## 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, $\mathcal{D}\subset \mathcal{T}$ 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

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### 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?

Posted by: Urs Schreiber on November 29, 2004 6:14 AM | Permalink | Reply to this

### Re: Splitness

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 $h\gt 3$.

Posted by: Jacques Distler on November 29, 2004 3:51 PM | Permalink | PGP Sig | Reply to this

### Re: Splitness

We just had the class where I had half an hour to show 4-5 different parameterizations of the bosonic genus g moduli spaces for larger g’s.

I really don’t understand how more or less any finite-dimensional supermanifold of this category can be non-split, in the sense that you could not project it to the “bosonic” space. All the odd dimensions are sort of “infinitesimal”.

Do you know a peaceful non-pathological example of a supermanifold that is non-split? Will you construct it from charts with transition functions that are nonlinear in the odd dimensions? Like “z” in one patch is “z-theta1.theta2” in the other patch? Cannot you show that there is a field redefinition in the individual patches that always kills it? The only thing that could happen is that you would get some monodromies, but I feel it can’t happen for these supermoduli spaces.

Posted by: Lubos Motl on November 29, 2004 4:47 PM | Permalink | Reply to this

### Re: Splitness

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

Absolutely.

Consider a $(1|2)$-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 $(z',\theta'_1,\theta'_2)$. On the overlap between the two patches, we have the transition functions

(1)\array{\arrayopts{\colalign{right left}} z' &= \frac{1}{z}\left(1 + F(z) \theta_1 \theta_2\right)\\ \theta'_1 &= z^{-a} \theta_1\\ \theta'_2 &= z^{-b} \theta_2}

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

(2)$F(z) =\sum_{k=2}^{a+b-2} \alpha_k z^{-k}$

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 $(n,2)$. The sheaf of odd functions on $M$, $\mathcal{O}_{M(1)}$ is isomorphic to the parity-reversal, $\Pi\mathcal{F}$, of some rank-2 locally-free sheaf, $\mathcal{F}$ on $M_{\text{red}}$. (In the above example, $\mathcal{F}=\mathcal{O}(-a)\oplus\mathcal{O}(-b)$.) The obstruction to splitness is a class in $V=H^1(M_{\text{red}}, T_M\otimes \wedge^2(\mathcal{F}))$. 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.

Posted by: Jacques Distler on November 29, 2004 9:44 PM | Permalink | PGP Sig | Reply to this

### Re: Splitness

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?

You put the components - X(z), psi(z), but also the gravitinos - to a fixed worldsheet bosonic metric. Then the analysis of the gravitino fields gives you the fermionic dimensions of the moduli space fibered at the given bosonic geometry, and then you finish the “base”. This seems to give you the natural projection to the bosonic manifold very directly.

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

Posted by: Lubos Motl on November 29, 2004 8:52 PM | Permalink | Reply to this

### Re: Splitness

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.

Posted by: Jacques Distler on November 29, 2004 11:02 PM | Permalink | PGP Sig | Reply to this

### Re: Splitness

Hi Jacques,

thanks, but indeed, I am not confusing the worldsheet and the moduli space. ;-)

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 - it means the integral over the times of the interactions, splitting of p+ into various intermediate strings, and the p+ twists.

This just gives you the same amplitude, for any loop level, and you may always view the LC variables as parameters on the bosonic moduli space.

Best
Lubos

Posted by: Luboš Motl on November 30, 2004 6:35 AM | Permalink | Reply to this

### Re: Splitness

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 $h\geq2$? 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.
Posted by: Jacques Distler on November 30, 2004 7:54 AM | Permalink | PGP Sig | Reply to this

### Re: Splitness

No, I have certainly not done 2-loop S-matrix calculations in light-cone gauge Green-Schwarz.

It’s easy to agree that this must kind of be possible.

But what I don’t understand is why you think that it does not shed the light on the existence of a formula that you look for. It is completely the SAME formula.

Every history of the worldsheets in the light cone gauge, with some division of P+ etc., corresponds, up to some analytical continuation, to one point of the bosonic covariant worldsheet.

It’s irrelevant whether you consider Green-Schwarz or RNS - these just give you different ways how to arrive at the same *integrand*, but the integral over the bosonic moduli is totally in RNS and GS.

My doubts are probably caused by my misunderstanding why you want a formula of some “sort” - is not it enough to have a correct formula of “any sort”?

Posted by: Lubos Motl on November 30, 2004 8:54 AM | Permalink | Reply to this

### Light-cone Green Schwarz

No, I have certainly not done 2-loop S-matrix calculations in light-cone gauge Green-Schwarz.

