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Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups in the singular case
 In preparation
"... Abstract. We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)Verlinde bundles over Teichmüller space, is asymptotically faithful, that is the intersection over all levels of the kerne ..."
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Abstract. We prove that the sequence of projective quantum SU(n) representations of the mapping class group of a closed oriented surface, obtained from the projective flat SU(n)Verlinde bundles over Teichmüller space, is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus is at least 3. For the genus 2 case, this intersection is exactly the order two subgroup, generated by the hyperelliptic involution, in the case of even degree and n = 2. Otherwise the intersection is also trivial in the genus 2 case. 1.
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). ..."
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Cited by 14 (4 self)
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We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan’s property (T).
A spin decomposition of the Verlinde formulas for type A modular categories, preprint
, 2001
"... additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular cate ..."
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Cited by 6 (0 self)
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additional algebraic features. The interest of this concept is that it provides a Topological Quantum Field Theory in dimension 3. The Verlinde formulas associated with a modular category are the dimensions of the TQFT modules. We discuss reductions and refinements of these formulas for modular categories related with SU(N). Our main result is a splitting of the Verlinde formula, corresponding to a brick decomposition of the TQFT modules whose summands are indexed by spin structures modulo an even integer. We introduce here the notion of a spin modular category, and give the proof of the decomposition theorem in this general context. Given a simple, simply connected complex Lie group G, the Verlinde formula [37] is a combinatorial function VG: (K, g) ↦ → VG(K, g) associated with G (here the integers K and g are respectively the level and the genus). In conformal field theory this formula gives the dimension
Reducibility of quantum representations of mapping class groups
"... In this paper we provide a general condition for the reducibility of the ReshetikhinTuraev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a nontrivial genus one partition function, we prove that the quant ..."
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Cited by 2 (1 self)
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In this paper we provide a general condition for the reducibility of the ReshetikhinTuraev quantum representations of the mapping class groups. Namely, for any modular tensor category with a special symmetric Frobenius algebra with a nontrivial genus one partition function, we prove that the quantum representations of all the mapping class groups built from the modular tensor category are reducible. In particular for SU(N) we get reducibility for certain levels and ranks. For the quantum SU(2) ReshetikhinTuraev theory we construct a decomposition for all even levels. We conjecture this decomposition is a complete decomposition into irreducible representations for high enough levels. 1 One of the features of three dimensional TQFT, as defined first by Witten, Atiyah and Segal in [W], [At] and [S], and further made precise by Reshetikhin and Turaev in [RT1], [RT2], is that it provides finite dimensional representations of mapping class groups of compact orientable surfaces, possibly with marked points. More precisely, a TQFT based on a modular category C with ground field k associates finite dimensional kvector
A COMBINATORIAL REALIZATION OF THE HEISENBERG ACTION ON THE SPACE OF CONFORMAL BLOCKS
, 708
"... Abstract. In this article we construct a combinatorial counterpart of the action of a certain Heisenberg group on the space of conformal blocks which was studied by AndersenMasbaum [1] and BlanchetHabeggerMasbaumVogel [5]. It is a nonabelian analogue of the description of the Heisenberg action ..."
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Cited by 1 (1 self)
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Abstract. In this article we construct a combinatorial counterpart of the action of a certain Heisenberg group on the space of conformal blocks which was studied by AndersenMasbaum [1] and BlanchetHabeggerMasbaumVogel [5]. It is a nonabelian analogue of the description of the Heisenberg action on the space of theta functions. 1.
External edge condition and group cohomologies associated with the quantum ClebschGordan condition, preprint
"... Abstract. In this article we investigate the structure of a twisted cohomology group of the first homology of a trivalent graph with a coefficient associated with its quantum ClebschGordan condition. By using this investigation we shall show the existence of the external edge cocycles defined in [3 ..."
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Abstract. In this article we investigate the structure of a twisted cohomology group of the first homology of a trivalent graph with a coefficient associated with its quantum ClebschGordan condition. By using this investigation we shall show the existence of the external edge cocycles defined in [3] and a finite algorithm to construct such cocycles. Such cocycles are used for a combinatorial realization of the Heisenberg action on the space of conformal blocks studied by Andersen and Masbaum [1] and Blanchet, Habegger, Masbaum and Vogel [2]. Moreover we shall give a characterization of their cohomology classes in our combinatorial setting. 1.
