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October 31, 2005
Sheaves of CDOs
Posted by Urs Schreiber
As has been pointed out here and here, in order to better understand the remarks on the recently mentioned Čech-cohomology approach to the pure spinor string it helps to look at some more (for physicists) or less (for mathematicians) recent literature.
More precisely, there is a series of math papers
V. Gorbounov, F. Malikov, V. Schechtman, A. Vaintrob
Chiral De Rham Complex
&
Gerbes of Chiral Differential Operators
math.AG/9803041
math.AG/9901065
math.AG/9906117
math.AG/0003170
math.AG/0005201
some essence of which has been translated this year to physics language and applications in
E. Witten
Two-Dimensional Models With Supersymmetry: Perturbative Aspects
hep-th/0504078 .
October 27, 2005
Plasma-Ball Duals of Black Holes
Posted by Urs Schreiber
Ofer Aharony and Micha Berkooz are visiting Hamburg. Yesterday they gave talks on recent work.
October 20, 2005
ZMP Colloquium, Thursday: Nekrasov on Pure Spinor Superstring
Posted by Urs Schreiber
As mentioned before, I am attending a mathematical physics conference at Hamburg University. Here are some impressions.
October 18, 2005
Kalkkinen: Nonabelian Gerbes from twisted SYM
Posted by Urs Schreiber
As I had briefly mentioned last time Jussi Kalkkinen had worked out a BRST-cohomology-like formulation of the cocycle description of a nonabelian gerbe in hep-th/0510069.
In a followup preprint which appeared today
Jussi Kalkkinen
Non-Geometric Magnetic Flux and Crossed Modules
hep-th/0510135
he now makes the suggestive relation to BRST symmetry of physical theories, namely Super Yang-Mills, more explicit and presents some arguments concerning the relation of nonabelian gerbes to the physics of M5-branes.
October 11, 2005
Synthetic Differential Geometry and Surface Holonomy
Posted by Urs Schreiber
L. Breen and W. Messing in their famous math.AG/0106083 had noted that what is called synthetic differential geometry with its use of combinatorial differential forms is naturally suited for talking about connections on higher order structures such as gerbes in terms of ‘finite’ morphisms between these structures.
Synthetic differential geometry goes back to category-theoretic ideas by Lawvere and was developed mainly by Anders Kock. There is a textbook
Anders Kock
Synthetic Differential Geometry
London Mathematical Society Lecture Notes Series 51
Cambridge University Press (1981)
as well as a series of more recent papers which discuss things like gauge theory
A. Kock
Combinatorics of Curvature, and the Bianchi Identity
Theory and Application of Categories 2 (1996), 69-89
and distribution theory
A. Kock
Categorical Distribution Theory; Heat Equation
to appear in Cahiers de Top. et Geom. Diff. Categorique.
preprint available here
from the synthetic point of view.
L. Breen and W. Messing have reformulated and generalized this framework to a scheme-theoretic context in
L. Breen, W. Messing
Combinatorial Differential Forms
math.AG/0005087 .
One can safely include synthetic/combinatorial differential geometry in the list of concepts which are very simple and easy to handle in their pedestrian version, but which are powerful and far-reaching enough to admit mind-bogglingly complex generalizations. Breen and Messing discuss the generalized setup. Kock mostly cares about the more pedestrian version.
Combinatorial differential forms make again an appearance in a recent paper by Jussi Kalkkinen which further investigates aspects of Breen&Messing’s work on gerbes with connection:
Jussi Kalkkinen
Topological Quantum Field Theory on Non-Abelian Gerbes
hep-th/0510069 .
The paper discusses, motivated by similar construction in physics, how to enlarge the ‘field content’ of local data of a (nonabelian) gerbe by odd-graded ‘ghost’ fields such that odd graded BRST-like nilpotent operators generate the infinitesimal version of gauge transformations on this data.
The approach used is different but not totally unrelated to the construction presented in section 13 of hep-th/0509163.
Incidentally, I have recently been thinking about how to use synthetic differential forms in order neatly relate smooth -holonomy -functors to their associated -forms. This is a kind of technical issue with probably little interest for physically inclined people, but it seems that there is a conceptually nice mechanism at work which relates ‘macroscopic’ -functors to their ‘infinitesimal’ parts.
Some preliminary notes which review material from the theory of smooth (‘diffeological’) spaces as well as synthetic differential geometry and uses them in order to analyse smooth -holonomy -functors can be found here:
Notes: Holonomy on Smooth Path Spaces .
I imagine that much more can be done with synthetic differential geometry in the context of -holonomy, but it should be a first step.
October 5, 2005
What is “the” Gerbe of a 2-Bundle?
Posted by Urs Schreiber
For some time I was puzzled by how exactly gerbes and 2-bundles fit together conceptually. It is known from studying their properties that they encode the same information (when appropriate qualifications are added). But the underlying conceptual reason for that has been unclear to me, for the following reason:
The obvious guess was that a gerbe is to a 2-bundle like a sheaf of sections is to a bundle. Principal 2-bundles have categories of 2-sections over open patches that happen to be groupoids. Hence it was tempting to speculate that these groupoids over each open set form a gerbe.
If done correctly this should even be true. But the trouble is that the stack in groupoids obtained this way can not be the one that we want to call ‘the gerbe of the 2-bundle’. The reason is that taking the collection of groupoids (to state it carefully) obtained this way, feeding it into the standard machinery and producing cocycles or whatnot from it, we do not get back to the 2-bundle data that we started with.
Now I have thought a little harder. It now seems to me that there is another natural groupoid structure on the set of 2-sections (local 2-trivializations, really) of a 2-bundle. And this does seem to be the right one.
Discussing this requires drawing some diagrams. I have done this in these notes: