### Mathai on T-Duality IV: Distler on CFT Checks

#### Posted by urs

Due to lack of time, I have so far only reproduced the first half of Varghese Mathai’s talk ($\to $) on “topological T-duality”, which was really just a review of some basics. Maybe I will find the time to type my notes on the noncommutative and non-associative aspects.

But meanwhile we had some discussion about the relation of this formalism to full CFT over at Jacques Distler’s Musings.

Jacques Distler rightly emphasized that one should not lose sight of the question how the axioms of topological T-duality harmonize with the CFT-duality they are supposed to approximate.

If there is detailed literature on this question, then I am not aware of it. Of course that does not mean much. But it is clear what should in principle be done:

1) For each pair of topologically T-dual backgrounds, one should check if we indeed have a full T-duality for the sigma-models with target these backgrounds.

2) Better yet, it would be good to have a general theorem, or something like that, which tells us (if true) that for any pair of topological T-duals a pair of corresponding dual 2D CFTs exists.

As I said, I believe there are some vague hints for part 2), but of course that looks like a major project.

On the other hand, 1) can be approached in the manner of consistency checks by randomly sampling some tractable setups.

In his latest entry, Jacques Distler does precisely this for the case where one background is the Hopf fibration

with $n$ units of Kalb-Ramond $H$-flux.

Topological T-duality predicts that the T-dual of this background has one unit of $H$-flux on the ${S}^{1}$-bundle over ${S}^{2}$ whose Chern class is $n$ times the generator of the integral cohomology ${H}^{2}({S}^{2},\mathbb{Z})$ of ${S}^{2}$.

That’s for instance equation (1.1) in

Peter Bouwknegt, Keith Hannabuss, Varghese Mathai
*T-duality for principal torus bundles*

hep-th/0312284 .

In section 3.2 of this paper it is discussed how this ${S}^{1}$-bundle in the present example is the Lens space

and here it carries a single unit of $H$-flux.

From the CFT point of view this can alternatively be understood as orbifolding the $\mathrm{SU}(2)$ WZW model at level $n$ on $\mathrm{SU}(2)$ by ${\mathbb{Z}}_{n}$.

Therefore, it should be true that the spectrum of level $n$ $\mathrm{SU}(2)$-WZW is equivalent to that of its ${\mathbb{Z}}_{n}$ orbifolded version.

This we can check, in principle, since WZW models are pretty much under control.

Jacques Distler works out the computation at level 2, where it indeed comes out as expected. For higher level the situation is more involved, but apparently an old paper by Gaberdiel

M.R. Gaberdiel
*Abelian duality in WZW models*

hep-th/9601016

claims to show precisely the equivalence we expect to see.

Another aspect of this result, using a mixture of string-theoretic and K-theoretic reasoning, is also given on pp. 15-16 of

Juan Maldacena, Gregory Moore, Nathan Seiberg
*D-brane Charges in Five-brane backgrounds*

hep-th/0108152 .

Actually, as Jacques Distler notes, Gaberdiel

[…] makes the claim much more generally, not just for $\mathrm{SU}(2)$, but for any simple, connected, simply-connected Lie group, $G$. And not just for ${\mathbb{Z}}_{n}$ , but for any finite subgroup of the Cartan torus.

That looks good, because this seems to be precisely the context that section 2.3 of the Bouwknegt-Hannabuss-Mathai paper cited above applies to.