Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2012
    • (edited Sep 7th 2012)

    I would like to understand the following question in an \infty-topos / in homotopy type theory:

    Given an object VV with an action by a group GG, and given another object XX. How can we naturally construct [V,X]G[V,X]\sslash G, the quotient of the space of maps VGV \to G by GG acting by precomposition?

    This seems basic, but I am being a bit dense, maybe.

    I am suspecting that this object is BG[VG,X×BG] BG\sum_{\mathbf{B}G}[V\sslash G, X \times \mathbf{B}G]_{\mathbf{B}G}. At least this exbits some GG-action, so if it’s not the one I am after then a second question I have: which GG action is this?

    More in detail:

    The GG-action on VV is exhibited by a fiber sequence

    V VG BG. \array{ V &\to& V \sslash G \\ && \downarrow \\ && \mathbf{B}G } \,.

    Similarly we get the fiber sequence

    Q x:BG[VG,X×BG] /BG BG \array{ Q &\to& \sum_{x : \mathbf{B}G} [V \sslash G , X \times \mathbf{B}G ]_{/\mathbf{B}G} \\ && \downarrow \\ && \mathbf{B}G }

    where QQ is the stand-in notation for whatever the fiber of the morphism on the right is. This exhibits an action of GG on Q.

    What is QQ?

    So I can say what Q\flat Q is, the discrete type of global points of QQ, by this line of reasoning:

    Q [pt BG,[VG,X×BG] BG] /BG [pt BG× BGVG,X×BG] BG [V,X×BG] BG [V,X]. \begin{aligned} \flat Q & \simeq \flat [ pt_{\mathbf{B}G} , [V\sslash G, X \times \mathbf{B}G]_{\mathbf{B}G} ]_{/\mathbf{B}G} \\ & \simeq \flat [ pt_{\mathbf{B}G} \times_{\mathbf{B}G} V\sslash G, X \times \mathbf{B}G ]_{\mathbf{B}G} \\ & \simeq \flat [ V , X \times \mathbf{B}G ]_{\mathbf{B}G} \\ & \simeq \flat [V,X] \end{aligned} \,.

    That matches my initial guess that Q[V,X]Q \simeq [V,X].

    I am currently hesitant as to how to proceed in the first step above without the \flats. It seems clear, but I need to think about it more.

    But so it looks like this gives a canonical construction of a GG-action on [V,X][V,X]. Can we also check that indeed it is the expected action by pre-composition with the GG-action on VV?

    I thought of this consistency check here:

    For a GG-action on VV, its invariants are Γ BG(VG)\Gamma_{\mathbf{B}G}(V\sslash G). So maybe I can check what Γ BG( BG[VG,X×BG] BG)\Gamma_{\mathbf{B}G}( \sum_{\mathbf{B}G} [V\sslash G, X \times \mathbf{B}G]_{\mathbf{B}G} ) is. I compute:

    Γ BG( BG[VG,X×BG] BG) [BG,[VG,X×BG] BG] BG [BG× BGVG,X×BG] BG [VG,X×BG] BG [VG,X] \begin{aligned} \Gamma_{\mathbf{B}G}( \sum_{\mathbf{B}G} [V\sslash G, X \times \mathbf{B}G]_{\mathbf{B}G} ) & \simeq \flat [ \mathbf{B}G , [V\sslash G, X \times \mathbf{B}G]_{\mathbf{B}G}]_{\mathbf{B}G} \\ & \simeq \flat [ \mathbf{B}G \times_{\mathbf{B}G} V \sslash G, X \times \mathbf{B}G]_{\mathbf{B}G} \\ & \simeq \flat [ V \sslash G , X \times \mathbf{B}G]_{\mathbf{B}G} \\ & \simeq \flat[ V \sslash G, X ] \end{aligned}

    which is the right answer, I suppose, the maps VXV \to X that are invariant under precomposing the GG-action on VV.

