Mini-Workshop: Gerbes, Twisted K-Theory and Conformal Field Theory

Abstract

The Mini-Workshop Gerbes, Twisted K-Theory and Conformal Field Theory organized by Branislav Jurčo (München), Jouko Mickelsson (Helsinki) and Christoph Schweigert (Hamburg) was held August 14th – August 20th, 2005. The idea of this mini-workshop was to bring together people working on different aspects of gerbes (and related topics) to exchange different points of view, make the interactions more intense and eventually establish new collaborations. Because of different backgrounds of participants, the organizers asked Jarah Evslin and Hisham Sati to prepare some expository lectures on $M$-theory, supergravity and superstrings oriented towards a more mathematical audience. Also Christoph Schweigert prepared an introduction on the use of gerbes and gerbe modules in the description of $D$-branes in Wess–Zumino–Witten models and Dale Husemöller reported on the point of view of stacks and gerbes in the form used in algebraic geometry following Deligne's exposition of nonabelian cohomology in LN 900.

As equivalence classes of line bundles are geometric realizations of the second integral cohomology classes, stably equivalence classes of bundle gerbes are geometric realizations of the third integral cohomology classes. Similarly to line bundle that can be equipped with connection, gerbe can be equipped with connection and curving ($B$-field). Taking this extra differential geometric structures into account the relevant cohomology is the Deligne cohomology. Gerbes can be equipped with modules, which are vector bundles twisted with the third integral cohomology class in questions. These are the structures important in string theory when we want to describe $D$-branes in topologically nontrivial backgrounds. $D$-brane charges are conjecturally classified (at least in some situations) by the corresponding twisted $K$-theory. This links very nicely $K$-theory to conformal filed theory. Similarly higher versions of gerbes and their nonabelian generalizations are expected to be useful for description of higher rank forms gauge theories and the conjectural $M$-theory. As usually, this interaction between geometry, topology and theoretical physics makes the subject attractive to researches with different backgrounds.

Gerbe and gerbe modules turn out to be a crucial input in the description of the Wess–Zumino term in the presence of boundaries. The quantization of the position of D-branes in string backgrounds with three-form fluxes actually arises as a consequence of integrality properties of these modules. This was discussed in talks of Christoph Schweigert and Krzyzstof Gawedzki who explained how gerbes help to solve WZW conformal sigma models and how they provide an uniform and effective treatment of classical and quantum theories without and with boundaries. Closely related to WZW theory is the Chern–Simons gauge theory. The precise relation between these two theories in the language of multiplicative bundle gerbes, 2-bundle gerbes and the corresponding Deligne cohomology classes was described in the talk of Michael Murray.

In the same way as vector bundles give rise to class in the topological K-theory of the manifold, gerbe modules give rise to a class in twisted K-theory. Twisted K-theory has been proposed as the recipient of D-brane. Equivariant twisted K-theory ${}^{\tau }{K}_G(G)$ on a compact Lie group $G$ can be endowed with the structure of a ring. Due to a result of Freed Hopkins and Teleman this ring turns out to be the Verlinde algebra, familiar from conformal field theory. More generally equivariant twisted K-theory ${}^{\tau }{K}_H(G)$ in the case when $G/H$ is hermitian symmetric can be related to the $\mathcal{N}=2$ chiral ring. This extension was discussed in the talk of Sakura Schäfer-Nameki. Volker Braun in his talk showed some explicit computations in twisted $K$-theory relevant for $N=1$ supersymmetric Wess–Zumino–Witten models. Explicit construction of twisted equivariant $K$-classes using Dirac operators related to supersymmetric WZW models was given in talk of Jouko Mickelsson. A nice application of multiplicative bundle gerbes was given by Alan Carey. The multiplicative structure gives rise to the fusion product on bundle gerbe $D$-branes. Quantization functor relates these to the twisted equivariant $K$-classes.

Another aspect is the action of dualities on gerbes and gerbe modules and the related twisted $K$-classes. In the case of circle and torus bundle, there has been a lot of activity recently. In their talks Jarah Evslin and Hisham Sati introduced the concept of duality how it is emerging in $M$-theory, supergravity and superstring theory. They explained how anomaly cancellation and consistency of quantum field theories under different dualities lead to the description of filed configurations in terms of generalized cohomology theories (e.g. $K$-theory and elliptic cohomology). Talk of Varghese Mathai was devoted to global aspects of $T$-duality and relations to noncommutative geometry. $T$-duality for orbispaces from the point of view of of stacks and gerbes introduced in talk of Dale Husemoller was presented by Ulrich Bunke in his lecture. Peter Bouwknegt gave an overview of the topic of generalized geometry and its relation to gerbes, $T$-duality and mirror symmetry.

Nonabelian gerbes, their relation to nonabelian cohomology and their application to quantum theory was another topic presented in some lectures. Nonabelian generalizations of bundle gerbes, their modules and differential geometry along with an application to anomaly of five branes were presented by Paolo Aschieri followed by the talk of Branislav Jurčo about simplicial description of nonabelian bundle gerbes with connection and curving. Simplicial description leads naturally to classification of bundle gerbes related to crossed modules. Closely related talk was given by Danny Stevenson who described the string gerbe based on the recent simplicial construction of the string group. He as well as Urs Schreiber in his talk on nonabelian holonomy used the groupoid point of view to gerbes. A nonabelian bundle gerbe can be thought as an example of a 2-bundle which comes together with the concept of higher algebraic structures and categorification, subjects of the lecture of Hendryk Pfeiffer.

We are very glad that there was a lot of interactions across the various groups of researchers. Physicists and mathematicians enjoyed both the talks and the lively after talks discussion. Also mini-workshop benefited a lot from the activity and curiosity of young participants.