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    • CommentRowNumber1.
    • CommentAuthordomenico_fiorenza
    • CommentTimeNov 24th 2010
    • (edited Nov 25th 2010)

    Hi all,

    sorry for having been absent lately, I’ve been fully absorbed by the preprint with Jim and Urs. now that’s over and my mind seems to be able again to follow nforum discussions :)

    I’ve been thinking of oo-Chern-Simons theory. At present we are presenting it in nLab as a morphism H(Σ,A conn)B ndimΣU(1)\mathbf{H}(\Sigma,A_{conn})\to \mathbf{B}^{n-dim \Sigma}U(1). This is fine but does not make explicit an important point: the relation to extended cobordism.

    Let me sketch it (in a simplified situation where I will not consider differential refinements). We have a cocycle c:AB nU(1)c:A\to \mathbf{B}^n U(1). if we now consider an nn-representation of B nU(1)\mathbf{B}^n U(1), e.g., the fundamental one, then we can see cc as the datum of an nn-vector bundle over AA. Now, it is likely that nnVect(A)(A) is a symmetric monoidal (,n)(\infty,n)-category, and it is hopeful that the nn-vector bundle corresponding to cc is a fuly dualizable object. So by the cobordism hypothesis we get a representation of Bord nBord_n with values in nnVect(A)(A). In particular to a closed connected nn-manifold Σ\Sigma it will correspond a 00-vector bundle over AA, i.e. a complex valued function on AA constant over the isomorphism classes of objects. Integrating this over AA produces the invariant associated to Σ\Sigma.

    We can associate a cobordism invariant to Σ\Sigma also in another way: first we push the given nn-vector bundle forward to the point, i.e. we take the nn-vector space of its sections. This is hopefully fully dualizable, so we have a representation Bord nnBord_n\to nVect. And so an invariant associated to Σ\Sigma. It is reasonable to expect that these two invariants are the same.

    There is one more point of view on this: namely, we can consider bordism with values in AA. if a representation Bord n(A)nBord_n(A)\to nVect is given, then to a morphism c:ΣAc:\Sigma\to A will correspond a (ndimΣ)(n-dim\Sigma)-vector space V cV_c. this gives a (ndimΣ)(n-dim\Sigma)-vector bundle over H(Σ,A)\mathbf{H}(\Sigma,A). if the nn-vector space V cV_c is the fundamental representation of B ndimΣU(1)\mathbf{B}^{n-dim\Sigma} U(1), then the (ndimΣ)(n-dim\Sigma)-vector bundle over H(Σ,A)\mathbf{H}(\Sigma,A) is induced by a principal B ndimΣU(1)\mathbf{B}^{n-dim\Sigma} U(1)-bundle, i.e., it corresponds to a morphism H(Σ,A)B ndimΣU(1)\mathbf{H}(\Sigma,A)\to \mathbf{B}^{n-dim\Sigma} U(1), which is where we started from.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2010
    • (edited Nov 26th 2010)

    Hi Domenico,

    thanks for starting/getting back to this discussion. With our the Friday seminar out of the way, I have now again some resources for \infty-Chern-Simons theory. Let me think a bit about what you just said (and catch my bus to catch my train), then I get back to you.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 26th 2010
    • (edited Nov 26th 2010)


    here are some general thoughts

    1. It is remarkable how the (infinity,n)-category of cobordisms is built by first building it simply as an nn-fold simplicial set and then applying a completion operation. This makes its nn-categorical nature rather tractable: nn-cells are simply little nn-cubes with an embedded manifold sitting inside, with boundary components sitting on the boundary of the cube. Composition is just the evident attaching of cubes. This is (intentionally) a slightlyover simplified description, but the point is that it is not very much oversimplified in fact. All the \infty-categorical subtlety is in completion the nn-fold simplicial space defined this way to an nn-fold complete Segal space.

