Decomposition of $\mathcal{N}=1$ superconformal minimal models and their fractional quantum Hall wavefunctions

TL;DR

This work provides an explicit coset decomposition of $ ext{N}=1$ superconformal minimal models into a $ ext{Z}_m$ parafermion sector, an Ising sector, and a free $U(1)$ boson, together with a unique decomposition of the supercurrent $G$ into constituent operators. Using free-field methods, it constructs ground-state wavefunctions for fractional quantum Hall states built from the neutral $ ext{N}=1$ sector, organizing the polynomials by clustering sectors and yielding explicit expressions for the summands; in the $c= frac{3}{2}$ limit, the dominant clustering reduces to $ ext{Pf}^3(z_{ij}^{-1})igl(igl| ext{prod} z_{ij}^3igr|igr)$, corresponding to three Majorana fermions. The paper also shows how the $n$-point correlators of the supercurrent $G$ can be computed via free-field realizations and cluster decompositions, enabling a polynomial organization of the wavefunctions in terms of central charge $c$. Finally, it discusses generalizations to $S_3$ minimal models and interprets clustering as a diagnostic of topological order, while outlining future work to distinguish topological sectors via conformal blocks and quasiparticle data.

Abstract

$\mathcal{N}=1$ superconformal minimal models are the first series of unitary conformal field theories (CFTs) extending beyond Virasoro algebra. Using coset constructions, we characterize CFTs in $\mathcal{N}=1$ superconformal minimal models using combinations of a parafermion theory, an Ising theory and a free boson theory. Supercurrent operators in the original theory also becomes sums of operators from each constituent theory. If we take our $\mathcal{N}=1$ superconformal theories as the neutral part of the edge theory of a fractional quantum Hall state, we present a systematic way of calculating its ground state wavefunction using free field methods. Each ground state wavefunction is known previously as a sum of polynomials with distinct clustering behaviours. Based on our decomposition, we find explicit expressions for each summand polynomial. A brief generalization to $S_3$ minimal models using coset construction is also included.

Figures (2)

• Figure 1: Every line represents a contraction between different pair of fields, $l_{i}$ labels a loop. By Wick's theorem, summing over all the possible products of loops gives the $n$ point amplitude.
• Figure 2: Orange line represents contraction between fields $b,c$, blue line represents contraction between fields $\partial\beta,\gamma$. $l_{i}$ labels a loop. For each loop, we add two graphs because they have the same underlying graph but different propagators. By Wick's theorem, summing over all the possible products of loops gives the $2n$ point amplitude of supercurrent operators.

Theorems & Definitions (23)

• Theorem 1
• Lemma 3.1.1
• proof
• Lemma 3.1.2
• proof
• proof
• Lemma 3.1.3
• proof
• Theorem 2
• proof