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January 25, 2010

6th Workshop on Categories, Logic and Foundations of Physics

Posted by Urs Schreiber

Readers of this blog will know about the workshop series on Categories, Logic and Foundations of Physics, that Bob Coecke and Andreas Döring have been running since two years ago. It has become tradition that we report on this here a bit on the nnCafé as witnessed by previous entries

Categories, Logic and Phyiscs in London

Categories, Logic and Phyiscs in London, II

Categories, Logic and Foundations of Physics in Oxford

on the first three events. John had also reported on

Categories, Quanta and Concepts at the Perimeter Institute

which sounds alike, in title and organizers, but was maybe not part of the official series. We seem to have skipped the announcement of parts 4 and 5, for some reason. But now here is number 6:

Categories, Logic and Foundations of Physics (CLP 6),
Oxford University Computing Laboratory
Tuesday, 9th March 2010, 12:00 - 18:20.
(website)

SPEAKERS AND SCHEDULE

- 12.00-12.50 Martin Hyland (Cambridge)

- 14:00-14:50 Urs Schreiber (Utrecht)

Gauge fields in an (,1)(\infty,1)-topos

The familiar theory of smooth Spin(n)Spin(n)-principal bundles with connnection has a motivation from physics: for the quantum mechanics of a spinning point particle to make sense, the space it propagates in has to have a Spin-structure. Then the dynamics of the particle is encoded in a smooth differential refinement of the corresponding topological Spin(n)Spin(n)-principal bundle to a smooth bundle with connection.

It has been known since work by Killingback and Witten that when this is generalized to the quantum mechanics of a spinning 1-dimensional object, the Spin-structure of the space has to lift to a String-structure, where the String-group is the universal 3-connected cover of the Spin group. Contrary to the Spin-group, the String-group cannot be refined to a (finite dimensional) Lie group. Therefore the question arises what a smooth differential refinement of a String-principal bundle would be, that encodes the dynamics of these 1-dimensional objects.

It turns out that this has a nice answer not in ordinary smooth differential geometry, but in “higher” differential geometry: String(n)String(n) naturally has the structure of a smooth 2-group – a differentiable group-stack. This allows to refine a topological String-principal bundle to a genralization of a differentiable nonabelian gerbe: a smooth principal 2-bundle. In the talk I want to indicate how the theory of smooth principal bundles with connection finds a natural generalization in such higher differential geometry, and in particular provides a good notion of connections on smooth String-principal bundles.

- 14:50-15.40 Pawel Blasiak (Krakow)

Graph Model of the Heisenberg-Weyl algebra

The Heisenberg-Weyl algebra, underlying most physical realizations of Quantum Theory, is considered from a combinatorial point of view. We construct a concrete model of the algebra in terms of graphs which endowed with intuitive concepts of composition and decomposition provide a rich bi-algebra structure. It will be shown how this encompass the Heisenberg-Weyl algebra, thereby providing a straightforward interpretation of the latter as a shadow of natural constructions on graphs. In this way, by focusing on algebraic structure of Quantum Theory we intend to draw attention to genuine combinatorial underpinning of its formalism. We will also discuss some combinatorial methods suitable for this graphical calculus.

- 15:40-16:10 Why n-categories? Panel discussion with Tom Leinster, Urs Schreiber and any other nn-Category Café server who shows up:

- 16:40-17.30 Boris Zilber (Oxford)

On Model Theory, noncommutative geometry and physics

Studying possible relations between a mathematical structure and its description in a formal language Model Theory developed a hierarchy of a ‘logical perfection’. On the very top of this hierarchy we discovered a new class of structures called Zariski geometries. A joint theorem by Hrushovski and the speaker (1993) indicated that the general Zariski geometry looks very much like an algebraic variety over an algebraically closed field, but in general is not reducible to an algebro-geometric object. Later the present speaker established that a typical Zariski geometry can be explained in terms of a possibly noncommutative ‘co-ordinate’ algebra. Moreover, conversely, many quantum algebras give rise to Zariski geometries and the correspondence ‘Co-ordinate algebra - Zariski geometry’ for a wide class of algebras is of the same type as that between commutative affine algebras and affine varieties. General quantum Zariski geometries can be approximated (in a certain model-theoretic sense) by quantum Zariski geometries at roots of unity. The latter are of a finitary type, where Dirac calculus has a well-defined meaning. We use this to give a mathematically rigorous calculation of the Feynman propagator in a few simple cases. References: On model theory, non-commutative geometry and physics (survey), author’s web-page 2009

- 17:30-18:20 Bertfried Fauser (Birmingham)

Advanced graphical calculus - Hopf, Frobenius, Schur and some motivation from group theory

Graphical calculus has become a tool in quantum information theory, especially the Frobenius algebra structure for modelling the copying of classical information and rewriting rules. In my talk I will try to provide a more general picture including a Hopf algebra structure, Hopf algebra cohomology and the operation of composition. The underlying isoclasses of vector spaces will be countably infinite. The development will be motivated by findings in group theory, particularly the theory of group chraraters, which will serve as running example. If time permits I will also address conformal fields.

Posted at January 25, 2010 6:15 PM UTC

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formalization of path integral quantization in higher category theory

Maybe we can start a discussion with this:

everybody interested in (higher) category theory and physics has to take note of the remarkable

The title doesn’t quite indicate what is so remarkable about this, as it refers to something like an application of a central conceptual development in there:

the authors present, more or less, a grand general proposal for a fully systematic notion of formalizing for discrete nn-dimensional theories such as Dijkgraaf-Witten theory

  • the notion of classical nn-dimensional field theory;

  • the notion of path integral quantization of these to an extended nn-dimensional TFT.

