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November 28, 2004

Splitness

It seems to me, judging by a comment by Luboš Motl, I ought to elaborate a bit on the issue of “splitness” of supermoduli space, and how it bears on the generalization of D’Hoker and Phong’s formulæ to higher genus.

A super-Riemann surface is a (1 ∣1 )-dimensional complex supermanifold with an odd distribution, a rank-(0 ∣1 ) sub-bundle, 𝒟⊂𝒯 of its tangent bundle, such that 𝒟,{𝒟,𝒟} span 𝒯. The supermoduli space of inequivalent compact SRSs of genus h≥2 has dimension (3 h−3 ∣2 h−2 ).

A supermanifold is said to be split if it can be covered in coordinate charts, such that the transition functions are at most linear in the odd coordinates. The archetype of a split supermanifold is a vector bundle over an ordinary bosonic manifold, where the fiber directions are taken to be odd. An SRS, Σ̂, is, perforce, a split supermanifold, and it has a projection onto an ordinary Riemann surface, simply by forgetting the odd coordinate.

Conversely, the (spin) moduli space of ordinary Riemann surfaces sits as a subspace of dimension (3 h−3 ∣0 ) inside the supermoduli space. Given Σ and a spin structure on it, you can construct an SRS Σ̂, as the total space of the spin bundle on Σ, taking the fiber direction to be odd.

But what of supermoduli space? Is it split? If not, then there is no hope of writing a formula for the string measure as a purely bosonic integral over the ordinary moduli space.

In the most favourable circumstance, the supermoduli space might take the form of a fiber bundle over the ordinary (spin) moduli space, with the aforementioned embedding being the zero section. What D’Hoker and Phong showed at genus-2, is that the supermoduli space is indeed split, and admits a projection to a space isomorphic to the ordinary moduli space. But it’s not the trivial one you might have hoped for. Rather, it involves taking the commuting entries in the super-period matrix, which differ from the entries in the ordinary period matrix by pieces quadratic in the odd moduli.

Still, once you’ve shown that the supermoduli space is split, you can integrate over the odd directions and obtain a purely bosonic integral over the ordinary moduli space. The same construction likely holds true at genus-3.

The challenge for higher genus is not merely to find coordinates for the h>3 moduli space, but to show that the supermoduli space is split and to find coordinates for it that are adapted to that splitness. Those, ultimately, will be the “good coordinates” in which to express the (purely bosonic) string measure for h>3 .

Posted by distler at 12:18 AM | Permalink | Followups (14)

November 25, 2004

Dark Exoticism

Sonia Paban points me to a recent paper with a very heterodox view on Dark Matter. The jumping off point is the observation by the INTEGRAL γ-ray satellite of the “511 KeV line” from the center of our galaxy.

“511 KeV?” I hear you say, “Why … that’s … the mass of the electron!” Indeed it is. These γ-rays originate from e +e −→2 γ annihilation. Now, there are lots of point sources of positrons in astrophysics: supernovæ, blackholes, … . But what INTEGRAL finds is that the galactic bulge is a diffuse source of 511 KeV γ-rays. That suggests that the center of our galaxy harbours a diffuse source of positrons.

What could it be? Dark matter is clustered in the center of our galaxy. Boem et al postulate that the dark matter is a scalar of mass 1-100 MeV. And there’s a process, involving the exchange of some very heavy intermediate particle, by which these can annihilate into e +e −. The cross-section must be incredibly tiny; otherwise we’d have seen them already. Indeed, the whole scenario sounds like it ought to have been already ruled out in a half dozen different ways. But Boem and Fayet claim it squeaks by existing collider and astrophysical limits.

Anyway, since these scalars are so light, the positrons that are produced by their annihilation lose energy through ionization and annihilate, nearly at rest, before they leave the galactic core.

All in all, a pretty far-fetched scenario, but one sure to provoke discussion as we sit down to our Thanksgiving turkeys.

Posted by distler at 12:08 AM | Permalink | Followups (2)

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 ))8 π 4 Ψ 9 (Ω (3 ))∏ 1 ≤i≤j≤3 dΩ ij (3 )

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)Ψ 2 h−1 (2 h+1 )k(Ω (h))=∏ δevenΘ[δ](0 ,Ω (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. Ψ 18 (Ω (3 )) is actually the square of a modular form,

(3)Ψ 18 (Ω (3 ))=Ψ 9 (Ω (3 )) 2

Whereas Ψ 6 (Ω (1 )) and Ψ 10 (Ω (2 )) have no zeroes in the interior of the moduli space, Ψ 9 (Ω (3 )) has a simple zero along the hyperelliptic locus. Fortunately, ∏ 1 ≤i≤j≤3 dΩ ij (3 ) also vanishes there, so the chiral bosonic string measure,

(4)1 64 π 18 Ψ 9 (Ω (3 ))∏ 1 ≤i≤j≤3 dΩ ij (3 )

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 )), 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 can be written as a sum of products of even Θ-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 2:58 AM | Permalink | Followups (6)

November 21, 2004

Poke a Stick in it

I thought about posting something about politics. Say, about how Rep. Istook (R, Oklahoma) managed to insert a paragraph into the 1300 page Omnibus Spending Bill, granting Congressional Committee Chairmen and their staff the right to examine the tax returns of anyone in the country, free of the restraints of privacy regulations. Or about how the House Republicans just voted to allow indicted felons to retain their Committee Chairmanships (how convenient: they could turn around and scrutinize the tax returns of the Prosecutor who indicted them).

Nah, I thought, just another typical week on Capitol Hill. I’ll only really get worried when the Republican Caucus votes to allow convicted felons to retain their Chairmanships and the Appropriations Bill contains language authorizing them to order FBI surveillance of whomever they choose.

Nope, if I want to cho