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Christoph Schweigert

Publications and source records attributed to Christoph Schweigert.

At least 91 records · Page 5Linked to original sources
arXiv2011

Equivariance In Higher Geometry

We study (pre-)sheaves in bicategories on geometric categories: smooth manifolds, manifolds with a Lie group action and Lie groupoids. We present three main results: we describe equivariant descent, we generalize the plus construction to our setting and show that the plus construction yields a 2-stackification for 2-prestacks. Finally we show that, for a 2-stack, the pullback functor along a Morita-equivalence of Lie groupoids is an equivalence of bicategories. Our results have direct applications to gerbes and 2-vector bundles. For instance, they allow to construct equivariant gerbes from local data and can be used to simplify the description of the local data. We illustrate the usefulness of our results in a systematic discussion of holonomies for unoriented surfaces.

math.AT↗
arXiv2011

Hopf algebras and finite tensor categories in conformal field theory

In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category. The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories. After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about representation categories of chiral symmetry algebras that are not semisimple. We focus on generalizations of the Verlinde formula, for which certain finite-dimensional complex Hopf algebras are used as a tool, and on the structural importance of the presence of a Hopf algebra internal to finite tensor categories.

hep-th↗
arXiv2010

A Geometric Construction for Permutation Equivariant Categories from Modular Functors

Let G be a finite group. Given a finite G-set X and a modular tensor category C, we construct a weak G-equivariant fusion category, called the permutation equivariant tensor category. The construction is geometric and uses the formalism of modular functors. As an application, we concretely work out a complete set of structure morphisms for Z/2-permutation equivariant categories, finishing thereby a program we initiated in an earlier paper arXiv:0812.0986 [math.CT].

math.QA↗
arXiv2010

The classifying algebra for defects

We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects - a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations give the defect transmission coefficients. We show in particular that the structure constants of the classifying algebra are traces of operators on spaces of conformal blocks and that the defect transmission coefficients determine the defect partition functions.

hep-th↗
arXiv2010

Hopf algebras and Frobenius algebras in finite tensor categories

We discuss algebraic and representation theoretic structures in braided tensor categories C which obey certain finiteness conditions. Much interesting structure of such a category is encoded in a Hopf algebra H in C. In particular, the Hopf algebra H gives rise to representations of the modular group SL(2,Z) on various morphism spaces. We also explain how every symmetric special Frobenius algebra in a semisimple modular category provides additional structure related to these representations.

math.QA↗
arXiv2010

Module categories for permutation modular invariants

We show that a braided monoidal category C can be endowed with the structure of a right (and left) module category over C \times C. In fact, there is a family of such module category structures, and they are mutually isomorphic if and only if C allows for a twist. For the case that C is premodular we compute the internal End of the tensor unit of C, and we show that it is an Azumaya algebra if C is modular. As an application to two-dimensional rational conformal field theory, we show that the module categories describe the permutation modular invariant for models based on the product of two identical chiral algebras. It follows in particular that all permutation modular invariants are physical.

math.CT↗
arXiv2009

On the Rosenberg-Zelinsky sequence in abelian monoidal categories

We consider Frobenius algebras and their bimodules in certain abelian monoidal categories. In particular we study the Picard group of the category of bimodules over a Frobenius algebra, i.e. the group of isomorphism classes of invertible bimodules. The Rosenberg-Zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the Picard group, which however is typically not surjective. We investigate under which conditions there exists a Morita equivalent Frobenius algebra for which the corresponding homomorphism is surjective. One motivation for our considerations is the orbifold construction in conformal field theory.

math.CT↗
arXiv2009

Defect lines, dualities, and generalised orbifolds

Defects are a useful tool in the study of quantum field theories. This is illustrated in the example of two-dimensional conformal field theories. We describe how defect lines and their junction points appear in the description of symmetries and order-disorder dualities, as well as in the orbifold construction and a generalisation thereof that covers exceptional modular invariants.

math-ph↗
arXiv2009

The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S^2 x S^1. Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.

hep-th↗
arXiv2009

Some remarks on defects and T-duality

The equations of motion for a conformal field theory in the presence of defect lines can be derived from an action that includes contributions from bibranes. For T-dual toroidal compactifications, they imply a direct relation between Poincare line bundles and the action of T-duality on boundary conditions. We also exhibit a class of diagonal defects that induce a shift of the B-field. We finally study T-dualities for S^1-fibrations in the example of the Wess-Zumino-Witten model on SU(2) and lens spaces. Using standard techniques from D-branes, we derive from algebraic data in rational conformal field theories geometric structures familiar from Fourier-Mukai transformations.

