When does a conformal rescaling preserve Yang's parallel-displacement gauge field and the pure-space (Stephenson–Kilmister–Yang) condition?

Yang's two-page 1977 paper 'Conformal mapping of gauge fields' defines a conformal mapping of gauge fields between conformally related metrics $g'=Kg$ (same potential components in the same parametrization) and proves that sourcelessness is preserved exactly in four dimensions, while a certain orthogonality class is always preserved. The paper explicitly emphasizes two non-preservation facts: the conformal image of the parallel-displacement gauge field of a space $S$ is in general not the parallel-displacement gauge field of the rescaled space $S'$ (flat-space-versus-sphere example), and a pure space need not remain pure. Flatness of the image connection is automatic when the original connection is flat, since gauge field strengths are metric-independent; the substantive question is whether the two parallel-displacement gauge fields coincide up to gauge. Degenerate classes are partially classified (conformally flat pure spaces are exactly those of constant scalar curvature), but no found work determines when the parallel-displacement gauge field or purity is preserved. The classification problem remains open.

Background

A Yang–Mills gauge field on a manifold with metric $g$ is given by Lie-algebra-valued potentials $b^i_\mu$ with field strengths $f^i_{\mu\nu}$; its source $J^k_\mu$ is the gauge-covariant divergence of the field strength with one index raised by $g$, and a sourceless (vacuum) field satisfies $J^k_\mu=0$. Yang's parallel-displacement gauge field (PDGF) is the gauge field obtained from parallel transport of vectors on a pseudo-Riemannian $n$-manifold: its gauge group is $GL(n)$, its potentials are the Christoffel symbols (one symbol for each frame index and each parameter index $\mu$), its field strengths are the components of the Riemann curvature tensor, and its source is the Ricci-type contraction $R_{\mu\nu;}{}^{\nu}$. A space whose PDGF is sourceless is called a pure space; the condition is $R_{\mu\nu;\lambda}=R_{\mu\lambda;\nu}$, where the semicolon denotes the Levi-Civita covariant derivative. Historically this third-order equation was proposed as a gravitational field equation by Stephenson and Kilmister (1953) and Kilmister and Newman (1961), and it is now called the Stephenson–Kilmister–Yang (SKY) equation; Yang's 1974 integral formalism revived it by defining pure spaces as sourceless PDGFs and showing that four-dimensional Einstein spaces (Ricci tensor proportional to the metric) are pure. The converse fails: Thompson (1975) showed the condition forces constant scalar curvature, that it is equivalent to constant scalar curvature together with vanishing Cotton tensor (the conformally sensitive tensor built from the Schouten tensor $P_{\mu\nu}=\frac{1}{n-2}(R_{\mu\nu}-\frac{R}{2(n-1)}g_{\mu\nu})$ as $C_{\mu\nu\lambda}=\nabla_\lambda P_{\mu\nu}-\nabla_\nu P_{\mu\lambda}$, equivalent to the vanishing divergence of the Weyl conformal tensor (the trace-free part of the Riemann curvature tensor, which is itself invariant under conformal rescaling)), and that within the conformally flat class ($n\geq 4$) purity is equivalent to constant scalar curvature; Pavelle (1975) exhibited unphysical spherically symmetric pure solutions. Later SKY literature (Szczyrba 1987; Maluf 1988/1991; Shen 2009; Chen–Izumi–Tung 2013) studies dynamics, degrees of freedom, and matter couplings, not conformal rescaling. Camenzind (1975) introduced a matter-current term as a source for the SKY equation; adding it defines the Stephenson–Kilmister–Yang–Camenzind (SKYC) theory later developed by Chen–Izumi–Tung (2013). The 1977 paper defines the conformal mapping of gauge fields: if $S'$ has metric $g'=Kg$ in the same parametrization, a gauge field on $S$ maps to the field on $S'$ with the identical potential components $b^i_\mu$. Since gauge field strengths are metric-independent, the field strengths of the image equal those of the original. Theorem 1 (arbitrary $n$): the sources transform as $(J^k_\mu)'=K^{-1}J^k_\mu+\frac{n-4}{2K^2}f^k{}_{\mu\beta}\partial^\beta K$, so sourcelessness is preserved exactly for $n=4$ (any signature). Theorem 2: gauge fields satisfying the algebraic orthogonality identity $f_{i\mu\nu}f_{j\lambda}{}^\nu=a^2G_{ij}g_{\mu\lambda}+aC^k{}_{ij}f_{k\mu\lambda}$ (where $a$ is a scalar amplitude, $C^k{}_{ij}$ are the gauge-group structure constants, and $G_{ij}$ is the gauge-group metric) map to orthogonal fields with amplitude $a'=a/K$. The paper then emphasizes that the conformal image of the PDGF of $S$ is in general not the PDGF of $S'$ (example: flat $S$ versus a conformal sphere $S'$, related by stereographic projection), and that a pure space need not remain pure under $g\to Kg$. The untouched target is the classification of when these two structures are nevertheless preserved. Two reductions delimit the problem: if literal equality of potential components (not just gauge equivalence) is demanded, the standard Christoffel transformation law forces $K$ to be a constant; and for conformally flat backgrounds Thompson's theorem reduces purity preservation to the rescaled metric having constant scalar curvature, a Yamabe-type problem about which metrics in a conformal class have constant scalar curvature.

