### 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, $\mathcal{D}\subset \mathcal{T}$ 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 $h\gt 3$.

Posted by distler at November 28, 2004 12:18 AM
## Re: Splitness

That’s very interesting. Are there any expectations on what happens for higher genus? Do we have to expect that split superspaces over 2D surfaces are the exception?