groups, called quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel'd's quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A family of quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, such as Drinfel'd's quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with a solution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of the quantum Yang-Baxter equation. Solutions of the Hom-Yang-Baxter equation can be obtained from modules of suitable quasi-triangular Hom-bialgebras.Comment: 21 page

Geometric Aspects of the Moduli Space of Riemann Surfaces

This is a survey of our recent results on the geometry of moduli spaces and Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces with very good properties, study their curvatures and boundary behaviors in great detail. Based on the careful analysis of these new metrics, we have a good understanding of the Kahler-Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable. Another corolary is a proof of the equivalences of all of the known classical complete metrics to the new metrics, in particular Yau's conjectures in the early 80s on the equivalences of the Kahler-Einstein metric to the Teichmuller and the Bergman metric.Comment: Survey article of our recent results on the subject. Typoes corrrecte

Topological Censorship

All three-manifolds are known to occur as Cauchy surfaces of asymptotically flat vacuum spacetimes and of spacetimes with positive-energy sources. We prove here the conjecture that general relativity does not allow an observer to probe the topology of spacetime: any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic, asymptotically flat spacetime satisfying the null energy condition, every causal curve from \scri^- to {\scri}^+ is homotopic to a topologically trivial curve from \scri^- to {\scri}^+. (If the Poincar\'e conjecture is false, the theorem does not prevent one from probing fake 3-spheres).Comment: 12 pages, REVTEX; 1 postscript figure in a separate uuencoded file. Our earlier version (PRL 71, 1486 (1993)) contained a secondary result, mistakenly attributed to Schoen and Yau, regarding ``passive topological censorship'' of a certain class of topologies. As Gregory Burnett has pointed out (gr-qc/9504012), this secondary result is false. The main topological censorship theorem is unaffected by the erro

Zooming in on local level statistics by supersymmetric extension of free probability

We consider unitary ensembles of Hermitian NxN matrices H with a confining potential NV where V is analytic and uniformly convex. From work by Zinn-Justin, Collins, and Guionnet and Maida it is known that the large-N limit of the characteristic function for a finite-rank Fourier variable K is determined by the Voiculescu R-transform, a key object in free probability theory. Going beyond these results, we argue that the same holds true when the finite-rank operator K has the form that is required by the Wegner-Efetov supersymmetry method of integration over commuting and anti-commuting variables. This insight leads to a potent new technique for the study of local statistics, e.g., level correlations. We illustrate the new technique by demonstrating universality in a random matrix model of stochastic scattering.Comment: 38 pages, 3 figures, published version, minor changes in Section

Gravitational Instantons from Gauge Theory

A gauge theory can be formulated on a noncommutative (NC) spacetime. This NC gauge theory has an equivalent dual description through the so-called Seiberg-Witten (SW) map in terms of an ordinary gauge theory on a commutative spacetime. We show that all NC U(1) instantons of Nekrasov-Schwarz type are mapped to ALE gravitational instantons by the exact SW map and that the NC gauge theory of U(1) instantons is equivalent to the theory of hyper-Kaehler geometries. It implies the remarkable consequence that ALE gravitational instantons can emerge from local condensates of purely NC photons.Comment: 4 pages with two columns; comments and references added, to appear in Phys. Rev. Let

Global stability of spacetimes with supersymmetric compactifications

This paper proves the stability, with respect to the evolution determined by the vacuum Einstein equations, of the Cartesian product of high-dimensional Minkowski space with a compact, Ricci-flat Riemannian manifold that admits a spin structure and a nonzero parallel spinor. Such a product includes the example of Calabi-Yau and other special holonomy compactifications, which play a central role in supergravity and string theory. The stability proved in this paper provides a counter example to an instability argument by Penrose

Non-ancient solution of the Ricci flow

For any complete noncompact K$\ddot{a}$hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.Comment: seven pages, latex fil