Analytic Direct Calculation of the Topological Charge Structure for Calorons on S¹×R³

Introduction: Topological solitons are classical solutions to field equations whose stability is guaranteed by topological invariants known as topological charges. Among these, calorons are defined as topological solitons in SU(N) gauge theories on the compactified space S¹×R³, where one spatial direction is periodic, representing the inverse temperature. Due to the direct-product structure of this background space, the total topological charge Q of a caloron exhibits a nontrivial internal structure, conventionally split into Q = q1 + q2 [1]. However, a rigorous mathematical procedure to directly derive this exact formula from explicit gauge configurations of caloron solutions has not been widely established.

Methods: In this study, we perform a rigorous analytic evaluation of the topological charge and its internal structure directly utilizing explicit caloron configurations. By meticulously evaluating the boundary integrals and the behavior of the fields under general gauge configurations, we clarify the specific conditions under which the internal components of the charge vary, thereby providing a systematic mathematical classification of the topological sectors.

Results: While the conventional Nye–Singer formula [1] holds under restricted gauge conditions, its mathematical interpretation and correspondence with explicit integration in general gauge configurations have been ambiguous. Our analysis explicitly investigates the singularities of the integrand, the regularizing effects, the contributions in the monopole limit, and the precise behavioral changes under varying holonomy parameters [2]. Through these steps, we successfully uncover the detailed mathematical mechanism governing the direct derivation of the caloron's topological charge.

Conclusions: Clarifying the mathematical framework of calorons under general gauge choices provides foundational insights into the nonperturbative vacuum structure of finite-temperature gauge theories, which is deeply related to the quark confinement mechanism.

[1] T. M. W. Nye and M. A. Singer, “An L²-Index Theorem for Dirac Operators on S¹×R³,” J. Funct. Anal. 177 (2000) 203–218 [arXiv:math/0009144 Search...].
[2] A. Nakamula and G. Sumiyama, “Topology of Calorons Re-examined,” [arXiv:hep-th/2601.08543].