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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.
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 :)
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
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.
started adding something to the Definition-section at string field theory
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.
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.)
I have added some more references on the CSFT tachyon vacuum to String field theory - References - Bosonic CSFT
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
added a pointer to the recent article by Branislav Jurco on superstring field theory.
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.
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.)
added pointer to today’s
Prodded by an alert from Jim Stasheff, I have added this recent reference:
added pointer to today’s
added pointer to
added these pointers:
Harold Erbin, String Field Theory – A Modern Introduction, 2020 (pdf)
Harold Erbin, String theory: a field theory perspective, 2020 (pdf)
added this pointer:
(what I was really looking for is a modern review that would touch on Witten’s old suggestion of defining the star-product for closed strings, as originally indicated in Fig 20 of his “Non-commutative geometry and string field theory” doi:10.1016/0550-3213(86)90155-0)
added todasy’s arXiv number and publication data to:
added pointer to today’s
pointer to
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