Boundary conformal field theories, limit sets of Kleinian groups and holography

Abstract

In this paper, based on the available mathematical works on geometry and topology of hyperbolic manifolds and discrete groups, some results of Freedman et al (hep-th/9804058) are reproduced and broadly generalized.Among many new results the possibility of extension of work of Belavin,Polyakov and Zamolodchikov to higher dimensions is investigated. Known in physical literature objections against such extension are removed and the possibility of an extension is convinsingly demonstrated

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

“…This does not exclude the possibility that such connection might exist and requires further study. In the present work we establish such connection (reduction) by reobtaining W-K results from formalism developed earlier for 2+1 gravity [1][2][3][4]. This formalism is based on some properties of Riemann surfaces.…”

“…It may not be exaggeration to say that dynamics of 2+1 gravity is just an interpretation of Nielsen -Thurston theory of surface homeomorphisms in terms of concepts known in physics [1][2][3][4]. Apparently, the reverse task of enriching mathematics with concepts known from physics had been accomplished by Witten [5].…”

“…Moduli spaces M g are connected with Teichmüller spaces T g of Riemann surfaces of genus g ≥ 2 in such a way that M g = T g /Γ with Γ being the mapping class group of S. Both M g and T g have been discussed extensively in our earlier published papers on 2+1 gravity [1][2][3][4]. The concept of moduli had been initially proposed by Riemann who stated that isomorphism classes of closed Riemann surfaces of genus g ≥ 2 are parametrized by 3g-3 complex parameters (or by 6g-6 real parameters).…”

“…As it was demonstrated in Refs. [1][2][3][4], this theory provides natural mathematical framework for description of dynamics of 2+1 gravity. In these references the canonical (fixed genus g) partition function for 2+1 gravity was obtained.…”

“…[3] and Section 5. An image of the closed geodesic on S, when lifted to H 2 , is just a segment of a circle whose both ends lie on S 1 ∞ .…”