Bootstrapping Chiral CFTs at Genus Two

TL;DR

This work presents an analytic bootstrap for 2d chiral CFTs by leveraging genus-two partition functions expressed as Siegel modular forms. By computing genus-2 conformal blocks in Schottky coordinates and enforcing both crossing symmetry and modular invariance, the authors relate the partition function to a finite set of light data and derive bounds on the spectrum and on three-point function squares for central charges up to 72. The strategy combines factorization, Siegel operator constraints, and conformal-block matching to fix partition functions case-by-case (c=8–48) and to map the allowed light-data regions, identifying extremal theories that saturate bounds. The results extend known constraints at low c and provide a systematic, analytic framework for constraining meromorphic CFTs using higher-genus consistency data. The approach offers a complementary perspective to numerical bootstrap, highlighting the power of modular and geometric constraints in shaping CFT spectra and interactions.

Abstract

Genus two partition functions of 2d chiral conformal field theories are given by Siegel modular forms. We compute their conformal blocks and use them to perform the conformal bootstrap. The advantage of this approach is that it imposes crossing symmetry of an infinite family of four point functions and also modular invariance at the same time. Since for a fixed central charge the ring of Siegel modular forms is finite dimensional, we can perform this analytically. In this way we derive bounds on three point functions and on the spectrum of such theories.

Figures (9)

• Figure 1: Upper and lower bounds on $C_{111}$ for the $c=32$ theory.
• Figure 2: Upper bound on $\langle c_{111}^2\rangle=\frac{C_{111}}{N_1^3}$ for $c=32$.
• Figure 3: The allowed region for the light data $N_1$, $C_{111}$, and $C_{222}$, satisfying the unitarity constraints for the $c=40$ theory.
• Figure 4: Upper bound on $C_{111}$ for $c=40$.
• Figure 5: Upper bound on $C_{222}$ for $c=40$.