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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 $\mathbf{H}(\Sigma,A_{conn})\to \mathbf{B}^{n-dim \Sigma}U(1)$. 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 $c:A\to \mathbf{B}^n U(1)$. if we now consider an $n$-representation of $\mathbf{B}^n U(1)$, e.g., the fundamental one, then we can see $c$ as the datum of an $n$-vector bundle over $A$. Now, it is likely that $n$Vect$(A)$ is a symmetric monoidal $(\infty,n)$-category, and it is hopeful that the $n$-vector bundle corresponding to $c$ is a fuly dualizable object. So by the cobordism hypothesis we get a representation of $Bord_n$ with values in $n$Vect$(A)$. In particular to a closed connected $n$-manifold $\Sigma$ it will correspond a $0$-vector bundle over $A$, i.e. a complex valued function on $A$ constant over the isomorphism classes of objects. Integrating this over $A$ produces the invariant associated to $\Sigma$.
We can associate a cobordism invariant to $\Sigma$ also in another way: first we push the given $n$-vector bundle forward to the point, i.e. we take the $n$-vector space of its sections. This is hopefully fully dualizable, so we have a representation $Bord_n\to n$Vect. And so an invariant associated to $\Sigma$. 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 $A$. if a representation $Bord_n(A)\to n$Vect is given, then to a morphism $c:\Sigma\to A$ will correspond a $(n-dim\Sigma)$-vector space $V_c$. this gives a $(n-dim\Sigma)$-vector bundle over $\mathbf{H}(\Sigma,A)$. if the $n$-vector space $V_c$ is the fundamental representation of $\mathbf{B}^{n-dim\Sigma} U(1)$, then the $(n-dim\Sigma)$-vector bundle over $\mathbf{H}(\Sigma,A)$ is induced by a principal $\mathbf{B}^{n-dim\Sigma} U(1)$-bundle, i.e., it corresponds to a morphism $\mathbf{H}(\Sigma,A)\to \mathbf{B}^{n-dim\Sigma} U(1)$, which is where we started from.
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.
Domenico,
here are some general thoughts
It is remarkable how the (infinity,n)-category of cobordisms is built by first building it simply as an $n$-fold simplicial set and then applying a completion operation. This makes its $n$-categorical nature rather tractable: $n$-cells are simply little $n$-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 $n$-fold simplicial space defined this way to an $n$-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 $n$-cube with escibed $n$-manifold. Then we should get a morphism of $n$-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).
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 $n$-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.
The previous two points combined bring me back to a construction that we may have talked about before: looking at an $n$-morphism in $(\infty,n)Cob$ just along one direction makes it look like a cospan
$\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}$(making all $n$ directions explicit would show that this is an $n$-cube of cospans!)
Homming this into our target space object $A_{conn}$ produces a span
$[\Sigma_{in}, A_{conn}] \leftarrow [\Sigma, A_{conn}] \to [\Sigma_{out}, A_{conn}]$of spaces of field configurations of our theory. Given our $\infty$-Chern-Simons action functional $[\Sigma,A_{conn}] \to \mathbf{B}^{n-dim \Sigma } U(1)$, regarding it as a cocycle and passing to the $\infty$-bundles that it classifies gives a span
$E(\Sigma_{in}) \stackrel{i}{\to} E(\Sigma) \stackrel{o}{\to} E(\Sigma_{out})$of bundles over the above span of spaces of field configurations.
Now, these are principal bundles, not $n$-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 $\mathbf{H}$ be the ambient $\infty$-topos (if we think of the expressions $[\Sigma,A]$ 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 $\infty Grpd$. i am not sure yet what the right way to go is. But anyway.)
Then the over-(infinity,1)-toposes $\mathbf{H}/[E(\Sigma_{in})]$ etc. would play the role of the $n$-vector spaces of states over $\Sigma_{in}$, etc. The quantum propagation along our $n$-morphism in the given direction should then be the integral transform
$o_! i^* : \mathbf{H}/[E(\Sigma_{in})] \to \mathbf{H}/[E(\Sigma_{out})]$That would give a fairly immediate description of our extended QFT along one of the $n$ directions. My hope would be that just doing this same process in an $n$-fold iterated way gives a morphism of $n$-fold simplicial sets, which under some completion then gives the desired $(\infty,n)$-functor.
I need to think about this.
