Spinon bases in supersymmetric CFTs

TL;DR

This paper introduces a supersymmetric spinon basis for organizing the spectra of certain superconformal field theories, linking them to lattice models with exclusion rules and algebraic structures like Yangian symmetry.

Contribution

It generalizes the spinon basis to supersymmetric theories, constructs multi-spinon bases, and connects CFT spectra to lattice models with specific exclusion principles.

Findings

Developed supersymmetric spinon basis for $ ext{N}=2$ and $ ext{N}=4$ superconformal theories.

Established a Yangian-like algebraic framework for the $c=1$ case.

Connected finitized spectra to supersymmetric lattice models with exclusion rules.

Abstract

We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with $\mathcal{N}=2$ or (non-linear) $\mathcal{N}=4$ superconformal symmetry. Generalising the spinon basis of the $SU(n)_1$ WZW theories, we introduce supersymmetric spinons $(\phi^-, \phi^{+})$, which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection $\{ n_j \}$ of (discrete) momenta. The state-content for given choice of $\{ n_j \}$ is determined through a generalised exclusion principle, similar to Haldane's `motif' rules for the $SU(n)_1$ theories. In the simplest case, which is the ${\cal N}=2$ superconformal theory with central charge $c=1$, we develop an algebraic framework similar to the Yangian symmetry of the $SU(2)_1$ theory. It includes an operator $H_2$, akin to a CFT Haldane-Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the $\mathcal{N}=2$ superconformal CFTs, this finitisation precisely leads to the so-called M$_k$ supersymmetric lattice models with characteristic order-$k$ exclusion rules on the lattice. Finitising the $c=2$ CFT with non-linear ${\cal N}=4$ superconformal symmetry similarly gives lattice model partition sums for spin-full fermions with on-site and nearest neighbour exclusion.