String theory is arguably the most exciting research area in modern mathematical physics. It is known to the general public as the "Theory Of Everything", thanks to its great success in unifying Relativity and Quantum Field Theory, yielding Quantum Gravity theory. The impact of string theory is not just felt in physics, but it also has profound interactions with a broad spectrum of modern mathematics, including noncommutative geometry, K-theory and index theory.
The theory of D-branes forms an important part of string theory. It arises as the T-dual of open strings on a circle bundle, where the open strings in the dual theory are no longer free to move everywhere in space, but are endowed with Dirichlet boundary conditions so that the endpoints are free to move only on a submanifold known as a D-brane. Such D-branes come with (Chan-Paton) vector bundles, and therefore their charge determines an element of K-theory, as was argued by Minasian-Moore. In the presence of a nontrivial B-field but whose Dixmier-Douady class is a torsion element of H^3(M, Z), Witten argued that D-branes no longer carry honest vector bundles, but they have a twisted or gauge bundle. In the presence of a nontrivial B-field whose Dixmier-Douady class is a general element of H^3(M, Z), it was proposed in [BM] that D-brane charges in type IIB string theories are measured by the twisted K-theory that was described earlier by Rosenberg, and the twisted bundles on the D-brane world-volumes were elements in this twisted K-theory. In [BCMMS], using bundle gerbes and their modules, a geometric interpretation of elements of twisted K-theory was obtained, and the the Chern-Weil representatives of the Chern character was studied. This was further generalized to the equivariant and the holomorphic cases in [MS]. The relevance of the equivariant case to conformal field theory was highlighted by the remarkable result of Freed, Hopkins and Teleman that the twisted G-equivariant K-theory of a compact connected Lie group G (with mild hypotheses) is graded isomorphic to the Verlinde algebra of G, with a shift given by the dual Coxeter number and the curvature of the B-field, where we recall that Verlinde algebra of a compact connected Lie group G is defined in terms of positive energy representations of the loop group of G, and arises naturally in physics in Chern-Simons theory which is defined using quantum groups and conformal field theory.
One recent development is the study in [BEM], [BEM2] of T-duality in the presence of background H-fluxes. Here the T-duality isomorphisms in twisted K-theory and twisted cohomology and the character formulae relating these are formulated and proved. Briefly, T-duality defines an isomorphism between the twisted K-theory of the total space of a circle bundle, to the twisted K-theory of the total space of a "T-dual" circle bundle with "T-dual" twist, and with a change of parity. Similar statements hold for twisted cohomology. One interesting consequence is that we can construct fusion products in twisted K-theory and twisted cohomology, whenever the twist is a non-trivial decomposable cohomology class. Another interesting consequence of our work is that it gives convincing evidence that a type IIA string theory A on a circle bundle of radius R in the presence of an H-flux, and a type IIB string theory B on a "T-dual" circle bundle of radius 1/R in the presence of a "T-dual" H-flux, are equivalent in the sense that the string states of string theory A are in canonical one to one correspondence with the string states of string theory B. This is a fundamental property of type II string theories that was predicted only in special cases earlier.
[BHM] studies the more general case of T-duality for principal torus bundles. The new phenomenon that occurs here is that not all H-fluxes are T-dualizable, and this paper works out the precise class of T-dualizable H-fluxes. The isomophisms in twisted K-theory and twisted cohomology also follow in this case. We also define the action of the T-duality group O(n, n, Z) on T-dual pairs of principal torus bundles, where n is the rank of torus bundle, where GO(n, n, Z) is the subgroup of GL(2n, Z) that preserves the bilinear pairing. All of T-dual pairs in a given orbit of O(n, n, Z) define physically equivalent type II string theories.
Time permitting, we will also describe results in [MR], where we give a complete characterization of T-duality on principal 2-torus-bundles with H-flux. As noticed in [BHM] for instance, principal torus bundles with H-flux do not necessarily have a T-dual which is a torus bundle. A big puzzle has been to explain these mysterious ``missing T-duals.'' Here we show that this problem is resolved using noncommutative topology. It turns out that every principal 2-torus-bundle with H-flux does indeed have a T-dual, but in the missing cases (which we characterize), the T-dual is non-classical and is a bundle of noncommutative tori. This suggests an unexpected link between classical string theories and the ``noncommutative'' ones, obtained by ``compactifying'' along noncommutative tori. More generally, in [MR2], we give a complete characterization of T-duality for general principal torus-bundles with H-flux, generalizing the results in [MR] to higher rank torus bundles. The striking new feature in the case when the rank of the torus bundle is greater than or equal to 3 is that not every such torus bundle has a T-dual, either classical or nonclassical. The simplest example is the rank 3 torus over a point.
We mention that the most general case of T-duality on principal torus-bundles with H-flux is covered in [BHM2], [BHM3], where the T-dual is in general a bundle of nonassociative tori.
Dixmier-Douady theory and bundle gerbes
Twisted K-theory and geometric realizations
Twisted Chern character and twisted cohomology
T-duality in the presence of background H-fluxes I
T-duality in the presence of background H-fluxes II
[BHM3] P. Bouwknegt, K. Hannabuss and V. Mathai, T-duality for principal torus bundles and dimensionally reduced Gysin sequences, (2004) 21 pages [hep-th/0412268]
[BHM2] P. Bouwknegt, K. Hannabuss and V. Mathai, Nonassociative tori and applications to T-Duality, (2004) 29 pages [hep-th/0412092]
[MR2] V. Mathai and J. Rosenberg, On mysteriously missing T-duals, H-flux and the T-duality group, (2004) 4 pages. [hep-th/0409073]
[MR] V. Mathai and J. Rosenberg, T-Duality for torus bundles via noncommutative topology, Communications in Mathematical Physics, 253 (2005) 705-721. [hep-th/0401168]
[BEM2] P. Bouwknegt, J. Evslin and V. Mathai, On the Topology and Flux of T-Dual Manifolds, Physical Review Letters 92, 181601 (2004) [hep-th/0312052]
[BHM] P. Bouwknegt, K. Hannabuss and V. Mathai, T-Duality for principal torus bundles, Journal of High Energy Physics, 03 (2004) 018, 10 pages. [hep-th/0312284]
[MSa] V. Mathai and H. Sati, Some relations between twisted K-theory and E8 gauge theory, Journal of High Energy Physics, 03 (2004) 016, 21 pages [hep-th/0312033]
[MMS] V. Mathai, M.K. Murray and D. Stevenson, Type I D-branes in an H-flux and twisted KO-theory, Journal of High Energy Physics, 11 (2003) 053, 23 pages [hep-th/0310164]
[BEM] P. Bouwknegt, J. Evslin and V. Mathai, T-duality: Topology Change from H-flux, Communications in Mathematical Physics, 249 no. 2 (2004) 383 - 415 [hep-th/0306062]
[MS] V. Mathai and D. Stevenson, Chern character in twisted K-theory: equivariant and holomorphic cases, Communications in MathematicalPhysics, 236, no. 1 (2003), 161-186.
[BCMMS] P. Bouwknegt, A. Carey, V. Mathai, M. Murray and D. Stevenson, Twisted K-theory and K-theory of bundle gerbes, Communications in Mathematical Physics, 228, no. 1, (2002) 17-49.
[BM] P. Bouwknegt and V. Mathai, D-Branes, B-Fields and twisted K-theory, Journal of High Energy Physics, 007 (2000) 7 pages.
T-duality for principal torus bundles
Updated 14/09/2005