Applying Symanzik' program to the Abelian Higgs-Yang-Mills field in the continuum

Vortrag von Dr. Isao Sauzedde

Datum: 01.04.26  Zeit: 17.15 - 18.45  Raum: Y27H12

Dynkin's duality between Markov fields and Markov paths relate Gaussian free fields with Brownian paths. 
Symanzik's program for the Phi^4 field consists in using this relation to write the 2k-points functions of the Phi^4 fields as the expectation of the intersection local time of k Brownian bridges. In this talk, I will present this program in a different framework, that of the Abelian Higgs-Yang-Mills field in a planar domain, for which a complex scalar field interacts with a vector (gauge) field. In this framework, we will see that the role played by the intersection local time is played instead by the Amperean area of the paths, a random variable associated with the path which I will present during the talk. 
Due to the fact we consider two fields interacting with each other, it is necessary to take some partition functions into account. I will indeed explain how the partition function of a Gaussian field, in presence of an external magnetic field, can be written as an expectation of magnetic flux over a Brownian loop soup.

The talk is freely based on two papers, one of which is joint with P. Perruchaud (ArXiv ids: 2402.00767 and 2412.16781). I will try to focus on presenting the big picture and I will stay rather informal at some stages.