Recursion representation of the Neveu-Schwarz superconformal block

TL;DR

The paper develops Recursion relations for four-point NS superconformal blocks in the $N=1$ SCFT by expressing block coefficients as sums over poles in the intermediate weight $\Delta$ and the central charge $c$, and by computing the residues of these poles. Using NS chiral vertex operators and the NS supermodule structure (including the Kac determinant and fusion rules), the authors derive explicit polynomials $P^{rs}_{c}$ and residue factors $A_{rs}(c)$ to construct recurrence relations for the $x$-expansion coefficients $F^{f}_{c,\Delta}$. They show that large-$c$ asymptotics yield leading terms $f^{f}_{\Delta}$, enabling efficient computation of coefficients via the recurrences, with separate, mixing recurrences for even and odd NS blocks. The framework extends Zamolodchikov-type recursion to the NS sector and outlines future work toward Ramond sectors and $q$-expansions, offering a practical tool for NS conformal blocks and bootstrap checks in $N=1$ theories. All key formulas involve the pole structure $\Delta_{rs}(c)$ and $c_{rs}(\Delta)$, the fusion polynomials $P^{rs}_{c}$, and the residue functions $\mathcal{R}^{f}_{c, rs}$ and $\tilde{\mathcal{R}}^{f}_{\Delta, rs}$.

Abstract

Four-point super-conformal blocks for the N = 1 Neveu-Schwarz algebra are defined in terms of power series of the even super-projective invariant. Coefficients of these expansions are represented both as sums over poles in the "intermediate" conformal weight and as sums over poles in the central charge of the algebra. The residua of these poles are calculated in both cases. Closed recurrence relations for the block coefficients are derived.