Generating functions for intersection products of divisors in resolved F-theory models

TL;DR

This paper introduces an efficient algorithm and a Mathematica package for computing intersection numbers of divisors in elliptic fibrations relevant to F-theory, significantly reducing computational complexity.

Contribution

The authors develop a symbolic algorithm and software for calculating intersection products in resolved F-theory models, improving efficiency over previous methods.

Findings

Successfully computed generating functions for all Tate models with classical groups up to rank twenty.

Demonstrated a substantial reduction in computation time compared to earlier approaches.

Highlighted the growth of computational complexity with the rank of the gauge group.

Abstract

Building on the approach of 1703.00905, we present an efficient algorithm for computing topological intersection numbers of divisors in a broad class of elliptic fibrations with the aid of a symbolic computing tool. A key part of our strategy is organizing the intersection products of divisors into a succinct analytic generating function, namely the exponential of the K\"ahler class. We use the methods of 1703.00905 to compute the pushforward of this function to the base of the elliptic fibration. We implement our algorithm in an accompanying Mathematica package IntersectionNumbers.m that computes generating functions of intersection products for resolutions of F-theory Tate models defined over smooth base of arbitrary complex dimension. Our algorithm appears to offer a significant reduction in computation time needed to compute intersection numbers as compared to previously explored implementations of the methods in 1703.00905; as an illustration, we explicitly compute the generating functions for all F-theory Tate models with simple classical groups of rank up to twenty and highlight the growth of the computation time with the rank of the group.