Musings

I think the general expectation/fear was that the higher-genus supermoduli spaces would not be split and that, therefore, there would be no purely “bosonic” formula for the chiral superstring measure.

The obstruction to splitness can be expressed in terms of the sheaf cohomology of the ordinary bosonic moduli space. I wouldn’t know how to compute whether the obstruction vanishes. And I’m not sure how D’Hoker and Phong’s explicit demonstration that it vanishes at $h=2$ should affect our expectations for $h>3$.

Do you know a peaceful non-pathological example of a supermanifold that is non-split?

Absolutely.

Consider a $(1\mid 2)$-dimensional supermanifold. I will cover it with two charts. In one patch, the coordinates will be $(z,{\theta }_1,{\theta }_2)$. In the other patch, the coordinates will be $(z\prime ,\theta {\prime }_1,\theta {\prime }_2)$. On the overlap between the two patches, we have the transition functions

where $a,b$ are integers such that $a+b\ge 4$ and

The overall scale of the coefficients ${\alpha }_k$ is irrelevant, but as long as they are not all identically zero, we have a nonsplit supermanifold. You can’t get rid of the quadratic term in the ${\theta }_i$ in the transition functions by a holomorphic change of coordinates. So we have a ${\mathbb{P}}^{a+b-4}$ worth of inequivalent nonsplit supermanifolds.

All of these nonsplit supermanifolds have $M_{\text{red}}={\mathbb{P}}^1$, the Riemann sphere. ($M_{\text{red}}\hookrightarrow M$ is the “locus” where all the odd coordinates vanish.)

You can see the obvious generalization of this construction to the arbitrary case of a supermanifold, $M$, of dimension $(n\mathrm{,2})$. The sheaf of odd functions on $M$, ${𝒪}_{M(1)}$ is isomorphic to the parity-reversal, $\Pi ℱ$, of some rank-2 locally-free sheaf, $ℱ$ on $M_{\text{red}}$. (In the above example, $ℱ=𝒪(-a)\oplus 𝒪(-b)$.) The obstruction to splitness is a class in $V=H^1(M_{\text{red}},T_M\otimes {\wedge }^2(ℱ))$. The space of inequivalent supermanifolds is $\mathbb{P}(V)$.

When the odd dimension is greater than 2, there are further obstructions of a similar sheaf-cohomological nature. All these obstructions vanish in the differentiable case, but – of course – we are interested in the chiral superstring measure, so we want to think of the supermoduli space as a complex supermanifold.

As long as the odd dimensionality is greater than one, non-split complex supermanifolds are as common as dirt.

The only thing that could happen is that you would get some monodromies, but I feel it can’t happen for these supermoduli spaces.

I’m not sure what that’s supposed to mean.

Does not the splitness simply follow from the fact that you can still consider superstring theory in terms of fields defined on the ordinary bosonic worldsheet, sort of in components?

No, of course not.

It is almost tautological that the worldsheet is a split supermanifold. Any supermanifold with but a single odd dimension is split.

The supermoduli space, however, has odd dimension $2h-2$. The zero modes of the gravitino are the odd tangent vectors to the supermoduli space. We are interested in the transition functions between different patches in supermoduli space.

This seems to give you the natural projection to the bosonic manifold very directly.

No, setting the gravitini to zero gives you $M_{\text{red}}$, which is the ordinary (spin) moduli space, sitting as a submanifold of $M$. As I have explained, that tells you nothing about whether $M$ is split.

Or do you have some more specific reason why you’re afraid that it won’t be split?

I have no idea whether it is split or not, for general genus. But I guarantee you that, if you pick your favourite coordinates for the supermoduli space and examine the transition-functions between patches, they will invariably look non-split. (You can already see this in low-genus examples, with which you are doubtless familiar.) But a clever change-of-coordinates, as found by D’Hoker and Phong, may remove the apparent non-splitness.

We are discussing the existence or nonexistence of such a clever change of coordinates.

There is one more trivial reason why the amplitudes must be calculable at the end as something integrated over the bosonic moduli space. You can simply see it if you go to Green-Schwarz light cone gauge, and make the integral in the variables over there

Hmmm.

1. Do you really know how to do Green-Schwarz for genus $h\ge 2$? I sorta, kinda know it’s possible, but have yet to see any explicit calculations.
2. The “chiral superstring measure” being discussed here is the contribution of a particular (even) spin structure to the NSR path integral. I don’t think any GS calculation (which does not depend on any choice of spin structure on $\Sigma$) can shed light on the (non)existence of a formula of the sort we are seeking.