## July 14, 2009

### Ben Zvi’s Lectures on Topological Field Theory II

#### Posted by John Baez

guest post by Orit Davidovich and Alex Hoffnung

Hi again! The following is a second set of notes which largely follows the second of David Ben-Zvi’s talks at a workshop on topological field theories, held at Northwestern University in May 2009. This post follows our previous post found here. We’ll again give a brief introduction, and then send you over to a PDF file for the full set of notes.

The first part of this lecture by David Ben-Zvi was dedicated to a discussion of the character and action maps that relate the open and closed sectors of our topological field theory. Attention is restricted to the case of representations of the form $A=\mathbb{C}[G/K]$ where $K\subset G$. In what follows ‘char’ and ‘act’ are defined via a pull-back and push-forward construction.

The second part of this lecture was dedicated to a discussion of topological field theories arising from gauge theory with symmetry group $G$, where $G$ is a complex reductive group. The first problem one encounters is that moduli spaces of fields on space-time manifolds, i.e., $G$-bundles with connections, are no longer discrete and do not allow for the same counting invariants that could be defined when $G$ was finite. This problem is solved by going up one dimension, namely, constructing $3$-dimensional TFTs.

One still encounters the issue of defining an appropriate theory of functions on moduli spaces of fields which are generally stacks. Following David, we discuss two natural categorified theories of ‘functions’. One is quasi-coherent sheaves and the other is $\mathcal{D}$-modules on $X$. Some of their properties that are essential for constructing the $3$d TFT are discussed.

Posted at July 14, 2009 6:00 PM UTC

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### Re: Ben Zvi’s Lectures on Topological Field Theory II

Posted by: Toby Bartels on July 15, 2009 12:44 AM | Permalink | Reply to this

### Re: Ben Zvi’s Lectures on Topological Field Theory II

Thanks for catching that Toby.

Posted by: Alex Hoffnung on July 15, 2009 7:10 AM | Permalink | Reply to this

### Re: Ben Zvi’s Lectures on Topological Field Theory II

Fixed.

Posted by: John Baez on July 15, 2009 8:49 AM | Permalink | Reply to this

### Re: Ben Zvi’s Lectures on Topological Field Theory II

I am stil trying to better understand how much of geometric $\infty$-function theory can be done in an arbitrary $(\infty,1)$-topos $\mathbf{H}$, or else what the necessary conditions on $\mathbf{H}$ are such that it does support a theory of geometric $\infty$-functions.

Here is a thought. I’ll formulate it first in the smooth context of the $(\infty,1)$-topos of smooth $\infinity$-stacks, but the concept won’t really depend on much structure in there and hence make sense more generally.

We know that D-modules, treated in the lecture notes on David Ben-Zvi’s talk above, are the same as representations of the tangent Lie algebroid #.

A representation on some complex $E$ of the tangent Lie algebroid $T X$ I like to think of as fibration sequence

$E \to E//T X \to T X$

in some suitable context of $L_\infty$-algebroids, being the differential analog of the action groupoid fibration

$V \to V//G \to \mathbf{B} G$

of the action of a group $G$.

Dually, when encodind the $L_\infty$-algebroids here by their (“Chevalley-Eilenberg”) function algebras, this should be encoded in a cofibration sequence

$\wedge^\bullet E \leftarrow CE(E//T X) \leftarrow \Omega^\bullet(X)$

Now, in the context of my recent thoughts at Question on synthetic differential forms this makes me want to consider the following:

let $X$ be a smooth $\infty$-stack. Let $C^\infty_{loc}(X)$ be its cosimplicial algebra of local smooth functions as described at $(\infty,1)$-quantity. We know (up to a sign for the monoidal structure that still needs to be checked) that under the dual Dold-Kan correspondence $C^\infty_{loc}(\Pi(X))$ is the deRham differential graded algebra $\Omega^\bullet(X)$ on $X$.

So that should mean that we can say D-module here by looking at cofiber sequences under $C^\infty_{loc}(\Pi(X))$

$E \leftarrow E//\Pi_{inf}(X) \leftarrow C^\infty(\Pi(X))$

in the $(\infty,1)$-category of cosimplicial smooth function algebras. (Here the object on the right is the one from above, the rest is some new notation meant only to indicate the pieces in a cofiber sequence).

For each $X$ these should form a $(\infty,1)$-category $Ext(C^\infty(\Pi(X)))$ and that should supposedly play a role similar to the $(\infty,1)$-category $QC(X)$ of quasi-coherent sheaves or other geometric $\infty$-functions.