It’s easy to agree that this must kind of be possible.

I would have a lot more confidence in light-cone Green-Schwarz, for $h\geq2$, if I saw even one calculation done in that formalism.

It may well be perfectly fine. Or there may be subtleties people haven’t noticed yet.

But what I don’t understand is why you think that it does not shed the light on the existence of a formula that you look for. It is completely the SAME formula.

They’re not really comparable.

1. We would hope that the integrals would be the same. We have no reason to expect the integrands to be the same.

2. As I’m sure you know, the light-cone formulation requires at least 2 external states. So there isn’t “a” formula in the light-cone case. There’s a different formula for each $n$-point function. In the RNS case (modulo some subtleties to do with the supermoduli associated to the punctures), the chiral vacuum amplitude we’ve been discussing summarizes the “hard” (but universal) part of the calculation. The “easy” (but specific) part involves using known Green’s functions for genus-$h$ (which, for the fermions, depend on spin-structure).

Certainly, the existence of your Green-Schwarz gedankencalculation leads one to hope that the desired formula exists, but a gedankencalculation is no substitute for an actual calculation.

Posted by: Jacques Distler on November 30, 2004 11:51 AM | Permalink | PGP Sig | Reply to this

### Re: Light-cone Green Schwarz

Sorry, Jacques, but I think that I disagree. Maybe *you* don’t have any reason to expect the integrands to be the same, but I think that I have. Otherwise I would not have written it.

The relation between RNS and GS is simply re-fermionization, and it holds for a fixed geometry of the worldsheet background. Therefore it’s not just the final result, but even the integrands agree pointwise, for each point in the moduli space of the Riemann surfaces, as long as you define them on both sides.

The only thing that you must do in advance is to sum the contribution of all spin structures, assuming that there are many (like in RNS). But once you do it, you get a pointwise agreement with GS, I believe.

I am also not quite getting the comment about “at least 2 external states”. Of course that you need at least two external strings, otherwise the amplitude is zero, is not it? When you write about the “hard universal” part of the calculation, do you want to suggest that the 0-point function is nonzero? You know, this is called the vacuum energy, and it certainly cancels to all loops in 10-dimensional supersymmetric situations. Also, the 1-point functions are tadpoles, and they vanish, too.

Did you want to suggest some inequivalence between the RNS and GS? Or inconsistency of string theory, or why are you exactly questioning all these things?

OK, g-loop superstring calculations may always be interesting to see them explicitly. On the other hand, it would probably not revolutionize the field. If you wish, Matrix string theory is a nonperturbative definition of these string theories, and the only thing that one needs to believe which implies that it makes sense is that the large N limit exists, and everything just seems to say Yes.

Posted by: Lubos Motl on November 30, 2004 4:43 PM | Permalink | Reply to this

### Re: Light-cone Green Schwarz

The relation between RNS and GS is simply re-fermionization, and it holds for a fixed geometry of the worldsheet background. Therefore it’s not just the final result, but even the integrands agree pointwise, for each point in the moduli space of the Riemann surfaces, as long as you define them on both sides.

They live on different (super)moduli spaces (one of which can be embedded as a subspace of the other), so I don’t even know what you mean by this statement.

And I certainly don’t believe that you know what “re-fermionization” means on a genus-$h$ surface, at an arbitrary point in its supermoduli space.

[I] am also not quite getting the comment about “at least 2 external states”.

I assume you know how light-cone gauge (bosonic, RNS or GS) is set up on a higher-genus surface. You need at least two punctures to be able to define a global “time” on the surface.

The only thing that you must do in advance is to sum the contribution of all spin structures, assuming that there are many (like in RNS). But once you do it, you get a pointwise agreement with GS, I believe.

A statement which may be true or false, if the supermoduli space is split, but which does not even make sense if it is not split.

Did you want to suggest some inequivalence between the RNS and GS? Or inconsistency of string theory, or why are you exactly questioning all these things?

I don’t know whether light-cone GS makes sense (or encounters some subtleties) on a higher genus surface. But, if it does make sense, I’m sure it reproduces the integrated RNS answers.

Or inconsistency of string theory, or why are you exactly questioning all these things?

Nowhere, ever did I suggest that there is any inconsistency in (RNS perturbative) string theory. Whether or not the supermoduli space is split, the theory is consistent.

RNS perturbation theory is a lot more subtle in the non-split case. But that is all.

Posted by: Jacques Distler on November 30, 2004 5:35 PM | 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 11:01 AM