Spherical T-duality and the spherical Fourier–Mukai transform

Abstract

In [3,4], we introduced spherical T-duality, which relates pairs of the form (P, H) consisting of an oriented S 3 -bundle P → M and a 7-cocycle H on P called the 7-flux. Intuitively, the spherical T-dual is another such pair (P ,Ĥ) and spherical T-duality exchanges the 7-flux with the Euler class, upon fixing the Pontryagin class and the second Stiefel-Whitney class. Unless dim(M ) ≤ 4, not all pairs admit spherical T-duals and the spherical T-duals are not always unique. In this paper, we define a canonical Poincaré virtual line bundle P on S 3 × S 3 (actually also for S n × S n ) and the spherical Fourier-Mukai transform, which implements a degree shifting isomorphism in K-theory on the trivial S 3 -bundle. This is then used to prove that all spherical T-dualities induce natural degreeshifting isomorphisms between the 7-twisted K-theories of the pairs (P, H) and (P ,Ĥ) when dim(M ) ≤ 4, improving our earlier results.2010 Mathematics Subject Classification. Primary 81T30.In [3,4], we introduced a new kind of duality for string theory (M-theory), termed spherical T-duality, for 7D spacetimes that are compactified as SU(2)-bundles with 7-flux over 4D manifolds,In [3] we dealt only with principal SU(2)-bundles with 7-flux, whereas in [4] we dealt with the more general case of (oriented) SU(2)-bundles with 7-flux that were not necessarily principal bundles. We showed that a principal SU(2)-bundle with 7-flux, had a unique spherical Tdual principal SU(2)-bundle with T-dual 7-flux, where the 7-flux gets exchanged with the 2nd Chern number, and there is an equivalence of 7-twisted cohomologies and 7-twisted K-theories (modulo an extension problem), see [3]. On the other hand, a non-principal (oriented) SU(2)-bundle with 7-flux can have infinitely many spherical T-duals that are also non-principal (oriented) SU(2)-bundles with 7-flux, and once again there is an equivalence of 7-twisted cohomologies and 7-twisted K-theories (modulo an extension problem), see [4]. The problem in this case is that non-principal (oriented) SU(2)-bundles on simply-connected 4manifolds are classified by the 2nd Chern number (or Euler number) as well as the Pontryagin number and the 2nd Stiefel-Whitney class. So if we fix the the Pontryagin number, and the 2nd Stiefel-Whitney class, we again get a unique spherical T-dual in the non-principal case also. Since the initial version of this paper was posted on the arxiv, the interesting paper [13] appeared on the arxiv, which formulated a generalization of spherical T-duality to possibly nonorientable sphere bundles, and proved that it induces an isomorphism on a class of twisted cohomology theories that includes algebraic K-theory. In [3,4], we argued that the 7-twisted cohomology and the 7-twisted K-theory which featured in our main theorems classify certain conserved charges in type IIB supergravity. We concluded that spherical T-duality provides a one to one map between conserved charges in certain topologically distinct compactifications and also a novel electromagnetic duality on the fluxes.I...

Computations in higher twisted $K$-theory

“…In the classical case, Bouwknegt, Evslin and Mathai show that the T-duality transformation induces an isomorphism on twisted K-theory, which was an unexpected result as the T-duality transformation often leads to significant differences between the topologies of the circle bundles in question [15,16]. They then generalise this to the case of spherical T-duality [12,13,14] in which the relevant cohomology class is of degree 7, which led to the result that the spherical T-duality transform induces an isomorphism on higher twisted K-theory. In the first of this series of papers, the authors also provide some insight into how higher twisted K-theory fits in with the D-brane picture.…”

“…Our final computations are for a class of examples which arise in spherical T-duality; the total spaces of SU (2)-bundles. Bouwknegt, Evslin and Mathai prove that the spherical T-duality transformation induces a degree-shifting isomorphism on the 7-twisted K-theory groups of these bundles [12,13,14], but they do not consider the 5-twisted K-theory of these SU (2)-bundles. We place restrictions on the base space M in order to ensure that the 5-twists of the total space of the bundle correspond exactly to its integral 5-classes, and then use the Atiyah-Hirzebruch spectral sequence to compute the 5-twisted K-theory in Section 5.6.…”

“…It is of great interest in both string theory and mathematical gauge theory to investigate SU (2)-bundles over manifolds M . The link with string theory appears through spherical T-duality; a generalisation of T-duality investigated in a series of papers by Bouwknegt, Evslin and Mathai [12,13,14]. As shown by the authors, this form of duality provides a map between certain conserved charges in type IIB supergravity and string compactifications.…”