    That’s how far I got for the moment. Will further mull over this. Beware that there might be major blunders in the above, these thoughts are only about as old as it took to type this post.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2012

    Can’t you just use the fact that fibers preserve exponentials? I assume that by [VG,X×BG] BG[V\sslash G, X\times B G]_{B G} you mean the exponential in the slice H/BG\mathbf{H}/B G, so that then BG\sum_{B G} denotes taking the total space. Since taking fibers preserves this exponential, the fiber of this space over pt:BGpt:B G is [V,X][V,X], since VV is the fiber of VGV\sslash G and XX is the fiber of X×BGX\times B G.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2012
    • (edited Sep 7th 2012)

    Ah, thanks! I wasn’t actively aware of this statement that taking fibers preserves the exponential.

    Phew, that’s awesome.

    You know, I found this statement rather curious, that homming out of VGV\sslash G in the slice gives one [V,X]G[V,X]\sslash G. It’s remarkable, because the naive hom out of it gives just the invariants, wich is entirely unlike [V,X]G[V,X]\sslash G So then it took me a bit to realize that it is the right answer if regarded as the global sections over BG\mathbf{B}G. It’s kind of magic that it all works out so nicely in the slice.

    By the way, this is important for doing not just gauge theory, but gravitational gauge theory in cohesive homotopy type theory. There the configuration type (the “integrated off-shell BRST complex”) is not [Σ,BG conn][\Sigma, \mathbf{B}G_{conn}], as for GG-Yang-Mills theory, but is [Σ,BG conn]Aut(Σ)[\Sigma, \mathbf{B}G_{conn}]\sslash\mathbf{Aut}(\Sigma).

    Next I want to understand “multisymplectic BRST” in cohesive HoTT. For that I need to understand [Σ,BG conn]Aut(Σ)[\Sigma, \mathbf{B}G_{conn}]\sslash\mathbf{Aut}(\Sigma) as the space of sections of a bundle over Σ\Sigma.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2012
    • (edited Sep 7th 2012)

    A comment on that aspect of “multisymplectic BRST”, just for the record, on the off-chance that anyone reading this here cares about it:

    So from the above I am thinking: for Σ\Sigma a worldvolume and XX a target space, the “integrated extended BRST phase space” of the relativistic Σ\Sigma-model on XX seems to want to be the dual jet bundle of

    ΣAut(Σ)× BAut(Σ)p *X ΣBAut(Σ) \array{ \Sigma \sslash \mathbf{Aut}(\Sigma) \times_{\mathbf{B} \mathbf{Aut}(\Sigma)} p^* X \\ \downarrow \\ \Sigma \sslash \mathbf{B} \mathbf{Aut}(\Sigma) }

    in H /BAut(Σ)\mathbf{H}_{/\mathbf{B}\mathbf{Aut}(\Sigma)}, where p:H /BAut(Σ)Hp : \mathbf{H}_{/\mathbf{B}\mathbf{Aut}(\Sigma)} \to \mathbf{H} is the étale geometric morphism.

    The graded coordinates on the Aut(Σ)\sslash \mathbf{Aut}(\Sigma)-piece would be the diffeomorphism ghosts in the exended multisymplectic picture. Passing to the space of global sections would give the ordinary (unextended) off-shell BRST phase space .

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeSep 7th 2012

    It’s kind of magic that it all works out so nicely in the slice.

    Yes, I agree!

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2012

    I have started making a brief note on this at infinity-action.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2012

    I have added a bit more to infinity-action meant to serve as exposition of the cartesian closed structure on GG-\infty-actions:

    • brief remark on the trivial action and inverse images of étale geometric morphisms, just so that we can point to it;

    • elementary discussion of cartesian closure of discrete 0-group \infty-actions aka GG-sets in Conjugation actions (for completeness)

    • brief remark on the above-mentioned case, which I find deserves to be named General covariance

    I had wanted to expand on that. But those Zombie-processes kept me from doing more.