We relate commutative algebras in braided tensor categories to braid-reversed tensor equivalences, motivated by vertex algebra representation theory. First, for C a braided tensor category, we give a detailed construction of the canonical algebra in C⊠C^rev: if C is semisimple but not necessarily finite or rigid, then ⨁X∈Irr(C) X′⊠X is a commutative algebra, with X′ a representing object for Hom_C(∙⊗_C X,1_C). Conversely, let A=⨁i∈I U_i ⊠ V_i be a simple commutative algebra in U⊠V with U semisimple and rigid but not necessarily finite, and V rigid but not necessarily semisimple. If the unit objects of U and V form a commuting pair in A, we show there is a braid-reversed equivalence between subcategories of U and V sending U_i to (V_i)*. When U and V are module categories for simple vertex operator algebras U and V, we glue U and V along U⊠V via a map τ: Irr(U)→Obj(V) such that τ(U)=V to create A=⨁X∈Irr(U) X′⊗τ(X). Thus under certain conditions, τ extends to a braid-reversed equivalence between U and V if and only if A is a simple conformal vertex algebra extending U⊗V. As examples, we glue Kazhdan-Lusztig categories at generic levels to obtain new vertex algebras extending the tensor product of two affine vertex algebras, and we prove braid-reversed equivalences between certain module categories for affine vertex algebras and W-algebras at admissible levels.

A tautological system, introduced in \cite{LSY}\cite{LY}, arises as a regular holonomic system of partial differential equations that govern the period integrals of a family of complete intersections in a complex manifold X , equipped with a suitable Lie group action. In this article, we introduce two formulas -- one purely algebraic, the other geometric -- to compute the rank of the solution sheaf of such a system for CY hypersurfaces in a generalized flag variety. The algebraic version gives the local solution space as a Lie algebra homology group, while the geometric one as the middle de Rham cohomology of the complement of a hyperplane section in X . We use both formulas to find certain degenerate points for which the rank of the solution sheaf becomes 1. These rank 1 points appear to be good candidates for the so-called large complex structure limits in mirror symmetry. The formulas are also used to prove a conjecture of Hosono, Lian and Yau on the completeness of the extended GKZ system when X is $\P^n$.

No abstract uploaded!

No abstract uploaded!

We study the relation between different spaces of vector-valued polynomials and analytic functions over dual-isomorphic Banach spaces. Under conditions of regularity on$E$and$F$, we show that the spaces of$X$-valued$n$-homogeneous polynomials and analytic functions of bounded type on$E$and$F$are isomorphic whenever$X$is a dual space. Also, we prove that many of the usual subspaces of polynomials and analytic functions on$E$and$F$are isomorphic without conditions on the involved spaces.