In February there is going to be a workshop and school dedicated to exploring the interactions of Quantum Gravity, Higher Gauge Theory, and Topological Field Theory. I’m excited about the chance to share ideas and hopefully create some new mathematics.
The conference will take place in Lisbon, Portugal, and yours truly will be giving one of the mini-courses for the school (the topic is going to be the classification of extended 2D tqfts, something near and dear to my heart). Of course that makes me really excited, but I am also excited about the other topics too and I think the mix of ideas will be invigorating. For more info look below the break.
This is happening really fast. The conference was just officially announced this week and the deadline for registering is Jan 8th, so act fast!
Here is the conference website, and a blog post by one of the organizers.
The description from the website:
Higher gauge theory is a fascinating generalization of ordinary abelian and non-abelian gauge theory, involving (at the first level) connection 2-forms, curvature 3-forms and parallel transport along surfaces. This ladder can be continued to connection forms of higher degree and transport along extended objects of the corresponding dimension. On the mathematical side, higher gauge theory is closely tied to higher algebraic structures, such as 2-categories, 2-groups etc., and higher geometrical structures, known as gerbes or n-gerbes with connection. Thus higher gauge theory is an example of the categorification phenomenon which has been very influential in mathematics recently.
There have been a number of suggestions that higher gauge theory could be related to (4D) quantum gravity, e.g. by Baez-Huerta (in the QG^2 Corfu school lectures), and Baez-Baratin-Freidel-Wise in the context of state-sums. A pivotal role is played by TQFTs in these approaches, in particular BF theories and variants thereof, as well as extended TQFTs, constructed from suitable geometric or algebraic data. Another route between higher gauge theory and quantum gravity is via string theory, where higher gauge theory provides a setting for n-form fields, worldsheets for strings and branes, and higher spin structures (i.e. string structures and generalizations, as studied e.g. by Sati-Schreiber-Stasheff). Moving away from point particles to higher-dimensional extended objects is a feature both of loop quantum gravity and string theory, so higher gauge theory should play an important role in both approaches, and may allow us to probe a deeper level of symmetry, going beyond normal gauge symmetry.
Thus the moment seems ripe to bring together a group of researchers who could shed some light on these issues. Apart from the courses and lectures given by the invited speakers, we plan to incorporate discussion sessions in the afternoon throughout the week, for students to ask questions and to stimulate dialogue between participants from different backgrounds.
and a provisional list of speakers:
- Paolo Aschieri (Alessandria)
- Benjamin Bahr (Cambridge)
- Aristide Baratin (Paris-Orsay)
- John Barrett (Nottingham)
- Rafael Diaz (Bogotá)
- Bianca Dittrich (Potsdam)
- Laurent Freidel (Perimeter)
- John Huerta (California)
- Branislav Jurco (Prague)
- Thomas Krajewski (Marseille)
- Tim Porter (Bangor)
- Hisham Sati (Maryland)
- Christopher Schommer-Pries (MIT)
- Urs Schreiber (Utrecht)
- Jamie Vicary (Oxford)
- Konrad Waldorf (Regensburg)
- Derek Wise (Erlangen)
- Christoph Wockel (Hamburg)
It looks great. I can’t wait!
Secret Blogging Seminar 
It is maybe noteworthy that there are quite a few more examples of higher gauge theories that are older and better studied than some of those mentioned in the abstract. Even if we take the Kalb-Ramond B-field for granted and count the 11d supergravity C-field as part of the story of higher spin structures that is being mentioned, there is notably the RR-field in type II supergravities, which has curvature forms in arbitrary odd/even degree and is modeled by differential K-theory. Of course this is still in higher _abelian/stable_ gauge theory and one might argue that true “gauge theory” proper starts only with higher Chern-Weil theory and nonabelian higher structure Lie group(oid)s.
But there is also the old AKSZ theory which is (somewhat secretly, admittedly) higher nonabelian gauge theory (in fact higher Chern-Simons theory) with higher structure Lie groupoids coming from higher symplectic Lie algebroids and action functionals coming from the transgssion of these higher symplectic structures.
And if we admit action functionals on spaces of higher connections that are not necessarily of generalized Yang-Mills or generalized Chern-Simons form (and this is understood in some of the attempts that are being mentioned in the abstract) then 10- and 11-dimensional supergravity themselves are higher gauge theories: specifically 11d sugra is the gauge theory for a certain super Lie 6-group (as noticed by d’Auria and Fre, if somewhat implicitly).
Passing to the quantization of any of this is of course a widely open problem (though for abelian higher gauge theories there are remarkable results by Freed and others) so that the relation of any of this to would-be quantum gravity is largely a matter of belief (or of definition) but as far as just the notion of higher gauge theory as such is concerned my impression is that the considerrable scope of examples and classes of examples that have been studied is not often fully appreciated.