12,109 research outputs found
Algebraic Structures in Euclidean and Minkowskian Two-Dimensional Conformal Field Theory
We review how modular categories, and commutative and non-commutative
Frobenius algebras arise in rational conformal field theory. For Euclidean CFT
we use an approach based on sewing of surfaces, and in the Minkowskian case we
describe CFT by a net of operator algebras.Comment: 21 pages, contribution to proceedings for "Non-commutative Structures
in Mathematics and Physics" (Brussels, July 2008
Positive Stationary Solutions and Spreading Speeds of KPP Equations in Locally Spatially Inhomogeneous Media
The current paper is concerned with positive stationary solutions and spatial
spreading speeds of KPP type evolution equations with random or nonlocal or
discrete dispersal in locally spatially inhomogeneous media. It is shown that
such an equation has a unique globally stable positive stationary solution and
has a spreading speed in every direction. Moreover, it is shown that the
localized spatial inhomogeneity of the medium neither slows down nor speeds up
the spatial spreading in all the directions
Gapless edges of 2d topological orders and enriched monoidal categories
In this work, we give a precise mathematical description of a fully chiral
gapless edge of a 2d topological order (without symmetry). We show that the
observables on the 1+1D world sheet of such an edge consist of a family of
topological edge excitations, boundary CFT's and walls between boundary CFT's.
These observables can be described by a chiral algebra and an enriched monoidal
category. This mathematical description automatically includes that of gapped
edges as special cases. Therefore, it gives a unified framework to study both
gapped and gapless edges. Moreover, the boundary-bulk duality also holds for
gapless edges. More precisely, the unitary modular tensor category that
describes the 2d bulk phase is exactly the Drinfeld center of the enriched
monoidal category that describes the gapless/gapped edge. We propose a
classification of all gapped and fully chiral gapless edges of a given bulk
phase. In the end, we explain how modular-invariant bulk conformal field
theories naturally emerge on certain gapless walls between two trivial phases.Comment: 26 pages, 8 figures, An explanation of the appearance of boundary
CFT's on a chiral gapless edge, which is based on a generalized "no-go
theorem", is added. Final versio
Modular invariance for conformal full field algebras
Let V^L and V^R be simple vertex operator algebras satisfying certain natural
uniqueness-of-vacuum, complete reducibility and cofiniteness conditions and let
F be a conformal full field algebra over the tensor product of V^L and V^R. We
prove that the q_\tau-\bar{q_\tau}-traces (natural traces involving
q_\tau=e^{2\pi i\tau} and \bar{q_\tau}=\bar{e^{2\pi i\tau}}) of geometrically
modified genus-zero correlation functions for F are convergent in suitable
regions and can be extended to doubly periodic functions with periods 1 and
\tau. We obtain necessary and sufficient conditions for these functions to be
modular invariant. In the case that V^L=V^R and F is one of those constructed
by the authors in \cite{HK}, we prove that all these functions are modular
invariant.Comment: 54 page
A relation between chiral central charge and ground state degeneracy in 2+1-dimensional topological orders
A bosonic topological order on -dimensional closed space may
have degenerate ground states. The space with different shapes
(different metrics) form a moduli space . Thus the
degenerate ground states on every point in the moduli space form a complex vector bundle over . It was
suggested that the collection of such vector bundles for -dimensional closed
spaces of all topologies completely characterizes the topological order. Using
such a point of view, we propose a direct relation between two seemingly
unrelated properties of 2+1-dimensional topological orders: (1) the chiral
central charge that describes the many-body density of states for edge
excitations (or more precisely the thermal Hall conductance of the edge), (2)
the ground state degeneracy on closed genus surface. We show that for bosonic topological orders. We explicitly
checked the validity of this relation for over 140 simple topological orders.
For fermionic topological orders, let ()
be the degeneracy with even (odd) number of fermions for genus- surface with
spin structure . Then we have and
for .Comment: 8 pages. This paper supersedes Section XIV of an unpublished work
arXiv:1405.5858. We add new results on fermionic topological orders and some
numerical check
Full field algebras
We solve the problem of constructing a genus-zero full conformal field theory
(a conformal field theory on genus-zero Riemann surfaces containing both chiral
and antichiral parts) from representations of a simple vertex operator algebra
satisfying certain natural finiteness and reductive conditions. We introduce a
notion of full field algebra which is essentially an algebraic formulation of
the notion of genus-zero full conformal field theory. For two vertex operator
algebras, their tensor product is naturally a full field algebra and we
introduce a notion of full field algebra over such a tensor product. We study
the structure of full field algebras over such a tensor product using modules
and intertwining operators for the two vertex operator algebras. For a simple
vertex operator algebra V satisfying certain natural finiteness and reductive
conditions needed for the Verlinde conjecture to hold, we construct a bilinear
form on the space of intertwining operators for V and prove the nondegeneracy
and other basic properties of this form. The proof of the nondegenracy of the
bilinear form depends not only on the theory of intertwining operator algebras
but also on the modular invariance for intertwining operator algebras through
the use of the results obtained in the proof of the Verlinde conjecture by the
first author. Using this nondegenerate bilinear form, we construct a full field
algebra over the tensor product of two copies of V and an invariant bilinear
form on this algebra.Comment: 66 pages. One reference is added, a minor mistake on the invariance
under \sigma_{23} of the bilinear form on the space of intertwining operators
is corrected and some misprints are fixe
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