Problem Statement

Let $S$ be a connected $n$-dimensional pseudo-Riemannian manifold with metric $g$ (any signature, any $n\geq 2$), let $K:S\to\mathbb{R}_{>0}$ be smooth and positive, as in Yang (1977), let $g'=Kg$ with the same parametrization, and let the conformal mapping of gauge fields act by keeping potential components $b^i_\mu$ fixed, as in Yang (1977). Establish the conformal behavior of the parallel-displacement gauge field under this mapping. Component (1), PDGF preservation: determine necessary and sufficient conditions on $(g,K)$ such that the conformal image of the parallel-displacement gauge field of $(S,g)$ coincides, as a gauge field (that is, up to a local $GL(n)$ gauge transformation), with the parallel-displacement gauge field of $(S,g')$. If literal equality of potential components is demanded instead of gauge equivalence, the standard transformation law for Christoffel symbols forces $K$ to be constant, so the gauge-class question is the substantive one; flatness of the image connection is automatic when the original connection is flat, since gauge field strengths are metric-independent. Component (2), purity preservation: classify the pairs $(g,K)$ for which both $g$ and $Kg$ are pure, i.e., both satisfy the Stephenson–Kilmister–Yang condition $R_{\mu\nu;\lambda}=R_{\mu\lambda;\nu}$, equivalently both have constant scalar curvature and vanishing Cotton tensor (for $n=2$ the Cotton tensor vanishes identically by convention and purity reduces to constant scalar curvature). For conformally flat $g$ this reduces by Thompson's theorem to $Kg$ having constant scalar curvature (a Yamabe-type problem); the general non-conformally-flat case — in particular, whether nonconstant $K$ can preserve purity for non-conformally-flat pure backgrounds — is the open target. Note that Yang's four-dimensional sourcelessness invariance does not decide this question, because the conformal image of the PDGF is not the PDGF of the rescaled space.

