Results 1  10
of
356
On fusion categories
 Annals of Mathematics
"... Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Gr ..."
Abstract

Cited by 220 (34 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we extend categorically the notion of a finite nilpotent group to fusion categories. To this end, we first analyze the trivial component of the universal grading of a fusion category C, and then introduce the upper central series ofC. For fusion categories with commutative Grothendieck rings (e.g., braided fusion categories) we also introduce the lower central series. We study arithmetic and structural properties of nilpotent fusion categories, and apply our theory to modular categories and to semisimple Hopf algebras. In particular, we show that in the modular case the two central series are centralizers of each other in the sense of M. Müger. Dedicated to Leonid Vainerman on the occasion of his 60th birthday 1. introduction The theory of fusion categories arises in many areas of mathematics such as representation theory, quantum groups, operator algebras and topology. The representation categories of semisimple (quasi) Hopf algebras are important examples of fusion categories. Fusion categories have been studied extensively in the literature,
Quantum Field Theory of ManyBody Systems
, 2004
"... condensation Extended objects, such as strings and membranes, have been studied for many years in the context of statistical physics. In these systems, quantum effects are typically negligible, and the extended objects can be treated classically. Yet it is natural to wonder how strings and membrane ..."
Abstract

Cited by 157 (3 self)
 Add to MetaCart
condensation Extended objects, such as strings and membranes, have been studied for many years in the context of statistical physics. In these systems, quantum effects are typically negligible, and the extended objects can be treated classically. Yet it is natural to wonder how strings and membranes behave in the quantum regime. In this chapter, we will investigate the properties of one dimensional, stringlike, objects with large quantum fluctuations. Our motivation is both intellectual curiosity and (as we will see) the connection between quantum strings and topological/quantum orders in condensed matter systems. It is useful to organize our discussion using the analogy to the well understood theory of quantum particles. One of the most remarkable phenomena in quantum manyparticle systems is particle condensation. We can think of particle condensed states as special ground states where all the particles are described by the same quantum wave function. In some sense, all the symmetry breaking phases examples of particle condensation: we can view the order parameter that characterizes a symmetry breaking phase as the condensed wave function of certain “effective particles. ” According to this point of view, Landau’s theory [Landau (1937)] for symmetry breaking phases is really a theory of “particle ” condensation. The theory of particle condensation is based on the physical concepts of long range order, symmetry
Quandle cohomology and statesum invariants of knotted curves and surfaces
 TRANS. AMER. MATH. SOC
, 1999
"... The 2twist spun trefoil is an example of a sphere that is knotted in 4dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a statesum invariant for classical knots and knotted surfaces is developed via a cohomolo ..."
Abstract

Cited by 144 (36 self)
 Add to MetaCart
(Show Context)
The 2twist spun trefoil is an example of a sphere that is knotted in 4dimensional space. Here this example is shown to be distinct from the same sphere with the reversed orientation. To demonstrate this fact a statesum invariant for classical knots and knotted surfaces is developed via a cohomology theory of racks and quandles (also known as distributive groupoids). A quandle is a set with a binary operation — the axioms of which model the Reidemeister moves in the classical theory of knotted and linked curves in 3space. Colorings of diagrams of knotted curves and surfaces by quandle elements, together with cocycles of quandles, are used to define statesum invariants for knotted circles in 3space and knotted surfaces in 4space. Cohomology groups of various quandles are computed herein and applied to the study of the statesum invariants of classical knots and links and other linked surfaces. Nontriviality of the invariants are proved for variety of knots and links, including the trefoil and figureeight knots, and conversely, knot invariants are used to prove nontriviality of cohomology for a variety of quandles.
A modular functor which is universal for quantum computation
 Comm. Math. Phys
"... Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based o ..."
Abstract

Cited by 121 (18 self)
 Add to MetaCart
(Show Context)
Abstract: We show that the topological modular functor from Witten–Chern–Simons theory is universal for quantum computation in the sense that a quantum circuit computation can be efficiently approximated by an intertwining action of a braid on the functor’s state space. A computational model based on Chern–Simons theory at a fifth root of unity is defined and shown to be polynomially equivalent to the quantum circuit model. The chief technical advance: the density of the irreducible sectors of the Jones representation has topological implications which will be considered elsewhere. 1.
Spin foam models
 Classical and Quantum Gravity
, 1998
"... 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 e ..."
Abstract

Cited by 89 (2 self)
 Add to MetaCart
(Show Context)
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 2dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higherdimensional 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 networks describe ‘quantum 3geometries’, we describe how spin foams describe ‘quantum 4geometries’. We conclude by presenting a spin foam model of 4dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
Abstract

Cited by 87 (2 self)
 Add to MetaCart
(Show Context)
A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
Vertex Operator Algebras, the Verlinde Conjecture and Modular Tensor Categories
, 2005
"... ..."
(Show Context)
CATEGORY THEORY FOR CONFORMAL BOUNDARY CONDITIONS
, 2001
"... We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur ind ..."
Abstract

Cited by 75 (18 self)
 Add to MetaCart
(Show Context)
We study properties of the category of modules of an algebra object A in a tensor category C. We show that the module category inherits various structures from C, provided that A is a Frobenius algebra with certain additional properties. As a byproduct we obtain results about the FrobeniusSchur indicator in sovereign tensor categories. A braiding on C is not needed, nor is semisimplicity. We apply our results to the description of boundary conditions in twodimensional conformal field theory and present illustrative examples. We show that module categories give rise to NIMreps of the fusion rules, and discuss a possible relation with the representation theory of vertex operator algebras.
Finite tensor categories
 Moscow Math. Journal
"... These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We wil ..."
Abstract

Cited by 75 (12 self)
 Add to MetaCart
These are lecture notes for the course 18.769 “Tensor categories”, taught by P. Etingof at MIT in the spring of 2009. In these notes we will assume that the reader is familiar with the basic theory of categories and functors; a detailed discussion of this theory can be found in the book [ML]. We will also assume the basics of the theory of abelian categories (for a more detailed treatment see the book [F]). If C is a category, the notation X ∈ C will mean that X is an object of C, and the set of morphisms between X, Y ∈ C will be denoted by Hom(X, Y). Throughout the notes, for simplicity we will assume that the ground field k is algebraically closed unless otherwise specified, even though in many cases this assumption will not be needed. 1. Monoidal categories 1.1. The definition of a monoidal category. A good way of thinking