The n-Category Café

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.