The n-Category Café

Wow, it’s really coming together at last, isn’t it?

I guess it is.

There are a bunch of further puzzle pieces which I think I have an idea how to fill in, but which need more time to work out in careful detail.

Let me use this opportunity to list a few, in case anyone feels like looking into them:

- The Lie version of Willerton’s interpretation of Freed-Hopkins-Teleman. Simon Willerton explained that a very nice point of view of FHT is obtained in the toy version of finite groups $G$ by (slightly, but not much, rephrased by me here) understanding it as being about the space of sections of the Chern-Simons 3-bundle over the groupoid $\mathbf{B} G$ transgressed to the groupoid $Funct(\mathbf{B}\mathbb{Z}, \mathbf{B} G)$.

I think this is a very good point of view. It’s precisely the statement that $n$-transport is quantized to quantum $n$-transport by transgressing to cobordisms and taking sections there.

And I think it does generalize to the full Lie version involving Kac-Moody loop groups proper.

There are two ways to do this: using Lie $n$-groups or Lie $n$-algebras.

Using Lie $n$-groups, I discussed in The 2-monoid of observables on $String(G)$ (pdf, blog) how $$\mathrm{Funct}(\mathbf{B}\mathbb{Z}, \mathbf{B} String_G)$$ yields the groupoid whose reps are twisted equivariant vector bundles on $G$.

The corresponding Lie $\infty$-algebraic description I began describing in loop Lie algebroids (pdf (section 2.3), blog).

The induced description of the fusion product I indicated in Fusion and String Field Star Product (blog).

All this is directly relevant for tackling Chern-Simons and WZW theory. But the underlying mechanism is much more general. This is really all about understanding the kinematics of the quantization functor $quantize : classical n-transport \to quantom$n$-transport$, namely understanding what it does on $d \lt n$-morphisms.

-the relation between $n$-functorial QFT and AQFT There exist two different axiom systems for defining what quantum field theory actually is:

a) transport $n$-functors on cobordism $n$-categories

b) cosheaves of local algebras.

I think both points of viewes are two sides of one single thing and it would be very useful to relate both approaches more closely

In Local Nets from 2-Transport (pdf, blog) I describe what I think is going on: AQFT is the image of quantum $n$-transport under postcomposition with a functor that forms endomorphism algebras:

I wish I’d find more time pushing this further, since there is a wealth of insights bound to be hidden here. I made some remarks about various relations here and at other places.

- deriving the BV-master equation and the perturbative BV path integral from $n$-transport – there are indications that BV-formalism is secretly the Lie $\infty$-algebra version of a theory of canonical measures (I, II, III) (“integrals without integrals”) on $\infty$-groupoids. Various aspects of BRST-BV quantization I think I can already interpret and derive that way (I, II, III,IV), but I am still not sure about the path integral itself (but that’s not too surprising…)

- integration of Lie $\infty$-algebras using fundamental $\infty$-groupoids of generalized smooth spaces and the tautologification of smooth $n$-transport – I am thinking that the existing presciptions for integrating Lie algebroids (Weinstein, et al) and $L_\infty$-algebra (Getzler, Henriques) are best thought of as forming the fundamental $\infty$-groupoids $$\mathbf{B} G = \mathbf{B} exp(g) := \Pi_\infty(X_{CE(g)})$$ of generalized smooth spaces $X_{CE(g)}$. I describe this in Differential forms and smooth spaces.

To the extent that this is right and works, it would be just a repackaging of what’s already being done. But it should be helpful for proving the

obvious central conjecture about $n$-transport: the $\infty$-category of smooth $\infty$-functors from $\infty$-paths in a smooth space $X$ to smooth $\infty$-groups $G = exp(g)$ is equivalent to the corresoponding $\infty$-groupoid of $g$-valued differential forms ($g$ an $L_\infty$-algebroid) on $X$.

It feels like this should collaps to a big general abstract nonsense tautology after realizing that we are talking about smooth morphisms $$\Pi_\infty(X) \to \Pi_\infty(X_{CE(g)}) \,.$$ But I am really not sure yet about this.

But if we could do this, it should help to take all of Lie $\infty$-connections and integrate it up systematically.