Spinon bases in supersymmetric CFTs

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 $(ϕ^-, ϕ^{+})$, 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.