Andre Henriques

Crystals and coboundary categories

Following an idea of A. Berenstein, we define a commutor for the category of crystals of a finite dimensional complex reductive Lie algebra. We show that this endows the category of crystals with the structure of a coboundary category. Similar to the case of braided categories, there is a group naturally acting on multiple tensor products in coboundary categories. We call this group the cactus group and identify it as the fundamental group of the moduli space of marked real genus zero stable curves.

Presentations of noneffective orbifolds

It is well known that an effective orbifold M (one for which the local stabilizer groups act effectively) can be presented as a quotient of a smooth manifold P by a locally free action of a compact Lie group K. We use the language of groupoids to provide a partial answer to the question of whether a noneffective orbifold can be so presented. We also note some connections to stacks and gerbes.

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

What Chern-Simons theory assigns to a point

Significance There are two main classes of 3D topological field theories: Turaev–Viro theories, associated to fusion categories, and Reshetikhin–Turaev theories, associated to modular tensor categories. Since the groundbreaking work by Lurie on the cobordism hypothesis, it has been a major open question to know which topological field theories (TFTs) extend down to points. Turaev–Viro theories can be extended down to points. But for most Reshetikhin–Turaev theories, including Chern–Simons theories, this was believed to be impossible (unless one puts them on the boundary of a 4D TFT). The present paper achieves two things: It shows that Reshetikhin–Turaev theories extend down to points, and it puts Turaev–Viro theories and Reshetikhin–Turaev theories on an equal footing by providing a unified language, bicommutant categories, that applies to both. We answer the questions, “What does Chern–Simons theory assign to a point?” and “What kind of mathematical object does Chern–Simons theory assign to a point?” Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group 𝛀G. We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel’d center of that representation category of 𝛀G is equivalent to the category of positive energy representations of the free loop group LG.† The abovementioned conjectures are known to hold when the gauge group is abelian or of type A1. Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside Bim(R), the category of bimodules over a hyperfinite 𝐼𝐼𝐼1 factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type An.

Bicommutant categories from fusion categories

Bicommutant categories are higher categorical analogs of von Neumann algebras that were recently introduced by the first author. In this article, we prove that every unitary fusion category gives an example of a bicommutant category. This theorem categorifies the well-known result according to which a finite dimensional

Automata, groups, limit spaces, and tilings

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