Improving Modular Bootstrap Bounds with Integrality

TL;DR

This paper introduces a new method to incorporate integrality constraints into numerical modular bootstrap, leading to tighter bounds on operator dimensions and eliminating non-physical solutions.

Contribution

The authors develop an efficient approach to impose integrality in modular bootstrap, improving bounds and refining the solution space in conformal field theories.

Findings

Improved bounds on operator gaps at c=3 with integrality assumption.

Elimination of non-physical solutions, leaving only discrete saturating points.

Slightly better upper bounds on scaling dimension gap near c=1.

Abstract

We implement methods that efficiently impose integrality -- i.e., the condition that the coefficients of characters in the partition function must be integers -- into numerical modular bootstrap. We demonstrate the method with a number of examples where it can be used to strengthen modular bootstrap results. First, we show that, with a mild extra assumption, imposing integrality improves the bound on the maximal allowed gap in dimensions of operators in theories with a $U(1)^c$ symmetry at $c=3$, and reduces it to the value saturated by the $SU(4)_1$ WZW model point of $c=3$ Narain lattices moduli space. Second, we show that our method can be used to eliminate all but a discrete set of points saturating the bound from previous Virasoro modular bootstrap results. Finally, when central charge is close to $1$, we can slightly improve the upper bound on the scaling dimension gap.