Phase space singularities of nonlinear field theories: interaction of gauge symmetries, conservation laws and topology

par Igor Khavkine (Prague)

vendredi 28 nov. 2025, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Bat. Braconnier)

A classical field theory is often described by a non-linear PDEs with gauge symmetries. Its space of solutions, endowed with a suitable (typically infinite dimensional) differential as well as symplectic/Poisson structures, constitutes the theory's phase space. The phase space is a central object in the geometric approach to field theory as a mechanical system and to its quantization. As for any nonlinear system, the space of solutions can fail to be a smooth manifold, typically featuring conical singularities along a singular locus. In this context, the solutions belonging to this singular locus are called "linearization unstable". Early explicit examples from General Relativity noted that such solutions require the simultaneous existence of some non-trivial symmetries and topological conditions (e.g., compactness). I will use a general geometric approach to PDEs to identify such "linearization instabilities" and show how they are associated to certain cohomology classes, naturally explaining the previous observations.
Based on Ann. Henri Poincaré 16, 255–288 (2015).
https://doi.org/10.1007/s00023-014-0321-9