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. For a cool video on string theory, based on Brian Greene's bestseller book entitled "The elegant universe", click here. 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 pioneers of string theory were M.B. Green and J.H. Schwarz and the leading figure in the field is E. Witten.

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. For a link describing the mathematics behind D-branes, cf. superstrings. 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 H³(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 H³(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.

Type I D-branes in the presence of an H-flux are studied in [MMS], where a geometric interpretation of H²(M,Z₂) is given in terms of stable isomorphisms of real bundle gerbes, and the twisted KO theory is interpreted geometrically in terms of real projective vector bundles.

One development is the novel discovery in [BEM], [BEM2] of T-duality isomorphisms in twisted K-theory and twisted cohomology and the character formulae relating these. 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 a spacetime and its T-dual spacetime can be topologically different! To quote Edward Witten, "There was a long history of speculation that in quantum gravity, unlike Einstein's classical theory, it might be possible for the topology of spacetime to change." Thus our research work can be viewed as realizing this speculation in the context of T-duality in String Theory with background H-flux.

Another outcome of our work is that we can construct fusion type products in twisted K-theory and twisted cohomology, whenever the twist is a 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.

In [MR], 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.

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 also define the action of the T-duality group GO(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 upto sign. All of T-dual pairs in a given orbit of GO(n, n, Z) define physically equivalent type II string theories.

February 2008 - January 2012, Editor in charge of "Global Analysis, Noncommutative Geometry, and the Mathematics of String Theory", Proceedings of the American Mathematical Society.

Recent invited talks in String Theory

Invited speaker, 1st PRIMA conference, UNSW, Sydney, July 6-10, 2009.

Invited speaker, NSF/CBMS Regional Conference in the Mathematical Sciences: Topology, C*-algebras, and String Duality. Texas Christian University, Fort Worth, Texas, USA May 18-22, 2009.

Invited speaker, Center of Excellence (COE) Program on "Exploring New Science by Bridging Particle-Matter Hierarchy", Tohoku University, Sendai, Japan, 13-15 December 2007.

I gave a keynote lecture at the special session on Mathematical Physics, at the 51st Annual Meeting of the Australian Mathematical Society, 25-28 September 2007, La Trobe University, Melbourne.

I gave a couple of plenary lectures at the Nishinomiya-Yukawa Memorial Symposium held at Nishinomiya, Japan, 11-12 November 2006, and at Kyoto, Japan, 13-15 November 2006.

I organized an IGA satellite workshop to "Mathematics of String Theory 2006" 27 July 2006.

I co-organized a major international AMSI workshop entitled, Mathematics of String Theory 2006 which was held at ANU, Canberra, 13-23 July, 2006.

From March - June, 2006, I took up a post as Senior Research Fellow at the Erwin Schrodinger Institute for Mathematical Physics in Vienna, Austria. I gave a course of advanced graduate lectures entitled "K-theory applied to Physics". About half of the course was on applications of K-theory to String Theory.

I am an invited participant at the KITP (Santa Barbara) program, "Mathematical Structures in String Theory", October 17 - November 12, 2005. The link to the video and slides of my talk is here.

I gave a talk entitled "Global aspects of T-duality in string theory" at the MFO, Oberwolfach (Germany) Mini-Workshop "Gerbes, Twisted K-Theory and Conformal Field Theory", 14-20 August 2005.

I gave an invited lecture series entitled, "Twisted K-theory and global aspects of T-duality in the presence of background flux" at the 2005 summer school entitled, "Geometric and Topological methods for Quantum Field Theory" which was held at Villa de Leyva, Colombia, July 7 - July 19, 2005.

I've been an invited speaker, in May 2004 to the Erwin Schrodinger Institute for a conference entitled "Mathematical aspects of branes in Calabi-Yau spaces", in June 2004 at Luminy, Marseille for a conference entitled "Geometry of index theory and quantum field theory", in July 2004 as a principal lecturer at Bialowieza, Poland for a conference entitled "Workshop on Geometric Methods in Physics", in September 2004 to the Oberwolfach Mathematische Institut for a conference entitled "Noncommutative Geometry".

I was invited to give a lecture series entitled "K-theory and Physics", February 22 until March 3, 2003, at IUPUI, Indianapolis and also the neighbouring universities in Bloomington and Wabash.

I was an invited speaker at the "String Theory Conference 2002" in Hangzhou, China August 12-15, 2002 and in Beijing, China August 17-19, 2002. I also co-organized an international workshop on String theory in January 2002, and am a co-organizer for an international String theory conference in 2003.

References

[BM] P. Bouwknegt and V. Mathai,
T-Duality as a Duality of Loop Group Bundles,
Journal of Physics A: Fast Track Communications,
42 no.16 (2009) 162001, 8 pages, [0902.4341]

[BMRS2] J. Brodzki, V. Mathai, J. Rosenberg and R. Szabo,
Noncommutative correspondences, duality and D-branes in bivariant K-theory
Advances in Theoretical and Mathematical Physics,
13 no. 2 (2009) 497-552. [0708.2648]

[BHM4] P. Bouwknegt, K. Hannabuss and V. Mathai,
C*-algebras in tensor categories,
Clay Mathematics Proceedings,
12 (2009) 36 pages, to appear. [math.QA/0702802]

[BMRS3] J. Brodzki, V. Mathai, J. Rosenberg and R. Szabo,
D-branes, KK-theory and duality on noncommutative spaces
Journal of Physics: Conference Series,
103 (2008) 012004, 13 pages. [0709.2128]

[BMRS] J. Brodzki, V. Mathai, J. Rosenberg and R. Szabo,
D-Branes, RR-Fields and Duality on Noncommutative Manifolds,
Communications in Mathematical Physics,
277, no.3 (2008) 643-706. [hep-th/0607020]

[MR3] V. Mathai and J. Rosenberg,
T-duality for torus bundles with H-fluxes via noncommutative topology, II:
the high-dimensional case and the T-duality group,
Advances in Theoretical and Mathematical Physics,
10 no. 1 (2006) 123-158, [hep-th/0508084]

[BHM2] P. Bouwknegt, K. Hannabuss and V. Mathai,
Nonassociative tori and applications to T-Duality,
Communications in Mathematical Physics,
264 no. 1 (2006) 41-69 [hep-th/0412092]

[BEJMS] P. Bouwknegt, J. Evslin, B. Jurco, V. Mathai and H. Sati,
Flux compactification on projective spaces and the S-duality puzzle,
Advances in Theoretical and Mathematical Physics,
10 no. 3 (2006) 345-394. [hep-th/0501110]

[MR2] V. Mathai and J. Rosenberg,
On mysteriously missing T-duals, H-flux and the T-duality group,
in Differential Geometry and Physics,
editors: Mo-Lin Ge and Weiping Zhang,
Nankai Tracts in Mathematics, Volume 10 (2006) 350-358, World Scientific.
[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]

[BHM3] P. Bouwknegt, K. Hannabuss and V. Mathai,
T-duality for principal torus bundles and dimensionally reduced Gysin sequences,
Advances in Theoretical and Mathematical Physics,
9 no. 5 (2005) 749-773. [hep-th/0412268]

[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]

[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]

[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]

[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.