Constraints on 2d CFT partition functions

TL;DR

This work develops a systematic, SDP-based implementation of the linear functional method to bound the lowest nontrivial conformal-weight gap in two-dimensional CFTs from modular invariance. It treats both bosonic Virasoro theories and N=(2,2) superconformal theories, including Calabi–Yau compactifications, and yields tighter bounds on the spectral gap, with Δ_B approaching c/6 for large central charge and Δ_B → 1/2 in the large Hodge-number limit. For CY_3s with no enhanced symmetry, the authors show a universal non-BPS bound Δ ≤ 0.6 on the total weight of the lowest non-BPS state. The results constrain the existence of consistent spectra and have implications for the geometry of Calabi–Yau manifolds via the elliptic genus and topological data, while also clarifying the limitations of the linear-functional approach in enforcing integrality and spin structure. Overall, the paper advances modular-invariance-based bootstrap bounds and highlights both the power and bounds of SDP-based functional methods in 2D CFTs and their geometric applications.

Abstract

Modular invariance is known to constrain the spectrum of 2d conformal field theories. We investigate this constraint systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We also consider generalized partition functions of N = (2,2) superconformal theories and discuss the application of our results to Calabi-Yau compactifications. For Calabi-Yau threefolds with no enhanced symmetry we find that there must always be non-BPS primary states of weight 0.6 or less.

Figures (5)

• Figure 1: The situation for $\Delta_1' < \Delta_1 < \Delta"_1$: For $\Delta_1'$, $v_{0}$ is within the cone $C_{\Delta'_1}$ so it is impossible to find a separating plane. For $\Delta_1"$, $v_{0}$ is outside of the cone, and we can find a separating plane $\rho$.
• Figure 2: $\Delta_{B}$ as a function of $c$. The bottom line is $2\gamma=(c-1)/12$, which is the smallest possible linear functional bound.
• Figure 3: The slope of $\Delta_{B}$ asymptotes to $1/6$ for large $c$.
• Figure 4: $\Delta_{B}(h^\mathrm{tot})$ for various $n_{{{\mathcal{D} } }}$.
• Figure 5: The bound $\Delta_1(h^\mathrm{tot})$ obtained from the Lambert $W$-function.