# Talbot 2010: Twisted K-Theory and Loop Groups

## Mentored by Constantin Teleman.

### May 23 to May 29, 2010. Breckenridge, Colorado

Immediately after the workshop, Daniel Berwick-Evans and Jesse Wolfson combined their notes from the workshop. These notes are, in unedited form, linked here:

Unedited notes by Daniel Berwick-Evans and Jesse Wolfson.

Also listed below is the talk schedule from the workshop. Some of the speakers graciously edited the above notes--the links provided below are the edited notes from the talks, in which the speakers made various corrections to the unedited notes above.

 Talk Title Speaker Description Day One Survey One Constantin Teleman, UC Berkeley. 2d TQFTs, string topology, finite group gauge theory, $K_G(G)$, relation to holomorphic bundles Introduction to K-Theory Jesse Wolfson, Northwestern. Generalized cohomology. K-theory: Vector bundles, BU, Bott periodicity, Thom isomorphism, Gysin maps. Chern character, topological Riemann-Roch. Edited notes posted here. More K-theory Chris Kottke, MIT. Fredholm and GL(res) models of BU. $C^*$ algebra K theory and K-homology (quickly). Clifford algebras and odd K-theory. Dirac operators, Gysin map as the index. Hopefully, relation between Dirac and Dolbeault, and $\operatorname{Spin}^c$ vs complex orientations. Edited notes posted here. Twisted K-theory Mehdi Sarikhani Khorami, Wesleyan Units and twistings in cohomology theories. Special twistings of K from line bundles and gradings. K (by graded lines) with examples. $C^*$ algebra model for twisted K via Azumaya algebras. Chern isomorphism (statement only). Edited notes posted here. More twistings Braxton Collier, University of Texas Distinction between twistings and their characteristic class (Example in computations). Category of twistings, Projective bundle model, gerbes. Gradings? Chern-Weil construction of Chern character? (informal) Could extend this to an optional discussion. Day Two Equivariant K-theory and its twisted versions Mio Alter, University of Texas. Vector bundle and $C^*$ algebra models in untwisted case. Completion theorem. Atiyah-Segal construction of twisted equivariant K-theory. $C^*$ (Azumaya) algebra construction. Edited notes posted here. Equivariant Chern character Owen Gwilliam, Northwestern. Generalities. Delocalised Chern character. Examples. version. Computation of twisted complex $K_G(G) \otimes \C$. (Time permitting but unlikely: the graded case, computation for $G=SO(3)$.) Edited notes posted here. $K_G(G)$ Daniel Halpern-Leistner, UC Berkeley. Computation in simply connected case (or free $\pi_1$) case. General (additive) description in terms of affine Weyl group. Beginnings of TFT structure: transgression of twistings from $H^4$ and the Pontrjagin product. K-theory of Topological Stacks Ryan Grady, Notre Dame. K-theory of topological stacks. Example of local-but-not global quotient stacks: gerbes as topological stacks. Stack description of twisted K. Computation of $S^1$-equivariant $K_G(G)$, induction from $\operatorname{Rep}(G\times S^1)$ and appearance of the Kac numerator. Edited notes posted here. Day Three Loop groups and their PER's Harold Williams, UC Berkeley. Case of a torus. Affine Weyl group, flag variety. Borel-Weil construction of irreps, for compact groups and for loop groups by holomorphic induction on co-adjoint orbits. Parametrization of irreps by affine Weyl action on weights. Character formulae Dario Beraldo, UC Berkeley Weyl-Kac and Kirillov formulae for compact groups and for loop groups. [Mention: Character of the Spin representation.] Edited notes posted here. Dirac family construction of K-classes Sander Kupers, Utrecht Construction of $K_T(T)$ classes by spectral flow, description in terms of Heisenberg-like groups based on a torus. Dirac operator on a compact group; algebraic description via Peter-Weyl, Dirac family on $g$ and K-class. Outline of the construction for loop groups. Edited notes posted here and here. Day Four 2-tier TQFT structure AJ Tolland, SUNY Stony Brook. Construction of TQFT operations via correspondence spaces. Trace on twisted $K_G(G)$ and topological Peter-Weyl. Verlinde formula as index formula in twisted K-theory. Higher twistings of K and more general index formula. Edited notes posted here. Survey 2: Known and unfinished business Constantin Teleman, UC Berkeley. Open-closed 2d field theories Matt Young, SUNY Stony Brook. After Kontsevich, Costello, Hopkins-Lurie, Discuss Hochschild homology, etc. Edited notes posted here. B-model of a singularity as open-closed theory Kevin Lin, UC Berkeley Landau-Ginzburg potentials, category of branes, Frobenius property $K_G(G)$ as open closed theory Constantin Teleman, UC Berkeley Day Five Chern-Simons theory as 3-2-1 theory Hiro Tanaka, Northwestern. (after Witten, Reshetikhin-Turaev etc). Discussion of braided tensor categories. Edited notes posted here. Chern-Simons for a torus via the categorified group ring Konrad Waldorf, UC Berkeley Edited notes posted here. Chern-Simons for nonabelian G via conformal nets Chris Douglas, UC Berkeley Elliptic cohomology Nick Rozenblyum, MIT. Informal introduction, restriction to the Tate curve and K-theory of loop space. Twistings of elliptic cohomology from H^4. Equivariant elliptic cohomology and loop group reps Toly Preygel, MIT.