      So I am thinking it might be useful to mimic this 2-step approach for defining our extended QFT: we should be able to get away with describing just how to propgate field in one direction along an nn-cube with escibed nn-manifold. Then we should get a morphism of nn-fold simplicial sets from that and just send it through the completion operation.

      (This is just a hunch for a strategy, not a detailed plan. I am just trying to see to which extent we can proceed by divide and conquer).

    2. We should see that we stick to general abstract mechanisms as much as possible. Experience shows that that’s a good thing. This makes me have the following attitude towards nn-vector spaces etc: there ought to be a nice general stract formulation of \infty-Chern-Simons theory along the lins of linear algebras as described at integral transforms on sheaves.

    3. The previous two points combined bring me back to a construction that we may have talked about before: looking at an nn-morphism in (,n)Cob(\infty,n)Cob just along one direction makes it look like a cospan

      Σ inΣΣ out \Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}

      (making all nn directions explicit would show that this is an nn-cube of cospans!)

      Homming this into our target space object A connA_{conn} produces a span

      [Σ in,A conn][Σ,A conn][Σ out,A conn] [\Sigma_{in}, A_{conn}] \leftarrow [\Sigma, A_{conn}] \to [\Sigma_{out}, A_{conn}]

      of spaces of field configurations of our theory. Given our \infty-Chern-Simons action functional [Σ,A conn]B ndimΣU(1)[\Sigma,A_{conn}] \to \mathbf{B}^{n-dim \Sigma } U(1), regarding it as a cocycle and passing to the \infty-bundles that it classifies gives a span

      E(Σ in)iE(Σ)oE(Σ out) E(\Sigma_{in}) \stackrel{i}{\to} E(\Sigma) \stackrel{o}{\to} E(\Sigma_{out})

      of bundles over the above span of spaces of field configurations.

      Now, these are principal bundles, not nn-vector bundles yet. But there is a way that allows us to think of an object over these bundles as presenting a section of an associated vector bundle. (I think we discussed this groupoid-cardinality approach before. Let me know if it is not clear what I am thinking of.)

      So let H\mathbf{H} be the ambient \infty-topos (if we think of the expressions [Σ,A][\Sigma,A] as internal homs, then this is still the original \infty-topos that we started in. If they are instead taken to be external homs, then this is now Grpd\infty Grpd. i am not sure yet what the right way to go is. But anyway.)

      Then the over-(infinity,1)-toposes H/[E(Σ in)]\mathbf{H}/[E(\Sigma_{in})] etc. would play the role of the nn-vector spaces of states over Σ in\Sigma_{in}, etc. The quantum propagation along our nn-morphism in the given direction should then be the integral transform

      o !i *:H/[E(Σ in)]H/[E(Σ out)] o_! i^* : \mathbf{H}/[E(\Sigma_{in})] \to \mathbf{H}/[E(\Sigma_{out})]

      That would give a fairly immediate description of our extended QFT along one of the nn directions. My hope would be that just doing this same process in an nn-fold iterated way gives a morphism of nn-fold simplicial sets, which under some completion then gives the desired (,n)(\infty,n)-functor.

    I need to think about this.

  1. Hi Urs,

    I very much agree with your over-toposes point of view. Yet I think it should not be push-pull, but push-tensor-pull.

