Constraints on Flavored 2d CFT Partition Functions

TL;DR

This work analyzes flavored (charged) two-dimensional CFT partition functions under modular invariance, deriving a simple proof of the transformation law via background gauge fields and applying modular bootstrap to constrain charged spectra for abelian and non-abelian currents. By combining semidefinite programming with the extremal functional method, the authors obtain improved bounds on the lightest charged state Δ_*, the mass-to-charge ratio r_*, and the lowest charge Q_*; in several cases they extract exact or highly constrained spectra, including integer occupation numbers at c=8 (the E8 lattice) and a unique neutral-flavored partition function at c=3. A key insight is that flavor information can reveal structure invisible to flavor-blind analyses, including representation-dependent constraints and potential paths to identifying underlying CFTs. The results illuminate the interplay between modular invariance, currents, and spectrum organization, and open avenues for extending the flavored bootstrap to higher spins, larger symmetry groups, and connections to classifications like Schellekens’ at c=24. Overall, the work demonstrates the power and limitations of flavored modular bootstrap as a tool for detailed spectral reconstruction in 2d CFTs.

Abstract

We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e. when the partition functions are "flavored". We begin with a new proof of the transformation law for the modular transformation of such partition functions. Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory. We improve previous upper bounds on the state with the greatest "mass-to-charge" ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory. We apply the extremal functional method to theories that saturate such bounds, and in several cases we find the resulting prediction for the occupation numbers are precisely integers. Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise.

Figures (16)

• Figure 1: Bounds of the dimension of lightest charged state assuming the theory has U(1) symmetry. The extrapolated gaps at $n_D \rightarrow \infty$ with the trend line.
• Figure 2: Bound of mass-to-charge ratio as a function of $c$; a trend line $\propto c^{-1/2}$ is shown for comparison. The extrapolation ("$n_D=\infty$" points) and error bars are computed by performing a fit as a function of $n_D$ and extrapolating to $n_D\rightarrow \infty$ as described in the text.
• Figure 3: Upper bound on $\frac{8 G_N m}{Q}$ (that is, there exists a state below the bound) as the number $n_D$ of derivatives used in the semidefinite programming analysis increases, for the specific case $c=10^5$. The bound value is still changing rapidly at $n_D = 41$.
• Figure 4: We obtain an upper bound, shown here, on the smallest nonzero charge $Q_*$; the bound is $Q_* \le 1$ for all $c$.
• Figure 5: Bound on the charge gap $Q_*$ and the scalar dimension gap $\Delta_*$ at $c=2$. The region near the kink in the left plot is magnified and shown in the right plot.