Semicontinuity of 4d N=2 spectrum under renormalization group flow

TL;DR

This paper proves that the spectrum of 4d N=2 SCFTs, defined by hypersurface singularities, is semicontinuous under RG flow, providing insights into the structure and endpoints of such flows.

Contribution

It establishes the semicontinuity of the N=2 spectrum under RG flow using singularity theory, linking mathematical deformation results to physical RG behavior.

Findings

Spectrum is semicontinuous under RG flow.

No dangerous irrelevant deformations occur.

Provides necessary conditions for RG flow endpoints.

Abstract

We study renormalization group flow of four dimensional N=2 SCFTs defined by isolated hypersurface three-fold singularities. We define the spectrum of N=2 theory as the set of scaling dimensions of the parameters on the Coulomb branch, which include Coulomb branch moduli, mass parameters and coupling constants. We prove that the spectrum of those theories is semicontinous under the RG flow on the Coulomb branch using the mathematical result about the singularity spectra under deformation. The semicontinuity behavior of N=2 spectrum implies a theorem under relevant and Coulomb branch moduli deformation, the absence of dangerous irrelevant deformations and can be taken as the necessary condition for the ending point of a RG flow. This behavior is also true for (c,c) ring deformation of two dimensional Landau-Ginzburg model with (2,2) supersymmetry.