# M$_k$ models: the field theory connection

## TL;DR

This paper explores the connections between M$_k$ lattice fermion models with supersymmetry and various quantum field theories, analyzing their critical and gapped regimes, and deriving new conformal field theory character formulas.

## Contribution

It establishes links between lattice models and supersymmetric quantum field theories, providing new insights into their spectra, boundary conditions, and vacuum structures.

## Key findings

- Finite size spectra analyzed for open chains with boundary conditions
- Identification of gapped regimes with massive ${
m N}=2$ supersymmetric QFTs
- Derivation of novel CFT character formulas from kink pictures

## Abstract

The M$_k$ models for 1D lattice fermions are characterised by ${\cal N}=2$ supersymmetry and by an order-$k$ clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of ${\cal N}=2$ supersymmetric conformal field theory (CFT) - we analyse finite size spectra on open chains with a variety of supersymmetry preserving boundary conditions. Specific staggering perturbations lead to a gapped regime corresponding to massive ${\cal N}=2$ supersymmetric QFT with Chebyshev superpotentials. At `extreme staggering' we uncover a simple physical picture with degenerate supersymmetric vacua and mobile kinks. We connect this kink-picture to the Chebyshev QFTs and use it to derive novel CFT character formulas. For clarity the focus in this paper is on the simplest models, M$_1$, M$_2$ and M$_3$.