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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)dμ δ(Ω (3))=Θ[δ](0,Ω (3))Ξ 6(Ω (3))8π 4Ψ 9(Ω (3)) 1ij3dΩ ij (3) 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 h1(2 h+1)k2^{h-1}(2^h+1)k, from products of the even theta functions,

(2)Ψ 2 h1(2 h+1)k(Ω (h))= δevenΘ[δ](0,Ω (h)) \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=1,k=2), (h=2,k=1)(h=2,k=1) and (h=3,k=1/2)(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. Ψ 18(Ω (3))\Psi_{18}(\Omega^{(3)}) is actually the square of a modular form,

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

Whereas Ψ 6(Ω (1))\Psi_{6}(\Omega^{(1)}) and Ψ 10(Ω (2))\Psi_{10}(\Omega^{(2)}) have no zeroes in the interior of the moduli space, Ψ 9(Ω (3))\Psi_{9}(\Omega^{(3)}) has a simple zero along the hyperelliptic locus. Fortunately, 1ij3dΩ ij (3)\prod_{1\leq i\leq j\leq 3}d\Omega^{(3)}_{ij} also vanishes there, so the chiral bosonic string measure,

(4)164π 18Ψ 9(Ω (3)) 1ij3dΩ ij (3) \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, Ξ 6(Ω (3))\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 Ξ 6(Ω (3)) 2\Xi_6(\Omega^{(3)})^2 can be written as a sum of products of even Θ\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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4 Comments & 2 Trackbacks

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 h3h\leq3:

(1)h(h+1)/2>3h3 h(h+1)/2 \gt 3h-3

for h>3h\gt 3. The problem of determining which “period matrices” (points in the Siegel upper half-plane) are actually period matrices of a genus-hh 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>3h\gt 3.

There are other coordinates you could use for h>3h\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>3h\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
Read the post Multiloop Amplitudes
Weblog: Musings
Excerpt: Yet more D'Hoker and Phong. And I attempt to penetrate the mist surrounding a paper of Nathan Berkovits.
Tracked: January 17, 2007 5:19 PM

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