Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

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

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/474

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

Post a New Comment