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101
Modular forms and quantum invariants of 3manifolds
 Asian J. Math
, 1999
"... 1. Introduction. The WittenReshetikhinTuraev (WRT) invariant of a compact connected oriented 3manifold M may be formally defined by [16] ..."
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Cited by 35 (1 self)
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1. Introduction. The WittenReshetikhinTuraev (WRT) invariant of a compact connected oriented 3manifold M may be formally defined by [16]
Derived categories and zerobrane stability
 JHEP
"... We define a particular class of topological field theories associated to open strings and prove the resulting Dbranes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0brane on an ..."
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Cited by 30 (1 self)
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We define a particular class of topological field theories associated to open strings and prove the resulting Dbranes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0brane on any Calabi–Yau threefold must become unstable along some path in the Kähler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation The idea that a Dbrane is simply some subspace of the target space where open strings are allowed to end is clearly too simple. Even at zero string coupling, we are faced with carefully analyzing the nonlinear σmodel of maps from the string worldsheet into the target space. It is wellknown that the nonlinear σmodel modifies the usual rules of classical geometry
Nuclear and Trace Ideals in Tensored *Categories
, 1998
"... We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed ..."
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Cited by 28 (10 self)
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We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called "probabilistic relations". The compact closed structure associated with the category of relations does not generalize directly, instead one obtains nuclear ideals. Most tensored categories have a large class of morphisms which behave as if they were part of a compact closed category, i.e. they allow one to transfer variables between the domain and the codomain. We introduce the notion of nuclear ideals to analyze these classes of morphisms. In compact closed tensored categories, all morphisms are nuclear, and in the tensored category of Hilbert spaces, the nuclear morphisms are the HilbertSchmidt maps. We also introduce two new examples of tensored categories, in which integration plays the role of composition. In the first, mor...
Braiding operators are universal quantum gates
 New J. Phys
, 2004
"... doi:10.1088/13672630/6/1/134 Abstract. This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of theYang–Baxter equation is a universal gate for quantum computing, in the presence of local unitary tra ..."
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Cited by 28 (10 self)
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doi:10.1088/13672630/6/1/134 Abstract. This paper explores the role of unitary braiding operators in quantum computing. We show that a single specific solution R (the Bell basis change matrix) of theYang–Baxter equation is a universal gate for quantum computing, in the presence of local unitary transformations.We show that this same R generates a new nontrivial invariant of braids, knots and links. Other solutions of the Yang– Baxter equation are also shown to be universal for quantum computation. The paper discusses these results in the context of comparing quantum and topological points of view. In particular, we discuss quantum computation of link invariants, the relationship between quantum entanglement and topological entanglement, and the structure of braiding in a topological quantum field theory.
DuistermaatHeckman measures and moduli spaces of flat bundles over surfaces, Geom
 Funct. Anal
"... Abstract. We introduce Liouville measures and DuistermaatHeckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces. 1. ..."
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Cited by 21 (5 self)
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Abstract. We introduce Liouville measures and DuistermaatHeckman measures for Hamiltonian group actions with group valued moment maps. The theory is illustrated by applications to moduli spaces of flat bundles on surfaces. 1.
The Small Scale Structure of SpaceTime: A Bibliographical Review
, 1995
"... This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1 ..."
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Cited by 19 (0 self)
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This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1
Edge coloring models and reflection positivity
 Journal of the American Mathematical Society
, 2007
"... The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states ” or “colors”. We think of G as a crystal in which either the edges or ..."
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Cited by 18 (0 self)
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The motivation of this paper comes from statistical physics as well as from combinatorics and topology. The general setup in statistical mechanics can be outlined as follows. Let G be a graph and let C be a finite set of “states ” or “colors”. We think of G as a crystal in which either the edges or the vertices
Quantum invariants of Seifert 3manifolds and their asymptotic expansions
 Invariants of knots and 3manifolds (Kyoto
, 2002
"... 3–manifolds and a rational surgery formula ..."
A Combinatorial Approach to Topological Quantum Field Theories and Invariants of Graphs
, 1993
"... : The combinatorial state sum of Turaev and Viro for a compact 3manifold in terms of quantum 6jsymbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techn ..."
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Cited by 15 (3 self)
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: The combinatorial state sum of Turaev and Viro for a compact 3manifold in terms of quantum 6jsymbols is generalized by introducing observables in the form of coloured graphs. They satisfy braiding relations and allow for surgeries and a discussion of cobordism theory. Application of these techniques gives the dimension and an explicit basis for the vector space of the topological quantum field theory associated to any Riemann surface with arbitrary coloured punctures. * Supported by DFG, SFB 288 "Differentialgeometrie und Quantenphysik" 1 email: karowski@vax1.physik.fuberlin.dbp.de 2 email: schrader@vax1.physik.fuberlin.dbp.de 1 1. Introduction Since the early days of topological quantum field theories there was the question whether such field theories have a lattice formulation analogous to lattice gauge theory. The reason is that one would like to work in a context with mathematically well defined quantities instead of more or less formal functional integrals. This qu...
Knot theory and quantum gravity in loop space: A premier, hepth/9301028, to appear
 in the proceedings of the Vth Mexican school on particles and
, 1993
"... These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar’s variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques o ..."
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Cited by 13 (0 self)
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These notes summarize the lectures delivered in the V Mexican School of Particle Physics, at the University of Guanajuato. We give a survey of the application of Ashtekar’s variables to the quantization of General Relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. We discuss the role that the Jones Polynomial plays as a generator of nondegenerate quantum states of the gravitational field. 1 Quantum Gravity: why and how? I wish to thank the organizers for inviting me to speak here. This may well be a sign of our times, that a person generally perceived as a “General Relativist ” would be invited to speak at a Particle Physics School. It just reflects the higher degree of interplay these two fields have enjoyed over the last years. In these lectures we will see more reasons for this enhanced interplay. We will see several notions from Gauge Theories, as Wilson Loops for instance, playing a central role in gravitation. An even greater interplay takes place with Topological Field Theories. We will see the important role that the ChernSimons form, the Jones Polynomial and other notions of knot theory seem to play in General Relativity. The quantization of General Relativity is a problem that has defied resolution for the last sixty years. In spite of the long time that has been invested in trying to solve it, we believe that several people do not necessarily fully appreciate the reasons of our failure and the magnitude of the problem. It is a general perception –especially among particle physicists – that “General Relativity is nonrenormalizable ” and that is the basic problem with the theory. This statement is misleading in three ways: a) The fact that a theory is nonrenormalizable does not necessarily mean that the theory has an intrinsic problem or is “bad ” in any way. It merely says that perturbation