The String Coffee Table

Before I start reproducing the notes I have taken, let me briefly outline some central points.

Overview.

The study of T-duality performed by mathematicians like Bouwknegt, Bunke ($\to$), Hannabus, Mathai, Rosenberg, Schick, and others concentrates on

$\bullet$ target space

and

$\bullet$ topological aspects

of T-duality. This means, first of all, that no 2D CFTs are ever mentioned in this sort of game. Instead, the deal is to consider topological spaces (or their noncommutative generalizations) representing target space, together with certain data on these modeling the “$H$-flux”, i.e. encoding abelian gerbes, in general with connection, on target space.

Given such a setup, the central object of interest is the twisted ($\to$, $\to$) K-theory ($\to$) of target space.

Physically, this is interesting because the RR fields and the D-brane charges of type II string theory are really elements of this twisted K-theory. (Essentially, K-theory tells us about the kinds of D-branes we have in the absence of Kalb-Ramond flux, while twisted K-theory takes into account that in the presence of a nontrivial Kalb-Ramond gerbe coupled to the string, the D-branes support not ordinary bundles, but twisted bundles - gerbe modules ($\to$).

But the interest of most of these mathematicians in the game called topological T-duality is pretty remote from such physical concepts. What they are concerned with is that there happens to be an interesting involution on twisted K-theory and this happens to be called (topological) T-duality.

The main emphasis of V. Mathai’s work in this context is on the $C^*$-algebraic formulation.

So in this context $C^*$-algebras are considered as representing generalized spaces. When the algebra is commutative it is always that of continuous functions on some topological space, and non-isomorphic topological spaces have non-isomorphic function algebras. So they sit inside the category of $*$-algebras. But, in addition, there are of course noncommutative $*$-algebras. By re-formulating all topological notions algebraically in terms of $*$-algebras one obtains a way of talking about noncommutative - and even nonassociative - topology.

In particular, while K-theory is most naively regarded as being about equivalence classes of vector bundles over topological spaces, this, too, is very naturally reformulated in purely algebraic terms ($\to$).

So there is a well-known way for defining K-theory, and also twisted K-theory, on generalzed target spaces.

In fact, there is a rather simple algebraic way how T-duality itself acts on a given $C^*$-algebra, representing a target space with $H$-flux on it, such that the result represents the T-dual background.

The importance of this fact is that T-duality forces one to consider noncommutative - and, as Varghese Mathai and collaborators claim, also nonassociative - topologies. That’s because, for non-vanishing $H$-flux, the T-dual of the $C^*$-algebra representing that background need not, in general, be a $C^*$-algebra which corresponds to an ordinary topological space (with $H$-flux).

Noncommutative T-duals have been known in the physics literature before (usually under the term “nongeometric backgrounds”), but the nonassociative T-duals - which Mathai claims are forced upon us if we take T-duality seriously

P. Bouwknegt, K. Hannabus & V. Mathai
Nonassociative tori and applications to T-duality
hep-th/0412092

- meet apparently with some reservation among physicists.

However, there is an upcoming paper

I. Ellwood & A. Hashimoto
Comment on $H$-flux and the non-associative torus
(to appear on arXiv)

which claims to rederive at least parts of these results using world-sheet analysis of scattering of string winding on a 3-torus background with cohomologically nontrivial $H$-flux.

In the next entry I’ll try to reproduce some of my notes taken in Mathai’s talks.