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Statesum construction of twodimensional openclosed TQFTs
 In preparation
"... We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smoo ..."
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We present a state sum construction of twodimensional extended Topological Quantum Field Theories (TQFTs), socalled openclosed TQFTs, which generalizes the state sum of Fukuma–Hosono–Kawai from triangulations of conventional twodimensional cobordisms to those of openclosed cobordisms, i.e. smooth compact oriented 2manifolds with corners that have a particular global structure. This construction reveals the topological interpretation of the associative algebra on which the state sum is based, as the vector space that the TQFT assigns to the unit interval. Extending the notion of a twodimensional TQFT from cobordisms to suitable manifolds with corners therefore makes the relationship between the global description of the TQFT in terms of a functor into the category of vector spaces and the local description in terms of a state sum fully transparent. We also illustrate the state sum construction of an openclosed TQFT with a finite set of Dbranes using the example of the groupoid algebra of a finite groupoid.
The history of qcalculus and a new method
, 2000
"... 1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8 ..."
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Cited by 10 (8 self)
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1.1. Partitions, generalized Vandermonde determinants and representation theory. 5 1.2. The Frobenius character formulae. 8
On Yetter’s invariant and an extension of the DijkgraafWitten invariant to categorical groups
 Theory Appl. Categ
"... We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the ..."
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Cited by 10 (0 self)
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We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined
Invariants of spin threemanifolds from ChernSimons theory and finitedimensional Hopf algebras
 Adv. Math
"... Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed threemanifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum group situations correspon ..."
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Cited by 6 (2 self)
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Abstract. A version of Kirby calculus for spin and framed threemanifolds is given and is used to construct invariants of spin and framed threemanifolds in two situations. The first is ribbon ∗categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the halfinteger level ChernSimons theories conjectured to give spin TQFTs by Dijkgraaf and Witten [10]. In particular, the spin invariants constructed by Kirby and Melvin [21] are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finitedimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg’s noninvolutory invariant of framed manifolds associated to that Hopf algebra.
A State Sum Model for (2+1) Lorentzian Quantum Gravity
, 2000
"... Thesis submitted to the University of Nottingham ..."
Renormalization of discrete models without background
 Nucl. Phys
"... Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The grou ..."
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Cited by 5 (2 self)
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Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The group of scale transformations is replaced by the groupoid of changes of cellular decompositions. We introduce cellular moves which generate this groupoid and allow to define a renormalization groupoid flow. We proceed to test our approach on several models. Quantum BF theory is the simplest example as it is almost topological and the renormalization almost trivial. More interesting is generalized lattice gauge theory for which a qualitative picture of the renormalization groupoid flow can be given. This is confirmed by the exact renormalization in dimension two.
Quantum hyperbolic state sum invariants of 3– manifolds
"... Any triple (W, L, ρ), where W is a compact closed oriented 3manifold, L is a link in W and ρ is a flat principal Bbundle over W (B is the Borel subgroup of upper triangular matrices of SL(2, C)), can be encoded by suitable distinguished and decorated triangulations T = (T, H, D). For each T, for e ..."
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Cited by 5 (0 self)
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Any triple (W, L, ρ), where W is a compact closed oriented 3manifold, L is a link in W and ρ is a flat principal Bbundle over W (B is the Borel subgroup of upper triangular matrices of SL(2, C)), can be encoded by suitable distinguished and decorated triangulations T = (T, H, D). For each T, for each odd integer N ≥ 3, one defines a state sum KN(T), based on the FaddeevKashaev quantum dilogarithm at ω = exp(2πi/N), such that KN(W,L, ρ) = KN(T) is a welldefined complex valued invariant. The purely topological, conjectural invariants KN(W,L) proposed earlier by Kashaev correspond to the special case of the trivial flat bundle. Moreover, we extend the definition of these invariants to the case of flat bundles on W \ L with not necessarily trivial holonomy along the meridians of the link’s components, and also to 3manifolds endowed with a Bflat bundle and with arbitrary nonspherical parametrized boundary components. As a matter of fact the distinguished and decorated triangulations are strongly reminiscent of the way one represents the classical refined scissors congruence class ̂ β(F), belonging to the extended Bloch group, of any given finite volume hyperbolic 3manifold F by using any hyperbolic ideal triangulation of F. We point out some remarkable specializations of the invariants; among these, the so called Seiferttype invariants, when W = S 3: these seem to be good candidates in order to fully reconstruct the Jones polynomials in the main stream of quantum hyperbolic invariants. Finally, we try to set our results against the heuristic backgroud of the Euclidean analytic continuation of (2+1) quantum gravity with negative cosmological constant, regarded as a gauge theory with the noncompact group SO(3, 1) as gauge group.
Problems on invariants of knots and 3manifolds Edited by T. Ohtsuki Preface
, 2003
"... The workshop and seminars on “Invariants of knots and 3manifolds ” was held at ..."
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The workshop and seminars on “Invariants of knots and 3manifolds ” was held at