All one would need to set this up is

a) a site $C$ with products and equipped with a cosimplicial object

$\Delta_C : \Delta \to C$

such that $\Delta^0_C = {*}$. This realizes the standard $k$-simplices in $C$ and thus induces a notion of path $\infty$-groupoid

$\Pi : C \to Sh_\infty(C)$

$Pi(X) := hocolim_k RHom(\Delta^k_C,X) \,.$

The other thing needed is that the cartesian spaces $\mathbb{R}^k$ required to say what the co-presheaves $C^\infty(C)$ are co-tested on are naturally represented in $Sh_\infty(C)$.

Posted by: Urs Schreiber on August 15, 2009 7:26 PM | Permalink | Reply to this

### Re: Ben Zvi’s Lectures on Topological Field Theory II

Urs - that’s very interesting. A couple of comments. First, in the fiber sequence for a TX action, was the base supposed to be B(TX), the classifying space of the tangent algebroid (which is the quotient of X by the formal neighborhood of the diagonal) rather than TX?

Second, as you imply, you can define D-modules in any context where you have a notion of tangent complex, and a notion of stable sheaf (eg stabilization of stacks). I know tangent complexes can be defined in huge generality (see eg Toen-Vezzosi’s HAG2), eg when you start with a category of commutative ring objects in some symmetric monoidal ∞-category. I would assume you have tangent complexes in your smooth context as well? or equivalently, that you can talk about formal neighborhoods of the diagonal, or about de Rham complexes? (Sorry, I assume this is what you’re explaining here, but I didn’t follow the details)

Third, what you end up defining sounds to me (probably naively) like representations of the path groupoid, which are not what I would think of as analogs of D-modules — representations of the path (oo)-groupoid are local systems, which are very special D-modules. D-modules can have interesting singularities, ie are not locally constant - their topological avatar are constructible sheaves. Constructible sheaves with respect to a fixed stratification are representations of the sub(oo-)groupoid of the path groupoid that preserves this stratification (and there are various refinements you are aware of in papers of Treumann and Wise etc). In any case I don’t think the path groupoid is the groupoid formally integrating the tangent algebroid (ie for which modules are the same as those for the tangent algebroid) - the path groupoid is the analog of a Lie group with Lie algebra TX, but Lie algebra modules are equivalent to representations of the formal group, not the (a) corresponding group, unless you impose strong finiteness (which kills the singularities in D-modules).. Well anyway it’s possible for your purposes local systems suffice? they do in the homotopy category of spaces (but not in the category of smooth manifolds).

Posted by: David Ben-Zvi on August 16, 2009 12:15 AM | Permalink | Reply to this

### Re: Ben Zvi’s Lectures on Topological Field Theory II

I would assume you have tangent complexes in your smooth context as well?

Okay, so that was the crux of the question here. Let me try to say this more in detail:

For $X$ a manifold let $\Pi(X)$ be its smooth path $\infty$-groupoid regarded as a simplicial sheaf.

Let $C^\infty(-)$ be the map that reads in a sheaf on $Diff$ and spits out a product-preserving copresheaf on cartesian spaces, i.e. a Moerdijk-Reyes smooth algebra.

Then applying $C^\infty(-)$ degreewise yields a cosimplicial copresheaf

$C^\infty(\Pi(X)) \,.$

Under the dual Dold-Kan correspondence this maps to the complex that computes smooth singular cohomology of $X$.

Since $\Pi(X)$ is in fact a simplicial concrete sheaf, it makes sense to speak of open neighbourhoods of degenarate elements and we may also consider the analogous construction but with all functions replaced by just local functions (germs at totally degenerate cells). That yields the cosimplicial copresheaf

$C^\infty_{loc}(\Pi(X))$

that I mentioned. Under dual Dold-Kan both yield cochain complexes that should be equivalent to the deRham complex of $X$.

Analogously, for $G$ a Lie group and $\mathbf{B} G$ the corresponding simplicial sheaf, we get that the cosimplicial copresheaf

$C^\infty_{loc}(\mathbf{B}G)$

is the one that computes local Lie group cohomology and is hence, under dual Dold-Kan, weakly equivalent to the Chevalley-Eilenberg complex $CE(Lie(G))$.

So based on this what I was suggesting is this:

to any simplicial sheaf $A$ assign the cosimplicial copresheaf $C^\infty_{loc}(A)$ of local Moerdijk-Reyes function algebras in the above sense.

That should be a model for the $L_\infty$-algebroid corresponding to $A$.

