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Genus 5+ supermoduli space is not split

In multiloop diagrams, superstring theory is more than bosonic string theory with a cherry on a pie

Three weeks ago, I discussed the possible shapes of the two-dimensional torus. They were parameterized by the parameter \(\tau\in\CC\) with an extra symmetry, "modular invariance" \(SL(2,\ZZ)\), identifying countable sets of values of \(\tau\).



The torus may be considered as a genus-one Riemann surface (a sphere with \(h=1\) added handle to it) and it plays the role – aside from other roles – of the 1-loop "thickened" Feynman diagram in string theory (with purely closed strings). There exist diagrams with an arbitrary number of loops and Edward Witten and Ron Donagi have prepared a surprise for us today.




They wrote a 57-page paper

Supermoduli Space Is Not Projected
whose title is so informative that it should have been introduced by a spoilers alert.




What's the beef?

The moduli space of the shapes of a torus gives us one \(\tau\), so it's one-complex-dimensional (or two-real-dimensional). Analogously, we may find out that for \(h\gt 1\) handles, the number of complex parameters is \(\dim {\mathfrak M}_h = (3h-3)\). You may be surprised that the formula gives zero instead of one for \(h=1\). But this irregularity is expected for low enough genera. You may feel happier if I tell you that this deviation is correlated with the existence of the (conformal) isometry group \(G\) of the torus – \(U(1)^2\) which corresponds to two real or one complex parameters – and the difference\[

\dim_\CC({\mathfrak M}_h)-\dim_\CC(G_h) = 3h-3 = -\frac {3\chi}{2}

\] behaves regularly even for small values of \(h\). The dependence on the Euler characteristic \(\chi\) is no coincidence because the identity above may be proved as an index theorem (constraining the difference of the number of zero modes of ghosts \(c\) and antighosts \(b\) on the Riemann surface).

The truly interesting and fully consistent theory isn't the \(D=26\) bosonic string theory but the \(D=10\) superstring theory. The superstring contains some extra fermionic excitations which may be conveniently described using the world sheet supersymmetry. Effectively, this may be achieved by adding some Grassmann dimension(s) to the list of the world sheet dimensions, \(\tau\) and \(\sigma\) or \((z,\bar z)\).

You could think that the Grassmann numbers are boring. They just multiply the number of coefficients in all functions by \(2^N\), a finite number, because every function may be Taylor-expanded into powers of the Grassmann dimensions and the number of terms is finite because the second power of a Grassmann variable is already zero.

So you may conclude that the Grassmann variables are just small decorations. Effectively, we may still describe the supermoduli space of Riemann surfaces essentially as a regular moduli space for the non-supersymmetric Riemann surfaces from bosonic string theory; and add the innocent extra fermionic dimensions as fibers (although not necessarily in a trivial bundle) to turn it into a supermanifold at the very end. (Note that the number of complex fermionic parameters labeling the fermionic dimensions of the supermoduli space goes like \((2h-2)\).)

This idea pretty accurately describes what we mean by a "split supermanifold": the essence of the construction is some purely bosonic manifold, a base, and the fermionic dimensions are peaceful additions that don't destroy this base. There is a related notion of a "projected supermanifold" which is a somewhat weaker condition: split supermanifolds are a proper subset of the projected supermanifolds. I am not able to explain what the exact difference between the classes is.

What Witten and Donagi show is that \({\mathfrak M}_h = {\mathfrak M}_{h,0}\) is non-projected for \(h\geq 5\) – and, without a proof, they suspect that even for \(h\geq 3\). If you add \(n\) marked points aside from the handles, the moduli space \({\mathfrak M}_{h,n}\) acquires some extra dimensions and structure and they may prove that for \(n=1\), the supermoduli space is non-projected already for \(h\geq 2\).

Their proofs exploit a separate discussion of even and odd spin structures and the quantification of "leading obstructions", some other notions of algebraic geometry, and lots of fibers and commutative diagrams with exact columns etc. (a representative stuff that is arguably enough to claim that someone's thinking is closer to mathematics than to physics, despite his being great in both). All the required background is presented in the paper; it is largely self-contained.

The paper implies that certain methods to organize the multiloop calculations in superstring theory – those that try to squeeze the superstring calculations into the straitjacket of the bosonic string theory methods and that don't respect the independence of the superstring and its structures – may start to fail for \(h\geq 5\) and perhaps a bit earlier. The bosonic string theory framework is just a useful toy model that is related to what we need for the superstring; however, the superstring calculations must be done from scratch instead of assuming that the bosonic part is always a useful beginning of the calculation.

I think that a heuristic explanation why the bosonic calculation isn't a usable "skeleton" of the superstring calculation is to say that products of an even number of the Grassmann dimensions on the supermoduli space, expressions such as \(\theta_1\theta_2\theta_3\theta_4\), can get mixed with finite changes of the bosonic dimensions of the supermoduli space, so the Grassmann moduli can no longer be considered "infinitesimal" and therefore "fiber-like".

