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130
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 99 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
Intersection theory, integrable hierarchies and topological field theory
, 1992
"... In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevic ..."
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Cited by 99 (5 self)
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In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevich naturally appear as τfunctions of integrable hierarchies related to topological minimal models.
Simulation of topological field theories by quantum computers
 Comm.Math.Phys.227
"... Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has ..."
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Cited by 81 (15 self)
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Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models ” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is twofold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm. 1.
The generally covariant locality principle  A new paradigm for local quantum physics
 COMMUN.MATH.PHYS
, 2001
"... A new approach to the modelindependent description of quantum field theories will be introduced in the present work. The main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity, thus giving rise to the concept of a locally cova ..."
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Cited by 68 (13 self)
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A new approach to the modelindependent description of quantum field theories will be introduced in the present work. The main feature of this new approach is to incorporate in a local sense the principle of general covariance of general relativity, thus giving rise to the concept of a locally covariant quantum field theory. Such locally covariant quantum field theories will be described mathematically in terms of covariant functors between the categories, on one side, of globally hyperbolic spacetimes with isometric embeddings as morphisms and, on the other side, of ∗algebras with unital injective ∗endomorphisms as morphisms. Moreover, locally covariant quantum fields can be described in this framework as natural transformations between certain functors. The usual HaagKastler framework of nets of operatoralgebras over a fixed spacetime backgroundmanifold, together with covariant automorphic actions of the isometrygroup of the background spacetime, can be regained from this new approach as a special case. Examples of this new approach are also outlined. In case that a locally covariant quantum field theory obeys the
TwoDimensional Topological Quantum Field Theories And Frobenius Algebras
 J. Knot Theory Ramifications
, 1996
"... We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either ..."
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Cited by 60 (2 self)
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We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either "annihilator algebras"  algebras whose socle is a principal ideal  or field extensions. The relationship between twodimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable twodimensional topological quantum field theories. Keywords: topological quantum field theory, frobenius algebra, twodimensional cobordism, category theory 1. Introduction Topological Quantum Field Theories (TQFT's) were first described axiomatically by Atiyah in [1]. Since then, much work has been done ...
and Category: is quantum gravity algebraic
 Journal of Mathematical Physics
, 1995
"... ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretati ..."
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Cited by 51 (3 self)
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ABSTRACT: We investigate the possibility that the quantum theory of gravity could be constructed discretely using algebraic methods. The algebraic tools are similar to ones used in constructing Topological Quantum Field theories. The algebraic structures are related to ideas about the reinterpretation of quantum mechanics in a general relativistic context. I.
On algebraic structures implicit in topological quantum field theories
 J. Knot Theory Ramifications
, 1999
"... In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamenta ..."
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Cited by 48 (2 self)
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In the course of the development of our understanding of topological quantum field theory (TQFT) [1,2], it has emerged that the structures of generators and relations for the construction of low dimensional TQFTs by various combinatorial methods are equivalent to the structures of various fundamental objects in abstract algebra.