The Geometry of the Modular Bootstrap

TL;DR

This paper investigates the geometric structure of the modular bootstrap, revealing convex and non-convex regions in the space of partition function coefficients, and develops methods to bound and constrain these coefficients using semi-definite programming.

Contribution

It introduces a geometric interpretation of the modular bootstrap via convex hulls and Hankel matrices, and proposes an analytic approach to enforce integrality constraints.

Findings

Bounds on the gap and twist-gap derived

Kinks in the coefficient space identified

Integrality conditions significantly restrict allowed regions

Abstract

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on $R^+{\otimes}\mathbb{Z}$, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as constant computation time with increasing central charge. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above $\Delta^*_{gap}$, where $\frac{c{-}1}{12}<\Delta^*_{gap}< \frac{c}{12}$, all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.