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June 30, 2008
Block on L-∞ Module Categories
Posted by Urs Schreiber
Jim Stasheff and Aaron Bergman kindly pointed out to me (here) work by Jonathan Block,
Jonathan Block
Duality and equivalence of module categories in noncommutative geometry I
arXiv:math/0509284
Part II: Mukai duality for holomorphic noncommutative Tori
arXiv:math/0604296
The way I would say it is that what Jonathan Block studies here are modules and bimodules for actions of -algebroids on -vector bundles, even though this is not quite the way he puts it.
The Manifold Geometries of QFT, I
Posted by Urs Schreiber
Spent today over at the Max-Planck Institute for Math in Bonn, close by the Hausdorff Institute, attending the first day of the conference
The manifold geometries of quantum field theory.
Here are some notes on what I have heard, concerning a) perturbative AQFT, b) algebraic AdS/CFT and c) rigorous path integrals for Chern-Simons theory.
Lerman on Orbifolds
Posted by David Corfield
Eugene Lerman gave us his views on orbifolds in a discussion beginning here. Now you can read a whole paper of his on the subject Orbifolds as Stacks?:
Abstract. The goal of this survey paper is to argue that if orbifolds are groupoids, then the collection of orbifolds and their maps need to be thought of as a 2-category. This 2-category may be either taken to be the weak 2-category of groupoids, bibundles and equivariant maps between bibundles or the strict 2-category of geometric stacks represented by proper étale Lie groupoids. While nothing in this paper is strictly speaking new, it is hoped that differential geometers unfamiliar with groupoids, bibundles and stacks would find it a useful introduction to the subject.
June 28, 2008
Michael Polanyi and Personal Knowledge
Posted by David Corfield
There was a discussion over at the Secret Blogging Seminar about the differences between mathematics and the natural sciences, which interested me greatly as someone who has frequently looked to the philosophy of science for ideas about how to treat mathematics. By and large Anglophone philosophy has chosen to treat these disciplines very differently, and has overlooked opportunities to elaborate their similarities, such as furthering George Polya’s Bayesian treatment of mathematics.
One way to lessen the difference between the disciplines is to bring to centre stage the personal involvement of scientist and mathematician in their respective theories. In that each is a member of a tradition of long standing, each has to struggle against some intransigent reality, and to convince their colleagues that their perspective on this reality is a good one, they can be seen to have much in common. For some, however, the distinction between empirical evidence and whatever support a mathematician receives trumps any such consideration.
Michael Polanyi in Personal Knowledge, written in 1958, while reflecting on this latter difference, seeks to understand it in the context of a general account of participation in a wide range of practices:
The acceptance of different kinds of articulate systems as mental dwelling places is arrived at by a process of gradual appreciation, and all these acceptances depend to some extent on the content of relevant experiences; but the bearing of natural sciences on facts of experience is much more specifiable than that of mathematics, religion or the various arts. It is justifiable, therefore, to speak of the verification of science by experience in a sense which would not apply to other articulate systems. The process by which other systems than science are tested and finally accepted may be called, by contrast, a process of validation.
Our personal participation is in general greater in a validation than in a verification. The emotional coefficient of assertion is intensified as we pass from the sciences to the neighbouring domains of thought. But both verification and validation are everywhere an acknowledgement of a commitment: they claim the presence of something real and external to the speaker. As distict from both of these, subjective experiences can only be said to be authentic, and authenticity does not involve a commitment in the sense in which both verification and validation do. (p. 202)
June 27, 2008
AQFT from Lattice Models (?)
Posted by Urs Schreiber
In AQFT from -functorial QFT (blog, arXiv) I had discussed how -functors on -paths in pseudo-Riemannian spaces give rise to local nets of algebras (or rather, more generally, of monoids) which are taken in algebraic quantum field theory (AQFT) as the definition of the local observables of QFT and indeed of QFT itself.
Now I am thinking about more examples in 2 dimensions.
Eli Hawkins on Geometric Quantization, II
Posted by Urs Schreiber
Today Eli gave the second of his two talks on -algebraic geometric quantization at HIM, based on his Groupoid approach to quantization.
I had reported on the first talk here and summarized some related results by Landsman and Ramazan here. One nice thing Eli explained today was how his approach encompasses the one by Landsman and Ramazan.
Recall that the basic idea here is this:
given a Poisson manifold (possibly but not necessarily symplectic), thought of as the phase space of a physical system, one wants to construct the -algebra quantizing (deforming) the Poisson Lie algebra of functions on phase space.
In more standard geometric quantization one would assume the Poisson structure to be actually symplectic, then try to build a Hilbert space from this data, cut it down a bit using a “polarization”, and finally find the above quantum deformed algebra as a subalgebra of the bounded operators of that Hilbert spaces.
In contrast, here the Hilbert space plays a secondary role or does not even appear explicitly. This is motivated by observations such as those by Landsman and Ramazan that in large classes of examples the quantum algebras turn out to be groupoid algebras of certain groupoids naturally associated with the original Poisson manifold.
Eli Hawkins’ approach aims to completely clarify this situation in that it explains in general which groupoid algebra is the right one. In brief words, the situation is simple and nice:
Every Poisson manifold naturally carries the corresponding Poisson Lie algebroid . If this integrates, then the integrating Lie groupoid (the source-simply connected cover or one of its quotients) is necessarily a symplectic groupoid (a groupoid with multiplicative symplectic structure on its space of morphisms) with the space of objects.
Using ordinary prequantization we may happen to get a line bundle on the space of morphisms (a line bundle with connection whose curvature is the given symplectic form) and furthermore – that’s Eli Hawkins’ big contribution here – there is a natural notion of polarization on the groupoid here, such that, finally, the quantum algebra in question is the groupoid convolution algebra of polarized sections of this line bundle.
