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.

Jacques Distler wrote:

Personally, I’m rather dubious of the approach, in that T-duality is more naturally described in terms of the loop space of $X$, rather than $X$ itself (remember, it exchanges momenta and windings). It’s neat, but somehow seemingly accidental, that it reduces to a simple mapping of the K-theories of $X$ and its T-dual.

Since, as you say, the approach to T-duality as followed by Mathai (and others) knows nothing about loop space, let alone about CFT (which is really where the duality lives), it is clearly just some sort of “shadow” of the full thing in terms of topological notions on target space.

(The question about the apparent “continuous orbifolding” which I mentioned is probably related to that, since clarifying it seems to call for a a setup that knows about the T-dual CFTs.)

I am not sure, though, how accidental the mapping between K-theories on T-dual target spaces is. Since these classes are where the RR-charges live, it seems necessary that such an isomorphism exists.

But that’s more a gut feeling than a precise statement.

It is no secret that “topological T-duality theorists” (like for instance also Ulrich Bunke math.GT/0501487) pretty mich just take the K-theory setup as given and are happy doing their math in that framework, without actively worrying about the connection to physics/CFT. The “abstract T-duality” definition which I mention at the end of part III certainly certifies this.

I would like to see the connection of the topological T-duality technology to CFT. As I tried to indicate before, there is an interesting analogy, which might point in the right direction.

So on the one hand side we see in the topological T-duality approach the Fourier-Mukai transformation on the correspondence space of the torus and its dual. In as far as the sheaves on $M \times \mathbf{T}$ that we are dealing with are modules for the structure sheaf, the Fourier-Mukai correspondence is a bimodule, and its operation on these sheaves by pullback can be understood alternatively in terms of the tensor product of modules.

Now, this is rather reminiscent of the way T-duality is realized in the Frobenius-algebraic approch to (R)CFT ($\to$).

As will be detailed in upcoming work that has been announced in cond-mat/0404051, T-duality (like other dualities) is induced by the action of certain bimodules internal to the representation category of the chiral vertex algebra.

I am hoping that this can be used to make the connection between T-duality in full CFT and the topological T-duality on target space studied by Mathai, Bunke et. al.

But that’s just me.