The Computational Complexity of the Weak Gravity Conjecture

TL;DR

This paper explores the computational challenges of verifying the Weak Gravity Conjecture in theories with multiple gauge fields, revealing exponential complexity growth that hinders explicit model construction.

Contribution

It introduces an algorithm to test the convex hull condition of the Weak Gravity Conjecture and demonstrates the exponential increase in computational complexity with more gauge fields.

Findings

Algorithm for convex hull construction in gauge theories

Exponential growth in computational time with gauge field number

Fundamental obstruction to explicit model realization

Abstract

The Weak Gravity Conjecture imposes stringent constraints on effective field theories to allow for an ultraviolet completion within quantum gravity. While substantial evidence supports the conjecture across broad classes of string theory-derived effective field theories, constructing low-dimensional models realizing it explicitly remains highly non-trivial. In this work, we illustrate how the presence of multiple gauge fields in an effective field theory significantly complicates the bottom-up implementation of the Weak Gravity Conjecture. To this end, we introduce a general algorithm that constructs the convex hull associated with a given set of superextremal states and verifies whether it satisfies the Convex Hull version of the Weak Gravity Conjecture. We show that the computational time of this construction grows exponentially with the number of gauge fields, thereby revealing a fundamental obstruction to concrete, algorithmic realizations of the conjecture in theories with many gauge fields.