Results 1  10
of
46
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 ..."
Abstract

Cited by 97 (7 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 77 (12 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 53 (11 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 37 (9 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 27 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 18 (8 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
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
Full field algebra
 HKo3] [HKr] [HL1] [HL2] [HL3] [HL4] [HL5] [K1] Y.Z. Huang and
"... We introduce the notions of openclosed field algebra and openclosed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an openclosed field algebra over V canonically gives an algebra over a Cextension of Swiss ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
We introduce the notions of openclosed field algebra and openclosed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an openclosed field algebra over V canonically gives an algebra over a Cextension of Swisscheese partial operad. We also give a tensorcategorical formulation and constructions of openclosed field algebras over V. 0
Extended structures in topological quantum field theory (preprint
, 1993
"... An n dimensional quantum field theory typically deals with partition functions and correlation functions of n dimensional manifolds and quantum Hilbert spaces of n−1 dimensional manifolds. One of the novel ideas in topological field theories is to extend these notions to manifolds of dimension n −2 ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
An n dimensional quantum field theory typically deals with partition functions and correlation functions of n dimensional manifolds and quantum Hilbert spaces of n−1 dimensional manifolds. One of the novel ideas in topological field theories is to extend these notions to manifolds of dimension n −2 and lower. Such extensions inevitably lead to the introduction of categories. These ideas are very much “in the air”. Some of the people involved are Kazhdan, Segal, Lawrence, Kapranov, Voevodsky, Crane, and Yetter. Mostly this has been considered for 3 dimensional theories, but recently such ideas have also appeared in relation to the 4 dimensional Donaldson invariants (see [Fu], for example). Our motivation comes from a detailed understanding of classical topological field theories, which we also extend to manifolds of codimension two and higher. In the particular case of gauge theory with finite gauge group we define extensions of the usual “path integral ” for the extended classical theory [F1]. For an nmanifold this is the usual path integral, and for an (n − 1)manifold we recover the quantum Hilbert space. The result of this integration for an (n − 2)manifold is a 2Hilbert space.