31 research outputs found
Lagrangian subcategories and braided tensor equivalences of twisted quantum doubles of finite groups
We classify Lagrangian subcategories of the representation category of a
twisted quantum double of a finite group. In view of results of 0704.0195v2
this gives a complete description of all braided tensor equivalent pairs of
twisted quantum doubles of finite groups. We also establish a canonical
bijection between Lagrangian subcategories of the representation category of a
twisted quantum double of a finite group G and module categories over the
category of twisted G-graded vector spaces such that the dual tensor category
is pointed. This can be viewed as a quantum version of V. Drinfeld's
characterization of homogeneous spaces of a Poisson-Lie group in terms of
Lagrangian subalgebras of the double of its Lie bialgebra. As a consequence, we
obtain that two group-theoretical fusion categories are weakly Morita
equivalent if and only if their centers are equivalent as braided tensor
categories.Comment: 26 pages; several comments and references adde
Bicategories for boundary conditions and for surface defects in 3-d TFT
We analyze topological boundary conditions and topological surface defects in
three-dimensional topological field theories of Reshetikhin-Turaev type based
on arbitrary modular tensor categories. Boundary conditions are described by
central functors that lift to trivializations in the Witt group of modular
tensor categories. The bicategory of boundary conditions can be described
through the bicategory of module categories over any such trivialization. A
similar description is obtained for topological surface defects. Using string
diagrams for bicategories we also establish a precise relation between special
symmetric Frobenius algebras and Wilson lines involving special defects. We
compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong
on boundary conditions and surface defects in abelian Chern-Simons theories and
in Turaev-Viro type TFTs, respectively.Comment: 34 pages, some figures. v2: references added. v3: typos corrected and
biliography update
Character D-modules via Drinfeld center of Harish-Chandra bimodules
The category of character D-modules is realized as Drinfeld center of the abelian monoidal category of Harish-Chandra bimodules. Tensor product of Harish-Chandra bimodules is related to convolution of D-modules via the long intertwining functor (Radon transform) by a result of Beilinson and Ginzburg (Represent. Theory 3, 1–31, 1999). Exactness property of the long intertwining functor on a cell subquotient of the Harish-Chandra bimodules category shows that the truncated convolution category of Lusztig (Adv. Math. 129, 85–98, 1997) can be realized as a subquotient of the category of Harish-Chandra bimodules. Together with the description of the truncated convolution category (Bezrukavnikov et al. in Isr. J. Math. 170, 207–234, 2009) this allows us to derive (under a mild technical assumption) a classification of irreducible character sheaves over ℂ obtained by Lusztig by a different method.
We also give a simple description for the top cohomology of convolution of character sheaves over ℂ in a given cell modulo smaller cells and relate the so-called Harish-Chandra functor to Verdier specialization in the De Concini–Procesi compactification.United States. Defense Advanced Research Projects Agency (grant HR0011-04-1-0031)National Science Foundation (U.S.) (grant DMS-0625234)National Science Foundation (U.S.) (grant DMS-0854764)AG Laboratory HSE (RF government grant, ag. 11.G34.31.0023)Russian Foundation for Basic Research (grant 09-01-00242)Ministry of Education and Science of the Russian Federation (grant No. 2010-1.3.1-111-017-029)Science Foundation of the NRU-HSE (award 11-09-0033)National Science Foundation (U.S.) (grant DMS-0602263
Cyclotomic integers, fusion categories, and subfactors
Dimensions of objects in fusion categories are cyclotomic integers, hence
number theoretic results have implications in the study of fusion categories
and finite depth subfactors. We give two such applications. The first
application is determining a complete list of numbers in the interval (2,
76/33) which can occur as the Frobenius-Perron dimension of an object in a
fusion category. The smallest number on this list is realized in a new fusion
category which is constructed in the appendix written by V. Ostrik, while the
others are all realized by known examples. The second application proves that
in any family of graphs obtained by adding a 2-valent tree to a fixed graph,
either only finitely many graphs are principal graphs of subfactors or the
family consists of the A_n or D_n Dynkin diagrams. This result is effective,
and we apply it to several families arising in the classification of subfactors
of index less then 5.Comment: 47 pages, with an appendix by Victor Ostri
The fusion algebra of bimodule categories
We establish an algebra-isomorphism between the complexified Grothendieck
ring F of certain bimodule categories over a modular tensor category and the
endomorphism algebra of appropriate morphism spaces of those bimodule
categories. This provides a purely categorical proof of a conjecture by Ostrik
concerning the structure of F.
As a by-product we obtain a concrete expression for the structure constants
of the Grothendieck ring of the bimodule category in terms of endomorphisms of
the tensor unit of the underlying modular tensor category.Comment: 16 page
From boundary to bulk in logarithmic CFT
The analogue of the charge-conjugation modular invariant for rational
logarithmic conformal field theories is constructed. This is done by
reconstructing the bulk spectrum from a simple boundary condition (the analogue
of the Cardy `identity brane'). We apply the general method to the c_1,p
triplet models and reproduce the previously known bulk theory for p=2 at c=-2.
For general p we verify that the resulting partition functions are modular
invariant. We also construct the complete set of 2p boundary states, and
confirm that the identity brane from which we started indeed exists. As a
by-product we obtain a logarithmic version of the Verlinde formula for the
c_1,p triplet models.Comment: 35 pages, 2 figures; v2: minor corrections, version to appear in
J.Phys.
Open-closed field algebras
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 the
Swiss-cheese partial operad. We also give a tensor categorical formulation and
categorical constructions of open-closed field algebras over V.Comment: 55 pages, largely revised, an old subsection is deleted, a few
references are adde
The Nakayama automorphism of the almost Calabi-Yau algebras associated to SU(3) modular invariants
We determine the Nakayama automorphism of the almost Calabi-Yau algebra A
associated to the braided subfactors or nimrep graphs associated to each SU(3)
modular invariant. We use this to determine a resolution of A as an A-A
bimodule, which will yield a projective resolution of A.Comment: 46 pages which constitutes the published version, plus an Appendix
detailing some long calculations. arXiv admin note: text overlap with
arXiv:1110.454