Musings

Since the mathematical definition is supposed to be an attempt to generalize the physicists’ notion of T-duality, I figure that “right” means that, whenever both definitions apply, the mathematicians’ definition should give the same answer as the physicists’.

The simplest nontrivial case to look at is the Hopf bundle, $S^3\to S^2$, with $k$ units of $H$-flux on $S^3$. The topological T-dual of this is $S^3/\mathbb{Z}_k$, with one unit of $H$-flux.

This is nice example because both sides admit a (unique) CFT description and we can check whether the two CFTs are, in fact, isomorphic.

$S^3$ is also known as the group manifold, $SU(2)$. Putting $k$ units of $H$-flux amounts to setting the (quantized) coefficient of the Wess-Zumino term equal to $k$. There’s an RG fixed point of this system (a constant curvature metric on the $S^2$, with a particular value for the total area), called the $SU(2)$ WZW model at level-$k$. This is a conformal field theory with central charge, $c=\frac{3k}{k+2}$, and primary states for the current algebra have $h_j =\frac{j(j+1)}{k+2}$, for $j=0,1/2,\dots,k/2$.

The other side of the duality, $S^3/\mathbb{Z}_k$, with one unit of $H$-flux, is the same as the quotient by $\mathbb{Z}_k$ of $S^3$ with $k$ units of $H$-flux. We can take the $\mathbb{Z}_k$ to act by left-multiplication¹

This theory also has a CFT description, as the $\mathbb{Z}_k$ orbifold of the (original!) level-$k$ WZW model by the above $\mathbb{Z}_k$ action. Naturally, these two CFTs have the same central charge.

Acting on the states of the CFT, the $\mathbb{Z}_k$ acts as

and, on all operators (including the currents which generate the $SU(2)$ current algebra), it acts by conjugation

So … is the $SU(2)$ WZW model at level-$k$ isomorphic to its orbifold by $\mathbb{Z}_k$?

The first nontrivial case, $k=2$, works out quite nicely. $e^{2\pi i J^3_0}$ acts as $+1$ on all states with $j\in\mathbb{Z}$, and as $-1$ on all states with half-integral $j$. Thus the $[j=0]$ and $[j=1]$ representations of the current algebra are preserved, whereas the entire $[j=1/2]$ representation is projected out. (Note, in particular, that the currents, which are the first excited states of the $[j=0]$ representation, are preserved, so all of the states of the orbifold theory must organize themselves into representations of the current algebra. That’s because we’re orbifolding by an element of the center of the group.)

Of course, we’re not done: we need to add the twisted sector of the orbifold. In this case, the ground state of the twisted sector has $h=3/16$, and so reproduces exactly the same spectrum of states that we projected out.

So far, so good, but the prospects for higher $k$ look, at first glance, rather bleak. For $k\gt 2$, the currents, $J^\pm$, are projected out of the untwisted sector of the orbifold ($J^3$ is preserved) and I did not see how they could be regenerated in the twisted sector. So I was pretty much tempted to conclude that $\widehat{SU}(2)_k$ and $(\widehat{SU}(2)_k)/\mathbb{Z}_k$ could not be equivalent for $k\gt 2$.

But then I turned up an old paper of Gaberdiel’s, where he claims² to prove precisely that: “there exists a quantization” of the orbifold theory in which the spectrum coincides with that of the original theory. The last part of the paper, where he actually purports to construct the spectrum of the orbifolded theory, is still a little opaque to me.

I should be able to see the answer to my question (how are the currents, which were projected out of the untwisted sector, regenerated in the twisted sectors?), but I don’t. Can anyone help me out?

They prove that the spin-$j$ characters for $(\widehat{SU}(2)_{k=k_1 k_2})/\mathbb{Z}_{k_1}$ orbifold (where, instead of (1), one mods out by $g \mapsto e^{2\pi i \sigma_3/k_1} g$) can be written as

Here, $$\chi_{n'}^{U(1)}(q) = \frac{1}{\eta(\tau)}\sum_{r\in \mathbb{Z}+ \frac{n'}{2k_1}} q^{k_1 r^2}$$ are the characters for the $\hat{U}(1)_{k_1}$ current algebra, whose extended chiral algebra contains a pair of charged fields of dimension $k_1$ (a free scalar, compactified on a circle of radius $R=\sqrt{\alpha' k_1}$). The $\mathbb{Z}_{k}$ parafermion characters can be written as $$\begin{aligned} \chi_{j,n}^{\text{PF}}(q) = q^{\frac{(2j+1)^2}{4(k+2)}-\frac{n^2}{k}}\eta(\tau)^{-2} \sum_{r,s=0}^{\infty}&(-1)^{r+s}q^{s(s+1)/2+r(r+1)/2+sr(k+1)} \\ &\cdot\left[ q^{s(j-n)+r(j+n)}-q^{k+1-2j +s(k+1-j+n)+r(k+1-j-n)} \right] \end{aligned}$$ They are defined for $-2j\leq n\leq 2j$, $n+2j\in2\mathbb{Z}$ and are explicitly invariant under $n\to -n$. The $\widehat{SU}(2)_k$ characters, $\chi_j^{SU(2)}(q)$, are obtained as a special case of (4), for $k_1=1$.

The partition function of the orbifolded theory is $$Z(q,\overline{q}) = \sum_{j=0, 1/2,\dots,k/2} \chi_j(q) \overline{\chi_j^{SU(2)}(q)}$$

T-duality is $n'\to -n'$, which, we see, equates the characters (and hence partition functions) of the $(\widehat{SU}(2)_{k})/\mathbb{Z}_{k_1}$ theory with those of the $(\widehat{SU}(2)_{k})/\mathbb{Z}_{k_2}$ theory.

In the particular case, $k_1=1, k_2=k$, they relate the $\widehat{SU}(2)_{k}$ theory to the $(\widehat{SU}(2)_{k})/\mathbb{Z}_{k}$ theory.