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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 ().
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.
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.
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.
May 23, 2006
Bunke on H, Part II
Posted by Urs Schreiber
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Bunke filled the board with plenty of high-powered notation, I have used this second chance for taking a look at his (unpublished) notes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 -gerbes
(dvi),
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 ().
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.
May 10, 2006
Pantev on Langlands, II
Posted by Urs Schreiber
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 .
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.
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 -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.