A new hyperbolic sigma model (Andrew Charles Kenneth Swan, EPFL)

28.04.2025 16:15 – 18:00

Abstract:
Recently, Sabot and Tarr\`es introduced a new type of vertex reinforced jump process: the $\star$-VRJP. It is defined on a directed graph $G$, which is equipped with a special involution $\star: G \rightarrow G$ that sends each vertex $j$ to a conjugate vertex $j^\star$, and each edge $\langle ij \rangle$ to a reversed conjugate edge $\langle j^\star i^\star \rangle$. Much like the ordinary VRJP, the $\star$-VRJP is linearly reinforced according to the local time of the walker, but where the ordinary VRJP prefers to jump to where it has been, the $\star$-VRJP prefers to jump to the \emph{conjugate} of where it has been.
Also much like the ordinary VRJP, the $\star$-VRJP possesses a variety of remarkable integral identities through its ``magic formula" and random Sch\"odinger representation. In the case of the VRJP, through its connection with the hyperbolic sigma model, the existence of these identities is seen to be a consequence of supersymmetric localisation: this naturally raises the question if there exists a ``$\star$-sigma model" counterpart to the $\star$-VRJP to give a similar supersymmetric explanation. In this talk, I will introduce this new $\star$-hyperbolic sigma model, the $\mathbb{H}^{2n+1|4m}_\star$-model, which is, in a sense, a complexification of the ordinary $\mathbb{H}^{n|2m}$-model, and will present several new isomorphism theorems which connect it to the $\star$-VRJP. Joint work with Sabot and Tarr\`es.

Lieu

Conseil Général 7-9, Room 1-15, Séminaire Math Physics

entrée libre