Christopher Schommer-Pries

Central extensions of smooth 2–groups and a finite-dimensional string 2–group

We provide a model of the String group as a central extension of finite-dimensional 2‐groups in the bicategory of Lie groupoids, left-principal bibundles, and bibundle maps. This bicategory is a geometric incarnation of the bicategory of smooth stacks and generalizes the more naive 2‐category of Lie groupoids, smooth functors and smooth natural transformations. In particular this notion of smooth 2‐group subsumes the notion of Lie 2‐group introduced by Baez and Lauda [5]. More precisely we classify a large family of these central extensions in terms of the topological group cohomology introduced by Segal [56], and our String 2‐group is a special case of such extensions. There is a nerve construction which can be applied to these 2‐groups to obtain a simplicial manifold, allowing comparison with the model of Henriques [23]. The geometric realization is an A1 ‐space, and in the case of our model, has the correct homotopy type of String.n/. Unlike all previous models [58; 60; 33; 23; 7] our construction takes place entirely within the framework of finitedimensional manifolds and Lie groupoids. Moreover within this context our model is characterized by a strong uniqueness result. It is a canonical central extension of Spin.n/. 57T10, 22A22, 53C08; 18D10 The String group is a group (or A1 ‐space) which is a 3‐connected cover of Spin.n/. It has connections to string theory, the generalized cohomology theory topological modular forms (tmf ), and to the geometry and topology of loop space. Many of these relationships can be explored homotopy theoretically, but a geometric model of the String group would help provide a better understanding of these subjects and their interconnections. Over the past decade there have been several attempts to provide geometric models of the String group; see Stolz [58], Stolz and Teichner [60], Jurco [33], Henriques [23] and Baez, Stevenson, Crans and Schreiber [7]. The most recent of these use the language of higher categories, and consequently string differential geometry also provides a test case for the emerging field of higher categorical differential geometry; see Waldorf [64] and Sati, Schreiber and Stasheff [51; 52].

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.

Tori detect invertibility of topological field theories

A once-extended d-dimensional topological field theory Z is a symmetric monoidal functor (taking values in a chosen target symmetric monoidal (infty,2)-category) assigning values to (d-2)-manifolds, (d-1)-manifolds, and d-manifolds. We show that if Z is at least once-extended and the value assigned to the (d-1)-torus is invertible, then the entire topological field theory is invertible, that is it factors through the maximal Picard infty-category of the target. Results are obtained in the presence of arbitrary tangential structures.