    Concretely, consider a span Σ inΣΣ out \Sigma_{in} \to \Sigma \leftarrow \Sigma_{out} and the associated cospan [Σ in,A conn][Σ,A conn][Σ out,A conn][\Sigma_{in}, A_{conn}] \leftarrow [\Sigma, A_{conn}] \to [\Sigma_{out}, A_{conn}] . Then oo-Chern-Simons action gives us a cocycle [Σ in,A conn]B ndimΣ inU(1)[\Sigma_{in}, A_{conn}]\to \mathbf{B}^{n-dim\Sigma_{in}}U(1), which we can pull-back to a cocycle [Σ,A conn]B ndimΣ inU(1)[\Sigma, A_{conn}]\to \mathbf{B}^{n-dim\Sigma_{in}}U(1). This is not the oo-Chern-Simons cocycle on [Σ,A conn][\Sigma, A_{conn}] (just look at the degree of delooping on the right hand side). Rather the oo-Chern-Simons cocycle [Σ,A conn]B ndimΣU(1)[\Sigma, A_{conn}]\to \mathbf{B}^{n-dim\Sigma}U(1) acts on H([Σ,A conn],B ndimΣ inU(1))\mathbf{H}([\Sigma, A_{conn}],\mathbf{B}^{n-dim\Sigma_{in}}U(1)), since B kU(1)\mathbf{B}^k U(1) is the kk-groupoid of morphisms of B k+1U(1)\mathbf{B}^{k+1}U(1).

    So, after having pulled back our oo-Chern-Simons cocycle from [Σ in,A conn][\Sigma_{in}, A_{conn}] to [Σ,A conn][\Sigma, A_{conn}] we act on it with the oo-Chern-Simons cocycle on [Σ,A conn][\Sigma, A_{conn}], and then, finally, we push it forward to [Σ out,A conn][\Sigma_{out}, A_{conn}].

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2010
    • (edited Nov 27th 2010)

    Good point, Domenico.

    But this ought to be related to what I said: I was pull-pushing along the total spaces of the bundles of these cocycles. That mimics a pull-tensor push.

    I need to think about this, because the setup we are talking about right now is a tad more involved than the bare-bones setup described at integral transforms on sheaves. But there it is shown how in the bare-bones setup every pull-tensor-push is equivalent to a pull-push.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeNov 27th 2010
    • (edited Nov 27th 2010)

    Here is another observation, coming from the discussion on higher order Hochschild homology and its relation to QFT in the other thread:

    Let me decompose the \infty-Chern-Simons action functional H(Σ,A conn) ndimΣU(1)\mathbf{H}(\Sigma, A_{conn}) \to \mathcal{B}^{n- dim\Sigma} U(1) again into its steps, where it reads

    H(Σ,A conn)H(Σ,B nU(1) conn)H(Π(Σ),B nU(1))Grpd(Π(Σ), nU(1))τ ndimΣ ndimΣU(1). \mathbf{H}(\Sigma, A_{conn}) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)) \simeq \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \stackrel{\tau_{\leq n - dim\Sigma}}{\to} \mathcal{B}^{n - dim \Sigma} U(1) \,.

    Let me disregard the very last step for the moment, the one that decategories at level ndimΣn - dim \Sigma to get the actual action. I want to look here ar the intermediate stepbefore, where we have H(Π(Σ),B nU(1))\mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)). Let’s see what we get if we replace the external hom here with the inernal one. Recalling the notation Π=LConstΠ\mathbf{\Pi} = LConst \Pi this is

    [Π(Σ),B nU(1)]=[LConstΠ(Σ),B nU(1)]. [\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)] = [LConst \Pi(\Sigma), \mathbf{B}^n U(1)] \,.

    This is curious, because comparing with the discussion at Hochschild cohomology, we see that under taking functions 𝒪\mathcal{O}, this is the higher order Hochschild holomogy of 𝒪B nU(1)\mathcal{O} \mathbf{B}^n U(1) over Π(Σ)\Pi(\Sigma) (notably if Σ=S 1\Sigma = S^1, it is the ordinary Hochschild homology of that \infty-algebra).

    Notably, if we let Σ\Sigma vary here over subsets of a larger Σ^\hat \Sigma, then the assignment

    Σ𝒪[LConstΠ(Σ),B nU(1)] \Sigma \mapsto \mathcal{O} [LConst \Pi(\Sigma), \mathbf{B}^n U(1)]

    is what Ginot et al in the article linked to at the entry on Hochschild cohomology show to be a locally constant factorization system on Σ^\hat \Sigma.

