The Brownian loop soup stress-energy tensor
Federico Camia (Science Division, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates, Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA, Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1111, Amsterdam, 1081 HV, The Netherlands); Valentino Foit (Science Division, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates); Alberto Gandolfi (Science Division, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates); Matthew Kleban (Center for Cosmology and Particle Physics, New York University, 726 Broadway, New York, NY, 10003, USA)
The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity λ > 0. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is c = 2λ. In the case of domains with boundary we identify a boundary operator that has properties consistent with the boundary stress-energy tensor. We show that this operator generates local deformations of the boundary and that it is related to a boundary operator that induces a Brownian excursion starting or ending at its insertion point.
