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July 9, 2012

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 BG conn\mathbf{B}G_{conn} for the smooth moduli stack of G-connections, for GG a simply connected simple Lie group, and write B nU(1) conn\mathbf{B}^n U(1)_{conn} for the smooth moduli nn-stack of n-form connections on smooth circle n-bundles ((n1)(n-1) bundle gerbes). Then, by FSS, there is an essentially unique morphism of smooth ∞-stacks

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

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

G[S 1,BG conn][S 1,c conn][S 1,B 3U(1) conn]exp(2πi S 1())B 2U(1) 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 GG-WZW-model B-field 2-bundle, its trangression to the mapping stack out of a compact oriented 2-dimensional manifold Σ 2\Sigma_2

[Σ 2,BG conn][Σ 2,c conn][Σ 2,B 3U(1) conn]exp(2πi Σ 2())BU(1) conn [\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 GG-Chern-Simons theory, and finally its transgression to the mapping stack out of a 3-dimensional Σ 3\Sigma_3

[Σ 3,BG conn][Σ 3,c conn][Σ 3,B 3U(1) conn]exp(2πi Σ 3())U(1) [\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 GG-Chern-Simons theory.

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

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

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3 Comments & 0 Trackbacks

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 Σ 2\Sigma_2 in the k=3k = 3 line on p. 1 of the handout and in the post, (and over Σ n\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.

Posted by: David Corfield on July 9, 2012 1:07 PM | Permalink | Reply to this

Re: Geometric quantization on moduli ∞-stacks

There were some typos in the long formulas in the post which I fixed,

Thanks! Sorry for causing this trouble. I have fixed one more degree-typo in the entry above. Also added more links (hopefully not introducing more typos…).

As you integrate over Σ 2 in the k=3 line on p. 1 of the handout and in the post, (and over Σ n\Sigma_n in the second line of the second box here), I’m not sure I’m getting transgression.

Yes, thanks, there were remaining degree-typos both in the handout pdf as well as in the nnLab entry, yes. I think I have fixed them now.

I’m not sure I’m getting transgression.

The rule for the degrees, at least, is that fiber integration in ordinary differential cohomology extends to a morphism from the moduli nn-stack of nn-connections on a kk-dimensional (compact, oriented) manfifold to that of (nk)(n-k)-connections on the point:

exp(2πi Σ k()):[Σ k,B nU(1) conn]B nkU(1) conn. \exp(2 \pi i \int_{\Sigma_k} (-)) : [\Sigma_k, \mathbf{B}^n U(1)_{conn}] \to \mathbf{B}^{n-k} U(1)_{conn} \,.

Thanks again for bothering with this!

Posted by: Urs Schreiber on July 9, 2012 2:06 PM | Permalink | Reply to this

Re: Geometric quantization on moduli ∞-stacks

Good. Consistency is restored to the universe.

Posted by: David Corfield on July 9, 2012 2:16 PM | Permalink | Reply to this

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