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nLab > Latest Changes: derived differential geometry

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2012
    • (edited Jan 12th 2012)

    finally created (a stub for) derived differential geometry

    expanded the old stub simplicial C-infinity ring a bit

    • CommentRowNumber2.
    • CommentAuthortrent
    • CommentTimeAug 18th 2015

    Added Dominic Joyce’s recent Derived Differential Geometry Aarhus Masterclass videos to the page.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2020

    I have added arXiv and webpage pointer to this item:

    On the webpage it says

    under contract to be published by Oxford University Press, hopefully to appear by the end of 2018

    But has it appeared yet?

    diff, v6, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2023

    have added pointer (here and elsewhere) to:

    diff, v7, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 31st 2023

    What do they mean right at the end (p.100) about the dynamics of M-theory? That so far work on Hypothesis H has concerned just the kinematics?

    There’s an old page kinematics and dynamics which probably needs some updating.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 31st 2023
    • (edited Jul 31st 2023)

    They mean the dynamics in the field of gravity (metric), which so far we have put aside.

    The issue is that addressed in Section 1.2 of Introduction to Hypothesis H:

    We decompose the full dynamics into a “pre-geometric part” that is independent of the dynamical metric and a self-duality constraint that couples everything to the field of gravity.

    I find it debatable to assert that the “pre-geometric part” is “non-dynamical” (it does satisfy ∼\sim half of the dynamical equations of motion!) but I can see why one might want to say so.

    In the comparable situation of Einstein-Maxwell theory, the analogous issue is the following:

    The full equations of motion of Einstein-Maxwell theory are:

    (0) dF=0,dG=Jd F = 0,\;\;\;\;d G = J

    (1) G=⋆FG = \star F

    (2) the Einstein-equation for the gravitational field with source FF

    Now one may take the point of view that the pre-geometric 0th equation of motion is “just kinematics”.

    The M-theoretic analog of that 0th equation is what Hypothesis H deals with, so far.

    We have indicated in various places what the full dynamics should be, under Hypothesis H, e.g. on slides 11-13 in Microscopic Brane Physics from Cohomotopy. With more man-hours in hands this could be and could have been developed, but for the time being this remains to be discussed. Luigi et al. is right that some higher derived geometry may be needed in fleshing this out.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 31st 2023

    OK, great, thanks!

1 to 7 of 7

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