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 $\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out}$ and the associated cospan $[\Sigma_{in}, A_{conn}] \leftarrow [\Sigma, A_{conn}] \to [\Sigma_{out}, A_{conn}]$. Then oo-Chern-Simons action gives us a cocycle $[\Sigma_{in}, A_{conn}]\to \mathbf{B}^{n-dim\Sigma_{in}}U(1)$, which we can pull-back to a cocycle $[\Sigma, A_{conn}]\to \mathbf{B}^{n-dim\Sigma_{in}}U(1)$. This is not the oo-Chern-Simons cocycle on $[\Sigma, A_{conn}]$ (just look at the degree of delooping on the right hand side). Rather the oo-Chern-Simons cocycle $[\Sigma, A_{conn}]\to \mathbf{B}^{n-dim\Sigma}U(1)$ acts on $\mathbf{H}([\Sigma, A_{conn}],\mathbf{B}^{n-dim\Sigma_{in}}U(1))$, since $\mathbf{B}^k U(1)$ is the $k$-groupoid of morphisms of $\mathbf{B}^{k+1}U(1)$.
So, after having pulled back our oo-Chern-Simons cocycle from $[\Sigma_{in}, A_{conn}]$ to $[\Sigma, A_{conn}]$ we act on it with the oo-Chern-Simons cocycle on $[\Sigma, A_{conn}]$, and then, finally, we push it forward to $[\Sigma_{out}, A_{conn}]$.
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.
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 $\mathbf{H}(\Sigma, A_{conn}) \to \mathcal{B}^{n- dim\Sigma} U(1)$ again into its steps, where it reads
$\mathbf{H}(\Sigma, A_{conn}) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{conn}) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)) \simeq \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \stackrel{\tau_{\leq n - dim\Sigma}}{\to} \mathcal{B}^{n - dim \Sigma} U(1) \,.$Let me disregard the very last step for the moment, the one that decategories at level $n - dim \Sigma$ to get the actual action. I want to look here ar the intermediate stepbefore, where we have $\mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1))$. Let’s see what we get if we replace the external hom here with the inernal one. Recalling the notation $\mathbf{\Pi} = LConst \Pi$ this is
$[\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)] = [LConst \Pi(\Sigma), \mathbf{B}^n U(1)] \,.$This is curious, because comparing with the discussion at Hochschild cohomology, we see that under taking functions $\mathcal{O}$, this is the higher order Hochschild holomogy of $\mathcal{O} \mathbf{B}^n U(1)$ over $\Pi(\Sigma)$ (notably if $\Sigma = S^1$, it is the ordinary Hochschild homology of that $\infty$-algebra).
Notably, if we let $\Sigma$ vary here over subsets of a larger $\hat \Sigma$, then the assignment
$\Sigma \mapsto \mathcal{O} [LConst \Pi(\Sigma), \mathbf{B}^n U(1)]$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 $\hat \Sigma$.
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 $\mathbf{H}(\Sigma, \mathbf{B}^n U(1)) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)_{conn})$ for dimensional reasons, because $dim \Sigma \leq n$. But this argument then fails in the internal hom, which is given by
$[\Sigma, \mathbf{B}^n U(1)_{conn}] : U \mapsto \mathbf{H}(\Sigma \times U, \mathbf{B}^n U(1)_{conn}) \,.$Not sure yet what that is telling us. But I thought I’d mention my thoughts anyway.
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 :(
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.
Hi Urs,
I was thinking about Thom work on cobordism. There, the module of $n$-dimensional oriented cobordims is realized as the $n$-th homotopy group of the Thom spectrum, i.e. as $\pi_n(MSO):=\lim \pi_{n+i}(MSO_i)$. 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, $\Pi(MSO)$.
It is not completely clear to me what kind of object the oo-Poincare’ groupoid of a spectrum should be, but I’m confident $\Pi(MSO)$ is the kind of object whose representations we are interested in when we consider a tqft.
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 $X$ with coefficients in a spectrum $E$ is $\Pi LConst E$ computed for the stabilized $\infty$-topos over $X$.
I am not sure if that really helps with your question, because over the point this amounts to saying that $\Pi(E) = E$ ! :-) But I mention it just in case that it makes you see more. I am, unfortunately, once again absorbed with preparing our friday seminar…
We need to sort out what kind of quantum structure we can naturally obtain from the $\infty$-Chern-Simons Lagrangian
$\mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn} \,.$There ought to be a factorization algebra which to $\Sigma$ assigns the collection of sections of the bundle over the space of fields $\mathbf{H}(\Sigma, \mathbf{B}G_{conn})$ that is classified by the action functional $\mathbf{H}(\Sigma, \mathbf{B}G_{conn}) \to \mathbf{H}(\Sigma, \mathbf{B}^n U(1)_{diff}) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), \mathbf{B}^n U(1)) \sime \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1))$.
Or maybe of the $(n-dim \Sigma)$-truncation of this, not sure.
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