We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory one can classically obtain 6-dimensional topological A and B models, the topological sector of loop quantum gravity in 4 dimensions, and Chern-Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective we find hints of an intriguing holographic link between non-supersymmetric Yang-Mills in

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.

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A = F-Vect, where F is a field. An object X ∈ A with two-sided dual X gives rise to a Frobenius algebra in A, and under weak additional conditions we prove a converse: There exists a bicategory E with ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

A. Ocneanu has observed that a mysterious orbifold phenomenon occurs in the system of the M∞-M ∞ bimodules of the asymptotic inclusion, a subfactor analogue of the quantum double, of the Jones subfactor of type A2n+1. We show that this is a general phenomenon and identify some of his orbifolds with the ones in our sense as subfactors given as simultaneous fixed point algebras by working on the Hecke algebra subfactors of type A of Wenzl. That is, we work on their asymptotic inclusions and show that the M∞-M ∞ bimodules are described by certain orbifolds (with ghosts) for SU(3)3k. We actually compute several examples of the (dual) principal graphs of the asymptotic inclusions. As a corollary of the identification of Ocneanu’s orbifolds with ours, we show that a non-degenerate braiding exists on the even vertices of D2n, n>2. 1

While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin ...

Abstract Recently, Hitoshi Murakami has shown that the quantum SU (2)- and SO(3)- invariants of 3-manifolds at roots of unity of prime order are algebraic integers. Unfortunately, his proof is by a very complicated computation. Here, a quite different and very simple proof is presented, based on the second author's method to show that the Turaev-Viro invariant is the square of the modulus of the ReshetikhinTuraev invariant.

: 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. INTRODUCTION The histories of mathematics and theoretical physics are so intimately interwoven that neither can really be understood in isolation from the other. There is no example of a fundamental advance in theoretical physics which did not involve a change in the mathematical structure in which the physical theory is formulated. On the other hand, a new construction in mathematics very often represents a distillation of a physical concept. Differential four dimensional manifolds with lorentzian metrics, for example, can be thought of as structures within which Einstein's falling elevators can coexist. Turning points in the development...

The correlators of two-dimensional rational conformal field theories that are obtained in the TFT construction of [FRS I, FRS II, FRS IV] are shown to be invariant under

Abstract: We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT’s is circumvented. We find compact formulae for the S ±1-matrices using the canonical, non degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford’s relations between the canonical forms and the moduli of integrals. We analyze the projective SL(2,Z)-action on the center of Uq(sl2) for q an l = 2m + 1-st root of unity. It appears that the 3m + 1-dimensional representation decomposes into an m + 1-dimensional finite representation and a 2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation of SL(2,Z) and the finite, m-dimensional representation, obtained from the truncated TQFT Since the seminal paper of Atiyah [A] on the abstract definition of a topological quantum field theory (TQFT) much progress has been made in finding non trivial examples and

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