Determinacy of classical non-Abelian gauge fields by their field strengths, and realizability of the duals of sourceless fields

Two structural questions about classical (unquantized) non-Abelian gauge fields were posed by T. T. Wu and C. N. Yang in 1975: whether the field strengths determine the gauge field, and whether the dual of a sourceless field is again the field strength of a gauge potential. The 'whether' parts are settled — 'no' for multiply connected regions by the Aharonov-Bohm effect; 'no' even locally by Wu and Yang's explicit SU(2) potentials with identical field strengths but different sources; and 'not always' for the dual by Gu Chao-hao and Yang's examples. What remains open is the general characterization problem: (1) classify all gauge-inequivalent potentials (and their sources) compatible with prescribed field-strength data, including the global obstructions arising in multiply connected regions — the 'gauge field copy' problem; and (2) give necessary and sufficient conditions under which the Hodge dual of a sourceless non-Abelian field is realizable as the field strength of some gauge potential — a structured case of the prescribed-curvature problem. The 1975-2025 literature (Deser-Wilczek; Calvo; Mostow; Doria; Mostow-Shnider; Tsarev; Gross; DeTurck-Talvacchia; Chan-Tsou; Sant'Anna et al.; Kycia-Šilhan; Nastase-Sonnenschein) yields generic uniqueness theorems, partial classifications, counterexamples, and inversion algorithms with failure modes, but no complete general characterization of either facet.

Background

Setting. In a Yang-Mills gauge theory with non-Abelian gauge group $G$ (Yang and Mills, 1954), the dynamical variable is the gauge potential $b^k_\mu$ — a vector field taking values in the Lie algebra of $G$, with $k$ labeling the generators $X_k$ and real structure constants $c^k_{ij}$ — from which the field strength

$$f^k_{\mu\nu}=\partial_\nu b^k_\mu-\partial_\mu b^k_\nu-c^k_{ij}\,b^i_\mu b^j_\nu$$

and the source $J^k_\mu$ (the gauge-covariant divergence of $f^k_{\mu\nu}$) are built. A gauge transformation is a spacetime-dependent rotation in the internal symmetry group acting on the index $k$; it reshuffles $b$ and $J$ while leaving the physical content invariant, so 'the gauge field' is the equivalence class of $b$ under such rotations. 'Unquantized' means classical: no path integral or operator ordering enters. For the Abelian group $U(1)$ this reduces to electromagnetism, with $b_\mu$ the vector potential and $f_{\mu\nu}$ the electromagnetic field.

Determinacy: what do the field strengths fix? In a simply connected region, an Abelian field strength determines the potential up to gauge (a Poincaré-lemma argument). In a multiply connected region — one containing loops that cannot be shrunk to a point, such as the exterior of a solenoid — this fails: the Aharonov-Bohm effect (1959) shows that quantum interference depends on the loop integral of the potential even where the field strength vanishes, so the field strengths do not describe the gauge field completely. Wu and Yang (1975) then showed that for $SU(2)$ the failure is worse: even locally, the field strengths do not determine the potential. Their example lives on the region of three-space excluding the origin and the negative $z$-axis: one potential has field strengths $f^l_{ij}=\epsilon_{ijk}x^k x^l(1-\Phi^2)/r^4$ and a nonzero source proportional to $\Phi(\Phi^2-1)$, while a gauge rotation of a sourceless monopole-type potential of strength $-(1-\Phi^2)$ produces identical field strengths; since sources transform covariantly, the two potentials cannot be gauge-related. Two potentials sharing field strengths but not related by a gauge transformation were later called gauge field copies. The paper deferred the systematic multiply-connected treatment to 'a later communication', which appeared as the companion paper published immediately after it (received the same day): there Wu and Yang formulated the intrinsic global description of gauge fields through nonintegrable phase factors — path-dependent exponentials of the potential around loops, i.e. holonomy — defined by patching overlapping regions, and identified gauge fields with connections on principal fiber bundles (the mathematical object whose local data are potentials patched together over a base space). That companion paper settles the conceptual question: the gauge field is determined by its phase factors, not by the field strengths alone. The classification of what the field strengths nevertheless fix is another matter, addressed below.

