## July 28, 2006

### Synthetic Transitions

#### Posted by Urs Schreiber

On the occasion of the availability of the new edition of Anders Kock’s book on synthetic differential geometry ($\to$) I want to go through an exercise which I wanted to type long time ago already.

I’ll redo the derivation of the transition laws for 2-connections ($\to$) using synthetic language. This greatly simplifies the derivation, to the extent that the equations in terms of differential forms become almost identical to the diagrammatic equations that we derive them from.

Posted at 7:58 PM UTC | Permalink | Followups (4)

## July 26, 2006

### Quillen’s Superconnections – Functorially

#### Posted by Urs Schreiber

As explained for instance in

Richard J. Szabo
Superconnections, Anomalies and Non-BPS Brane Charges
hep-th/0108043

a special case of Quillen’s concept of superconnections can be used to elegantly subsume both the gauge connection as well as the tachyon field on non-BPS D-branes into a single entity.

Assuming that this is not just a coincidence, one might ask what it really means. What notion of functorial parallel transport ($\to$) is encoded in these superconnections?

I’ll give an interpretation below. With hindsight, it is absolutely obvious. But I haven’t seen it discussed before, and - trivial as it may be - it deserves to be stated.

Posted at 8:11 PM UTC | Permalink | Followups (5)

### K-Theory for Dummies, II

#### Posted by Urs Schreiber

Before finishing the last entry I should review some basic facts about K-theory and D-branes, beyond of what I had in my previous notes ($\to$).

Apart from the Brodzki-Mathai-Rosenberg-Szabo paper ($\to$) I’ll mainly follow

T. Asakawa, S. Sugimoto, S. Terashima
D-branes, Matrix Theory and K-homology
hep-th/0108085

which is based in part on

Richard J. Szabo
Superconnections, Anomalies and Non-BPS Brane Charges
hep-th/0108043

and

Jeffrey A. Harvey, Gregory Moore
Noncommutative Tachyons and K-Theory
hep-th/0009030.

Posted at 10:42 AM UTC | Permalink | Followups (2)

## July 20, 2006

### Brodzki, Mathai, Rosenberg & Szabo on D-Branes, RR-Fields and Duality

#### Posted by Urs Schreiber

Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard J. Szabo
D-Branes, RR-Fields and Duality on Noncommutative Manifolds
hep-th/0607020 .

This is a detailed study of the concepts appearing in the title, using and extending the topological and algebraic machinery known from “topological T-duality” (I, II, III, IV, V). The motivation is to formulate everything in ${C}^{*}$-algebraic language, in order to get, both, a powerful language for the ordinary situation as well as a generalization to noncommutative spacetimes.

Warning: I am still editing this entry.

Posted at 10:36 PM UTC | Permalink | Followups (4)

## July 19, 2006

### 2-Palatini

#### Posted by Urs Schreiber

A few entries ago, I was claiming that other people are implicitly claiming that the field content of $D=11$ supergravity encodes precisely a 3-connection taking values in a certain Lie 3-algebra ($\to$). In my first attempt to make a couple of remarks on that, I ran out of time ($\to$). Here is the second attempt.

Posted at 5:20 PM UTC | Permalink | Followups (19)

## July 18, 2006

### Herbst, Hori & Page on Equivalence of LG and CY

#### Posted by Urs Schreiber

Yesterday, Kentaro Hori gave a talk on (unpublished) joint work with Manfred Herbst and David Page, another version of which I had heard a while ago in Vienna ($\to$), on

K. Hori, M. Herbst
Phases of $N=2$ theories in $1+1$ dimensions with boundary, I

## July 14, 2006

### Gomi on Reps of p-Form Connection Quantum Algebras

#### Posted by Urs Schreiber

Quantizing abelian self-dual $p$-form connections on $\left(2p+2\right)$-dimensional spaces gives rise to quantum observable algebras which are Heisenberg central extensions of the group of gauge equivalence classes of these connections, with the cocycle given by the Chern-Simons term in $2p+1$ dimensions (I, II, III ).

In

Kiyonori Gomi
Projective unitary representations of smooth Deligne cohomology groups
math.RT/0510187

the author spells out the technical details of the construction of unitray representations for a certain (“level 2”) cases of these central extensions (compare the discussion in II), effectively generalizing the construction of positive energy reps of Kac-Moody groups $\stackrel{^}{\mathrm{LU}}\left(1\right)/{ℤ}_{2}$ (corresponding to $p=0$) to higher $p$.

These reps should be the Hilbert spaces of states of the quantum theory of self-dual $p$-form fields. Their irreps would correspond to the superselection sectors.

Posted at 1:10 PM UTC | Permalink | Followups (6)

## July 13, 2006

### Seminar on 2-Vector Bundles and Elliptic Cohomology, VI

#### Posted by Urs Schreiber

In the 4th (and probably last) session of our seminar Birgit Richter talked in more detail about

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$ 0) elliptic curves and formal groups

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$1) “classical” elliptic cohomology (according to Landweber, Ochanine and Stong)

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$2) topological modular forms (due mainly to Hopkins)

and ran out of time before talking about

$\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}$3) other forms of elliptic cohomology (e.g. Kriz-Sati) ,

complementing my rough outline last time with more technical details.

Posted at 8:04 PM UTC | Permalink | Followups (1)

## July 11, 2006

### Freed, Moore, Segal on p-Form Gauge Theory, II

#### Posted by Urs Schreiber

On Heisenberg groups in the quantization of $p$-form gauge theory.

Posted at 10:36 AM UTC | Permalink | Followups (4)

## July 10, 2006

### Freed, Moore, Segal on p-Form Gauge Theory, I

#### Posted by Urs Schreiber

These preprints were sitting in a pile of unread papers waiting to be read one day. Now I have started reading

Daniel S. Freed, Gregory W. Moore, Graeme Segal
The Uncertainty of Fluxes
hep-th/0605198

and

Heisenberg Groups and Noncommutative Fluxes
hep-th/0605200.

Posted at 12:13 PM UTC | Permalink | Followups (6)

## July 8, 2006

### How many Circles are there in the World?

#### Posted by Urs Schreiber

On the right notion of torsor for suspended $U\left(1\right)$.

Posted at 12:17 PM UTC | Permalink | Followups (1)

## July 5, 2006

### Gomi on Chern-Simons terms and central extensions of gerbe gauge groups

#### Posted by Urs Schreiber

There is an interesting paper

K. Gomi
Central extensions of gauge transformation groups of higher abelian gerbes
hep-th/0504075.

Posted at 4:21 PM UTC | Permalink | Followups (5)