T-Duality: Topology Change from H-Flux

Abstract

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E 8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, Lens spaces, both circle bundles over RP n , and the AdS 5 × S 5 to AdS 5 × CP 2 × S 1 with background H-flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G 4 receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings. * 1 However there are torii that do not lift to F-theory on which we may T-dualize, for example, a 2-torus that supports G3 flux. S-dualizing, the obstruction to T-duality on a torus with H-flux is the controversial obstruction to S-duality in the presence of G1 flux.2 Strictly speaking, the various forms entering (1.1) are the pull-backs of forms to the correspondence space3 To simplify the notations we will use the same notation for a cohomology class [H], or for a representative H, throughout this paper. It should be clear which is meant from the context. 4 Throughout this paper the notationÊ will refer to the T-dual of the bundle E, and not to the dual bundle in the usual sense.

T-Duality for Torus Bundles with H-Fluxes via Noncommutative Topology

“…The case n = 1 was treated in [27,Theorem 4.12], from a purely C * -algebraic perspective, in [4], from a combined mathematical and physical perspective, and in [5] from a more physical point of view. In this case, G = R, T = T = R/Z, and N = Z.…”

“…Just as a complete characterization of T-duality on circle bundles with H-flux is given in [4] and [5], in this paper, we give a complete characterization of T-duality on principal T 2 -bundles with H-flux, Theorem 4.4.2. We also describe partial results for T-duality on general principal torus bundles with H-flux.…”

Non-Kähler SYZ Mirror Symmetry

“…T-duality and SYZ in the setting of generalized Calabi-Yau manifolds were studied in [BEM04,BHM04,FMT05,GSN07,CG10] where H-flux played the key role, and also from the bispinor perspective which includes RR fluxes in [GMPT07,GMPW09]. The emphasis of this paper is different: we study SYZ in the presence of RR-flux and in particular focusing on the balanced Hermitian metric condition on non-Kähler manifolds, which also appears as part of the Strominger system [Str86].…”

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

“…Thanks to Theorem 1.1 we conclude that if two stable C * -algebras B and B ′ are T-dual, such that there is an invertible element α ∈ KK 0 (B, ΣB ′ ), then their noncommutative motives HPf dg (B) and HPf dg (B ′ ) are isomorphic in Hmo 0 . Furthermore, Theorem 1.2 asserts that the invertible element α implements the twisted K-theory isomorphism (like (2)) that is expected from T-duality.…”

“…T-duality is an interesting phenomenon is string theory, some of whose mathematical aspects were studied in [2]. We are solely going to focus on axiomatic topological T-duality from [8], which builds upon the earlier work in [2]. Let B be a topological base space.…”