Now, by the properties of Moerdijk-Reyes algebras it should be true that this assignment $C^\infty_{loc}(-)$ sends pullbacks to pushouts.

$C^\infty(X_1 \times_Y X_2) \simeq C^\infty(X_1) \otimes_{C^\infty(Y)} C^\infty(X_2) \,.$

Given this I suggested to view the category of modules for $\Omega^\bullet(X)$ as the under-category of cosimplicial Moerdijk-Reyes algebras under $C^\infty_{loc}(X)$.

Generally, this might suggest to assign to any simplicial sheaf $A$ the under-category $Q(A) := C^\infty_{loc}(A)/CoSCoSh$.

I am playing with the thought that this $A \mapsto Q(A)$ produces a good general notion of “geometric $\infty$-functions”.

Posted by: Urs Schreiber on August 16, 2009 9:42 PM | Permalink | Reply to this

### toy and other examples for geometric functions

Following my desire to exhibit the bare-bones strructure necessary for a “geometric $\infty$-function theory” I thought it might be useful to create a page on which to start constructing and listing examples of structures that are or come usefully close to being “geometric $\infty$-functions”.

I didn’t get as far as I intended to. But it is late at night here by now and I need to call it quits. Blame all nonsense in the following on the late hour.

But as a start the entry

examples for geometric function objects

now contains a detailed discussion of the simple, somewhat tautological and a bit degenerate looking but actually already interesting case of “geometric functions” modeled simply by over $\infty$-catgeories.

This is – somewhat implicitly – the model of geometric $\infty$-functions that underlies John Baez’s notion of groupoidification. I thought maybe some of those who followed that and our journal club of geometric $\infty$-functions might find it useful to make that kind of exercise explicit.

My main motivation for exhibiting it in such detail was the observation that by just applying Isbell-duality to this one obtains something that already looks like it has considerably more structure (and more of the kind expected), but whose basic properties still follow directly by duality from those of the above examples. I have briefly indicated that at the entry, too, but that’s it for today. More tomorrow.

All comments are welcome. I am thinking that maybe we could even reactivate (or activate in the first place…) our “journal club” interaction activity a bit. Anyone still interested in chatting about this?

Posted by: Urs Schreiber on August 18, 2009 12:39 AM | Permalink | Reply to this

### Re: toy and other examples for geometric functions

There seems to be an interesting variant on that “bare-bones geometric function theory” given by over-categories

$C(A) := \mathbf{H}/A$

as described here:

this assignment yields functorial pull-push through spans. But not through spans-of-spans. But, unless I am mixed up, we do get the higher degree functorial pull-push if instead of the plain over-category $\mathbf{H}/A$ we take spans in that:

$C : A \mapsto Spans(\mathbf{H}/A) \,.$

Here one needs to say what “Spans(\mathbf{H}/A)” means for $A$ something an $\infty$-groupoid (in the case that $\mathbf{H} = \infty Grpd$). Of course Jacob Lurie did that in OTCTFT, around page 57, where it’s called $Fam_1(A)$.

I am thinking that the assignment

$C : A \mapsto Fam_1(A)$

with $Fam_1$ not just necessarily interpreted in $\mathbf{H} = \infty Grpd$ should yield a “geometric function theory” in the more-groupoidification-like sense of the discussion here which is not only functorial with respect to pull-push along spans, but also along spans-of-spans.

(Draw the obvious diagram on paper to see what I am thinking of, or else wait until I post a graphic tomorrow).

More generally then it should be true that the assignment

$C : A \mapsto Fam_n(A)$

yields something for which pull-push extends to an $(n+1)$-functor on $(n+1)$-fold spans-of-spans.

If this is correct and I am not mixed up, that should allow to extend the groupoidification/”geometric $\infty$-function theory” pull-push propagation to higher dimensions.

Of course such a higher dimensional abstract-nonsense-path-integral thing is precisely what Freed-Hopkins-Lurie-Teleman hint at in their latest, and using precisely these $Fam_n$-constructions. So I have to emphasize that what I just said, in the case anyone actually read it, means a use of the notion of the $Fam_n(-)$ construction very different in flavor to that in FHLT, even if motivated by the same goal.

Makes me wonder. But it’s too late today, I’ll continue wondering tomorrow morning.

Posted by: Urs Schreiber on August 18, 2009 10:09 PM | Permalink | Reply to this
Read the post Ben-Zvi's Lectures on Topological Field Theory III
Weblog: The n-Category Café
Excerpt: This is the third set of notes on David Ben-Zvi's lectures on topological field theory at Northwestern University.
Tracked: September 1, 2009 7:26 AM

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