## 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*

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.

## 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.

## 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…

## July 10, 2012

### Morton and Vicary on the Categorified Heisenberg Algebra

#### Posted by John Baez

Wow! It’s here!

- Jeffrey Morton and Jamie Vicary, The Categorified Heisenberg Algebra I: A Combinatorial Representation.

## 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 $\mathbf{B}G_{conn}$ 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.