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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 9th 2023

    Created:

    Idea

    A notion of connection for principal 2-bundles that does not impose the fake flatness condition.

    Related concepts

    References

    See also the references at adjusted Weil algebra.

    Definition of adjusted connections in terms of cocycle data:

    Examples for T-duality:

    Further references for T-duality:

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 10th 2023

    added the original references, with brief commentary

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeOct 10th 2023

    Thanks a lot for providing the original sources. Is the “modified Weil algebra” the same as “adjusted Weil algebra”?

    Is there a source that discusses adjusted connections in full generality of Lie ∞-groups?

    Rist–Saemann–Wolf only discuss Lie 2-algebras, whereas your papers seem to talk only about String-like extensions, if I understood them correctly.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 11th 2023

    We actually called it the Chern-Simons L L_\infty-algebra (p. 48), because it brings out the expected Chern-Simons-terms in the L L_\infty – which is what the whole “adjustment”-process is about.

    There is no sense of doing this for non-string-like extensions: These higher central extensions are what is classified by invariant polynomials transgressing to Chern-SImons elements. In accounts you may be looking at this may be hidden in the assumption of a bilinear invariant pairing: This is the binary invariant polynomial which classifies the String-like extension. Without such, there is no sense to adjust to any Chern-Simons terms.

    The delooping BG\mathbf{B}G of the smooth \infty-group Lie integrating an L L_\infty-algebra 𝔤\mathfrak{g} is a truncation of the stackification of

    exp(𝔤):(U,[k])Hom dgAlg(CE(𝔤),Ω dR,vert (U×Δ smth k)). exp(\mathfrak{g}) \,\colon\, (U, [k]) \,\mapsto\, Hom_{dgAlg}\big( CE(\mathfrak{g}),\, \Omega^\bullet_{dR, vert}( U \times \Delta^k_{smth} ) \big) \,.

    This is the stacky formulation of Henriques’s notion of integration of L L_\infty-algebras. For Lie 1-algebras and using 1-truncation this reduces to the “path method” for Lie integration, which is for instance implicit in Brylinski & McLaughlin 1996.

    The process we described enhances this Lie integration (in a way I have recalled when we discussed this last time) to a stack BG conn\mathbf{B}G_{conn} such that maps XBG connX \to \mathbf{B}G_{conn} are (cocycles for) GG-principal \infty-bundles on XX with non-fake flat \infty-connection given by the expected Chern-Simons terms.