What question did this study set out to answer?

The research explores the generic non-existence of global sections of smooth functions on compact moduli spaces.

January 25, 2026Open Access

ZiZhu Generic Continuous Weak Non-Existence of Sections Theorem

Key Points

The research explores the generic non-existence of global sections of smooth functions on compact moduli spaces.
Utilized static geometry and topological analysis for insights.
Focused on smooth functions within compact smooth moduli spaces.
Avoided any dynamical or temporal assumptions.
Demonstrated that finite complexity global sections are typically non-existent.
Linked the findings to physical phenomena in Higgs gauge theory and low-dimensional quantum systems.
Explained the significance of Goldstone modes and topological defects like monopoles.

Abstract

This paper investigates the generic structure of smooth functions on compact smooth moduli spaces, revealing that, in a typical sense, global sections of finite complexity do not exist. This conclusion carries direct physical significance: in Higgs gauge theory, it corresponds to the non-flatness of the vacuum moduli space SU(2)/U(1) ≃ S2, thereby explaining the appearance of Goldstone modes and topological defects such as monopoles; in low-dimensional quantum systems and anyon statistics, it corresponds to the multivaluedness of the state space and the intrinsic degeneracy of low-energy degrees of freedom. The approach of this work is built entirely on static geometry and topological analysis, relying neither on time nor on dynamical assumptions. It provides a unified, pre-dynamical structural perspective that clarifies the relationship between the local description of effective field theory and the global geometry.

ZiZhu Generic Continuous Weak Non-Existence of Sections Theorem

Key Points

Abstract

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