### Geometric Quantization on Moduli ∞-Stacks

#### Posted by Urs Schreiber

I am on the train to *Higher Structures VI* in Göttingen, doing last touches on my talk handout

*Higher geometric prerequantization on moduli infinity-stacks*talk handout pdf (4 pages) .

I have mentioned this before. Here is a quick idea of what this is about:

There are two formalizations of the notion of quantization: *geometric quantization* and *algebraic (deformation) quantization*. The latter is naturally
formulated in higher algebra in terms of ∞-cosheaves. The former should have
a natural formulation in higher geometry, in terms of ∞-sheaves = ∞-stacks.
Aspects of a formulation of such higher geometric quantization over smooth manifolds
have been introduced and studied by Chris Rogers, see the references listed *here*. Examples like the following
suggest that this is usefully generalized to higher geometric quantization over
∞-stacks in general and moduli ∞-stacks of higher gauge fields in particular.

For, write $\mathbf{B}G_{conn}$ for the smooth moduli stack of G-connections, for $G$ a simply connected simple Lie group, and write $\mathbf{B}^n U(1)_{conn}$ for the smooth moduli $n$-stack of n-form connections on smooth circle n-bundles ($(n-1)$ bundle gerbes). Then, by FSS, there is an essentially unique morphism of smooth ∞-stacks

$\mathbf{c}_{\mathrm{conn}} : \mathbf{B}G_{\mathrm{conn}} \to \mathbf{B}^3 U(1)_{conn}$

which refines the generating universal characteristic class $[c] \in H^4(B G, \mathbb{Z})$, and this as the following properties: its transgression to the loop mapping stack of $\mathbf{B}G_{conn}$

$G \to [S^1, \mathbf{B}G_{conn}] \stackrel{[S^1, \mathbf{c}_{\mathrm{conn}}]}{\to} [S^1, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to} \mathbf{B}^2 U(1)_{conn}$

modulates the $G$-WZW-model B-field 2-bundle, its trangression to the mapping stack out of a compact oriented 2-dimensional manifold $\Sigma_2$

$[\Sigma_2, \mathbf{B}G_{conn}] \stackrel{[\Sigma_2, \mathbf{c}_{\mathrm{conn}}]}{\to} [\Sigma_2, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_2}(-))}{\to} \mathbf{B} U(1)_{conn}$

restricts to the prequantum circle bundle of $G$-Chern-Simons theory, and finally its transgression to the mapping stack out of a 3-dimensional $\Sigma_3$

$[\Sigma_3, \mathbf{B}G_{conn}] \stackrel{[\Sigma_3, \mathbf{c}_{\mathrm{conn}}]}{\to} [\Sigma_3, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{\Sigma_3}(-))}{\to} U(1)$

is the action functional of $G$-Chern-Simons theory.

This suggests that we should regard $\mathbf{c}_{conn}$ itself as the
prequantum circle 3-bundle of $G$-Chern-Simons theory *extended* down to dimension 0.

What is a Hamiltonian vector field on a moduli stack such as $\mathbf{B}G_{conn}$? What is the Poisson bracket $L_\infty$-algebra? How does it act on prequantum 3-states? How to these trace/transgress to prequantum $(3-k)$-states on $[\Sigma_k,\mathbf{B}G_{conn}]$?

Answers to such questions I’ll indicate in my talk.

Posted at July 9, 2012 7:39 AM UTC
## Re: Geometric quantization on moduli ∞-stacks

There were some typos in the long formulas in the post which I fixed, so they’re the same as the handout.

As you integrate over $\Sigma_2$ in the $k = 3$ line on p. 1 of the handout and in the post, (and over $\Sigma_n$ in the second line of the second box here), I’m not sure I’m getting transgression. But I haven’t properly looked yet.