Modular Constraints on Conformal Field Theories with Currents

TL;DR

This work investigates how modular invariance constrains two-dimensional CFTs with currents by formulating a modular bootstrap problem and solving it via semidefinite programming. By analyzing twist-gap, scalar-gap, and overall-gap bounds across a range of central charges, the authors reveal sharp boundary behaviors where known RCFTs saturate the bounds, notably Deligne's exceptional series at level-one WZW, the Monster CFT, and the Baby Monster. The study extends to ${ m W}$-algebras, showing that the boundary structure at small $c$ depends on the chosen ${ m W}$-algebra and that several RCFTs are realized as boundary theories, with extremal spectra characterized by the extremal functional method. These results illuminate the landscape of unitary CFTs with extended chiral algebras and highlight accumulation phenomena near the rank of the symmetry algebra, providing a quantitative map between modular constraints and RCFT classifications.

Abstract

We study constraints coming from the modular invariance of the partition function of two-dimensional conformal field theories. We constrain the spectrum of CFTs in the presence of holomorphic and anti-holomorphic currents using the semi-definite programming. In particular, we find the bounds on the twist gap for the non-current primaries depend dramatically on the presence of holomorphic currents, showing numerous kinks and peaks. Various rational CFTs are realized at the numerical boundary of the twist gap, saturating the upper limits on the degeneracies. Such theories include Wess-Zumino-Witten models for the Deligne's exceptional series, the Monster CFT and the Baby Monster CFT. We also study modular constraints imposed by $\mathcal{W}$-algebras of various type and observe that the bounds on the gap depend on the choice of $\mathcal{W}$-algebra in the small central charge region.

Figures (12)

• Figure 1: Numerical bound on the twist gap $\Delta_t \equiv \Delta - |j|$ for the non-current operators in the presence of holomorphic currents.
• Figure 2: Numerical bound for the twist gap, in the presence of the conserved currents of spin $1 \le j \le j_{\textrm{max}}$ and the conserved currents of spin $2 \le j \le j_{\textrm{max}}$.
• Figure 3: Numerical bounds on the twist gap with various $\mathcal{W}$-algebras. We also assume the presence of the conserved currents of $j \ge 1$ in the spectrum.
• Figure 4: Numerical upper bounds on the scalar gap $\Delta_s$, the overall gap $\Delta_o$, and the twist gap $\Delta_t$.
• Figure 5: Numerical upper bounds on scalar gap, overall gap and twist gap zoomed in around $c=6$ and $7$.