The topological Maxwell theorem beyond real dimension three: multipole representations of harmonics, the topology of complex multipole configurations, and the smooth structure of the associated orbit spaces

A 1996 paper of V. I. Arnold showed that the classical Maxwell–Sylvester multipole representation of spherical harmonics in three real variables has exact topological content: the symmetric power $\mathrm{Sym}^{n}(\mathbb{RP}^{2})$ is diffeomorphic to $\mathbb{RP}^{2n}$, and two orbit spaces of Coxeter-type groups acting on $(\mathbb{CP}^{1})^{n}$ are identified with $\mathbb{RP}^{2n}$ and $S^{2n}$. The author explicitly left parts of this theory unproved or undeveloped: the smoothing of the real algebraic homeomorphisms beyond his Section 4 (Section 4, together with the nonvanishing-Jacobian argument of Theorem 7, itself completes the proof of the main identification $\mathrm{Sym}^{n}(\mathbb{RP}^{2})\cong\mathbb{RP}^{2n}$ as a diffeomorphism) and the detailed relation to the independent computation of symmetric powers of nonorientable surfaces by Dupont and Lusztig; higher-dimensional and representation-theoretic generalizations were not developed in the paper itself. Later work related the two constructions at survey level and advanced the quadratic-surface and complex cases (Katz 2007; Gichev 2026), but not the systematic generalization theory. The problem is to determine that theory: which multipole-representation and orbit-space statements survive for harmonic polynomials in $m\geq 4$ real variables (including Shubin's conjecture), what the global topology of the complex multipole correspondence is, and whether the remaining orbit-space identifications hold in the smooth category.

Background

A spherical harmonic of degree $n$ (called a spherical function in Arnold's terminology) is the restriction to the unit sphere $S^{2}\subset\mathbb{R}^{3}$ of a homogeneous polynomial $h$ of degree $n$ in the three variables $x,y,z$ satisfying the Laplace equation $\Delta h=0$; such polynomials are called harmonic, and the space of degree-$n$ harmonic polynomials in three variables has dimension $2n+1$. The classical theorem of Maxwell and Sylvester states that, up to a constant factor, every such $h$ is the iterated directional derivative $\nabla_{u_{1}}\cdots\nabla_{u_{n}}(1/r)$ of the Coulomb potential $1/r$ along $n$ constant vector fields, and that the unordered multiset of projectivized directions $[u_{i}]\in\mathbb{RP}^{2}$ (the space of lines through the origin in $\mathbb{R}^{3}$) is determined by $h$ uniquely up to a common rescaling of the $u_{i}$ and permutation. The directions $[u_{i}]$ are the multipoles or poles of $h$; in quantum mechanics the same data is called the Majorana sphere representation of a spin-$n/2$ state. Maxwell's construction makes this a topological statement: the unordered configuration space $\mathrm{Sym}^{n}(X)=X^{n}/S(n)$ of $n$ indistinguishable points of a space $X$ (the $n$-th symmetric power, the quotient of the $n$-fold product by the permutation group $S(n)$; also written $\mathrm{SP}^{n}$) is, for $X=\mathbb{RP}^{2}$, a closed $2n$-manifold, and Maxwell's theorem identifies $\mathrm{Sym}^{n}(\mathbb{RP}^{2})$ with the projectivized space $\mathbb{RP}^{2n}$ of degree-$n$ harmonics. Arnold's 1996 paper proves this in two stages: an explicit real algebraic homeomorphism built from Sylvester's algebraic decomposition $f=pg+(x^{2}+y^{2}+z^{2})h$, where a real homogeneous polynomial of degree $n$ decomposes uniquely as a product $p$ of $n$ real linear factors (its 'main axes') plus a multiple of the radius-squared quadratic form, followed (in Section 4, Theorem 7) by a nonvanishing-Jacobian argument showing the map is a local diffeomorphism. A remark in the paper states that in most other cases where two manifolds are said to 'coincide' only a real algebraic homeomorphism is provided and the pedantic checking that it can be smoothed is left to the reader. Independently of Maxwell, Dupont and Lusztig computed in 1971 the symmetric powers of all closed nonorientable surfaces using the Abel–Jacobi map (the classical homomorphism from divisors on a curve to its Jacobian variety) on the orientable double cover: for $\dim_{\mathbb{Q}}H_{1}(X)=g$ and $n\geq g$, $\mathrm{SP}^{n}(X)$ is diffeomorphic to a real projective $(2n-g)$-bundle over the $g$-torus, which for $X=\mathbb{RP}^{2}$ recovers $\mathrm{SP}^{n}(\mathbb{RP}^{2})\cong\mathbb{RP}^{2n}$. Arnold's paper additionally identifies orbit spaces of a larger group acting on $(\mathbb{CP}^{1})^{n}$: identifying each real direction with a pair of complex conjugate lines in $\mathbb{C}^{2}$ via the standard identification of $S^{2}$ with complex lines, one lets $B(n)$ be the group generated by permutations of the $n$ factors of $(\mathbb{CP}^{1})^{n}$ and by the antiholomorphic involutions that replace one line by its hermitian orthocomplement (the involution induced by the quaternionic structure $j$ of $\mathbb{C}^{2}$), and $D(n)$ its index-two subgroup using only even numbers of such replacements. Theorems 3 and 8 of the paper state $(\mathbb{CP}^{1})^{n}/B(n)\approx\mathbb{RP}^{2n}$ and $(\mathbb{CP}^{1})^{n}/D(n)\approx S^{2n}$, the latter suggested as an informal extension of Chevalley's theorem, the classical result that the orbit space of a finite real reflection group (Coxeter group) acting on $\mathbb{R}^{N}$ is an affine space, namely the space of invariant polynomials. Over the complex field the representation changes character: a polynomial is called completely reducible if it factors as a product of linear forms, and for a generic complex harmonic $h$ of degree $n$ the decomposition $h=p+r^{2}q$ with $p$ completely reducible of degree $n$ exists but is not unique (there are $(2n-1)!!$ such $p$ for generic $h$). In $m\geq 4$ real variables the fundamental solution of the Laplacian is $r^{2-m}$, and the direct analogue of Maxwell's construction — expressing every degree-$d$ harmonic polynomial as a sequence of $d$ directional derivatives of the potential $r^{2-m}$ parametrized by unordered projective directions — is recorded by Katz as apparently failing, without proof; a proposed $n$-dimensional variant (Shubin's conjecture, stated by Katz) replaces the monomial product of directional derivatives by a general polynomial differential operator. The survey of Degtyarev and Kharlamov positions the source paper as giving, via Maxwell, the higher-dimensional extension of the classical fact $\mathbb{CP}^{2}/\mathrm{conj}\cong S^{4}$ (the quotient of the complex projective plane by complex conjugation).

