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July 20, 2012

Extended higher Chern-Simons theories

Posted by Urs Schreiber

We have been busy working on a little article. Now we feel it is time to show our current version around, before doing some last polishing and then posting it to the arXiv:

  • Domenico Fiorenza, Hisham Sati, Urs Schreiber,

    Extended higher cup-product Chern-Simons theories

    (pdf, web)

Check out section 1 Introduction and overview, for the story (6 pages). The rest is details.

I am grateful to the organizers of String-Math 2012 for providing an environment in which this was finalized.

Posted at 8:55 PM UTC | Permalink | Followups (2)

July 19, 2012

Zero-Determinant Strategies in the Iterated Prisoner’s Dilemma

Posted by Mike Shulman

A note to those arriving from the article in the Chronicle of Higher Education: my opinion may not have been accurately represented in that article. Please read my whole post and judge for yourself.

Usually I blog about things because I am excited about them, but today is a little different. Not that I’m not excited about this post, but recently I’ve been too busy with other things to do much blogging. However, what excitement can’t do, annoyance can, and I’m getting tired of seeing this (amazing, surprising, insightful, groundbreaking) paper subject to hype and (what seems to me to be) misinterpretation.

  • William Press and Freeman Dyson, Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent, PNAS open access, March 2012.
Posted at 8:34 PM UTC | Permalink | Followups (19)

July 16, 2012

Notes from String-Math 2012

Posted by Urs Schreiber

This week I am in Bonn at String-Math 2012, the second in a new annual series of conferences on mathematical aspects of string theory (the first was in Philadelphia, last year).

I will post notes about some talks below in the comment section.

They will be a bit more terse than I originally intended, because at the same time I have to be finalizing an article with Hisham and Domencio, on Extended higher cup-product Chern-Simons theories before I go on vacation next week.

We’ll see how it works out. Now on to Frenkel’s talk…

Posted at 11:24 AM UTC | Permalink | Followups (33)

July 10, 2012

Morton and Vicary on the Categorified Heisenberg Algebra

Posted by John Baez

Wow! It’s here!

Posted at 9:16 AM UTC | Permalink | Followups (17)

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 7:39 AM UTC | Permalink | Followups (3)