Their results don't change anything about the well-definedness and finiteness of superstring theory to all orders – which has been proved, also in recent papers by Witten – but yes, I am somewhat surprised that in his recent papers, Witten wasn't "forced" to have noticed the non-splitness of the supermanifolds. I am sure that he would give us a "trivial" explanation for that. ;-)

Also, a thing that puzzles me is that some approaches to the superstring – in particular, Nathan Berkovits' pure spinor approach based on the Green-Schwarz superstring – don't have world sheet supersymmetry at all. For this reason, they have to work with the bosonic moduli space of the Riemann surfaces only. Because the RNS-like calculations for \(h\geq 5\) can't be reduced to a bosonic "base", the equivalence of the multiloop calculations in the pure spinor and RNS frameworks has to be rather hard to prove, I feel – or, what would be worse, the pure spinor approach even becomes defective for \(h\geq 5\) (or at least Berkovits' approach may require some translation to the RNS non-split supermanifolds which can make the pure spinor formulae for the multiloop amplitudes contrived and ugly). I will probably ask Nathan about it.

Pierre Deligne, the most recent winner of the Abel Prize, is the most gratefully thanked to mathematician (who has read the manuscript) in the acknowledgements.



IBM has hired unusual actors, namely individual atoms (well, carbon monoxide molecules), to record a one-minute movie, A Boy and His or Her Atom. The atoms were genuinely moved to the right places and pictures of each arrangement were taken to create the frames of the movie. The authors – who are often female – are mostly excited and their goal, as explained in the extra 5-minute explanatory video, is to prevent young folks from going to the law school. A difference between a dead rabbit and a dead lawyer is that one may recognize the braking distance (a trace of rubber) in front of the rabbit, IBM has determined.

Of course, they didn't do it just for the infantile movie. They're motivated by finding out how many atoms may behave as a reliable magnet in a "flash disk". It turns out that the current number of 1 million atom might be reduced down to something like 12 in their case. Of course that all movies ever recorded could be stored on your iPod.

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reader Lord Nelson said...

Deligne is my personal hero. Equally brilliant as Grothendieck but not psychotic (in clinical terms) and not extreme left (as far as I know) as his teacher.


reader dalpezzo said...

Lubos, if you want a formula for the dimension of M_g which works for all genera, you can use (3g-3)+ dim(Aut), where Aut is the group of automorphisms of your curve. Then dim(Aut) is 3 for g=0 (Aut=PGL_2(C)), 1 for g=1, and 0 for g>1 (a curve of genus g>1 has a finite number of automorphisms, actually, at most 84(g-1).)


reader Robert Rehbock said...

So much here ... So little time. Thank you for these great posts. I enjoy lawyer jokes. Many are true stories


reader Gene Day said...

Now that’s really cool!


reader Trimok said...

If you may detail your arguments, this would be great, and very interesting for me (references are welcome) . Thanks


reader Phil Jones said...

Thanks for reminding me how much more maths I need to learn.... :-)


reader Cliff said...

Awesome! What an incredible result. Im reading the paper this instant, and its efforts to be complete and understandable are much appreciated.


This definitely seems to be a prime example of how 21st century mathematical technology is necessary to gain a proper understanding of string theory.


There was another apparently very important result on superstrings pretty recently (thanking Witten in the acknowledgements) and I wasn't sure if you missed it but I was sure looking for your reaction. The basic claim was there is another superstring perturbatively equivalent to Type I but nonperturbatively inequivalent.
http://arxiv.org/abs/1304.1551


reader dalpezzo said...

Given Riemann surface (smooth, proper algebraic curve over C), you can ask for its group of automorphisms (biholomorphisms). This is a complex Lie group (depending on the particular curve, of course), and you can look at its dimension (which depends only on the topology, i.e. on the genus).

For P^1 ("Riemann sphere"), automorphisms are just Moebius transformations, i.e., elements of PGL_2(C), which is a (complex) Lie group of (complex) dimension 2x2-1=3. For an elliptic curve (1-dim'l complex torus), the automorphisms consist of the torus itself (acting as translations of the origin), plus a finite group (depending on the torus), so the complex dimension of the automorphism group is 1. For a higher genus curve, the automorphism group is always finite (so doesn't contribute anything to the dimension count): google Hurwitz automorphisms theorem. The moral is that when you count moduli you have to take care of automorphisms.


reader Dilaton said...

Yep, this is even more true for me :-D

This very nicely explained article quite smoothely matches and continues or extends what I have just read about superstrings in my demystified book such that I think I can at least see what it is talking about :-)


I like such articles a lot !


reader Stephen Crowley said...

how is it fair to say that Grothendieck is psychotic? Is there some legend I'm missing out on?