Results 1  10
of
23
Quantum invariants of Seifert 3manifolds and their asymptotic expansions
 Invariants of knots and 3manifolds (Kyoto
, 2002
"... 3–manifolds and a rational surgery formula ..."
Mapping Class Groups do not have Kazhdan’s Property (T)
, 2007
"... We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan’s property (T). ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan’s property (T).
Integral bases for TQFT modules and unimodular representations of mapping class groups
"... Abstract. We construct integral bases for the SO(3)TQFTmodules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
Abstract. We construct integral bases for the SO(3)TQFTmodules of surfaces in genus one and two at roots of unity of prime order and show that the corresponding mapping class group representations preserve a unimodular Hermitian form over a ring of algebraic integers. For higher genus surfaces the Hermitian form sometimes must be nonunimodular. In one such case, genus 3 and p = 5, we still give an explicit basis. 1.
MAPPING CLASS GROUP DYNAMICS ON SURFACE GROUP REPRESENTATIONS
"... Abstract. Deformation spaces Hom(π,G)/G of representations of the fundamental group π of a surface Σ in a Lie group G admit natural actions of the mapping class group ModΣ, preserving a Poisson structure. When G is compact, the actions are ergodic. In contrast if G is noncompact semisimple, the asso ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
Abstract. Deformation spaces Hom(π,G)/G of representations of the fundamental group π of a surface Σ in a Lie group G admit natural actions of the mapping class group ModΣ, preserving a Poisson structure. When G is compact, the actions are ergodic. In contrast if G is noncompact semisimple, the associated deformation space contains open subsets containing the FrickeTeichmüller space upon which ModΣ acts properly. Properness of the ModΣaction relates to (possibly singular) locally homogeneous geometric structures on Σ. We summarize known results and state open questions about these actions.
Hitchin’s connection, Toeplitz operators and symmetry invariant deformation quantization
, 2006
"... Abstract. We establish that Hitchin’s connection exist for any rigid holomorphic family of Kähler structures on any compact prequantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that Hitchin’s connection induces a unique formal ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
Abstract. We establish that Hitchin’s connection exist for any rigid holomorphic family of Kähler structures on any compact prequantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that Hitchin’s connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding BerezinToeplitz deformation quantizations. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetryinvariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetryinvariant deformation quantization. Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)connections on any compact surface. 1.
Quantum mechanics and nonabelian theta functions for the gauge group SU(2), preprint
"... Abstract. This paper outlines an approach to the nonabelian theta functions of the SU(2)ChernSimons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, a ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Abstract. This paper outlines an approach to the nonabelian theta functions of the SU(2)ChernSimons theory with the methods used by A. Weil for studying classical theta functions. First we translate in knot theoretic language classical theta functions, the action of the finite Heisenberg group, and the discrete Fourier transform. Then we explain how the nonabelian counterparts of these arise in the framework of the quantum group quantization of the moduli space of flat SU(2)connections on a surface, in the guise of the nonabelian theta functions, the action of a skein algebra, and the ReshetikhinTuraev representation of the mapping class group. We prove a Stonevon Neumann theorem on the moduli space of flat SU(2)connections on the torus, and using it we deduce the existence and the formula for the ReshetikhinTuraev representation on the torus from quantum mechanical considerations. We show how one can derive in a quantum mechanical setting the skein that allows handle slides, which is the main ingredient in the construction of quantum 3manifold invariants.
Analytic asymptotic expansions of the Reshetikhin–Turaev invariants of Seifert 3–manifolds for SU(2)
, 2005
"... We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic fo ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic formula thereby proving the socalled asymptotic expansion conjecture (AEC) due to J. E. Andersen [An1], [An2] for these manifolds. For the case of Seifert manifolds with base S 2 we actually prove a little weaker result, namely that the asymptotic formula has a form as predicted by the AEC but contains some extra terms which should be zero according to the AEC. We prove that these ‘extra ’ terms are indeed zero if the number of exceptional fibers n is less than 4 and conjecture that this is also the case if n≥4. For the case of Seifert fibered rational homology spheres we identify the Casson–Walker invariant in the asymptotic formula. Our calculations demonstrate a general method for calculating the large r asymptotics of a finite sum Σ r k=1f(k), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to Rozansky [Ro1], [Ro3], is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis.
On power subgroups of mapping class groups
, 2009
"... In the first part of this paper we prove that the mapping class subgroups generated by the Dth powers of Dehn twists (with D ≥ 2) along a sparse collection of simple closed curves on a orientable surface are right angled Artin groups. The second part is devoted to power quotients i.e. quotients by ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In the first part of this paper we prove that the mapping class subgroups generated by the Dth powers of Dehn twists (with D ≥ 2) along a sparse collection of simple closed curves on a orientable surface are right angled Artin groups. The second part is devoted to power quotients i.e. quotients by the normal subgroup generated by the Dth powers of all elements of the mapping class group. We show first that for infinitely many D the power quotient groups are nontrivial. On the other hand, if 4g + 2 does not divide D then the associated power quotient of the mapping class group of the genus g closed surface is trivial. Eventually, an elementary argument shows that in genus 2 there are infinitely many power quotients which are infinite torsion groups.
The NielsenThurston classification of mapping classes is determined by TQFT
 Kyoto University
"... Abstract. For each fixed n ≥ 2 we show how the NielsenThurston classification of mapping classes for a closed surface of genus g ≥ 2 is determined by the sequence of quantum SU(n)representations (ρk)k∈N. That this is the case is a consequence of the asymptotic faithfulness property proved in [A3]. ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Abstract. For each fixed n ≥ 2 we show how the NielsenThurston classification of mapping classes for a closed surface of genus g ≥ 2 is determined by the sequence of quantum SU(n)representations (ρk)k∈N. That this is the case is a consequence of the asymptotic faithfulness property proved in [A3]. We here provide explicit conditions on (ρk(φ))k∈N, which determines the NielsenThurston type of any mapping class φ. 1.