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July 1, 2008

The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Posted by Urs Schreiber

The highlight of today’s talks at The manifold geometries of QFT for me was a talk by Walter v. Suijlekom in which he made a connection between the Connes-Kreimer Hopf algebra of Feynman diagrams and the BV-formalism.

Building on the Connes-Kreimer fact that Feynman diagrams form a Hopf algebra, there is a certain Hopf quotient which one can form. The question is what this corresponds to physically. The answer Walter Suijlekom gives is: it corresponds to imposing the BV master equation!

I have taken rather detailed notes, with everything that was on the blackboard, here. For everything except the BV-stuff and relations to it look at his latest very readable article

Walter D. van Suijlekom
Renormalization of gauge fields using Hopf algebras
arXiv:0801.3170

Posted at July 1, 2008 7:39 PM UTC

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8 Comments & 1 Trackback

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

I look forward to the grand day when I will be able to understand this. So many pieces of the puzzle seem to be clicking into each other now. It’s like that mysterious point in a statistical mechanical system when the “correlations go to infinity”, or like in Back to the Future, where we are approaching 88 miles per hour and lightning is going to blast down from the church tower any moment now and blast us into the future!

Posted by: Bruce Bartlett on July 3, 2008 4:22 PM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Well said.

By the way, regarding puzzle pieces: did you notice how the Landman-fact seems to hang together with BV?

Every fiber-linear dual of a Lie algebroid is naturally a Poisson manifold. Now, those fibers really ought to be regarded as being in degree +1. After fiberwise dualization they ought to be regarded as being in degree -1. But then the Poisson structure needs to be regarded as a graded degree +1 Poisson structure. That’s precisely the starting point for BV!

In particular, take the archetypical case where the Lie algebroid to start with is the tangent Lie algebroid. Dualize to get the cotangent bundle. But now with the grading just described, regarded as the shifted cotangent bundle with the fibers in degree -1. Compare with the BV-BRST grading structure (for instance here). These are the antifields!

So what seems to happen is that you start with a Lie algebroid, dualize to get a Poisson manifold in the graded world. Then extend that in turn to a DGCA, which amounts to picking a differential, hence an odd vector field, hence the Hamiltonian generating it, hence the action functional satisfying the master equation. Hence, finally, another Lie nn-algebroid structure on the extended dual of the original one.

Posted by: Urs Schreiber on July 3, 2008 5:19 PM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Any report backs on Danny’s talk on The basic bundle gerbes on unitary groups, revisited, or Aldrovandi’s talk on 2-gerbes, or Wang’s talk on Differential Twisted K-theory and Applications?

Posted by: Bruce Bartlett on July 3, 2008 4:29 PM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Any report backs on Danny’s talk on The basic bundle gerbes on unitary groups, revisited, or Aldrovandi’s talk on 2-gerbes, or Wang’s talk on Differential Twisted K-theory and Applications?

Unfortunately I am commuting between Bonn and elsewhere and have trouble being here in the morning before 10am. As a result, I missed Danny’s and Aldrovandi’s talk and saw just the last few minutes of Wang’s.

But Danny says he will give me his notes Aldrovandi. (Hm, maybe I can make Aldrovandi give me his notes on Danny’s talk, then ;-)

I did take notes in the talk following that, Arthur Jaffe on supersymmetric quantum mechanics and its relation to cyclic cohomology by a generalization of the formula for the index of the supercharge to a certain QM correlator. You want to see those?

After Jaffe’s talk I ran back from the Max-Planck institute to the Hausdorff center to hear Masoud Khalkali continue his lecture on noncommutative geometry – also talking about cyclic cohomology (of course!).

After that I looked into our research with Hisham and Jim and with Konrad, then I had already missed most of tea and had to run already again for yet another part of Masoud’s lecture.

Now I just need to reply to my email and some blog comments, and then I am already ready to do some real work. :-)

Busy days, these days.

Posted by: Urs Schreiber on July 3, 2008 5:27 PM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

pic Cry, the beloved country, for the effort Urs goes through for this blog!

I did take notes in the talk following that, Arthur Jaffe on … supercharge to a certain QM correlator. You want to see those?

Sure, sounds great. Why do I feel like I have just performed a seedy drug deal of some kind?

Posted by: Bruce Bartlett on July 3, 2008 9:38 PM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Cry, the beloved country, for the effort Urs goes through for this blog!

I’ll second that!

Posted by: Eric on July 3, 2008 10:02 PM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Sure, sounds great.

Here.

Why do I feel like I have just performed a seedy drug deal of some kind?

It’s good smoke.

Posted by: Urs Schreiber on July 4, 2008 10:20 AM | Permalink | Reply to this

Re: The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism)

Via Jim Stasheff (thanks!), I am pointed to this article:

Domenico Fiorenza, Sums over Graphs and Integration over Discrete Groupoids

Abstract: We show that sums over graphs such as appear in the theory of Feynman diagrams can be seen as integrals over discrete groupoids. From this point of view, basic combinatorial formulas of the theory of Feynman diagrams can be interpreted as pull-back or push-forward formulas for integrals over suitable groupoids.

Seems to address the comment I made here.

Posted by: Urs Schreiber on July 20, 2008 8:12 PM | Permalink | Reply to this
Read the post Hopf Algebraic Renormalization
Weblog: The n-Category Café
Excerpt: The starting point and basic idea of combinatorial Hopf algebraic techniques in perturbative quantum field theory.
Tracked: October 23, 2008 7:54 PM

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