Duality. In four dimensions the dual of an antisymmetric tensor is $f^{k*}_{\mu\nu}=\tfrac{1}{2}\epsilon_{\mu\nu\xi\eta}f^{k\xi\eta}$ (with a metric-density factor in curved space), exchanging electric and magnetic roles. Sourceless Maxwell theory is invariant under $f\mapsto f^*$: the dual of a sourceless electromagnetic field is again an electromagnetic field, with a dual potential obtainable locally. Wu and Yang asked whether the same holds for non-Abelian fields; the answer, from examples in Gu Chao-hao and Yang (1975), is 'not always' — there exist sourceless non-Abelian fields whose dual is not the field strength of any gauge potential, so the electric-magnetic symmetry is generally lost. No general criterion for when the dual is realizable was given; this is the second open facet.

Later work. The determinacy question became the field-copy problem: Calvo (1977) reduced it to the solvability of an overdetermined differential-algebraic system for the difference of two potentials; Mostow (1980) proved that the curvature and all of its covariant derivatives determine the connection up to global gauge when the infinitesimal holonomy group (the Lie algebra generated by parallel transport around infinitesimal loops) has constant dimension and the base is simply connected, and analyzed the topological obstructions appearing otherwise; Doria (1981) classified the copies into quasi-Abelian types (whose shared field strength is Abelian in character) and fully non-Abelian types; Mostow and Shnider (1983) constructed counterexamples to some claimed existence results; index-theoretic conditions for 'false' copies were derived by Sant'Anna, da Costa and Doria (1997). The dual-realizability question is a structured instance of the prescribed-curvature problem — deciding whether a given Lie-algebra-valued 2-form is the curvature $db+b\wedge b$ of some connection: Tsarev (1982) gave a local criterion; Gross (1985) proved a Poincaré lemma for connection forms; DeTurck and Talvacchia (1987) analyzed the existence theory; Kycia and Šilhan (2025) gave an inversion algorithm for the covariant exterior derivative whose constraints can fail. Chan and Tsou (1999) review a loop-space program constructing non-Abelian dual potentials with a doubled gauge group — a different notion of duality, not an answer to the classical question — and Nastase and Sonnenschein (2024) still note, when defining non-Abelian helicities, that a dual non-Abelian potential is not generally defined. What no later work supplies is the complete classification of copies with their global obstructions, or necessary and sufficient conditions for dual realizability of sourceless fields; that joint characterization problem is the one posed here.

Problem Statement

Let $G$ be a non-Abelian Lie group with real structure constants $c^k_{ij}$, let $b^k_\mu$ be a gauge potential on a region of four-dimensional spacetime, and let

$$f^k_{\mu\nu}=\partial_\nu b^k_\mu-\partial_\mu b^k_\nu-c^k_{ij}\,b^i_\mu b^j_\nu,\qquad f^{k*}_{\mu\nu}=\tfrac{1}{2}\epsilon_{\mu\nu\xi\eta}f^{k\xi\eta}$$

be its field strength and dual field strength (with the appropriate metric-density factor if the metric is not flat), and let $J^k_\mu$ denote the source, the gauge-covariant divergence of $f^k_{\mu\nu}$ with respect to $b^k_\mu$. Two linked characterization problems arise from the two questions raised — and answered negatively — in Wu and Yang (1975) ('whether' the field strengths determine the gauge field; 'whether' the dual of a sourceless field is again a field strength, with the counterexamples of Gu and Yang), and were formulated as open problems in the subsequent literature (the gauge-field-copy problem; the prescribed-curvature/dual-realizability problem). Both remain open:

(1) Determinacy and field copies. Given field-strength data $f^k_{\mu\nu}$ on a region — simply or multiply connected — realized by at least one gauge potential, characterize the set of all gauge potentials $b^k_\mu$ realizing it, up to gauge transformation, together with their sources; in particular, determine for which data this set contains potentials with different sources (gauge field copies, in the terminology of the later literature), and classify the global obstructions that arise in multiply connected regions.

(2) Dual realizability. For a sourceless field ($J^k_\mu=0$, the vacuum Yang-Mills equations), give necessary and sufficient conditions under which the dual $f^{k*}_{\mu\nu}$ is the field strength of some gauge potential; equivalently, characterize the obstruction — local (a prescribed-curvature integrability condition) and global (patching/holonomy data on the region) — that makes Gu Chao-hao and Yang's examples fail.

Both facets share one objective: determine exactly what the field-strength data alone fix about a classical non-Abelian gauge field.

References

1. Tai Tsun Wu and Chen Ning Yang, "Some remarks about unquantized non-Abelian gauge fields", Physical Review D 12, 3843-3844 (1975). DOI: 10.1103/PhysRevD.12.3843, https://doi.org/10.1103/PhysRevD.12.3843 (source paper)
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