T-duality for circle bundles via noncommutative geometry

Abstract

Recently Baraglia showed how topological T-duality can be extended to apply not only to principal circle bundles, but also to non-principal circle bundles. We show that his results can also be recovered via two other methods: the homotopy-theoretic approach of Bunke and Schick, and the noncommutative geometry approach which we previously used for principal torus bundles. This work has several interesting byproducts, including a study of the Ktheory of crossed products by O(2) = Isom(R), the universal cover of O(2), and some interesting facts about equivariant K-theory for Z/2. In the final section of this paper, some of these results are extended to the case of bundles with singular fibers, or in other words, non-free O(2)-actions.

The Künneth Theorem in equivariantK–theory for actions of a cyclic group of order 2

“…This work was supported by NSF Grant DMS-1206159. It grew out of the author's puzzlement about certain aspects of equivariant K-theory that came up in joint work with Mathai Varghese [7] begun in March, 2012, with partial support from the University of Adelaide and the Australian Mathematical Sciences Institute. Some of the ideas in this paper go back to the author's joint work with Claude Schochet in the 1980's.…”

T-duality on nilmanifolds

“…Remark. One can start with a different complex structure and do the same calculations above to get a condition similar to (18). There is only one 6-dimensional 2-step nilpotent Lie algebra that admits no symplectic form: a 5 (R) × R = (0, 0, 0, 0, e 12 + e 34 , 0).…”

“…The following is a clear result. The next example is due to Mathai and Rosenberg [18] (see also [17]).…”

Pre‐NQ Manifolds and Correspondence Spaces: the Nilmanifold Example

“…Studying dualities from the sigma‐model point of view uses total spaces of Courant algebroids as target spaces . The mathematical study of T‐duality for principal torus bundles with H ‐flux in the geometric case and beyond showed the need for continuous fields of noncommutative and nonassociative tori . This is referred to as topological T‐duality.…”

“…[7][8][9][10][11][12][13] The mathematical study of T-duality for principal torus bundles with H-flux in the geometric case and beyond showed the need for continuous fields of noncommutative and nonassociative tori. [14][15][16][17][18] This is referred to as topological T-duality. In physics jargon, the latter are the nongeometric Tduals of manifolds with abelian gerbe structure.…”