## October 20, 2005

### ZMP Colloquium, Thursday: Nekrasov on Pure Spinor Superstring

#### Posted by urs

As mentioned before, I am attending a mathematical physics conference at Hamburg University. Here are some impressions.

First talk today was by Nikita Nekrasov on Berkovits’ pure spinor formulation of the superstring.

N. Nekrasov reviewed some well known facts about quantization of the superstring in GS, NSR and now also in the PS (‘pure spinor’) formulation.

The main issue he dealt with was a geometric interpretation of the pure spinor constraint. While the theory looks free at first sight, this constraint actually makes the fermionic part of the PS-string a nontrivial 2D sigma model on the space of pure spinors.

Nekrasov dicussed how the theory looks like a free theory locally, when we restrict attention to worldsheets which map to a contractible open subset of target space (the pure spinor space in this case). These locally free theories on open patches can then be glued together on double overlaps. This is governed by a certain 2-form, the details of which are too technical for the kind of sloppy report that I am giving. The upshot is that transitions on double overlaps don’t necessarily glue together on triple overlaps. Instead one finds an obstruction there which, being physicists, they call an anomaly. But the point is that what appears here is a gerbe-like structure. OPE’s are expressible in terms of the Courant bracket, which should be the algebroid version of that gerbe.

Even more strikingly, at the very end N. Nekrasov mentioned the following:

Let $Q$ be Berkovits’ BRST-like operator in the PS formalism. There is a certain operator called $G$ of odd gost grade such that the worldsheet energy-momentum operator is $Q$-exact.

(1)${T}_{i}=\left[Q,{G}_{i}\right]$

on each patch ${U}_{i}$. ${T}_{i}$ is well defined on all intersections, hence its Čech-differential $\delta$ should vanish

(2)$\left(\delta T{\right)}_{\mathrm{ij}}={T}_{i}-{T}_{j}=0\phantom{\rule{thinmathspace}{0ex}}.$

But the Čech-differential of $G$ does not vanish. It vanishes only up to another exact term

(3)$\left(\delta G{\right)}_{\mathrm{ij}}=\left[Q,{G}_{\mathrm{ij}}^{\left(1\right)}\right]\phantom{\rule{thinmathspace}{0ex}}.$

And the same holds for the new ${G}^{\left(1\right)}$

(4)$\left(\delta {G}^{\left(1\right)}{\right)}_{\mathrm{ijk}}=\left[Q,{G}_{\mathrm{ijk}}^{\left(2\right)}\right]\phantom{\rule{thinmathspace}{0ex}}.$

And once again the same holds for the new ${G}^{\left(2\right)}$

(5)$\left(\delta {G}^{\left(1\right)}{\right)}_{\mathrm{ijkl}}=\left[Q,{G}_{\mathrm{ijkl}}^{\left(3\right)}\right]\phantom{\rule{thinmathspace}{0ex}}.$

${G}_{\mathrm{ijkl}}^{\left(3\right)}$ finally is Čech-closed and hence well-defined globally:

(6)$\delta {G}^{\left(3\right)}=0\phantom{\rule{thinmathspace}{0ex}}.$

This was the end of the talk.

I cannot resist noting the following: We can suggestively group all the $G$ together in a cochain

(7)$\left(G,{G}^{\left(1\right)},{G}^{\left(2\right)},{G}^{\left(3\right)}\right)$

and define a Čech-BRST hyper-coboundary operator

(8)$D=\delta -Q$

on these cochains (up to some implicit signs and some technical details). Then the above should say (remember that I haven’t seen the gory details of what Nikita Nekrasov has mentioned) that we have a $D$-closed cochain

(9)$D\left(G,{G}^{\left(1\right)},{G}^{\left(2\right)},{G}^{\left(3\right)}\right)=0$

which could be addressed as an element of generalized Deligne hypercohomology (in degree 3).

Of precisely this kind (for different choices of $Q$) is the generalized Deligne hypercohomology which gives the infinitesimal cocycles of (fake flat) nonabelian $p$-gerbes with $p$-connection, as discussed in the last section here (hep-th/0509163).

Among other things, this means that there should be a nice 2-functorial reformulation of the PS-string.

Of course from every complex $\left(\Omega ,Q\right)$ of sheaves one gets a Čech hypercohomology $\left(\stackrel{˜}{\Omega },\delta ±Q\right)$. Not all of them need to be related to be related to gerbes, I guess (on the other hand, gerbes are very general…). But since in other places of the above talk the Courant bracket appeared, it is natural to conjecture that there is a gerbe hiding here. In fact, the Courant bracket should be a sign of the Courant algebroid, which again is known to be the algebroid version of a gerbe.

I got the impression that there is not yet a published preprint discussing this stuff concerning the ${G}^{\left(i\right)}$ operators. The last thing by Nikita Nekrasov on pure spinors which I see on the arXiv, which probably deals with most things mentioned in his talk today, is hep-th/0503075.

I am not going to discuss further talks, it seems, but Robert mentions them all over at atdotde.

Posted at October 20, 2005 2:57 PM UTC

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## 1 Comment & 5 Trackbacks

### Re: ZMS Colloquium, Thursday: Nekrasov on Pure Spinor Superstring

Hi there,
I am an undergraduate physics student in New Zealand.
I’m not sure if this is right place to post this but I was wondering if you ‘expert’ physicists could have a look around my weblog and if there are misunderstandings in the articles that I write, to correct me. I’m only a junior in University so I have a lot more to learn and I think your comments or criticism or feedback will help me learn A LOT more. Thanks!
http://precondition.blogspot.com

Posted by: Geon Oh on October 22, 2005 10:15 AM | Permalink | Reply to this
Read the post CDO and Pure Spinors
Weblog: Musings
Excerpt: Urs on Nikita on pure-spinors.
Tracked: October 26, 2005 10:32 PM
Read the post Nekrasov on Pure Spinors
Weblog: Musings
Excerpt: Nikita on pure spinors
Tracked: November 2, 2005 7:21 AM
Read the post Sheaves of CDOs
Weblog: The String Coffee Table
Excerpt: Literature on Cech-methods in CFT.
Tracked: November 2, 2005 10:20 AM
Read the post Nekrasov Lecture Online
Weblog: The String Coffee Table
Excerpt: Quicktime video of Nikita Nekrasov's lectures on the pure spinor superstring is available online.
Tracked: January 4, 2006 7:12 PM
Read the post Nekrasov Lecture Online II
Weblog: The String Coffee Table
Excerpt: The third session of Nikita Nekrasov's lecture in Jerusalem. Some news on the Cech description of the pure spinor string.
Tracked: January 5, 2006 7:18 PM

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