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T-Duality in K-theory and Elliptic Cohomology
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This page records some information on the results announced in the talk

Thomas Nikolaus , T-Duality in K-theory and Elliptic Cohomology , talk at String Geometry Network Meeting , Feb 2014, ESI Vienna (website )
The talk announced three results which are foundational for the topic of topological T-duality .

Contents
1. T-duality equivalences in universal elliptic cohomology ($tmf$ )
A famous result of the 1990s is that D-brane charges , hence the information associated with the boundary conditions of open superstrings , are cocycles in twisted K-theory . Since two target spaces related under T-duality are supposed to induce equivalent superstring sigma-models , they should in particular have the same K-theory . This is indeed the case, a fact proven in full generality in topological T-duality (see there for references).

But T-duality is supposed to induce an equivalence not just of these booundary data, but for the full string theory . By the seminal results on the Witten genus it is known that the partition function of the superstring is an elliptic genus , and by the seminal results on the string orientation of tmf it is known that this is just a shadow of an orientation in tmf -generalized (Eilenberg-Steenrod) cohomology theory (the “universal elliptic cohomology ”). Ever since it is natural to expect that full string theories are actually classified not just in K-theory but in elliptic cohomology hence in tmf .

A properly developed theory of elliptic cohomology is likely to shed some light on what string theory really means. (Witten 87, very last sentence ).

Now (Nikolaus 14 ) claims, under certain conditions, that indeed topological T-duality induces equivalences in tmf cohomology classes. The conditions are actually fairly mild, one sufficient condition is that the rank of the torus bundle is $\leq 6$ . (This is already the most interesting case in applications in physics .)

2. Precise definition of T-folds
See at T-duality 2-group

3. K-Cohomology of T-folds
See at T-duality 2-group

Created on March 13, 2014 at 07:18:24.
See the history of this page for a list of all contributions to it.