Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

TL;DR

This paper introduces a numerical algebraic geometry approach that overcomes limitations of symbolic methods, enabling the solution of complex problems in string and gauge theories through highly parallelizable computations.

Contribution

It presents a novel numerical paradigm that surpasses symbolic algebraic geometry in solving physics-related problems, demonstrating its effectiveness in string and gauge theories.

Findings

Successfully solved complex physics problems previously intractable by symbolic methods

Demonstrated the high parallelizability of the numerical approach

Provided new insights into string and gauge theories through computational solutions

Abstract

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gr\"{o}bner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.