hep-th↗
arXiv2009

Bundle Gerbes and Surface Holonomy

Hermitian bundle gerbes with connection are geometric objects for which a notion of surface holonomy can be defined for closed oriented surfaces. We systematically introduce bundle gerbes by closing the pre-stack of trivial bundle gerbes under descent. Inspired by structures arising in a representation theoretic approach to rational conformal field theories, we introduce geometric structure that is appropriate to define surface holonomy in more general situations: Jandl gerbes for unoriented surfaces, D-branes for surfaces with boundaries, and bi-branes for surfaces with defect lines.

math.DG↗
arXiv2008

Uniqueness of open/closed rational CFT with given algebra of open states

We study the sewing constraints for rational two-dimensional conformal field theory on oriented surfaces with possibly non-empty boundary. The boundary condition is taken to be the same on all segments of the boundary. The following uniqueness result is established: For a solution to the sewing constraints with nondegenerate closed state vacuum and nondegenerate two-point correlators of boundary fields on the disk and of bulk fields on the sphere, up to equivalence all correlators are uniquely determined by the one-, two,- and three-point correlators on the disk. Thus for any such theory every consistent collection of correlators can be obtained by the TFT approach of hep-th/0204148, hep-th/0503194. As morphisms of the category of world sheets we include not only homeomorphisms, but also sewings; interpreting the correlators as a natural transformation then encodes covariance both under homeomorphisms and under sewings of world sheets.

hep-th↗
arXiv2007

Kramers-Wannier dualities for WZW theories and minimal models

We study Kramers-Wannier dualities for Wess-Zumino-Witten theories and (super-)minimal models in the Cardy case, i.e. the case with bulk partition function given by charge conjugation. Using the TFT approach to full rational conformal field theories, we classify those dualities that preserve all chiral symmetries. Dualities turn out to exist for small levels only.

hep-th↗
arXiv2007

Gerbes and Lie Groups

We present a review of bundle gerbes, emphasizing their relations to Lie groups. Indeed, compact Lie groups do not only carry the structure of a Riemannian manifold, but also canonical families of bundle gerbes. We recall the construction of these bundle gerbes and their relation to loop groups. We discuss several algebraic structures for bundle gerbes with connection such as Jandl structures, gerbe modules and gerbe bimodules, and indicate their applications to Wess-Zumino terms in two-dimensional field theories.

math.DG↗
arXiv2007

Unoriented WZW Models and Holonomy of Bundle Gerbes

The Wess-Zumino term in two-dimensional conformal field theory is best understood as a surface holonomy of a bundle gerbe. We define additional structure for a bundle gerbe that allows to extend the notion of surface holonomy to unoriented surfaces. This provides a candidate for the Wess-Zumino term for WZW models on unoriented surfaces. Our ansatz reproduces some results known from the algebraic approach to WZW models.

hep-th↗
arXiv2007

Topological defects for the free boson CFT

Two different conformal field theories can be joined together along a defect line. We study such defects for the case where the conformal field theories on either side are single free bosons compactified on a circle. We concentrate on topological defects for which the left- and right-moving Virasoro algebras are separately preserved, but not necessarily any additional symmetries. For the case where both radii are rational multiples of the self-dual radius we classify these topological defects. We also show that the isomorphism between two T-dual free boson conformal field theories can be described by the action of a topological defect, and hence that T-duality can be understood as a special type of order-disorder duality.

hep-th↗
arXiv2007

Bi-branes: Target Space Geometry for World Sheet topological Defects

We establish that the relevant geometric data for the target space description of world sheet topological defects are submanifolds - which we call bi-branes - in the product M1 x M2 of the two target spaces involved. Very much like branes, they are equipped with a vector bundle, which in backgrounds with non-trivial B-field is actually a twisted vector bundle. We explain how to define Wess-Zumino terms in the presence of bi-branes and discuss the fusion of bi-branes. In the case of WZW theories, symmetry preserving bi-branes are shown to be biconjugacy classes. The algebra of functions on a biconjugacy class is shown to be related, in the limit of large level, to the partition function for defect fields. We finally indicate how the Verlinde algebra arises in the fusion of WZW bi-branes.

hep-th↗