Musings

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, $\mathcal{D}\subset \mathcal{T}$ of its tangent bundle, such that $\mathcal{D},\, \{\mathcal{D},\mathcal{D}\}$ span $\mathcal{T}$. The supermoduli space of inequivalent compact SRSs of genus $h\geq2$ has dimension $(3h-3|2h-2)$.

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, $\hat\Sigma$, 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 $(3h-3|0)$ inside the supermoduli space. Given $\Sigma$ and a spin structure on it, you can construct an SRS $\hat\Sigma$, as the total space of the spin bundle on $\Sigma$, 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 $h\gt 3$ 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 $h\gt 3$.

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 $\gamma$-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 $\gamma$-rays originate from $e^+e^-\to 2\gamma$ 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 $\gamma$-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.

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.

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 choose a controversial topic, I should write something about XHTML and MIME types, what we, here at Musings, like to call the “third rail” of Web Design. There are those who say serving XHTML as text/html is just fine and those who say it’s evil incarnate. Sort of a Red State/Blue State thing, but with much lower stakes.

Me, I figure it’s pointless to argue about whether people currently serving XHTML as text/html should switch to serving it as application/xhtml+xml. 99% of it is ill-formed tag soup, which would instantly produce a yellow screen of death if served with the correct MIME type.

But what about the remaining 1%? “I’ve got this XHTML thing down,” they say, “I could switch anytime I want. I just don’t feel the need to … yet.” Yes, pick up a case of the Patch at the pharmacy, and you’ll do fine.

Except …

Aside from the obvious difference that well-formedness now become an absolute requirement (and not an easy one to satisfy) there are lots of subtler differences between otherwise identical documents, served as text/html and application/xhtml+xml. You may not have heard about it, since most discussions get bogged down well before they get to the subtle stuff.

Fortunately, Gez Lemon has started putting together a test suite to illustrate the differences. Help him out by contributing your own examples of how switching MIME types affects the rendering of XHTML documents.

[Gez also has a service which lets you test serving up any page as application/xhtml+xml. It’s somewhat more advanced than the one I linked to above.]

After their breakthrough with the genus-2 superstring measure, D’Hoker and Phong plow along on the road to genus-3. Their observation was that, if one used super-period matrices as coordinates, one could show that the supermoduli space for genus-2 was is, in fact, a split-supermanifold. They could, therefore, integrate over the odd directions and obtain a measure on the ordinary moduli space which, for $h\leq 3$ is a domain in the Siegel upper half-plane.

The formula for the vacuum amplitude takes the form

Where the sum is over independent even spin structures, $c_{\delta,\delta'}$ are some constant phases , and $d\mu_{\delta}(\Omega)$ is the chiral measure which, at one-loop takes the familiar form,

At genus-2, they found, for an even spin structure, $\delta$,

Here $\Psi_{10}(\Omega)= \prod_{\delta \text{even}}\Theta[\delta](0,\Omega)^2$. The formula for $\Xi_6[\delta](\Omega)$ requires a bit of notation.

If we fix a marking of $\Sigma$, then a spin structure, $\delta$, can be identified with a $2h$-vector of half-characteristics,

$\delta$ is even or odd, depending on the sign of $e^{4\pi i\delta'\cdot\delta''}$. For a pair of spin structures, one can define a relative signature,

This is not quite modular-invariant. Under an $Sp(2h,\mathbb{Z})$ transformation of the form $\textstyle{\left( \array{I & B\\ 0 & I} \right)}$,

At genus-2, there are 10 even spin structures and 6 odd ones. It is a peculiar fact that the even spins structures are in 1-1 correspondence with partitions of the odd spin structures into two groups of 3. Denoting the odd spin structures by $\nu_i$, let

Using this, one can write $\Xi_6[\delta](\Omega)$ as

This formula depends heavily on peculiarities of $h=2$. What D’Hoker and Phong propose in their new paper is a new formula for $\Xi_6[\delta](\Omega)$, which does generalize to higher genus, and so provides a guess for the $h=3$ amplitude.

