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Oxford Quantum Talks Archive

Gage fields in an (oo,1)-topos

Urs Schreiber, University of Utrecht
Categories, Logic and Foundations of Physics VI, March 2010, University of Oxford

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" or "derived" 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 generalization 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.

[video] [streaming video]