## November 22, 2004

### More D’Hoker and Phong

Let’s pick up our discussion of D’Hoker and Phong with their followup paper on the genus-3 superstring measure. Their ansatz for the genus-3 chiral measure is similar to their genus-2 result,

(1)$d\mu_\delta(\Omega^{(3)})= \frac{\Theta[\delta](0,\Omega^{(3)})\Xi_6(\Omega^{(3)})}{8\pi^4 \Psi_9(\Omega^{(3)})}\prod_{1\leq i\leq j\leq 3}d\Omega^{(3)}_{ij}$

The denominator is closely related to the bosonic string chiral measure. At genus 1,2 & 3, one can construct modular forms, of weight $2^{h-1}(2^h+1)k$, from products of the even theta functions,

(2)$\Psi_{2^{h-1}(2^h+1)k}(\Omega^{(h)})= \prod_{\delta\, \text{even}}\Theta[\delta](0,\Omega^{(h)})$

The relevant cases are $(h=1,k=2)$, $(h=2,k=1)$ and $(h=3,k=1/2)$. The first two cases are recognizable as the denominators of the chiral bosonic string measure at genus-1,2.

The third case is peculiar. $\Psi_{18}(\Omega^{(3)})$ is actually the square of a modular form,

(3)$\Psi_{18}(\Omega^{(3)}) = \Psi_{9}(\Omega^{(3)})^2$

Whereas $\Psi_{6}(\Omega^{(1)})$ and $\Psi_{10}(\Omega^{(2)})$ have no zeroes in the interior of the moduli space, $\Psi_{9}(\Omega^{(3)})$ has a simple zero along the hyperelliptic locus. Fortunately, $\prod_{1\leq i\leq j\leq 3}d\Omega^{(3)}_{ij}$ also vanishes there, so the chiral bosonic string measure,

(4)$\frac{1}{64\pi^{18}\Psi_9(\Omega^{(3)})}\prod_{1\leq i\leq j\leq 3}d\Omega^{(3)}_{ij}$

has no poles in the interior of the moduli space.

The numerator is the tricky part. They discuss various candidates for the modular form of weight-6, $\Xi_6(\Omega^{(3)})$, and examine their behaviour under the degeneration of a genus-3 surface into the product of a genus-1 and a genus-2 surface. Eventually, the come to the conclusion that $\Xi_6(\Omega^{(3)})^2$ can be written as a sum of products of even $\Theta$-functions, and that the square-root of this sum behaves correctly under factorization, thus yielding a candidate for the genus-3 measure.

Posted by distler at November 22, 2004 2:58 AM

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### Re: More D’Hoker and Phong

That’s very interesting. Once you have this Ansatz for the 3-loop case, cannot you just guess how it generalizes to any number of loops?

Posted by: Lubos Motl on November 23, 2004 9:55 AM | Permalink | Reply to this

### Schottky Problem

The period matrices are only good coordinates for $h\leq3$:

(1)$h(h+1)/2 \gt 3h-3$

for $h\gt 3$. The problem of determining which “period matrices” (points in the Siegel upper half-plane) are actually period matrices of a genus-$h$ Riemann surface is called the Schottky problem. It’s been solved, but not in a way that leads to a useful parametrization of the moduli space for $h\gt 3$.

There are other coordinates you could use for $h\gt 3$, but the relation to the formulæ presented here (in terms of period matrices) is obscure.

Posted by: Jacques Distler on November 23, 2004 11:41 AM | Permalink | PGP Sig | Reply to this

### Re: Schottky Problem

I see, thanks for refreshing my memory. I should mention this in the class sometime next week, so it will be necessary to re-learn the different parameterizations of the higher genus surfaces.

Posted by: Lubos Motl on November 23, 2004 7:00 PM | Permalink | Reply to this

### Splitness

Just to emphasize something discussed in my previous post: D’Hoker and Phong proved the splitness of the supermoduli space for genus-2, and hence the existence of a formula of the sort we are discussing. The same thing should hold, presumably, for genus-3.

For genus $h\gt 3$, we still don’t really know that the supermoduli space is split, and hence whether there even should exist a formula written purely in terms of bosonic moduli.

Posted by: Jacques Distler on November 26, 2004 10:16 AM | Permalink | PGP Sig | Reply to this
Read the post Two-Loop Superstring Amplitudes
Weblog: Not Even Wrong
Excerpt: Eric D'Hoker and D.H. Phong this past week finally posted two crucial papers with results from their work on two-loop superstring amplitudes. The first one shows gauge slice independence of the two-loop N-point function, the second shows that, for N...
Tracked: January 29, 2005 11:02 AM