group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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.
The idea is due to
which in the course considers higher Snaith spectra and higher order algebraic K-theory.
A special case of this general story is discussed in some detail in
Peter Bouwknegt, Jarah Evslin, Varghese Mathai, Spherical T-Duality, Commun. Math. Phys. 337:909-954, 2015 (arXiv:1405.5844)
Peter Bouwknegt, Jarah Evslin, Varghese Mathai, Spherical T-duality II: An infinity of spherical T-duals for non-principal SU(2)-bundles, J.Geom.Phys.92:46-54, 2015 (arXiv:1409.1296)
Peter Bouwknegt, Jarah Evslin, Varghese Mathai, Spherical T-Duality and the spherical Fourier-Mukai transform (arXiv:1502.04444)
See also
The realization in M-theory at the level of rational super homotopy theory is derived in
Domenico Fiorenza, Hisham Sati, Urs Schreiber, Higher T-duality of super M-branes (arXiv:1803.05634)
Hisham Sati, Urs Schreiber, Higher T-duality in M-theory via local supersymmetry, Physics Letters B Volume 781, 10 June 2018, Pages 694-698 (arXiv:1805.00233, doi:10.1016/j.physletb.2018.04.058)
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