110 research outputs found
TFT construction of RCFT correlators III: Simple currents
We use simple currents to construct symmetric special Frobenius algebras in
modular tensor categories. We classify such simple current type algebras with
the help of abelian group cohomology. We show that they lead to the modular
invariant torus partition functions that have been studied by Kreuzer and
Schellekens. We also classify boundary conditions in the associated conformal
field theories and show that the boundary states are given by the formula
proposed in hep-th/0007174. Finally, we investigate conformal defects in these
theories.Comment: 78 pages, table of contents, several figures; v2: corrected
definition 2.20, added remark to section 4.
Calabi-Yau Frobenius algebras
We define Calabi-Yau and periodic Frobenius algebras over arbitrary base
commutative rings. We define a Hochschild analogue of Tate cohomology, and show
that the "stable Hochschild cohomology" of periodic CY Frobenius algebras has a
Batalin-Vilkovisky and Frobenius algebra structure. Such algebras include
(centrally extended) preprojective algebras of (generalized) Dynkin quivers,
and group algebras of classical periodic groups. We use this theory to compute
(for the first time) the Hochschild cohomology of many algebras related to
quivers, and to simplify the description of known results. Furthermore, we
compute the maps on cohomology from extended Dynkin preprojective algebras to
the Dynkin ones, which relates our CY property (for Frobenius algebras) to that
of Ginzburg (for algebras of finite Hochschild dimension).Comment: 39 pages; v3 has several corrections and some reorganizatio
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Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups
Let be an affine Weyl group, and let be a field of characteristic . The diagrammatic Hecke category for over is a categorification of the Hecke algebra for with rich connections to modular representation theory. We explicitly construct a functor from to a matrix category which categorifies a recursive representation , where is the rank of the underlying finite root system. This functor gives a method for understanding diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules which are "smaller" by a factor of . It also explains the presence of self-similarity in the -canonical basis, which has been observed in small examples. By decategorifying we obtain a new lower bound on the -canonical basis, which corresponds to new lower bounds on the characters of the indecomposable tilting modules by the recent -canonical tilting character formula due to Achar-Makisumi-Riche-Williamson
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