December 16, 2008

Super Version of 2-Plectic Geometry for Classical Superstrings?

Posted by John Baez

In our paper Categorified symplectic geometry and the classical string, Alex Hoffnung, Chris Rogers and I described a Lie 2-algebra of observables for the classical bosonic string. The idea was to generalize the usual Poisson brackets coming from symplectic geometry, which make the observables for a classical point particle into a Lie algebra. The key was to replace symplectic geometry by the next thing up the dimensional ladder: 2-plectic geometry.

Now I have a slight hankering to do the same thing for the classical superstring. Ideally this would be a formal exercise in ‘super-thinking’ — replacing everything in sight by its ‘super’ (meaning $\mathbb{Z}/2$-graded) analogue. But maybe it’s not. Either way, I have a lot of catching up to do. So, here are some basic questions.

First, is there a nice ‘superspace formulation’ of the classical superstring in 3, 4, 6 and 10 dimensions? That is, can we describe such a superstring as a map from a (super?) Riemann surface to some supermanifold? And can we write the action nicely in these terms?

For some reason the catchphrase ‘superspace formulation’ seems to get used a lot more often for supergravity and super-Yang-Mills theory that for superstrings. Why is that?

Second, does anyone know references where the phase space of a classical superparticle has been described as a ‘supersymplectic manifold’ or ‘super-Poisson manifold’? There seems to be at least a little work along these lines, e.g.:

Third — and now perhaps I’m pushing my luck — has anybody studied supersymmetric field theories using ‘super-multisymplectic geometry’, in analogy to the treatment of ordinary field theories (e.g. nonlinear sigma-models) using multisymplectic geometry?

I just don’t want to reinvent a wheel, or waste my time inventing a square one.

Posted at December 16, 2008 7:14 PM UTC

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Re: Super Version of 2-Plectic Geometry for Classical Superstrings?

I was about to call it quits for today when I saw this post. So just verz quickly:

the big technical problem with superstrings is that

- there is a formulation where the worldsheet is a supermanifold, but target space is not. This is called the NSR superstring. It’s quantization is tractable and leads to 2-dimensional superconformal field theory (SCFT) on the worldsheet. The big problem is that using this the supersymmetry on the effective target space is, while present, not manifest.

- there is a formulation where the worldsheet is taken to be an ordinary manifold, but target space is taken to be a supermanifold. This is the GS-superstring (Green-Schwarz). It has the advantage that target space supersymmetry is manifest, but the disadvantage that it is hard to quantize. Or impossible even. The thing is that this involves certain second class constraints which noboody, as far as I know, has managed to handle in the quantum theory.

- There are attempts to combine the advantages of the two approaches while at the same time getting rid of their disadvantages. One such attempt is Berkovits’ formulation of the superstring. I am not sure what the status of this is, precisely. It started with an extremely ad-hoc guess, which later was seen to have nice relations to the chiral deRham complex. The point here is that one “solves” the complicated dynamics of the fermions by, roughly, covering the complicated target space for the fermions (the “pure spinors” in this approach) by open subsets. Restricting maps to go just to one such subset one is left with a free worldsheet theory, which can directly be quantized. This way one obtains a presheaf of SCFTs over the pure spinor target. The whole problem has now been moved into gluing that back into one global structure.

This is the idea, which is quite beautiful I think. But I don’t know to which extent superstring quantization has been understood this way entirely.

For what you want to do, super-2-plectic geometry, the RNS superstring is the right approach.

Posted by: Urs Schreiber on December 16, 2008 9:03 PM | Permalink | Reply to this

Re: Super Version of 2-Plectic Geometry for Classical Superstrings?

Thanks for all this information, Urs!

For what you want to do, super-2-plectic geometry, the RNS superstring is the right approach.

I trust you, but can you say why this is better than the Green–Schwarz approach? Naively I’d be happier with a manifold mapped into a supermanifold than vice versa, for some reason.

(I’m not completely sure why — maybe because I understand the loop space of a supermanifold better than the superloop space of a manifold!)

Obviously the formulation that’s sufficiently tractable to give a 2d SCFT upon quantization is likely to be better. But I bet you have some more precise reason.

Posted by: John Baez on December 17, 2008 7:07 AM | Permalink | Reply to this

Re: Super Version of 2-Plectic Geometry for Classical Superstrings?

For some reason the catchphrase superspace formulation seems to get used a lot more often for supergravity and super-Yang-Mills theory that for superstrings. Why is that?

For search results that better fit your needs, try combinations like “super worldsheet”.

I am not sure where best to point you short of the standard superstring text books and the standard online lecture notes which you know yourself.

I see that there is a 1-page summary of the RNS superstring on p. 5 of D’Hoker-Phong: Lectures on Two-Loop Superstring. Maybe you want to chase references from there.

Posted by: Urs Schreiber on December 16, 2008 9:19 PM | Permalink | Reply to this

Re: Super Version of 2-Plectic Geometry for Classical Superstrings?

Appendix B p. 53 of D’Hoker-Phong Two-Loop Superstrings II summarizes useful stuff for $N=1$, $d=2$ supergeometry. This is what you want.

More details in their The geometry of string perturbation theory.

Posted by: Urs Schreiber on December 16, 2008 9:28 PM | Permalink | Reply to this

Re: Super Version of 2-Plectic Geometry for Classical Superstrings?

Thanks for these pointers. I know basic superstring references, but I’m looking for mathematically minded stuff. Not just for superstrings, but also superparticles. Do you know stuff that describes the phase space of a superparticle as some sort of ‘supersymplectic manifold’?

Posted by: John Baez on December 17, 2008 7:13 AM | Permalink | Reply to this

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