Actually, they come up with two rewritings of $\Xi_6[\delta](\Omega)$, but only one, written in terms of sums of squares of even $\Theta$-functions, has the requisite factorization properties (to be shown in a followup paper) to be a candidate for the genus-3 measure.

So what’s all this good for?

String perturbation theory is still the only detailed quantitative tool we have. It’s wonderful to be able to say that theory “A” is dual to theory “B.” But, if we want to extract numbers, we ultimately need to be able to compute in one or the other. And, until someone comes up with the string equivalent of lattice gauge theory, string perturbation theory is the only tool we’ve got.

There aren’t too many quantities that we need to know to two-loop (let alone 3-loop) precision, but there are some. And it’s not clear that the ones we’re interested in will be amenable to perturbation theory. But still…

Now, the formulæ above are for the vacuum amplitude in 10-d flat space. But, they (along with the genus-$h$ green’s functions) can be straightforwardly used to compute $h$-loop parity-conserving scattering amplitude. If you don’t like 10-d flat space, the formulæ naturally generalize to toroidal compactifications and orbifolds thereof. More general CFTs are harder, but you need to know the answer for flat space before you can tackle a more general background.

An interesting project would be to extend these formulæ to the odd spin structures, which do not contribute to the vacuum amplitude, but which do contribute to parity-violating amplitudes.

The 600 pound gorilla tackles scholarly citation indices. Hours of amusement. [Tip 'o the hat to Aaron Bergman.]

Evan takes a buzz saw to his X-philes list, with predictable results.

“Why,” you may ask, “is XHTML so hard?” It isn’t, really. With the right tools, it’s positively easy. Unfortunately, the “right tools” are not exactly readily available. You either have to be a programming whiz, a deranged physicist, or some equally atypical character, who’s willing to cobble together something half-way bulletproof.

Everyone else quickly realizes that slapping an XHTML DOCTYPE on an otherwise ill-formed bit of tag soup has all of the advantages (you appear progressive and au courant) without any of the disadvantages (actually having to figure out how to reliably ensure well-formedness) of XHTML.

Personally, none of this would matter, but for the fact that doing the old MathML thing actually requires well-formed code. With current technology, that’s still daunting for any who would follow in these footsteps.

Which is why Evan’s list remains short, and MathML-enabled weblogs remain scarce.

Addendum: To be fair, there are a few people working on the problem. But they’re still pretty far from having something even by a deranged physicist could use. [Hat tip to jgraham for the pointer to syncato.]

In a porkpie hat and powder-blue suit, he shuffled gingerly onstage. At age 75, the lion of “free jazz” seemed frail, but when he put his lips to the saxophone …

Accompanied by bassists, Tony Falanga and Greg Cohen, and his son, drummer Denardo Coleman, he played, seemingly effortlessly. There were languid echoes of cool jazz, insistent hard bop and the occasional snippet of funk. Falanga mostly bowed his instrument, while Cohen plucked his. Denardo self-effacingly accompanied on drums.

I remember listening to Ornette Coleman, as a youth, on Radio Canada’s late-night jazz show. There, in my darkened bedroom, Coleman’s music seemed like a message in a bottle, unlike anything I’d ever heard before.

Tonight was the first time I had the opportunity to see him in concert. Sitting in the 4^th row, I was spellbound, from the first notes, through the encore of Turnaround.

Longtime readers of this blog will know that I think that discrete symmetries (particularly, those that suppress proton decay and flavour-changing neutral currents) are a serious challenge for ideas about the Landscape and, more generally, applications of the Anthropic Principle to String Theory Vacua (see, for instance, these three and posts).

If supersymmetry is broken in the visible sector below the GUT scale, then there are, generically, dimension-4 operators in the supersymmetric theory which violate baryon number. Below the supersymmetry-breaking scale, these induce the usual dimension-6 baryon number operators, suppressed, not by $M_{\text{GUT}}^{-2}$, but by $M_{\tilde q}^{-2}$, where $M_{\tilde q}$ is the squark mass. To suppress these bad operators (note that the observed proton lifetime is at least 20 orders of magnitude longer than the anthropic bound), one generally invokes a discrete symmetry. For instance, R-parity arises as a $\mathbb{Z}_2$ global symmetry, under which the quark and lepton chiral multiplets are odd, while the Higgs chiral multiplets are even.

