The String Coffee Table

Update 07/12/05: See further discussion here.

The exact definition of what a closed string really is (not depending on ill-defined notions) comes from Graeme Segal and can apparently only be found in the book

U. Tillmann (ed.)
Topology, Geometry and Quantum Field Theory
Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal
London Mathematical Society Lecture Notes Series 308
Cambridge University Press (2004)

which is available here.

This book is a treat. Apart from Segal’s notes it contains Baas,Dundas & Rognes’ vector 2-bundles, Moore’s ideas on K-theory, open strings and Dirac-Ramond operators, Stolz & Teichner’s definition of elliptic object, Sullivan’s algebraic interpretation of open/closed strings and Witten’s discussion of ‘nonabelian gerbe theories’ in six dimensions, among other things.

All these topics should be parts of a grand picture, which has not fully emerged yet. Contrasting Segal’s definition of closed string CFT with Sullivan’s and Moore’s contributions shows that one thing that is not completely understood is the exact functorial defintion of open (possibly topological) strings.

This is emphasized in particular in

C. Lazaroiu
On the structure of open-closed topological field theory in two dimensions
hep-th/0010269

where aspects of an adaption of Segal’s definition to open strings are discussed.

The full cobordisms category $C$ of worldsheets of open/closed strings, however, surprisingly seems to have first been discussed recently in

N. Baas, R. Cohen & A. Ramírez
The topology of the category of open and closed strings
math.AT/0411080

(which I had recently mentioned on sci.physics.strings).

Apparently a Segal-like functor

on this, precisely defining open (topological) 2D field theory is the topic of Antonio Ramírez’ Ph.D. thesis, which is apparently still unpublished but aspects of which are disucssed in the recent

R. Cohen & A. Voronov
Notes on String Topology
math.GT/0503625

in section 3.3.

In this announcement of a seminar talk Ramírez writes:

The area of string topology began with a construction by Chas and Sullivan of previously undiscovered algebraic structure on the homology $H_*(LM)$ of the free loop space of an oriented manifold $M$. Among other results, Chas and Sullivan showed that $H_*(LM)$, suitably regraded, carries the structure of a graded-commutative algebra. The product pairing was subsequently extended by Cohen and Godin into a form of topological quantum field theory (TQFT). Open-closed string topology, first sketched by Sullivan, arises when considering spaces of paths in $M$ with endpoints constrained to lie on given submanifolds (the so-called D-branes). In this talk, I describe a way to extend the TQFT structure of string topology into an analogue of TQFT which incorporates open strings. The method of construction is homotopy theoretic, and it makes use of constrained mapping spaces from fat B-graphs (which I define) into the ground manifold $M$. I also discuss how certain spaces of fat B-graphs model the classifying space of constrained diffeomorphism groups of an open-closed cobordism.

I am assuming that a good understanding of this is a prerequisite for making the usual physics derivation of derived categories from open topological strings exact. Given that for instance the correct category for the A-model (some flavor of a triangulated Fukaya category) still seems to be elusive and that the stability conditions are not fully understood, this should be worthwhile.