What question did this study set out to answer?

This work aims to explore gauge configurations and their cohomological properties in four-dimensional space.

July 1, 20260 citationsOpen Access

A Cohomological Obstruction Invariant for Gauge Configuration Types in Dimension Four: An Initial Contact with Yang-Mills Theory

Key Points

This work aims to explore gauge configurations and their cohomological properties in four-dimensional space.
Developed a homotopy-theoretic framework for gauge configuration spaces.
Defined mod-2 cohomology using mappings into Eilenberg–MacLane spaces.
Introduced an invariant through Steenrod operations on characteristic classes.
The invariant vanishes for dimensions d < 4, indicating triviality.
Demonstrated nontriviality of the invariant in dimension d = 4 for natural classes.
Characterized the unique role of four dimensions in gauge theory properties.

Abstract

This work presents a homotopy-theoretic framework for encoding gauge configurationspaces and constructing dimension-sensitive cohomological invariants. Using HomotopyType Theory (HoTT) as a conceptual foundation, I model gauge symmetries intrinsicallyand define mod-2 cohomology via maps into Eilenberg–MacLane spaces. Within this framework,I introduce an invariant obtained by applying Steenrod operations to characteristicclasses. Under explicit assumptions, I show that this invariant vanishes in dimensions d < 4and is nontrivial in dimension d = 4 for natural classes of examples. This provides ahomotopy-cohomological characterization of the special role of four dimensions in gaugetheory.

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