Christopher L. Douglas

The balanced tensor product of module categories

The balanced tensor product M (x)_A N of two modules over an algebra A is the vector space corepresenting A-balanced bilinear maps out of the product M x N. The balanced tensor product M [x]_C N of two module categories over a monoidal linear category C is the linear category corepresenting C-balanced right-exact bilinear functors out of the product category M x N. We show that the balanced tensor product can be realized as a category of bimodule objects in C, provided the monoidal linear category is finite and rigid.

Dualizability and index of subfactors

In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with finite dimensional centers, the Haagerup L 2 -space and Connes fusion are functorial with respect to homor- phisms of finite index. Along the way, we describe a string diagram notation for maps between bimodules that are not necessarily bilinear.

Conformal Nets II: Conformal Blocks

Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the ‘bundle of conformal blocks’, a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.

Topological modular forms

Elliptic genera and elliptic cohomology by C. Redden Ellliptic curves and modular forms by C. Mautner The moduli stack of elliptic curves by A. Henriques The Landweber exact functor theorem by H. Hohnhold Sheaves in homotopy theory by C. L. Douglas Bousfield localization and the Hasse square by T. Bauer The local structure of the moduli stack of formal groups by J. Lurie Goerss-Hopkins obstruction theory by V. Angeltveit From spectra to stacks by M. Hopkins The string orientation by M. Hopkins The sheaf of E ring spectra by M. Hopkins The construction of tmf by M. Behrens The homotopy groups of tmf and of its localizations by A. Henriques Ellitpic curves and stable homotopy I by M. J. Hopkins and H. R. Miller From elliptic curves to homotopy theory by M. Hopkins and M. Mahowald 1 E ring spectra by M. J. Hopkins Glossary by C. L. Douglas, J. Francis, A. G. Henriques, and M. A. Hill

Fusion Rings of Loop Group Representations

We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.

On the Structure of the Fusion Ideal

We prove that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, we compute the fusion ring of G2 at all levels.

On the algebra of cornered Floer homology

Bordered Floer homology associates to a parametrized oriented surface a certain differential graded algebra. We study the properties of this algebra under splittings of the surface. To the circle we associate a differential graded 2-algebra, the nilCoxeter sequential 2-algebra, and to a surface with connected boundary an algebra-module over this 2-algebra, such that a natural gluing property is satisfied. Moreover, with a view toward the structure of a potential Floer homology theory of 3-manifolds with codimension-two corners, we present a decomposition theorem for the Floer complex of a planar grid diagram, with respect to vertical and horizontal slicing.

Conformal nets IV : The 3-category

Conformal nets are a mathematical model for conformal field theory, and defects between conformal nets are a model for an interaction or phase transition between two conformal field theories. In the preceding paper of this series, we introduced a notion of composition, called fusion, between defects. We also described a notion of sectors between defects, modeling an interaction among or transformation between phase transitions, and defined fusion composition operations for sectors. In this paper we prove that altogether the collection of conformal nets, defects, sectors, and intertwiners, equipped with the fusion of defects and fusion of sectors, forms a symmetric monoidal 3-category. This 3-category encodes the algebraic structure of the possible interactions among conformal field theories.