Lessons from the Ramond sector

TL;DR

This work develops Ramond-sector consistency tests for (1+1)d fermionic CFTs, integrating modular invariance with bosonization/fermionization dualities to tighten spectrum constraints. It introduces practical diagnostics—Ramond ground-state unitarity/integrality and Sugawara-bounded current counts—and applies them to rule out unphysical NS+ extremal functions while identifying an underlying CFT for a bootstrap kink. By extending the N=1 Maloney–Witten construction to include Ramond sectors and performing a full N=1 RNS modular bootstrap, the authors obtain stronger bounds on operator gaps and relevant deformations, revealing how SUSY preservation shapes RG flows. Collectively, these results sharpen the landscape of consistent fermionic CFTs, provide tools to fix or discard proposed spectra, and point toward deeper holographic and algebraic structure guiding fermionic modular bootstrap in two dimensions.

Abstract

We revisit the consistency of torus partition functions in (1+1)$d$ fermionic conformal field theories, combining traditional ingredients of modular invariance/covariance with a modernized understanding of bosonization/fermionization dualities. Various lessons can be learned by simply examining the oft-ignored Ramond sector. For several extremal/kinky modular functions in the bootstrap literature, we can either rule out or identify the underlying theory. We also revisit the ${\cal N} = 1$ Maloney-Witten partition function by calculating the spectrum in the Ramond sector, and further extending it to include the modular sum of seed Ramond characters. Finally, we perform the full ${\cal N} = 1$ RNS modular bootstrap and obtain new universal results on the existence of relevant deformations preserving different amounts of supersymmetry.

Figures (9)

• Figure 1: The bosonization/fermionization map and the isomorphism of Hilbert spaces. Even or odd refers to the charge under $(-1)^F$ or the dual $\mathbb{Z}_2$. On the left, defect means the defect Hilbert space quantized with twisted periodic boundary conditions (by $\mathbb{Z}_2$).
• Figure 2: Left: Lower bounds on the central charge $c$ for given numbers of spin-one conserved currents $n_J$, up to $n_J = 1000$. The dashed line shows the lower envelope \ref{['Envelopes']} for $n_J \ge 300$. Right: The same bounds for $n_J \le 20$, with the dashed line indicating $n_J = 7, 11, 15$, $c_\text{min} = 3$.
• Figure 3: Left: Lower bounds on the number of spin-two conserved currents $n_T$, for given numbers of spin-one conserved currents $n_J$, up to $n_J = 1000$. The black dashed line shows the lower envelope \ref{['Envelopes']} for $n_J \ge 300$. Right: The same bounds for $n_J \le 20$, with the dashed line indicating $n_J = 11$, $(n_T)_\text{min} = 42$.
• Figure 4: Lower bounds on the dimension of a connected manifolds, for given dimensions of the isometry group. The black dashed line shows the lower envelope \ref{['ClassicalEnvelope']}.
• Figure 5: Upper bounds on the gap in the NS spectrum of scalar super-Virasoro primaries, as the central charge is varied up to $c=25$. The range of $c$ in which the bound is below 2, marked by the dashed lines, indicates the necessary existence of a supersymmetry breaking relevant deformation.