Such discrete symmetries typically only occur on subspaces of the moduli space of very high codimension. In flux compactifications, not only must the vacuum lie on this invariant subspace, but the fluxes must be chosen to respect the symmetry as well (they appear as coupling constant in the low-energy theory). This drastically reduces the number of flux vacua.

For concreteness, let us work in the orientifold limit, with a Calabi-Yau 3-fold, $X$. The standard estimate of the number of flux vacua is roughly $L^{b_3(X)}$, where $L$ is a measure of the D3-brane charge that must be cancelled. If the desired discrete symmetry requires restricting to a subspace of the moduli space of complex dimension $n$, then the naïve estimate of the number of flux vacua respecting the symmetry is $L^{2(n+1)}$ (the extra 1 comes from $H^{(3,0)}\oplus H^{(0,3)}$). This is vastly smaller than $L^{b_3(X)}$.

There’s a very interesting recent paper which does a more careful estimate. What they find is that — in the class of examples they look at — the number of vacua respecting the discrete symmetry can be enhanced, for number-theoretic reasons, over the above naïve estimate by powers of $\log(L)$.

Where does number theory come in? Pick a point, $z$, in the complex structure moduli space and a value, $\phi$, for the axio-dilaton. View the equations $D_\alpha W=0$ as a set of (linear) equations for the fluxes. If the fluxes were real-valued, we could always solve those equations. But they’re integer-valued. For a generic point, there will be no integer-valued solutions for the allowed fluxes. For special points, the periods take values in some finite extension of $\mathbb{Q}$, and it is possible to adjust the (integer) fluxes, so as to find a solution. Consider the case where the field extension is of degree¹ $d$ and $\phi$ is of height² $H(\phi)$ in this extension. If the degree of the extension and the height, $H(\phi)$, are small enough, then one gets $$N_{\text{vacua}}\sim \left(\frac{L^{(1-d/4)}}{H(\phi)^{d/2}}\right)^{b_3}$$ as an estimate of the number of flux vacua at such a point.

At least in the simple examples they consider, summing over such points respecting the discrete symmetry produces a result enhanced by powers of $\log(L)$ over the naïve one.

They also address a related question: how many such vacua also satisfy $W_{\text{tree}}=0$ (i.e. are supersymmetric in 4D flat space, in the no-scale approximation)? Such vacua are non-generic³ as the system of equations in overdetermined. But their abundance is closely related to the statistical likelihood (if you believe in such statistical games) of low-scale supersymmetry breaking. Again, DeWolfe et al give an estimate $$N_{W=0\,\text{vacua}}\sim \left(\frac{L^{1/2}}{H(\phi)}\right)^{-d} \left(\frac{L^{(1-d/4)}}{H(\phi)^{d/2}}\right)^{b_3}$$ of the number of $W_{\text{tree}}=0$ vacua at a point where the periods lie in a degree-$d$ extension of the rationals.

The people have spoken and, by a margin of 3.7 million or so, have chosen their pumpkin. Yes, it’s a bit soft, slightly smelly, and should have been put out at the kerb by now. But there it is.

Kerry could still win Ohio (and hence the election) in a count of provisional ballots 10 days from now¹. But that would be almost worse. Both Houses of Congress are firmly in Republican control and the popular vote is unmistakably against him. Governing under such circumstances would be nigh on impossible.

This way, at least, a second Bush Administration will have to face up to the ballooning Budget deficit, the anæmic economy and the disaster they have created in Iraq. They, alone, will face responsibility for the mess.

Some discern hopeful signs that sanity (fiscal, diplomatic or strategic) will return in a 2^nd Bush term. I don’t see it. Inhale deeply; we’re headed for four very smelly years…