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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 9th 2011
• (edited Oct 25th 2012)

I have started adding references to string field theory , in particular those by Jim Stasheff et al. on the role of L-infinity algebra and A-infinity algebra. Maybe I find time later to add more details.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 9th 2011

Comment: I am very glad in last month or two Urs is getting so much more back into physics with fruit at very high level :)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeSep 9th 2011

I am very glad in last month or two Urs is getting so much more back into physics with fruit at very high level

Thanks. I am, too! :-)

Maybe it’s clear what the reason is, and what the reason was for being more quiet on physics for a long time: I needed that time, personally, to get some general theory into place. Now that I understand how infinity-Chern-Simons theory (schreiber) follows from “first principles”, I can go back and re-examine what I now understand as examples of this.

The Zwiebach $L_\infty$-action for closed string field theory is a potential candidates to fit into this story: the CSFT action looks entirely like it should be an example for an $\infty$-Chern-Simons theory where the underlying (derived) $L_\infty$-algebra is the one that Zwiebach identifies on the string’s BRST complex, where the invariant polynomial is the binary pairing that he uses, the string correlator. It is a 3-dimensional theory, or rather a $(0|3)$-dimensional theory, which makes it a bit more exotic: the integration in the action functional is the Berezinian integral over the three string diffeomorphis ghost modes $c_0$, $c_1$, $c_{-1}$.

I have to check some details on this, but it looks like this should be true. If so, it would actually make CSFT yet another example of an AKSZ sigma-model. Which would be somewhat remarkable

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeSep 10th 2011

I have added to Chern-Simons element in a new section Properties – canonical CS element the discussion that for an arbitrary $L_\infty$-algebra with quadratic invariant polynomial, the corresponding Chern-Simons element is of the general for as the closed string field theory Lagrangian.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeSep 10th 2011

started adding something to the Definition-section at string field theory

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeSep 11th 2011

I have added to string field theory in the Definition-section a list of details extracted from Zwiebach’s main article.

Then after that I added a detailed proof that his inner product is indeed an $L_\infty$-invariant polynomial.

I still need to add more details on the various gradings in Zwiebach’s article.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeSep 12th 2011
• (edited Sep 12th 2011)

Then after that I added a detailed proof that his inner product is indeed an L ∞-invariant polynomial.

Maybe I have to take that back: while it is true that the inner product satisfies the defining equation of an invariant polynomial on the configuration space, I am not sure anymore if it satisfies it on the unconstrained $L_\infty$-algebra.

What I mean is: for $\langle-,-\rangle \in W(\mathfrak{g})$ to be an invariant polynomial, we need $d_W \langle-,-\rangle = 0$. It seems I can show that $d_W \langle -,-\rangle$ indeed vanishes when restricted to those fields that Zwiebach allows in the configuration space, but not in general.

(All this assuming that I did not otherwise make some mistake with the various gradings and signs. By the nature of this exercise, it is easy to make such mistakes.)

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeSep 25th 2012

I have added some more references on the CSFT tachyon vacuum to String field theory - References - Bosonic CSFT

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 25th 2012
• (edited Oct 25th 2012)

Added to References - Bosonic string field theory - Closed SFT explicit pointers to where exactly one can find written out the mode expansion which shows that the closed string field theory action is an extension of the Einstein-Hilbert action coupled to the B-field and the dilaton.

(This is eventually to supplement the discussion at geometry of physics, where I have now decided to discuss Einstein-Yang-Mills theory in the section Chern-Simons-type gauge theories in the derivations

• cohesion $\to$ general Chern-Simons-type actions $\to$ closed string field theory $\to$ Einstein-axion theory $\stackrel{KK-reduction}{\to}$ Einstein-Yang-Mills $\to$ standard-model + gravity :-)
• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMay 13th 2013
• (edited May 13th 2013)

added a pointer to the recent article by Branislav Jurco on superstring field theory.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 11th 2014

Since I pointed to the entry string field theory from this PhysicsOverflow reply I went and created two minimum entries such as to un-gray links:

While both just contain a reference for the moment, in the first case this is already useful, I’d think: this is the reference that Witten highlighted at String2012 as being crucial but having been kind of missed by the community.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeApr 22nd 2015
• (edited Apr 22nd 2015)

Finally added a (lightning brief, for the moment) paragraph on open-closed string field theory here. Added also a remark that it gives “one half” of the axioms of an $\infty$-Lie-Rinehart pair

$\mathfrak{g}_{closed} \longrightarrow Der(A_{open}) \,.$

Does one also have the “other half”? Is this discussed anywhere?

(I feel like I knew this once, but seem to have forgotten.)

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeOct 5th 2018
• (edited Oct 5th 2018)

updated references on the supersymmetric case. Okawa 16 is a good review of the recent breakthrough in getting the RR-sector under control. One should add more comprehensive references on this, but I don’t have the leisure now

1. Edited a typo: “bosononic->bosonic” in “bosonic closed string field theory”

Alex Arvanitakis

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeMar 8th 2019

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeApr 29th 2019

Prodded by an alert from Jim Stasheff, I have added this recent reference:

• Hiroshi Kunitomo, Tatsuya Sugimoto, Heterotic string field theory with cyclic L-infinity structure (arXiv:1902.02991)
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeNov 12th 2019

• Hiroshi Kunitomo, Tatsuya Sugimoto, Type II superstring field theory with cyclic L-infinity structure (arxiv:1911.04103)
• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJan 28th 2020