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June 25, 2010

Alan Carey’s 60th Birthday Conference

Posted by Urs Schreiber

This week at MPI Bonn is (or has been) taking place a conference in honor of Alan Carey’s 60th birthday. on “noncommutative geometry and index theory, statistical models, geometric issues in quantum field theory. Hamiltonian anomalies and bundle n-gerbes”.

I missed most of it, but am on my way now for the last day at least.

This abstract here sounds interesting:

Speaker: Christian Bär (Potsdam)

Title: Renormalized integrals and a path integral representation of the heat kernel

Abstract: We propose a concept of renormalized integrals which generalizes integrals on measure spaces. This is a suitable framework for defining path integrals without having to construct an actual measure on path space. We show how this works for the heat kernel of a Laplace type operator on a compact Riemannian manifold. The hope is that this approach can circumvent problems with the standard approach using Wiener measure when one wants to deal with the Schrödinger equation instead of the heat equation.

On the following we once had a little bit of discussion here on the blog. I am hoping to learn more from the talk:

Speaker: Dennis Borisov (Yale)

Title: Higher dimensional infinitesimal groupoids of manifolds

Abstract: The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.

Am also very glad to see former colleague Danny Stevenson. Naturally, he speaks about nonabelian cohomology represented by gerbes. In – let’s seee – 15 minutes! I better run now, just arrived at the station in Bonn only…

Posted at June 25, 2010 7:57 AM UTC

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Nonabelian sheaf cohomology

In the first part of his talk Danny Stevenson gave a nice introduction to how the abelian sheaf cohomology and the nonabelian cohomology of a topological space XX are all unified into the intrinsic cohomology of the (,1)(\infty,1)-topos H=Sh (,1)(X)\mathbf{H} = Sh_{(\infty,1)}(X) (or its hypercompletion) over that space, as described at nnLab: cohomology.

He has been working for some time now on results that show how this intrinsic cohomology of H\mathbf{H} relates to that back in GrpdTop\infty Grpd \simeq Top. Not sure if he will make it to that in his talk, but roughly the statement is that for BAH\mathbf{B}A \in \mathbf{H} the (,1)(\infty,1)-sheaf induced by a suitably well behaved topological \infty-group AA, and for |A|Top\mathcal{B}|A| \in Top its geometric realization, the corresponding cohomologies computed in H\mathbf{H} coincide with those computed in TopTop in that H(X,A):=π 0H(X=*,BA)π 0Top(X,|A|)H(X,A) := \pi_0 \mathbf{H}(X= *, \mathbf{B}A) \simeq \pi_0 Top(X,\mathcal{B}|A|).

(This is not in the talk, though, I see now, hopefully I am allowed to say it here anyways.)

This generalizes the corresponding reesult by Baez-Stevenson and Baas, Kro et al. which we discussed here at some point, and which is the restriction of the above to the case that AA is a topological 2-group, in which case the cohomology in question classifies AA-principal 2-bundles on XX, or equivalently generalized AA-gerbes on XX.

Danny approaches this by representing AA as a simplicial topological group and looking at its Moore complex.

Posted by: Urs Schreiber on June 25, 2010 9:30 AM | Permalink | Reply to this

Re: Alan Carey’s 60th Birthday Conference

From Michael Murray’s talk – Bispaces and Bibundles:

  • Bibundles are not a new idea. They certainly goes back to work of Breen on bitorsors in 1990.
  • Also discussed by Aschieri, Cantini, and Jurco in 2005.
  • However when you look for examples there are not as many of them as there are principal bundles.
  • Our aim is to address this existence question.
  • It turns out that we need to use crossed modules instead of just Lie groups GG.
  • While my coworkers [David Roberts and Danny Stevenson] are keen crossed module and 2-group people I resisted this at first.
  • Let me take you through the reasons for adding this extra complexity.

Hmm, shades of the old Klein 2-geometry there.

Posted by: David Corfield on June 25, 2010 9:52 AM | Permalink | Reply to this

bibundles

Thanks, David!

Maybe you have (or somebody else has) five minutes add something to nnLab: bibundle.

(I don’t right now. Uli Bunke is talking….)

Posted by: Urs Schreiber on June 25, 2010 10:23 AM | Permalink | Reply to this

Re: bibundles

I made a start but it seems there is a range of competing definitions.

Posted by: David Corfield on June 25, 2010 12:24 PM | Permalink | Reply to this

differential cohomology

Uli Bunke gives a survey of differential cohomology in general and his work with Thomas Schick on models for differential K-theory in particular.

Notice by the way that recently he gave a precise formulation of Killingbacks’ old worldsheet Green-Schwarz anomaly cancellation of the heterotic string using differential K-theory. See nnLab: quantum anomaly – spinning particles and super-branes for the reference.

Posted by: Urs Schreiber on June 25, 2010 11:00 AM | Permalink | Reply to this

path integrals

Christian Bär made the following statement in his talk.

Remember the formula for the path integral of the non-relativistic quantum particle coupled to gravity and electromagnetism as given in standard physics textbooks, as a limit over finite-dimensional multiple integrals over spaces of piecewise straight (geodesic) paths supported on points (γ:{1,2,,N}X)(\gamma : \{1, 2, \cdots, N\} \to X).

For the special simple case of trivial gravitational and electromagnetic background field physics textbooks give this as

(1)x|e iHt|y=lim N(i2πmΔt) N2 Xe i 0 tdt12mγ˙ 2 n=1 N1dγ n \langle x|e^{- i H t}|y\rangle = \lim_{N \to \infty} \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{N}{2}} \int_X e^{i \int_0^t d t \frac{1}{2}m \dot{\gamma}^2} \prod_{n=1}^{N-1} d \gamma_n

And this has a straightforward generalizaton to non-trivial background fields. (I was going to point to nnLab: path integral, but this entry is still badly underdeveloped…)

Christian Bär’s statement was (that’s at least how I understood the message of his talk): for imaginary time he has a proof that this limit over NN does indeed exist and does indeed yield the expected value.

Posted by: Urs Schreiber on June 25, 2010 8:03 PM | Permalink | Reply to this

Re: Alan Carey’s 60th Birthday Conference

Happy Birthday, Alan!! It is many years since I have seen Alan, but it is still difficult to believe that he has reached this age.

Posted by: Kea on June 25, 2010 10:27 PM | Permalink | Reply to this

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