Musings

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,

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,

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,

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,

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.