Boundary condition and reflection anomaly in $2+1$ dimensions

TL;DR

This paper investigates the reflection symmetry boundary conditions in 2+1 dimensional Majorana fermion theories, revealing anomalies for a single fermion and their resolution with 16 fermions, and explores related scattering phenomena.

Contribution

It demonstrates the impossibility of reflection symmetric boundary conditions for one Majorana fermion and their feasibility for 16, highlighting anomaly cancellation and boundary behavior.

Findings

Single Majorana fermion exhibits reflection anomaly.

16 Majorana fermions can satisfy reflection symmetric boundary conditions.

Connection established between 2+1D boundary anomalies and 3+1D fermion-monopole scattering.

Abstract

It is known that the $2+1$d single Majorana fermion theory has an anomaly of the reflection, which is canceled out when 16 copies of the theory are combined. Therefore, it is expected that the reflection symmetric boundary condition is impossible for one Majorana fermion, but possible for 16 Majorana fermions. In this paper, we consider a reflection symmetric boundary condition that varies at a single point, and find that there is a problem with one Majorana fermion. The problem is the absence of a corresponding outgoing wave to a specific incoming wave into the boundary, which leads to the non-conservation of the energy. For 16 Majorana fermions, it is possible to connect every incoming wave to an outgoing wave without breaking the reflection symmetry. In addition, we discuss the connection with the fermion-monopole scattering in $3+1$ dimensions.