    I haven’t thought this fully through. But I am beginning to think now that we should be able to unify the AQFT and the FQFT perspective on \infty-Chern-Simons theory along such lines.

    But I don’t understand yet the following step in this would-be story: in the external hom we have H(Σ,B nU(1))H(Π(Σ),B nU(1) conn)\mathbf{H}(\Sigma, \mathbf{B}^n U(1)) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)_{conn}) for dimensional reasons, because dimΣndim \Sigma \leq n. But this argument then fails in the internal hom, which is given by

    [Σ,B nU(1) conn]:UH(Σ×U,B nU(1) conn). [\Sigma, \mathbf{B}^n U(1)_{conn}] : U \mapsto \mathbf{H}(\Sigma \times U, \mathbf{B}^n U(1)_{conn}) \,.

    Not sure yet what that is telling us. But I thought I’d mention my thoughts anyway.

  2. Hi Urs,

    that’s a very good point!

    concerning the last truncation step, I must say that from the very beginning I had mixed feelings about it: on one side it reproduced neatly classical constructions, but on the other it was a truncation, so this suggested the “real thing” had to be the object before truncation, which is much more canonical. And now it seems we are beginning to see why.

    Sorry to be so short, I’m in a hurry. Won’t be back before tomorrow evening :(

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2010

    Yes, I was also wondering about this.

    The truncation step is of course the integration step of the local Lagrangian over the surface to an actual local action functional. It is very nice how this comes out, but possibly, as you said, we want to be careful with applying this too early on.

    • CommentRowNumber9.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 2nd 2010
    • (edited Dec 2nd 2010)

    Hi Urs,

    I was thinking about Thom work on cobordism. There, the module of nn-dimensional oriented cobordims is realized as the nn-th homotopy group of the Thom spectrum, i.e. as π n(MSO):=limπ n+i(MSO i)\pi_n(MSO):=\lim \pi_{n+i}(MSO_i). The cobordism ring is then (saying this in a very rought way) the collection of all these homotopy groups. But then this suggests that a natural point of view on the cobordism ring is in terms of the oo-Poincare’ groupoid of the Thom spectrum, Π(MSO)\Pi(MSO).

    It is not completely clear to me what kind of object the oo-Poincare’ groupoid of a spectrum should be, but I’m confident Π(MSO)\Pi(MSO) is the kind of object whose representations we are interested in when we consider a tqft.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2010
    • (edited Dec 2nd 2010)

    Hi Domenico,

    yes, I need to think about this. But here is a quick comment. You write:

    It is not completely clear to me what kind of object the oo-Poincare’ groupoid of a spectrum should be,

    Maybe the discussion in the other thread Homology from the nPOV is relevant:

    There the idea is that the generalized homology of a space XX with coefficients in a spectrum EE is ΠLConstE\Pi LConst E computed for the stabilized \infty-topos over XX.

    I am not sure if that really helps with your question, because over the point this amounts to saying that Π(E)=E\Pi(E) = E ! :-) But I mention it just in case that it makes you see more. I am, unfortunately, once again absorbed with preparing our friday seminar…

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2010

    We need to sort out what kind of quantum structure we can naturally obtain from the \infty-Chern-Simons Lagrangian

    BG connB nU(1) conn. \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} \,.

    There ought to be a factorization algebra which to Σ\Sigma assigns the collection of sections of the bundle over the space of fields H(Σ,BG conn)\mathbf{H}(\Sigma, \mathbf{B}G_{conn}) that is classified by the action functional H(Σ,BG conn)H(Σ,B nU(1) diff)H(Π(Σ),B nU(1))simeGrpd(Π(Σ), nU(1))\mathbf{H}(\Sigma, \mathbf{B}G_{conn}) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{diff}) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)) \sime \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)).

    Or maybe of the (ndimΣ)(n-dim \Sigma)-truncation of this, not sure.