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May 30, 2006

2-Spectral Theory, Part II

Posted by Urs Schreiber

Continuing my ponderings from last time () I here talk about a necessary condition for a 2-linear map to have a basis of 2-eigenvectors in some sense.

I would like Hecke eigensheaves () to be realizations of this, but for the time being I restrict attention to a setup which is probably not quite general enough to accomodate 2-vectors that are complexes of coherent sheaves.

This entry has been typeset using the new math environments provided by Jacques Distler ().

Posted at 8:19 PM UTC | Permalink | Followups (1)

May 29, 2006

Some Links

Posted by Urs Schreiber

Here are some links I would like to record, without having the time to discuss them in detail.

Posted at 9:55 PM UTC | Permalink | Post a Comment

2-Spectral Theory, Part I

Posted by Urs Schreiber

I am doing some detective work in categorified linear algebra.

The goal is to understand the relation between Hecke-like operators () and dualities in RCFT (). The hypothesized connection between the two is a certain condition known in RCFT, which characterizes disorder operators that induce CFT dualities (Kramers-Wannier, T-duality, etc). This condition is reminiscent of the condition on ordinary operators to be (semi)-normal. Therefore it might imply a categorified spectral theorem (). Therefore this condition might characterize disorder operators that have a basis of categorified eigenvectors () - like the Hecke operator does.

Might.

Posted at 1:00 PM UTC | Permalink | Followups (20)

May 25, 2006

2-Metric Geometry

Posted by Urs Schreiber

Here is a reaction to a discussion taking place over at David Corfield’s weblog.

Don’t read this if you are not into this kind of game.

Posted at 7:00 PM UTC | Permalink | Followups (13)

May 23, 2006

Bunke on H, Part III

Posted by Urs Schreiber

Here is the second part of the talk by U. Bunke.

Posted at 7:42 PM UTC | Permalink | Post a Comment

Bunke on H, Part II

Posted by Urs Schreiber

Here is a transcript of the talk by U. Bunke which I mentioned in the last entry ().

Posted at 4:35 PM UTC | Permalink | Post a Comment

Bunke on H, Part I

Posted by Urs Schreiber

Regrettably, I did not find the time, yet, to write more about a couple of interesting talks we heard in Vienna ().

Among the unwritten reports is one on Simon Willerton’s talk () as well as one on U. Bunke’s talk.

Fortunately, though, Ulrich Bunke visits Hamburg today in order to give the same (I gather) talk again. So I get a second chance.

Since in Vienna Ulrich Bunke filled the board with plenty of high-powered notation, I have used this second chance for taking a look at his (unpublished) notes

U. Bunke, Th. Schick & M. Spitzweck
Sheaf theory for stacks in manifolds and twisted cohomology for S 1-gerbes
(dvi),

which are available on his website ().

Posted at 12:50 PM UTC | Permalink | Followups (2)

May 19, 2006

The FRS Theorem on RCFT

Posted by Urs Schreiber

I was asked to say more about the FRS theorem. Here is a rough account. For more details see the existing literature ().

Posted at 11:44 AM UTC | Permalink | Followups (22)

May 15, 2006

Hecke Eigensheaves and higher Eigenvectors

Posted by Urs Schreiber

Here is an addendum to the previous three entries (I, II, III).

Given the details on the nature of Hecke operators described in Pantev’s talk () I can now make the interpretation of these in terms of higher linear maps (), more precise.

Posted at 7:35 PM UTC | Permalink | Post a Comment

May 10, 2006

Pantev on Langlands, II

Posted by Urs Schreiber

Here are notes on the second part of Tony Pantev’s lecture () on Langlands duality.

Posted at 11:30 AM UTC | Permalink | Followups (5)

May 9, 2006

Pantev on Langlands, I

Posted by Urs Schreiber

I am in Vienna, at the Erwin Schrödinger institute (), attending a workshop titled Gerbes, Groupoids and QFT (). One series of talks is

T. Pantev
Langlands duality, D-branes and quantization

Here are some notes taken in the first lecture.

More detailed lecture notes are of course available. See for instance

E. Frenkel
Lectures on the Langlands Program and Conformal Field Theory
hep-th/0512172 .

Posted at 4:17 PM UTC | Permalink | Followups (13)

May 7, 2006

Eigenbranes and CatLinAlg

Posted by Urs Schreiber

One aspect of

A. Kapustin & E. Witten
Electric-Magnetic Duality And the Geometric Langlands Program
hep-th/0604151

is the action of certain line operators on the space of branes. The branes of interest in this context are “eigenbranes” under some such operators. The author points out that the eigenbrane condition is formally similar to the eigenvector condition familiar from linear algebra, and that similar structures show up in FRS formalism ().

Here I indicate how the general structure which seems to be at work here is categorified linear algebra.

Posted at 2:36 PM UTC | Permalink | Followups (10)

May 4, 2006

Formal HQFT

Posted by Urs Schreiber

Timothy Porter was so kind to draw my attention to

T. Porter & V. Turaev
Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras
math.QA/0512032

and

T. Porter
Formal Homotopy Quantum Field Theories, II: Simplicial Formal Maps
math.QA/0512034.

I’ll summarize some ideas of the former and comment on some of the interpretational issues in the latter. In particular, I argue that the structure appearing here is that of enriched 2-dimensional cobordisms equipped with flat 2-bundles with 2-connection.

Posted at 6:30 PM UTC | Permalink | Followups (1)