General properties of the boundary renormalization group flow for supersymmetric systems in 1+1 dimensions

TL;DR

This paper proves a gradient formula for boundary renormalization group flow in supersymmetric 1+1 dimensional systems, showing the boundary partition function decreases along the flow and establishing properties of boundary energy.

Contribution

It establishes a gradient formula relating boundary beta functions, the boundary partition function, and a positive-definite metric in supersymmetric systems, providing new insights into boundary RG flows.

Findings

Boundary partition function decreases along RG flow.

Boundary energy is nonnegative and decreases under RG flow.

Gradient formula links beta functions to boundary partition function.

Abstract

We consider the general supersymmetric one-dimensional quantum system with boundary, critical in the bulk but not at the boundary. The renormalization group flow on the space of boundary conditions is generated by the boundary beta functions \beta^{a}(\lambda) for the boundary coupling constants \lambda^{a}. We prove a gradient formula \partial\ln z/\partial\lambda^{a} =-g_{ab}^{S}\beta^{b} where z(\lambda) is the boundary partition function at given temperature T=1/\beta, and g_{ab}^{S}(\lambda) is a certain positive-definite metric on the space of supersymmetric boundary conditions. The proof depends on canonical ultraviolet behavior at the boundary. Any system whose short distance behavior is governed by a fixed point satisfies this requirement. The gradient formula implies that the boundary energy, -\partial\ln z/\partial\beta = -T\beta^{a}\partial_{a}\ln z, is nonnegative. Equivalently, the quantity \ln z(\lambda) decreases under the renormalization group flow.