References

1. Chen Ning Yang, 'Conformal mapping of gauge fields', Physical Review D 16, 330–331 (1977), DOI 10.1103/PhysRevD.16.330, https://doi.org/10.1103/PhysRevD.16.330
2. Chen Ning Yang, 'Integral formalism for gauge fields', Physical Review Letters 33, 445–447 (1974), DOI 10.1103/PhysRevLett.33.445, https://doi.org/10.1103/PhysRevLett.33.445
3. A. H. Thompson, 'Yang's Gravitational Field Equations', Physical Review Letters 34, 507–508 (1975), DOI 10.1103/PhysRevLett.34.507, https://doi.org/10.1103/PhysRevLett.34.507
4. A. H. Thompson, 'Geometrically Degenerate Solutions of the Kilmister-Yang Equations', Physical Review Letters 35, 320–322 (1975), DOI 10.1103/PhysRevLett.35.320, https://doi.org/10.1103/PhysRevLett.35.320
5. Richard Pavelle, 'Unphysical Solutions of Yang's Gravitational-Field Equations', Physical Review Letters 34, 1114 (1975), DOI 10.1103/PhysRevLett.34.1114, https://doi.org/10.1103/PhysRevLett.34.1114
6. G. Stephenson and C. W. Kilmister, 'A unified field theory of gravitation and electromagnetism', Il Nuovo Cimento 10, 230–235 (1953), DOI 10.1007/BF02786194, https://doi.org/10.1007/BF02786194
7. D. E. Littlewood, 'Conformal transformations and kinematical relativity', Mathematical Proceedings of the Cambridge Philosophical Society 49, 90–96 (1953), DOI 10.1017/s0305004100028085, https://doi.org/10.1017/s0305004100028085 (cited by Thompson (1975) as an early source of non-Einstein-type pure solutions)
8. C. W. Kilmister and G. Stephenson, 'Field equations in general relativity', Il Nuovo Cimento 1, 361–362 (1955), DOI 10.1007/BF02900647, https://doi.org/10.1007/BF02900647
9. C. W. Kilmister and D. J. Newman, 'The use of algebraic structures in physics', Mathematical Proceedings of the Cambridge Philosophical Society 57, 851–864 (1961), DOI 10.1017/s0305004100036008, https://doi.org/10.1017/s0305004100036008
10. Gu Chaohao, 'Conformally flat spaces and solutions to Yang-Mills equations', Physical Review D 21, 970–971 (1980), DOI 10.1103/PhysRevD.21.970, https://doi.org/10.1103/PhysRevD.21.970
11. Victor Szczyrba, 'Stephenson-Kilmister-Yang theory of gravity and its dynamics', Physical Review D 36, 351–374 (1987), DOI 10.1103/PhysRevD.36.351, https://doi.org/10.1103/PhysRevD.36.351
12. J. W. Maluf, 'The Stephenson-Kilmister-Yang Hamiltonian', Classical and Quantum Gravity 5, L81–L85 (1988), DOI 10.1088/0264-9381/5/5/002, https://doi.org/10.1088/0264-9381/5/5/002
13. José W. Maluf, 'On the absence of physical degrees of freedom in the Stephenson-Kilmister-Yang theory', Journal of Mathematical Physics 32, 1556–1559 (1991), DOI 10.1063/1.529265, https://doi.org/10.1063/1.529265
14. Jian Qi Shen, 'Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation', International Journal of Theoretical Physics 48, 1566–1582 (2009), DOI 10.1007/s10773-009-9929-9, https://doi.org/10.1007/s10773-009-9929-9
15. Pisin Chen, Keisuke Izumi, Nien-En Tung, 'Natural emergence of cosmological constant and dark radiation from the Stephenson-Kilmister-Yang-Camenzind theory of gravity', Physical Review D 88, 123006 (2013), DOI 10.1103/PhysRevD.88.123006, https://doi.org/10.1103/PhysRevD.88.123006
16. M. Camenzind, 'On the curvature dynamics for metric gravitational theories', Journal of Mathematical Physics 16, 1023–1031 (1975), DOI 10.1063/1.522654, https://doi.org/10.1063/1.522654 (origin of the matter-current term added to the SKY equation)
17. F. L. Teixeira, 'Differential form approach to the analysis of electromagnetic cloaking and masking', Microwave and Optical Technology Letters 49, 2051–2053 (2007), DOI 10.1002/mop.22640, https://doi.org/10.1002/mop.22640
18. I. Robinson and A. Trautman, 'Conformal geometry of flows in n dimensions', Journal of Mathematical Physics 24, 1425–1429 (1983), DOI 10.1063/1.525878, https://doi.org/10.1063/1.525878
19. K. L. Duggal, 'Relativistic fluids with shear and timelike conformal collineations', Journal of Mathematical Physics 28, 2700–2704 (1987), DOI 10.1063/1.527715, https://doi.org/10.1063/1.527715
20. Oscar Castillo-Felisola, 'Beyond Einstein: A Polynomial Affine Model of Gravity', arXiv:1902.09131 (2019), https://arxiv.org/abs/1902.09131