Bounds for State Degeneracies in 2D Conformal Field Theory

TL;DR

The paper shows that modular invariance in 2D CFTs implies a universal lower bound on the canonical entropy at β=2π and surveys why universal upper bounds on state degeneracies are hard to achieve in complete generality. By imposing infrared stability, discreteness, and a moderate total central charge with no relevant operators, the authors derive explicit upper bounds: a bound on the number of marginal operators N ≤ (c_tot/(48 - c_tot)) e^{4π} - 2 and an upper bound on the entropy at β=2π, σ ≤ π c_tot/12 + ln(48/(48 - c_tot)). These results illustrate how modular invariance can bound degeneracies under specific physical assumptions and open directions for refining bounds (e.g., via extended symmetries or elliptic genera) and applying them to related geometries like Calabi–Yau compactifications. The work emphasizes both the power and the limitations of current modular-invariance techniques in constraining the landscape of 2D CFTs.

Abstract

In this note we explore the application of modular invariance in 2-dimensional CFT to derive universal bounds for quantities describing certain state degeneracies, such as the thermodynamic entropy, or the number of marginal operators. We show that the entropy at inverse temperature 2 pi satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for fully general CFTs. We then restrict our attention to infrared stable CFT with moderately low central charge, in addition to the usual assumptions of modular invariance, unitarity and discrete operator spectrum. For CFT in the range c_left + c_right < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2 pi. Under the same conditions we also prove that a CFT can have a number of marginal deformations no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.