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101
Involutions On Moduli Spaces And Refinements Of The Verlinde Formula
, 1997
"... The moduli space M of semistable rank 2 bundles with trivial determinant over a complex curve \Sigma carries involutions naturally associated to 2torsion points on the Jacobian of the curve. For every lift of a 2torsion point to a 4torsion point, we define a lift of the involution to the determ ..."
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Cited by 12 (5 self)
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The moduli space M of semistable rank 2 bundles with trivial determinant over a complex curve \Sigma carries involutions naturally associated to 2torsion points on the Jacobian of the curve. For every lift of a 2torsion point to a 4torsion point, we define a lift of the involution to the determinant line bundle L. We obtain an explicit presentation of the group generated by these lifts in terms of the order 4 Weil pairing. This is related to the triple intersections of the components of the fixed point sets in M , which we also determine completely using the order 4 Weil pairing. The lifted involutions act on the spaces of holomorphic sections of powers of L, whose dimensions are given by the Verlinde formula. We compute the characters of these vector spaces as representations of the group generated by our lifts, and we obtain an explicit isomorphism (as group representations) with the combinatorialtopological TQFTvector spaces of [BHMV]. As an application, we describe a `brick...
Quantum computing and the Jones polynomial
 math.QA/0105255, in Quantum Computation and Information
"... This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluati ..."
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Cited by 12 (9 self)
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This paper is an exploration of relationships between the Jones polynomial and quantum computing. We discuss the structure of the Jones polynomial in relation to representations of the Temperley Lieb algebra, and give an example of a unitary representation of the braid group. We discuss the evaluation of the polynomial as a generalized quantum amplitude and show how the braiding part of the evaluation can be construed as a quantum computation when the braiding representation is unitary. The question of an efficient quantum algorithm for computing the whole polynomial remains open. 1
Partition Function of a Quadratic Functional and Semiclassical Approximation for Witten’s 3Manifold Invaqriant
, 1995
"... An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in topological quantum field theory, for which no method has p ..."
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Cited by 10 (0 self)
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An extension of the method and results of A. Schwarz for evaluating the partition function of a quadratic functional is presented. This enables the partition functions to be evaluated for a wide class of quadratic functionals of interest in topological quantum field theory, for which no method has previously been available. In particular it enables the partition functions appearing in the semiclassical approximation for the Witteninvariant to be evaluated in the most general case. The resulting k−dependence is precisely that conjectured by D. Freed and R. Gompf. 1
Knot Theory and the Heuristics of Functional Integration
 PHYSICA A
, 2000
"... This paper is an exposition of the relationship between the heuristics of Witten's functional integral and the theory of knots and links in threedimensional space. ..."
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Cited by 9 (9 self)
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This paper is an exposition of the relationship between the heuristics of Witten's functional integral and the theory of knots and links in threedimensional space.
ChernSimons theory, knot invariants, vertex models and threemanifold invariants, hepth/9804122
 in Frontiers of Field Theory, Quantum Gravity and Strings (Volume 227 in Horizons in World Physics
, 1999
"... ChernSimons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and explicitly solved. Expectation values of Wilson link operators yie ..."
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Cited by 8 (2 self)
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ChernSimons theories, which are topological quantum field theories, provide a field theoretic framework for the study of knots and links in three dimensions. These are rare examples of quantum field theories which can be exactly and explicitly solved. Expectation values of Wilson link operators yield a class of link invariants, the simplest of them is the famous Jones polynomial. Other invariants are more powerful than that of Jones. These new invariants are sensitive to the chirality of all knots at least upto ten crossing number unlike those of Jones which are blind to the chirality of some of them. However, all these invariants are still not good enough to distinguish a class of knots called mutants. These link invariants can be alternately obtained from two dimensional vertex models. The Rmatrix of such a model in a particular limit of the spectral parameter provides a representation of the braid group. This in turn is used to construct the link invariants. Exploiting theorems of Lickorish and Wallace and also those of Kirby, Fenn and Rourke which relate threemanifolds to surgeries on framed links, these link invariants in S 3 can also be used to construct threemanifold invariants.
