The String Coffee Table

In

Bert Schroer
String theory and the crisis in particle physics
physics/0603112

the main technical argument, apart from several sociological arguments ($\to$), is a certain subtlety in the commutators of string fields.

Bert Schroer interprets the nature of these commutators as saying that from the intrinsic point of view of quantum string field theory strings are pointlike, and this he regards as a fatal flaw, as far as I understand.

We had discussed this at great length a while ago with Bert Schroer himself here on the Coffee Table ($\to$).

The relevant literature ($\to$) is

E. Martinec
Strings and Causality
hep-th/9311129

as well as

H. Hata, H. Oda
Causality in Covariant String Field Theory
hep-th/9608128

and in particular

J. Dimock
Local String Field Theory
math-ph/0308007 .

At least superficially, there appears to be a certain disagreement between these results. According to the last paper, two free, bosonic string fields have vanishing commutator if the center of mass of the two strings described by the two fields are spacelike seperated.

The first two papers, however, seem to come to the conclusion that the excitations of the string (its spatial extension) also enters the string field commutator. In our last discussion, this was confirmed by Barton Zwiebach ($\to$).

I must admit that I have not taken the time to look at this closely enough to sort this out properly. I had asked J. Dimock about this ($\to$), who replied that there is no contradiction.

One obvious but easily overlooked fact at least seems to be important to note:

What is usually called the center of mass of the string is not in any way an intrinsic, invariant quantity.

What is usually called the “center of mass” is really the “center of coordinate density”, if you like. It depends on the parameterization of the string and hence on whether or not any gauges have been fixed.

Of course, when the string is quantized in the usual way, there is a natural and obvious choice of “center of mass mode” which does play the role that one would expect naively. But in particular if one is interested in “intrinsic” properties of quantum systems, one should probably be careful with interpreting any condition that involves the center of mass of two strings too literally.

In any case, even if we agree that two free string fields commute precisely when their respective “centers-of-mass” are spacelike seperated, I don’t quite see why this is supposed to be problematic.

I think the question is this: are we facing technical inconsistencies or just inconsistencies with our intuitive expectations of what technical results should look like?

I was pointed to

J. Baez, D. Wise, & A. Crans
Exotic Statistics for Loops in 4d $B F$ theory
gr-qc/0603085,

which is closely related in particular to

R. Szabo
Topological Field Theory and Quantum Holonomy Representation of Motion Groups
hep-th/9908051.

Here is a report on

Jose M. Isidro
Gerbes and Heisenberg’s Uncertainty Principle
hep-th/0512241.

Over on the category theory mailing list ($\to$) category theorists are debating in a long thread mainly whether but also how category theory has been sensibly applied in mathematical physics, in particular in string theory. In an attempt to substantiate such discussions I here try to list some information on applications of category theory in mathematical physics and string theory.

(The following list is clearly incomplete. If anyone thinks his favorite example or link to resources is missing, drop me a note.)

I am on vacation, and not supposed to be hanging around on the web. But here are two nice links.

Daniel Stevenson has written up some informal notes concerning the String(n)-2-group and the String-gerbe: pdf.

(Disclaimer: Please note that some constant in some cocycle equation in this text might need correction.)

André Henriques has turned his notes on integrating Lie $n$-algebras and how the String group arises as the integration of the String Lie 2-algebra into a paper:

André Henriques
Integrating L-infinity algebras
math.AT/0603563

Abstract:

Given an $n$-term $L_\infty$ algebra $L$, we construct a Kan simplicial manifold which we think of as the ‘Lie $n$-group’ integrating $L$. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group. In the case of the string Lie 2-algebra of Baez and Crans, this recovers the model of the string group introduced in math.QA/0504123.

Here is a first refinement of some ideas related to the representation theory of the $\mathrm{String}(n)$-2-group which I mentioned recently ($\to$).

I need to better understand the theory of vertex algebras. For my own personal convenience, here is a selection of some links.

Today one can find this nice article on the arXiv:

J. Mickelsson
Gerbes and Quantum Field Theory
math-ph/0603031.

Let $P\mathrm{Spin}(n)$ be the 2-group ($\to$) whose nerve is the group $\mathrm{String}(n)$ ($\to$).

I would like to understand if Stolz/Teichner’s conception ($\to$) of $\mathrm{String}(n)$-connections can be understood in terms of 2-connections in 2-bundles ($\to$) which are associated by way of a 2-representation ($\to$) to a principal $P\mathrm{Spin}(n)$-2-bundle ($\to$).

Here are some remarks.

In the context of functorial transport over supermanifolds, I mentioned recently that sorting out the true nature of supermanifolds leads to concepts that may look quite sophisticated, compared to the familiar carefree handling of Grassmann coordinates. Actually, I think they are quite elementary but most of us are not used to thinking in terms of them. Since supermanifolds are really just a simple example for way more general generalized spaces that are bound to play a major role in (formal) high energy physics - like for instance noncommutative geometries - it seems worthwhile to spend a couple of minutes sorting out the answer to the question in the title of this entry.

The following is an exercise in physics made difficult. It reminds me of this piece of wisdom I once heard.

Before you understand, a tree is a tree.
When you understand, a tree is no longer a tree.
After you have understood, a tree is a tree.

Currently I am staying at the Weizmann Institute where I am visiting Ofer Aharony and Micha Berkooz (I had reported about their last visit to Hamburg here). But at the moment I am still occupied with the internal mental post-production of the elliptic workshop. Here I want to list some literature on an interesting result by C. Bödigheimer which could be relevant for the program of realizing elliptic objects in terms of transport 2-functors.

Last time I tried to indicate, roughly, how a time-dependent euclidean (Wick-rotated) system of supersymmetric quantum mechanics defines a virtual vector bundle (with fibers the kernel minus the cokernel of the supercharges/Dirac operators) over the space which parameterizes the family of operators at different “times”, hence an element of the K-theory of that space. If this space consists of a single point (the “time-independent” case) then the dimension of the corresponding virtual vector space coincides with the partition function of the SQM system.

I ended by mentioning that in one dimension higher, for superconformal 2D euclidean field theories, the integer number called the partition function gets replaced by a modular form. Here I’ll try to review the mathematician’s description for what this means and how it comes about.

The following is essentially a transcript of Johannes Ebert’s talk at our workshop, which again roughly followed section 3.3 of WIAEO?.

Here are some notes taken from or induced by the talk given by Elke Markert at that workshop.

From Field Theories to Elliptic Objects?

This is a review of concepts and results in the relation of 1-dimensional euclidean field theory with K-theory and 2-dimensional euclidean field theory with elliptic objects.

I am currently staying at Schloss Mickeln, attending the workshop on Stolz&Teichner’s approach to elliptic cohomology which I mentioned recently.

There is little time to blog with lots of things to be said. Here I’ll just mention a single cool factoid.