Musings

Chiral Superstring Measure

After their breakthrough with the genus-2 superstring measure, D’Hoker and Phong plow along on the road to genus-3. Their observation was that, if one used super-period matrices as coordinates, one could show that the supermoduli space for genus-2 was is, in fact, a split-supermanifold. They could, therefore, integrate over the odd directions and obtain a measure on the ordinary moduli space which, for $h\le 3$ is a domain in the Siegel upper half-plane.

The formula for the vacuum amplitude takes the form $$A={\int }_{{ℳ}_h}\left(\det \mathrm{Im}\Omega {)}^{-5}\sum _{\delta ,\delta \text{'}}{c}_{\delta ,\delta \text{'}}d{\mu }_{\delta }\right(\Omega )\overline{d{\mu }_{\delta \text{'}}\left(\Omega \right)}$$ Where the sum is over independent even spin structures, $c_{\delta ,\delta \text{'}}$ are some constant phases , and $d{\mu }_{\delta }\left(\Omega \right)$ is the chiral measure which, at one-loop takes the familiar form, $$d{\mu }_{\delta }\left(\Omega \right)\propto \Theta \left[\delta \right](0,\Omega {)}^4/\eta (\Omega {)}^{12}$$ At genus-2, they found, for an even spin structure, $\delta$, $$d{\mu }_{\delta }\left(\Omega \right)=\frac{\Theta \left[\delta \right](0,\Omega {)}^4{\Xi }_6[\delta \left]\right(\Omega )}{16{\pi }^6{\Psi }_{10}\left(\Omega \right)}\prod _{1\le i\le j\le 2}d{\Omega }_{\mathrm{ij}}$$ Here ${\Psi }_{10}\left(\Omega \right)={\prod }_{\delta \text{even}}\Theta \left[\delta \right](0,\Omega {)}^2$. The formula for ${\Xi }_6\left[\delta \right]\left(\Omega \right)$ requires a bit of notation.

If we fix a marking of $\Sigma$, then a spin structure, $\delta$, can be identified with a $2h$-vector of half-characteristics, $$\delta =(\delta \text{'},\delta \text{'}\text{'}),\delta \text{'},delt\text{'}\text{'}\in \{0,1/2{\}}^h$$ $\delta$ is even or odd, depending on the sign of $e^{4\pi i\delta \text{'}\cdot \delta \text{'}\text{'}}$. For a pair of spin structures, one can define a relative signature, $$\langle {\delta }_1\mid {\delta }_2\rangle =e^{4\pi i(\delta {\text{'}}_1\cdot \delta \text{'}{\text{'}}_2-\delta {\text{'}}_2\cdot \delta \text{'}{\text{'}}_1)}$$ This is not quite modular-invariant. Under an $\mathrm{Sp}(2h,\mathbb{Z})$ transformation of the form $\left(\begin{array}{cc}I& B\\ 0& I\end{array}\right)$, $$\langle {\delta }_1\mid {\delta }_2\rangle \to \langle {\delta }_1\mid {\delta }_2\rangle e^{2\pi i(\delta {\text{'}}_1-\delta {\text{'}}_2{)}^t\text{diag}B}$$ At genus-2, there are 10 even spin structures and 6 odd ones. It is a peculiar fact that the even spins structures are in 1-1 correspondence with partitions of the odd spin structures into two groups of 3. Denoting the odd spin structures by ${\nu }_i$, let $$\delta ={\nu }_1+{\nu }_2+{\nu }_3={\nu }_4+{\nu }_5+{\nu }_6$$ Using this, one can write ${\Xi }_6\left[\delta \right]\left(\Omega \right)$ as $${\Xi }_6\left[\delta \right]\left(\Omega \right)=\sum _{1\le i\le j\le 3}\langle {\nu }_i\mid {\nu }_j\rangle \prod _{k=4,5,6}\Theta [{\nu }_i+{\nu }_j+{\nu }_k](0,\Omega {)}^4$$

This formula depends heavily on peculiarities of $h=2$. What D’Hoker and Phong propose in their new paper is a new formula for ${\Xi }_6\left[\delta \right]\left(\Omega \right)$, which does generalize to higher genus, and so provides a guess for the $h=3$ amplitude.

Actually, they come up with two rewritings of ${\Xi }_6\left[\delta \right]\left(\Omega \right)$, but only one, written in terms of sums of squares of even $\Theta$-functions, has the requisite factorization properties (to be shown in a followup paper) to be a candidate for the genus-3 measure.

So what’s all this good for?

String perturbation theory is still the only detailed quantitative tool we have. It’s wonderful to be able to say that theory “A” is dual to theory “B.” But, if we want to extract numbers, we ultimately need to be able to compute in one or the other. And, until someone comes up with the string equivalent of lattice gauge theory, string perturbation theory is the only tool we’ve got.

There aren’t too many quantities that we need to know to two-loop (let alone 3-loop) precision, but there are some. And it’s not clear that the ones we’re interested in will be amenable to perturbation theory. But still…

Now, the formulæ above are for the vacuum amplitude in 10-d flat space. But, they (along with the genus-$h$ green’s functions) can be straightforwardly used to compute $h$-loop parity-conserving scattering amplitude. If you don’t like 10-d flat space, the formulæ naturally generalize to toroidal compactifications and orbifolds thereof. More general CFTs are harder, but you need to know the answer for flat space before you can tackle a more general background.

An interesting project would be to extend these formulæ to the odd spin structures, which do not contribute to the vacuum amplitude, but which do contribute to parity-violating amplitudes.