# Two loop corrections to T-duality

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T-duality for Type II string theory is conveniently encoded in the Buscher rules, a set of transformations for the metric, Kalb-Ramond field, and dilaton which implements T-duality:

$\hat{g}_{\bullet \bullet} = \frac{1}{g_{\bullet \bullet}}$ $\hat{g}_{\mu \bullet} = \frac{B_{\mu \bullet}}{g_{\bullet \bullet}}$ $\hat{g}_{\mu \nu} = g_{\mu \nu} - \frac{1}{g_{\bullet \bullet}} \left( g_{\mu \bullet} g_{\nu \bullet} - B_{\mu \bullet} B_{\nu \bullet} \right)$ $\hat{B}_{\mu \bullet} = \frac{g_{\mu \bullet}}{g_{\bullet \bullet}}$ $\hat{B}_{\mu \nu} = B_{\mu \nu} - \frac{1}{g_{\bullet \bullet}} \left( g_{\mu \bullet} B_{\nu \bullet} - g_{\nu \bullet} B_{\mu \bullet} \right)$

The Buscher rules are consistent with the required vanishing of the beta functions, at least up to the one-loop level, as shown here. On the other hand, this paper (and others) suggest that the normal Buscher rules are inconsistent with the vanishing of the two loop beta function. The required corrections are known as the two-loop corrections. I have found a few papers which compute the corrections for specific backgrounds, but I am looking for a reference in which the two loop corrections are computed for an arbitrary background (with abelian isometry).

edited Apr 7, 2016

First question - no idea how to put LaTeX into a post :S

Hi Mark, welcome to PhysicsOverflow !

As described here

http://physicsoverflow.org/faq#a15446

You can for example use the TEX button in the middle of the toolbar of the editor to write LaTex.

@Dilaton As far as i can see the source for this question looks fine. Is there a bug somewhere that prevents the post from being displayed correctly? Could somebody more experienced maybe have a look?

@Heterotic yes it looks like a strange bug to me, I have reported it ...

It was in a <pre> environment. Fixed. Sorry for the delay with it.

Thanks for fixing that :)

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