Problem Statement

Determine the generalized topological Maxwell theory: the exact boundary within which the multipole representation of harmonic polynomials and its symmetric-product and orbit-space identifications survive beyond the real three-dimensional setting, and close the smooth-category gaps left open by the source. Concretely, with the terminology of the background and $r^{2}=x_{1}^{2}+\cdots+x_{m}^{2}$:

(1) Higher dimensions. For harmonic polynomials of degree $d$ in $m\geq 4$ real variables (harmonic meaning annihilated by the $m$-variable Laplacian, fundamental solution $r^{2-m}$): determine for exactly which pairs $(m,d)$ every degree-$d$ harmonic polynomial can be represented by directional derivatives of the potential $r^{2-m}$ parametrized by unordered projective directions, and, where representations exist, determine the uniqueness class and the fibers of the multipole map (the analogue, in apolarity theory, of secant-variety parametrization and its degeneracy) and the topology of the corresponding multipole configuration spaces — the analogue of $\mathrm{Sym}^{d}(\mathbb{RP}^{2})\cong\mathbb{RP}^{2d}$. This includes settling or refuting Shubin's conjecture: for every real homogeneous harmonic polynomial $P$ of degree $d$ in $m>2$ variables there exists a unique real homogeneous harmonic polynomial $P^{\clubsuit}$ such that $P=r^{2d+m-2}\,P^{\clubsuit}(\partial x_{1},\dots,\partial x_{m})\,r^{2-m}$, with $P$ and $P^{\clubsuit}$ proportional by a constant depending only on $d$ and $m$; and proving or disproving the assertion of Katz that the monomial-operator multipole representation is a three-dimensional phenomenon.

(2) Complex field. Determine the global structure of the complex Maxwell correspondence for the sphere: describe the incidence variety $\{(h,p):p\in\Lambda_{h}\}$ of harmonic polynomials of degree $n$ and their completely reducible associates over the space of harmonics — its cardinality over the generic stratum is $(2n-1)!!$ (Katz 2007, Theorems 6–7: the multipole map is onto with generic fiber of cardinality $(2n-1)!!$, and $\Psi_{Q}:\mathrm{Sym}^{n}(\mathbb{CP}^{2})\to\mathbb{CP}^{2n}$ is a finite ramified covering of degree $(2n-1)!!$ ramified over a codimension-one discriminant, with $K(\pi,1)$ complements for modified braid groups; the count was re-proved together with the center-of-mass identity by Gichev 2026) — including the monodromy of the covering over the generic stratum, its degeneration over the discriminant locus (where decompositions become non-generic or infinite in number), and the topology of the resulting complex multipole configuration spaces (Katz's modular spaces $M(k)$); in particular, determine what replaces the real identification $\mathrm{Sym}^{n}(\mathbb{RP}^{2})\cong\mathbb{RP}^{2n}$ over the complex numbers.

(3) Smooth completion of the orbit spaces. Prove, in the smooth category and with the quotient smooth structures, that $(\mathbb{CP}^{1})^{n}/D(n)$ is diffeomorphic to the standard sphere $S^{2n}$ and $(\mathbb{CP}^{1})^{n}/B(n)$ to $\mathbb{RP}^{2n}$, where $B(n)$ is the group of permutations of the $n$ factors together with the antiholomorphic involutions replacing a line by its hermitian orthocomplement and $D(n)\subset B(n)$ the index-two subgroup of elements involving an even number of such replacements — including at orbits with nontrivial isotropy (degenerate configurations with coinciding or mutually hermitian-orthogonal lines).

The three components share one objective: determine which parts of the topological content of Maxwell's theorem survive beyond real dimension three and over the complex field, and close the explicitly deferred smoothness verifications of the source. The boundary is sought within each of the three components over the relevant ranges of $(m,d,n)$, at both the generic and the degenerate (non-generic) strata.

References

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