Seiberg-witten Theory Complexity Reveals Finite Descriptions for Effective Field Theories Using O-minimality

The challenge of describing complex physical systems with simplified, effective theories drives current research in theoretical physics, and Martin Carrascal, Ferdy Ellen, Thomas W. Grimm, and David Prieto from the Institute for Theoretical Physics, Utrecht University, now investigate how to quantify the inherent complexity of these descriptions. They explore the geometric and logical requirements for accurately representing moduli spaces and couplings within effective field theories, using pure super-Yang-Mills theory and Seiberg-Witten elliptic curves as a concrete example.

The team demonstrates that complexity calculations benefit from focusing on local regions where new states emerge, employing different duality frames to maintain a finite overall complexity measure, a crucial step towards a general framework for quantifying the complexity of the entire space of effective field theories. This approach reveals how dualities and tame geometry combine to yield manageable descriptions of complex physical phenomena. This work introduces a complexity measure, denoted by F, and provides its explicit form for the Log-Noetherian structure RLN and its Pfaffian extension RLN,PF. The research demonstrates that organizing complexity computations in terms of local cells, covering near-boundary regions where new states appear, is crucial for maintaining finite complexity.

The team measured the format F, which characterizes the number of logical components needed to define a set or function, and relates to the frequency of derivative sign changes within a function. They show that any set within an effective o-minimal structure can be decomposed into a finite number of cells, and that the number of connected components on these cells is governed by a primitive recursive function of the format F. Measurements confirm that the format F provides an estimate of the volume of functions, and is linked to the number of logical components required to define a set. Experiments revealed that the complexity measure F can be applied to Seiberg-Witten theory, and also encompasses finite-loop Feynman amplitudes as functions of masses and external momenta. The research demonstrates that the format F characterizes the number of times basic functions, such as multiplication and addition, appear in defining a polynomial, and therefore quantifies the number of real coefficients needed to fix it. Data shows that the complexity measure F is a powerful tool for analyzing the geometric and logical structure of effective field theories, providing a framework for quantifying their complexity and ensuring finite descriptions.

Effective Field Theory Complexity via o-minimality This research develops a framework for quantifying the complexity of effective field theories, building on the mathematical concept of o-minimality.

The team demonstrates how to assess the geometric and logical complexity required to describe these theories, specifically focusing on pure super-Yang-Mills and its moduli space as a concrete example. By analysing the theory through Seiberg-Witten elliptic curves, they establish a method for patching together local descriptions of the effective field, ensuring a finite overall complexity. The key achievement lies in applying recent advances in effective o-minimality, particularly the introduction of Log-Noetherian functions, to assign a quantifiable complexity to period integrals and, consequently, to the effective field theories themselves. This approach avoids divergent complexity measures that would arise from attempting to describe the entire theory with a single, overarching framework, instead relying on a patchwork of local descriptions. The researchers acknowledge that the current framework, while robust, is limited by the size of the effectively o-minimal structure employed, and further work is needed to expand its capacity for handling more complex theories. Future research directions include exploring larger structures within effective o-minimality to accommodate a wider range of physical scenarios and refining the methods for assigning and tracking complexity in increasingly intricate field theories. 👉 More information 🗞 On the Complexity of Effective Theories — Seiberg-Witten theory 🧠 ArXiv: https://arxiv.org/abs/2512.11029 Tags: Rohail T. As a quantum scientist exploring the frontiers of physics and technology. My work focuses on uncovering how quantum mechanics, computing, and emerging technologies are transforming our understanding of reality. I share research-driven insights that make complex ideas in quantum science clear, engaging, and relevant to the modern world. Latest Posts by Rohail T.: Cognisnn Enables Neuron-Expandability and Dynamic-Configurability Via Random Graph Architectures in Spiking Neural Networks December 16, 2025 Magneto-arpes Reveals Momentum-Dependent Symmetry Breaking in CsV Sb, Confirming Exotic Charge Density Order December 16, 2025 Quantum Simulation of Fermion Dynamics Achieves Local Encoding with Flow Sets for Scalable Systems December 16, 2025