Contents

cohomology

# Contents

## Idea

Spherical T-duality (Bouwknegt-Evslin-Mathai 14a) is the name given to a variation of topological T-duality where the role of the circle $S^1$, or the circle group $U(1)$, is replaced by the 3-sphere $S^3$, or the special unitary group $SU(2)$. Where topological T-duality relates pairs consisting of total spaces of $U(1)$-principal bundles equipped with a cocycle in degree-3 ordinary cohomology, spherical T-duality relates pairs consisting of $SU(2)$-principal bundles (or just $S^3$-fiber bundles (Bouwknegt-Evslin-Mathai 14b)) equipped with cocycles in degree-7 cohomology. As for topological T-duality, under suitable conditions spherical T-duality lifts to an isomorphism of twisted K-theory classes of these bundles with twisting by the 7-class.

In the approximation of rational super homotopy theory, topological spherical T-duality has been derived for the M5-brane, not on 11d super Minkowski spacetime itself, but on its M2-brane-extended super Minkowski spacetime, and from there on the exceptional super spacetime; see FSS 18a, reviewed in FSS 18b.

## References

The idea is due to

which in the course considers higher Snaith spectra and higher order iterated algebraic K-theory.

A special case of this general story is discussed in some detail in