Combinatorial quantisation of Euclidean gravity in three dimensions
, 2000
"... In the ChernSimons formulation of Einstein gravity in 2+1 dimensions the phase space of gravity is the moduli space of flat Gconnections, where G is a typically noncompact Lie group which depends on the signature of spacetime and the cosmological constant. For Euclidean signature and vanishing c ..."
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Cited by 8 (8 self)
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In the ChernSimons formulation of Einstein gravity in 2+1 dimensions the phase space of gravity is the moduli space of flat Gconnections, where G is a typically noncompact Lie group which depends on the signature of spacetime and the cosmological constant. For Euclidean signature and vanishing cosmological constant, G is the threedimensional Euclidean group. For this case the Poisson structure of the moduli space is given explicitly in terms of a classical rmatrix. It is shown that the quantum Rmatrix of the quantum double D(SU(2)) provides a quantisation of that Poisson structure. MSC 17B37, 81R50, 81S10, 83C45 1
The YangMills heat flow on the moduli space of framed bundles on a surface
 math.SG/0211231. HUI LI Department of mathematics, Instituto Superior Tecnico, Av. Rovisco Pais 1049001 Lisboa Portugal Email address: hli@math.ist.utl.pt
"... Abstract. We study the analog of the YangMills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kähler quantiz ..."
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Cited by 7 (0 self)
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Abstract. We study the analog of the YangMills heat flow on the moduli space of framed bundles on a cut surface. Existence and convergence of the heat flow give a stratification of Morse type invariant under the action of the loop group. We use the stratification to prove versions of Kähler quantization commutes with reduction and Kirwan surjectivity. 1.
Formulas of Verlinde type for nonsimply connected groups
"... Abstract. We derive Verlinde’s formula from the fixed point formula for loop groups proved in the companion paper [FP], and extend it to compact, connected groups that are not necessarily simplyconnected. 1. ..."
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Cited by 6 (0 self)
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Abstract. We derive Verlinde’s formula from the fixed point formula for loop groups proved in the companion paper [FP], and extend it to compact, connected groups that are not necessarily simplyconnected. 1.
Quantum Topology and Quantum Computing
"... This paper is a quick introduction to key relationships between the theories of knots,links, threemanifold invariants and the structure of quantum mechanics. In section 2 we review the basic ideas and principles of quantum mechanics. Section 3 shows how the idea of a quantum amplitude is applied to ..."
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Cited by 6 (3 self)
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This paper is a quick introduction to key relationships between the theories of knots,links, threemanifold invariants and the structure of quantum mechanics. In section 2 we review the basic ideas and principles of quantum mechanics. Section 3 shows how the idea of a quantum amplitude is applied to the construction of invariants of knots and links. Section 4 explains how the generalisation of the Feynman integral to quantum fields leads to invariants of knots, links and threemanifolds. Section 5 is a discussion of a general categorical approach to these issues. Section 6 is a brief discussion of the relationships of quantum topology to quantum computing. This paper is intended as an introduction that can serve as a springboard for working on the interface between quantum topology and quantum computing. Section 7 summarizes the paper.
Quantum topology, hypergraphs and flag vectors, preprint qalg/9708001
, 1997
"... Each rule f that assigns a vector f(G) to an (n+1)graph G determines a class (or property) of nmanifold invariants. An invariant v = v(M) is in this class if, for any triangulated manifold G  = M, one has that v(M) is a linear function of f(G). This paper defines a flag vector f(G) for igraphs ..."
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Cited by 6 (6 self)
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Each rule f that assigns a vector f(G) to an (n+1)graph G determines a class (or property) of nmanifold invariants. An invariant v = v(M) is in this class if, for any triangulated manifold G  = M, one has that v(M) is a linear function of f(G). This paper defines a flag vector f(G) for igraphs, whose associated invariants might be quantum, and which is of interest in its own right. The definition (via the concept of shelling, and a ‘disjoint pair of optional cells ’ rule for the link) seems to apply to any finite combinatorial object, and so to any compact topological object that can be triangulated. It also applies to finite groups. 1