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 two-fold: 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.

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

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 are mathematically rigorous objects corresponding to chiral algebras in conformal field theory. Operads are mathematical devices to describe operations, that is, n-ary 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 (two-dimensional) “complex ” geometric object, in the sense in which classical algebraic structures (groups, algebras, Lie algebras and the like) are always implicitly based on (one-dimensional) “real ” geometric objects. In effect, the standard analogy between point-particle theory and string theory is being shown to manifest itself at a more fundamental mathematical level. 1

In a pair of papers, we construct invariants for smooth four-manifolds equipped with ‘broken fibrations’—the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov—generalising the Donaldson-Smith invariant for Lefschetz fibrations. The ‘Lagrangian matching invariants ’ are designed to be comparable with the Seiberg-Witten invariants of the underlying four-manifold; formal properties and first computations support the conjecture that equality holds. They fit into a field theory which assigns Floer homology groups to 3-manifolds fibred over S 1. The invariants are derived from moduli spaces of pseudo-holomorphic 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. 1

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 2-category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this

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

We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over V canonically gives an algebra over a C-extension of Swiss-cheese partial operad. We also give a tensor-categorical formulation and constructions of open-closed field algebras over V. 0

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 n-manifold 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 2-Hilbert space.

Abstract. We discuss the geometry of topological terms in classical actions, which by themselves form the actions of topological field theories. We first treat the Chern-Simons action directly. Then we explain how the geometry is best understood via integration of smooth Deligne cocycles. We conclude with some remarks about the corresponding quantum theories in 3 and 4 dimensions. Introduction. A fundamental property of a field theory is its locality. This roughly means that computations can be made locally in spacetime. Or, to make a global computation one can compute locally and combine the results. The easiest way to do this is to cut a spacetime along codimension one submanifolds. For example, in a 3 dimensional