Spherical T-Duality

TL;DR

This paper introduces spherical T-duality, a new mathematical framework relating principal $SU(2)$-bundles and 7-cocycles, which induces isomorphisms in twisted cohomologies and K-theories, with implications for string theory charges.

Contribution

It develops the concept of spherical T-duality, establishing its properties, limitations, and effects on cohomological invariants, expanding the understanding of dualities in higher-dimensional bundles.

Findings

Spherical T-duality exchanges 7-cocycles with second Chern classes.

Induces degree-shifting isomorphisms on 7-twisted cohomologies.

Provides charge identifications in certain string theory compactifications.

Abstract

We introduce spherical T-duality, which relates pairs of the form $(P,H)$ consisting of a principal $SU(2)$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$. Intuitively spherical T-duality exchanges $H$ with the second Chern class $c_2(P)$. Unless $dim(M)\leq 4$, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. Nonetheless, we prove that all spherical T-dualities induce a degree-shifting isomorphism on the 7-twisted cohomologies of the bundles and, when $dim(M)\leq 7$, also their integral twisted cohomologies and, when $dim(M)\leq 4$, even their 7-twisted K-theories. While spherical T-duality does not appear to relate equivalent string theories, it does provide an identification between conserved charges in certain distinct IIB supergravity and string compactifications.