ON REDUCIBILITY OF MAPPING CLASS GROUP REPRESENTATIONS: THE SU(N) CASE
, 902
"... We review and extend the results of [1] that give a condition for reducibility of quantum representations of mapping class groups constructed from ReshetikhinTuraev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rat ..."
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Cited by 1 (0 self)
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We review and extend the results of [1] that give a condition for reducibility of quantum representations of mapping class groups constructed from ReshetikhinTuraev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in modular categories. Given a modular categoryC, a rational conformal field theory can be constructed from a Frobenius algebra A inC. We show that ifC contains a symmetric special Frobenius algebra A such that the torus partition function Z(A) of the corresponding conformal field theory is nontrivial, implying reducibility of the genus 1 representation of the modular group, then the representation of the genus g mapping class group constructed fromC is reducible for every g ≥ 1. We also extend the number of examples where we can show reducibility significantly by establishing the existence of algebras with the required properties using methods developed by Fuchs, Runkel and Schweigert. As a result we show that the quantum representations are reducible in the SU(N) case, N> 2, for all levels k ∈ N. The SU(2) case was treated explicitly in [1], showing reducibility for even levels k ≥ 4.
A GAUGE THEORETIC PROOF OF THE ASYMPTOTIC FAITHFULNESS CONJECTURE.
, 2002
"... Abstract. We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the Verlinde bundle over Teichmüller space constructed independently by Axelrod, Della Pietra and Witten and by Hitchin is asymptotically faithful, that is th ..."
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Abstract. We prove that the sequence of projective representations of the mapping class group obtained from the projective flat connection in the Verlinde bundle over Teichmüller space constructed independently by Axelrod, Della Pietra and Witten and by Hitchin is asymptotically faithful, that is the intersection over all levels of the kernels of these representations is trivial, whenever the genus of the underlying surface is at least 3. For the genus 2 case, we prove that this intersection is exactly the order two subgroup, generated by the hyperelliptic involution. 1.
Spin ChernSimons and Spin TQFTs
, 2006
"... In [14] we constructed classical spin ChernSimons for any compact Lie group G: a gauge theory whose action depends on the spin structure of the 3manifold. Here we apply geometric quantization to the classical Hamiltonian theory and investigate the formal properties of the partition function in the ..."
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In [14] we constructed classical spin ChernSimons for any compact Lie group G: a gauge theory whose action depends on the spin structure of the 3manifold. Here we apply geometric quantization to the classical Hamiltonian theory and investigate the formal properties of the partition function in the Lagrangian theory, all in the case G = SO3. We find that the quantum theory for SO3 spin ChernSimons corresponds to the spin TQFT constructed by Blanchet and Masbaum [9] in the same way that the quantum theory for standard SU2 ChernSimons corresponds to the TQFT constructed by Reshetikhen and Turaev [19] or the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel [8]. 1 Knot invariants and physics The correlation between between knot invariants and quantum ChernSimons was first laid out in the seminal paper by Witten [21]. The crux of Witten’s argument is that the quantum theory is a topological quantum field theory, or TQFT. In particular, he applies the formal properties of the quantum partition function to show that the theory satisfies the defining axioms of a TQFT. Together with certain results from conformal field theory, Witten uses these axioms to show that the quantum partition function associated to a compact oriented 3manifold is the Jones polynomial of the corresponding knot. For a complete definition of a TQFT and an explanation of Witten’s results we suggest [5]. Mathematicians – in particular, Reshetikhin and Turaev [19] – took a different approach to knot and 3manifold invariants by constructing their own TQFT from the representation theory of quantum groups. So that their invariants matched Witten’s, their TQFT was necessarily isomorphic to quantum ChernSimons. The quartet of Blanchet, Habegger, Masbaum, and Vogel (BHMV) come at TQFTs from the other direction; their starting point is a generalized version of the Reshetikhin and Turaev invariant. They apply algebrocategorical techniques to the Kauffman bracket and an oriented bordism category [8]. As desired, the BHMV TQFT nicely matches Witten’s. We elaborate in section
The Jacobian
, 812
"... ABSTRACT. We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the FourierMukai transform, and their spaces of sections are naturally dual. ..."
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ABSTRACT. We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the FourierMukai transform, and their spaces of sections are naturally dual.