On the Complexity of Effective Theories -- Seiberg-Witten theory

TL;DR

This paper explores the complexity of effective field theories using tame geometry, focusing on Seiberg-Witten theory, and demonstrates how dualities help maintain finite complexity in describing moduli spaces.

Contribution

It introduces a framework employing effective o-minimality and dualities to quantify and control the complexity of effective theories, especially near boundary regions.

Findings

Duality frames are essential for finite complexity.

Local cell decomposition helps organize complexity near boundaries.

Tame geometry provides a promising approach for effective theory classification.

Abstract

Motivated by the idea that consistent quantum field theories should admit a finite description, we investigate the complexity of effective field theories using the framework of effective o-minimality. Our focus is on quantifying the geometric and logical information required to describe moduli spaces and quantum-corrected couplings. As a concrete setting, we study pure $\mathcal{N}=2$ super-Yang-Mills theory along its quantum moduli space using Seiberg-Witten elliptic curves. We argue that the complexity computation should be organized in terms of local cells that cover the near-boundary regions where additional states become light, each associated with an appropriate duality frame. These duality frames are crucial for keeping the global complexity finite: insisting on a single frame extending across all such limits would yield a divergent complexity measure. This case study illustrates how tame geometry uses dualities to yield finite-complexity descriptions of effective theories and points towards a general framework for quantifying the complexity of the space of effective field theories.