Moduli Space of Tori: Intuitive Understanding in T-Duality

[haushofer]

Hi,

recently I'm studying some stuff about T-duality in string theory, toroidal compactification and doubled geometry. Now I think I understand the moduli space of a torus, $T^2$, but apparently (see for instance Hull's "Doubled geometry and T-folds") one can write the moduli $\tau$ of $T^d$ as elements of

$$\tau \in \frac{O(d,d)}{O(d)\times O(d)}$$

So my question is, is there an intuitive way to see that the moduli space of $T^d$ is
$\frac{O(d,d)}{O(d)\times O(d)}$? Can I see those O(d)'s as acting on the cycles of the tori or something like that? Where does this O(d,d) come from? Thanks in forward!

 

[fzero]

You probably want to take a look at the review http://arxiv.org/abs/hep-th/9401139 by Giveon, Porrati, and Rabinovici. In section 2.4 they construct the moduli space of a bosonic string compactified on a torus and show that it is of the $$O(d,d)/(O(d)\times O(d))$$ form. This treatment uses the metric and B-field data, however, in the subsection 2.4.2 they relate these directly to the Kaehler and complex structure moduli for a 2-torus. In the case of a general complex torus, you can assemble the periods of (1,1) and (0,2) forms into the coset, but I don't have any references handy.

 

[suprised]

Hi,
So my question is, is there an intuitive way to see that the moduli space of $T^d$ d!

The intuitive way of understanding is in terms of the "Narain lattice". The string winding and momentum states have a mass given by the inner product of 2d dimensional lattice vectors p=(p_L,p_R), where p_L and p_R are left- and right-moving momenta which depend on the background geometry (metric and B-field); this inner product has a (d,d) lorentzian signature, ie, (+++..,----...), ie.,

m^2 ~ <p,p> = p_L^2 - p_R^2

The moduli space is then given by the continuous O(d,d) rotations of this lattice which are compatible with this inner product, divided out by an O(d) for each of the left- and right-moving sectors (since rotations that leave p_L^2 or p_R^2 invariant do not change the spectrum, thus act trivially and so do not belong to the moduli space).

Moreover, one should also divide out the subgroup O(d,d;Z) of discrete transformations, which correspond to T-dualities that too leave the spectrum invariant.

Note that this gives not the "mathematical" moduli space of a (multi-) torus but the stringy version of it. The difference is the B-field that exists in string theory and the O(d,d)/O(d)xO(d) structure only appears when the B-field is included.

 

[haushofer]

Hey, thanks for the link and the intuïtive explanation. Indeed, I had the idea that this coset space was the "mathematical" notion of the moduli space of T^d instead of the "stringy". I think with some reading everything will become clear; if not I'll come back! :D Enjoy your sunday ;)