The resulting groupoid -algebra can be regarded as a -algebraic deformation quantization of the original Poisson algebra. Notice that this is different from and really “stronger” than formal deformation quantization in terms of formal power series. See maybe my discussion here or, better, the nice introduction in the Landsman-Ramazan article.
In particular, the Landsman-Ramazan situation is recovered as follows:
recall that they observe that the -algebraic deformation quantization of any Poisson manifold arising as the fiberwise dual of a Lie algebroid is the groupoid algebra of the groupoid integrating .
Now, in Eli Hawkins setup we are to form the Poisson Lie algebroid over , integrate that, cut down functions on that to polarized ones and then form the convolution algebra of those. And, lo and behold, this does reproduce the direct prescription. The reason for that is the following nice
Fact. The Lie groupoid integrating the Poisson Lie algebroid over the dual of a Lie algebroid is the cotangent Lie groupoid of the Lie groupoid integrating .
Then it is clear that there is a choice of polarization which divides out the cotangent fibers and hence the polarized sections on are just the ordinary sections on .
(I recall more details below. This beautiful result is discussed on p. 32.)
Eli Hawkins has a wealth of concrete examples beyond this large class of examples. Today he only found time to say a bit about the Moyal space and the noncommutative torus. But look at his article for more.
June 26, 2008
Category Algebras
Posted by Urs Schreiber
Currently Masoud Khalkali over from Noncommutative Geometry Blog is giving an introductory lecture series on – right – noncommutative geometry, here at the Hausdorff institute.
He is following his notes
Masoud Khalkali
Very basic noncommutative geometry
arXiv:math/0408416 .
Today he started talking about noncommutative algebras arising as groupoid algebras (or “groupoid convolution algebras”, special cases of category algebras, see p. 58) motivating them by their ubiquitousness in noncommutative geometry:
The good news is that most of the noncommutative spaces which are currently in use in noncommutative geometry are constructed by this method.
p. 53
Here I want to use this opportunity to point out how category algebra works from the point of view of groupoidification along the lines of the discussion in An Exercise in Groupoidification: The Path Integral.
June 24, 2008
Tim Porter on Formal Homotopy Quantum Field Theories and 2-Groups
Posted by David Corfield
Guest post by Bruce Bartlett
I’d like to give something of a report-back on Tim Porter’s second talk at Barcelona, on Formal Homotopy Quantum Field Theories and 2-Groups (slides).
Firstly let me say that this was the first time I had the pleasure of meeting Tim. If anyone at the Café would like to know, Tim is one of those curious and charming breeds of Englishmen who was born in Wales but occasionally lapses into an Irish accent, and whose constitution requires for its good upkeep a steady diet of fine European cuisine (especially seafood), regular cups of rooibos tea, a daily dollop of French, bird sightings (you live in Buenos Aires or Birmingham? Tim will tell you the magical birds you can see there!), and frequent screenings of The Two Ronnies.
Is it Tim’s love of birds that caused him to title his gimungous pedagogical opus on cohomology, simplicial sets and crossed gadgetry, the Crossed Menagerie?
June 23, 2008
Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups
Posted by Urs Schreiber
guest post by Timothy Porter
At the meeting in Barcelona, on Thursday morning, we had Behrang Noohi talking on “Butterflies and morphisms between weak 2-groups”. This was a fun expanded version of his preprint, (see arXiv:math/0506313).
Aldrovandi on Non-Abelian Gerbes and 2-Bundles
Posted by Urs Schreiber
guest post by Bruce Bartlett
Just a quick report back on one of the talks given at the workshop on categorical groups in Barcelona - the one given on Thursday afternoon by Ettore Aldrovandi. Tomorrow I hope to report back on part II of Tim Porter’s talk (slides) on Friday. (The other talks on Friday were very important and impressive too of course, focusing on stable homotopy theory… but I don’t know enough to say anything intelligent there. I hope that a certain Professor Porter will report back on the morning talk by Behrang Noohi. [he did - urs])
June 22, 2008
Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids
Posted by Urs Schreiber
In the last entry I mentioned the work in
N.P. Landsman, B. Ramazan
Quantization of Poisson algebras associated to Lie algebroids
arXiv:math-ph/0001005
summarizing it with the remarkable slogan
Quantization is Lie integration.
true at least in all the cases where the classical Poisson manifold to be quantized is the dual of a Lie algebroid or of a coadjoint orbit inside of that.
This article states and proves a theorem (two versions, actually) which asserts that the -algebraic deformation quantization of such a Poisson manifold is the groupoid C-star algebra of one of the Lie groupoids integrating the given Lie algebroid.
June 20, 2008
Eli Hawkins on Geometric Quantization, I
Posted by Urs Schreiber
Recently I had mentioned Eli Hawkins’ Groupoid approach to quantization. Today at HIM he gave the first of a two-part lecture on this. This first one was on basics of geometric quantization. Next Friday we’ll here the corresponding groupoid version.
Here are some of the interesting aspects of today’s talk, including a remarkable slogan on the relation of quantization and Lie integration.
This Week’s Finds in Mathematical Physics (Week 266)
Posted by John Baez
In week266 of This Week’s Finds visit Io, the volcanic moon of Jupiter:
Then read about Pythagoras, the Pythagorean tuning system, the tetractys, and the categorical groups workshop in Barcelona.
June 19, 2008
Fundamental 2-Groups and 2-Covering Spaces
Posted by John Baez
guest post by David Roberts
This is a talk prepared for the Categorical Groups workshop in Barcelona. With the technology at hand, why let funding issues stop me from presenting it? You can see the slides here: