# nLab T-Duality in K-theory and Elliptic Cohomology

Contents

## Phenomenology

cohomology

### Theorems

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 boundary 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

category: reference

Last revised on April 7, 2020 at 21:55:25. See the history of this page for a list of all contributions to it.