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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 ..."
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Cited by 26 (8 self)
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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
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 ..."
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Cited by 25 (2 self)
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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.
Integral transforms and Drinfeld centers in derived algebraic geometry
"... Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derive ..."
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Compact objects are as necessary to this subject as air to breathe. R.W. Thomason to A. Neeman, [N3] Abstract. We study natural algebraic operations on categories arising in algebraic geometry and its homotopytheoretic generalization, derived algebraic geometry. We work with a broad class of derived stacks which we call stacks with air. The class of stacks with air includes in particular all quasicompact, separated derived schemes and (in characteristic zero) all quotients of quasiprojective or smooth derived schemes by affine algebraic groups, and is closed under derived fiber products. We show that the (enriched) derived categories of quasicoherent sheaves on stacks with air behave well under algebraic and geometric operations. Namely, we identify the derived category of a fiber product with the tensor product of the derived categories of the factors. We also identify functors between derived categories of sheaves with integral transforms (providing a generalization of a theorem of Toën [To1] for ordinary schemes over a ring). As a first application, for a stack Y with air, we calculate the Drinfeld center (or synonymously,
An analog of a modular functor from quantized Teichmüller theory
"... The program of the quantization of the Teichmüller spaces T (Σ) of Riemann surfaces Σ which was started in [Fo, CF] and independently in [Ka1] 1 is motivated by certain problems and conjectures from mathematical physics. One of the main aims of this program is to construct a ..."
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The program of the quantization of the Teichmüller spaces T (Σ) of Riemann surfaces Σ which was started in [Fo, CF] and independently in [Ka1] 1 is motivated by certain problems and conjectures from mathematical physics. One of the main aims of this program is to construct a
Quantum invariants of Seifert 3manifolds and their asymptotic expansions
 Invariants of knots and 3manifolds (Kyoto
, 2002
"... 3–manifolds and a rational surgery formula ..."
Pointed admissible Gcovers and Gequivariant cohomological Field Theories
 Compositio Math
"... Abstract. For any finite group G we define the moduli space of pointed admissible Gcovers and the concept of a Gequivariant cohomological field theory (GCohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. ..."
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Abstract. For any finite group G we define the moduli space of pointed admissible Gcovers and the concept of a Gequivariant cohomological field theory (GCohFT), which, when G is the trivial group, reduce to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the “quotient ” by Greduces a GCohFT to a CohFT. We also prove that a GCohFT contains a GFrobenius algebra, a Gequivariant generalization of a Frobenius algebra, and that the “quotient ” by G agrees with the obvious Frobenius algebra structure on the space of Ginvariants, after rescaling the metric. We then introduce the moduli space of Gstable maps into a smooth, projective variety X with G action. GromovWittenlike invariants of these spaces provide the primary source of examples of GCohFTs. Finally, we explain how these constructions generalize (and unify) the ChenRuan orbifold GromovWitten invariants of [X/G] as well as the ring H • (X,G) of Fantechi and Göttsche. 1.
Conformal field theory, boundary conditions and applications to string theory
 in: Nonperturbative QFT Methods and Their Applications
"... This is an introduction to twodimensional conformal field theory and its applications in string theory. Modern concepts of conformal field theory are explained, and it is outlined how they are used in recent studies of Dbranes in the strong curvature regime by means of CFT on surfaces with boundar ..."
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This is an introduction to twodimensional conformal field theory and its applications in string theory. Modern concepts of conformal field theory are explained, and it is outlined how they are used in recent studies of Dbranes in the strong curvature regime by means of CFT on surfaces with boundary. 1 1
Nonabelian anyons and topological quantum computation
 Reviews of Modern Physics
"... Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are partic ..."
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Cited by 10 (0 self)
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Contents Topological quantum computation has recently emerged as one of the most exciting approaches to constructing a faulttolerant quantum computer. The proposal relies on the existence of topological states of matter whose quasiparticle excitations are neither bosons nor fermions, but are particles known as NonAbelian anyons, meaning that they obey nonAbelian braiding statistics. Quantum information is stored in states with multiple quasiparticles,
On braided tensor categories of type BCD
 J. reine angew. Math
"... Abstract. We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep ` O(∞) ´ (formally), Rep ` O(N) ´ , Rep ` Sp(N) ´ or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by th ..."
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Abstract. We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep ` O(∞) ´ (formally), Rep ` O(N) ´ , Rep ` Sp(N) ´ or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. 1.
6j symbols for Uq(sl2) and nonEuclidean tetrahedra. math.QA/0305113
"... In this paper we study the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra Uq(sl2) for q a primitive root of unity. Because of the work of Finkelberg [7], these 6j symbols can also be defined in terms of fusion product of representations ..."
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In this paper we study the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra Uq(sl2) for q a primitive root of unity. Because of the work of Finkelberg [7], these 6j symbols can also be defined in terms of fusion product of representations of the affine Lie algebra ̂ sl2, defined using