The path integral is given by a colimit construction in (,n)(\infty,n)-categories over functors with values in nn-vector spaces.

We had talked here about various aspects of this before, but maybe it is good to extract the central message here in one concise statement.

To that end, I wrote what I think is a concise summary of this proposal with highlighting remarks on the crucial conceptual points at

All you researches in (higher) category theory and physics: what do you think?

Posted by: Urs Schreiber on January 27, 2010 4:55 PM | Permalink | Reply to this

Re: 6th Workshop on Categories, Logic and Foundations of Physics

And your picture fits happily with Sati’s?

Posted by: David Corfield on January 28, 2010 12:31 PM | Permalink | Reply to this

Re: 6th Workshop on Categories, Logic and Foundations of Physics

And your picture fits happily with Sati’s?

Sorry, can you say more precisely what you are wondering about?

Here is something I can say that relates the above entry to Hisham’s latest article:

  1. in the little query box at TQFT from Compact Lie Groups (and elsewhere) I indicate how the theory of differential nonabelian cohomology should relate, roughly, to what Freed, Hopkins, Lurie, Teleman are proposing as a general abstract formulation of quantization. (It’s not like we hadn’t thought about these things before ourselves, but they seem to have made some crucial progress on realizing it technically.)

  2. in a series of articles with Hisham Sati and Jim Stasheff, we discuss how this theory of differential nonabelian cohomology describes the various twisted background fields that the string and the NS5-brane and the M5-brane couple to.

    1. L L_\infty-algebra connections and applications to String- and Chern-Simons transport (the differential geometry and dg-algebra behind it)

    2. Fivebrane structures (the topology behind it and its relation to quantum anomalies)

    3. Differential twisted String and Fivebrane structures (the full picture: smooth differential refinements of the corresponding topological structures).

In his very latest, Hisham reviews these applications and talks about a host of further occurences of higher categorical structures in String and M-theory. As usual, he is leaving me a bit behind here, there is just so much higher category theory hidden in this, and Hisham already sees what all the connections should be, while I am still quite occupied with fully formalizing what we have unraveled so far. But eventually I’ll catch up.

One of the very interesting aspects, I think, of Hisham’s work, based on his joint work with Igor Kriz, is that they indicate how elliptic cohomology should arise as the theory of charges in what is called F-theory . I think their proposal is the first one that goes in the direction of “geometric models for elliptic cohomology” that actually identifies the elliptic curve in the model. The elliptic curve is strikingly absent from the geometric models that have been proposed so far – magnificent as these are!– the Stolz-Teichner proposal, the Baas-Dundas-Rognes proposal, the Bartels-Douglas-Henriques proposal. As far as I am aware at least, would be happy to be corrected!

Sati and Kriz say that the elliptic curve for the elliptic cohomology theory that occurs in String theory – the way K-theory does appear there – is the one of the torus fibration over 10-dimensional base space of Type II string theory that F-theory is about. To my shame I have to admit that I still have to study their construction in any detail, but this aspects sounds noteworthy to me.

So, anyway, there is a long distance to traverse from the general abstract description of quantization of (differential) cocycles that is indicated for finite toy models in the above entry, and the application to the full-fledged theory of sigma models for strings and 5-branes that Hisham is talking about.

But, yes, that distance should eventually be fully traversed. It’s my goal to see that happen. And maybe to help a bit make it happen, if possible. But in view of articles as the one discussed in the above entry, I am confident that we will see considerable progress on this understanding of the general abstract nonsense nature of the universe in our lifetetime.

Posted by: Urs Schreiber on January 28, 2010 1:06 PM | Permalink | Reply to this

Re: 6th Workshop on Categories, Logic and Foundations of Physics

I had nothing particular in mind. I was just interested to get a sense of how you think the abstract nonsense approach is going, which you’ve given me, thanks. So many insights seem to come your way, which tie in with others at the forefront of TQFT, that it feels like a huge theoretical revolution is quite close. So good to have your assessment that we may expect ‘considerable progress…in our lifetime’.

People who work in philosophy of science and are interested in revolutions/breakthroughs/theoretical transformations, call them what you will, generally study completed stories. It’s rather odd to be looking at one as its happening, with its accompanying difficulty of gaining an accurate sense of how far through the episode we are at the present time. It’s even excusable, although let us hope wrong, to be less than 100% confident that an abstract nonsense view will prevail.

With the benefit of hindsight it’s much easier to point to the important moments in a story.

Posted by: David Corfield on January 28, 2010 2:02 PM | Permalink | Reply to this

Re: 6th Workshop on Categories, Logic and Foundations of Physics

Well, it’s true that I have not been working in CLP lately, because I don’t have an internet account, I must move often, and I find it hard to work when I cannot afford meat or chocolate or cheese, and I have to face a lifetime of waitressing.

Posted by: Kea on February 2, 2010 9:52 PM | Permalink | Reply to this

Re: 6th Workshop on Categories, Logic and Foundations of Physics

Will the rest of the abstracts and titles be announced, here or on the conference website? There’s just one on the latter so far, for Bertfried Fauser’s talk.

Posted by: Jocelyn Paine on March 2, 2010 5:30 PM | Permalink | Reply to this

Re: 6th Workshop on Categories, Logic and Foundations of Physics

Will the rest of the abstracts and titles be announced, here or on the conference website?

Yes, soon.

Posted by: Urs Schreiber on March 2, 2010 6:13 PM | Permalink | Reply to this

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