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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 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 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)(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 0c_0, c 1c_1, c 1c_{-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 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 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 L_\infty-algebra.

    What I mean is: for ,W(𝔤)\langle-,-\rangle \in W(\mathfrak{g}) to be an invariant polynomial, we need d W,=0d_W \langle-,-\rangle = 0. It seems I can show that d W,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 KKreduction\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

    𝔤 closedDer(A open). \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

    diff, v55, current

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

    Alex Arvanitakis

    diff, v60, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMar 8th 2019

    added pointer to today’s

    diff, v61, current

    • 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)

    diff, v62, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 12th 2019

    added pointer to today’s

    • Hiroshi Kunitomo, Tatsuya Sugimoto, Type II superstring field theory with cyclic L-infinity structure (arxiv:1911.04103)

    diff, v64, current

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