### 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 $n$Café 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 $(\infty,1)$-topos**

The familiar theory of smooth $Spin(n)$-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 $Spin(n)$-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: $String(n)$ 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 $n$-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.

## 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

Topological Quantum Field Theories from Compact Lie Groups.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 $n$-dimensional theories such as Dijkgraaf-Witten theory

the notion of classical $n$-dimensional field theory;

the notion of path integral quantization of these to an extended $n$-dimensional TFT.

The path integral is given by a colimit construction in $(\infty,n)$-categories over functors with values in $n$-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?