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:
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 |
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| 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. |
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| 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. |
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| 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. |
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| 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 | |
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| 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. | |