$\newcommand{\g}{\mathfrak g}$ Let $G$ be a reductive group and $U_q(\g)$ the associated quantum group. One can argue that the classical limit of $U_q(\g)$ is $G$ or $\g$, with some Poisson structure, but here's a picture I like better about the category of modules over those:

The basic observation is that a $G$-module is the same as a $G$-equivariant quasi-coherent sheaf on a point, i.e. a quasi-coherent sheaf on the quotient stack $BG=pt/G$. This stack has a 2-shifted symplectic structure, and one can define a Poisson sigma model which to a topological surface $S$ attaches $$QC(Map(S,BG)).$$ So this thing is a categorified 2-dimensional topological field theory, or equivalently the 2-dimensional part of a would be 4-dimensional field theory. Note that $Map(S,BG)$ is identified with the moduli stack of $G$-local systems on $S$ so this thing really is the classical limit of Chern-Simons theory. This is in accordance with the old idea that Chern-Simons/Reshetikhin-Turaev really is secretly a 4, not 3, dimensional TFT.

Hence it is expected that Khovanov homology and its generalizations should come from a partially defined 5-dimensional TFT, which mathematically should be computed by a still conjectural braided monoidal 2-category structure on categorified quantum groups. Witten has a lot of ideas about the physical side of this story, which I honestly don't understand.

So I'm wondering if at least the classical part of this theory has a nice description similar to he one above, i.e. a nice geometrical construction from $G$ of a 2-dimensional TFT valued in (dg) 2-categories.

Edit: To be a bit more precise, I guess what i'm really looking for is some natural way of associating some geometric object $Z(M)$ to a topological 3-manifold $M$ in such a way that for a surface $\Sigma$, $$Z(S^1\times \Sigma)=Map(\Sigma,BG).$$ Then $QC(Z(\mathbb{R}\times \Sigma))$ will be a monoidal category, i.e. a 2 category, and $Z(\mathbb{R^3})$ should then automatically be a braided (in fact symmetric, probably) monoidal 2-category categorifying $G$-mod.

I feel like such a $Z$ should exist and be defined purely in terms of $G$ from some physical considerations, i.e. without assuming a priori the existence of a categorifcation of $G$ but rather with the aim of producing such a categorification from "first principles".

  • 2
    $\begingroup$ Two comments that might be obvious. 1. My understanding is that categorified quantum groups don't have a quantum parameter, so there's no way to talk about a classical limit. Namely, $\mathbf{Z}[q, q^{-1}]$ gets categorified to graded vector spaces. 2. In Witten's approach one of the key steps is S-duality (which gets rid of NS5 branes); in particular, the classical limit of the dual theory is different. $\endgroup$ – Pavel Safronov Jun 17 '16 at 19:07
  • 2
    $\begingroup$ Having said that, I think the corresponding 5d TFT is "twice categorified Chern-Simons". Namely, it's a $\mathbf{Z}_2$-graded theory which to $S^3$ assigns the $\mathbb{E}_4$-category $U_{\hbar}(\mathfrak{g})-\mathrm{mod}$ with $\mathrm{deg}(\hbar) = -2$. $\endgroup$ – Pavel Safronov Jun 17 '16 at 19:10

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.