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51
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.
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 76 (12 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.
Affine algebras, Langlands duality and Bethe ansatz
 Proc. International Congr. of Math. Physics
, 1995
"... By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a field F, and homomorphisms from the Galois group Gal(F/F) to the Langlands dual group G L. It was originally introduced in the case when F is a number ..."
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Cited by 53 (11 self)
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By Langlands duality one usually understands a correspondence between automorphic representations of a reductive group G over the ring of adels of a field F, and homomorphisms from the Galois group Gal(F/F) to the Langlands dual group G L. It was originally introduced in the case when F is a number field or the field of rational
Conformal Field Theory and Elliptic Cohomology
"... The purpose of the present paper is to address an old question (posed by Segal [37]) to find a geometric construction of elliptic cohomology. This question has recently become much more pressing due to the work of Mike Hopkins and Haynes Miller [19], who constructed exactly the “right”, or universal ..."
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Cited by 37 (9 self)
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The purpose of the present paper is to address an old question (posed by Segal [37]) to find a geometric construction of elliptic cohomology. This question has recently become much more pressing due to the work of Mike Hopkins and Haynes Miller [19], who constructed exactly the “right”, or universal, elliptic cohomology,
Vertex operator algebras and operads
, 1993
"... Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, nary operations for all n greater than or equal to 0, not just binary products. In this paper, a reformulation o ..."
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Cited by 27 (4 self)
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Vertex operator algebras are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, nary operations for all n greater than or equal to 0, not just binary products. In this paper, a reformulation of the notion of vertex operator algebra in terms of operads is presented. This reformulation shows that the rich geometric structure revealed in the study of conformal field theory and the rich algebraic structure of the theory of vertex operator algebras share a precise common foundation in basic operations associated with a certain kind of (twodimensional) “complex ” geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (onedimensional) “real ” geometric objects. In effect, the standard analogy between pointparticle theory and string theory is being shown to manifest itself at a more fundamental mathematical level. 1
Lagrangian matching invariants for fibred fourmanifolds: I
, 2008
"... In a pair of papers, we construct invariants for smooth fourmanifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the DonaldsonSmith invariant for Lefschetz fibrations. The ‘Lagrangian matching invariants ’ are designed to be ..."
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Cited by 19 (2 self)
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In a pair of papers, we construct invariants for smooth fourmanifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the DonaldsonSmith invariant for Lefschetz fibrations. The ‘Lagrangian matching invariants ’ are designed to be comparable with the SeibergWitten invariants of the underlying fourmanifold; formal properties and first computations support the conjecture that equality holds. They fit into a field theory which assigns Floer homology groups to 3manifolds fibred over S 1. The invariants are derived from moduli spaces of pseudoholomorphic sections of relative Hilbert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I is devoted to the symplectic geometry of these Lagrangians.
Pseudo limits, biadjoints, and pseudo algebras: categorical foundations of conformal field theory
 Mem. Amer. Math. Soc
"... The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjun ..."
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Cited by 18 (8 self)
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The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this
Ktheory in quantum field theory
 Current Develop. Math
"... Abstract. We survey three different ways in which Ktheory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted Ktheory, and we illustrate with some finite models. Part 2 is a review of pfaffians of ..."
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Cited by 15 (3 self)
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Abstract. We survey three different ways in which Ktheory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted Ktheory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, and the relationship to differential Ktheory. Part 3 is a geometric exposition of Dirac charge quantization, which in superstring theories also involves differential Ktheory. Parts 2 and 3 are related by the GreenSchwarz anomaly cancellation mechanism. An appendix, joint with Jerry Jenquin, treats the partition function of RaritaSchwinger fields. Grothendieck invented KTheory almost 50 years ago in the context of algebraic geometry, specifically in his generalization of the Hirzebruch RiemannRoch theorem [BS]. Shortly thereafter, Atiyah and Hirzebruch brought Grothendieck’s ideas into topology [AH], where they were applied to a variety of problems. Analysis entered after it was realized that the symbol of an elliptic operator determines an element of Ktheory. Atiyah and Singer then proved a formula for the index of such an operator (on a compact manifold) in terms of the Ktheory class of the symbol [AS1]. Subsequently, Ktheoretic ideas permeated other areas of linear analysis, algebra, noncommutative geometry, etc. One of the pleasant surprises of the past few years has been the relevance of Ktheory to superstring theory and related parts of theoretical physics. Furthermore, the story involves not only topological Ktheory, but also the Ktheory of C ∗algebras, the Ktheory of sheaves, and other forms of Ktheory. Not surprisingly, this new arena for Ktheory has inspired some developments in mathematics which are the subject of ongoing research. Our exposition here aims to explain three different ways in which topological Ktheory appears in physics, and how this physics motivates the mathematical ideas we are investigating. Part 1 concerns topological quantum field theory. Recall that an ndimensional topological theory assigns a complex number to every closed oriented nmanifold and a complex vector space to every closed oriented (n − 1)manifold. Continuing the superposition principle and ideas of locality to
Classical Chern Simons theory, part 1
, 1993
"... The formulations of Classical Mechanics by Lagrange and Hamilton are the modern foundation of classical physics [Ar]. Not only do these theories describe the motion of systems of particles, but Maxwell’s theory of electromagnetism, as well as other field theories, can also be formulated in Lagrangia ..."
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Cited by 12 (0 self)
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The formulations of Classical Mechanics by Lagrange and Hamilton are the modern foundation of classical physics [Ar]. Not only do these theories describe the motion of systems of particles, but Maxwell’s theory of electromagnetism, as well as other field theories, can also be formulated in Lagrangian and Hamiltonian terms. A Lagrangian field theory is defined by a local functional of the fields, called the lagrangian, and its integral over spacetime, 1 called the action. The classical solutions of the field theory are the critical points of the action. In particular, the minima satisfy the “least action principle ” of Maupertius. 2 The Hamiltonian theory is defined by a function, the hamiltonian, on phase space, or more generally on a symplectic manifold. The classical motion of the system is then described by Hamilton’s equations, whose solutions are integral curves of the symplectic gradient vector field of the hamiltonian. For many mechanical systems of particles, which should be regarded as 0 + 1 dimensional field theories, there is both a Lagrangian and Hamiltonian formulation. Then the relationship between them is expressed by the Legendre transform, if the lagrangian is nondegenerate. A typical example