Real Left-Invariant Spin Gaussian Fields

• Real left-invariant spin Gaussian fields are defined as the real parts of complex spin fields on SO(3), capturing spin-weighted data such as CMB polarization.
• They are constructed using harmonic analysis and homogeneous line bundles, ensuring left invariance in law while retaining spin-specific right-covariance.
• This framework enables explicit computation of excursion-set Lipschitz–Killing curvatures and chaos decompositions, which are essential for analyzing geometric and topological features in cosmological data.

Real left-invariant spin Gaussian fields are real-valued Gaussian random fields on $SO(3)$ obtained as the real part of complex spin random fields, and they provide a group-theoretic representation of spin-weighted data such as Cosmic Microwave Background polarization. In the representation-theoretic formulation of spin fields, a spin-$s$ random section over $S^2$ is encoded by a type-$s$ random function on $SO(3)$, with isotropy of the section corresponding to invariance in law under the left regular action of the group. Passing to the real part yields a real field on $SO(3)$ that is left-invariant in law but, for $s\neq 0$, not isotropic on $SO(3)$ in the usual scalar sense. This framework combines harmonic analysis on $SO(3)$, homogeneous line bundles, and Gaussian geometry, and underlies both exact formulas for excursion-set Lipschitz–Killing curvatures and Wiener–Itô chaos decompositions of level-set area (Baldi et al., 2013, Pistolato et al., 2024, Pistolato et al., 11 Jul 2025).

1. Group-theoretic and bundle-theoretic formulation

The foundational construction begins with a compact group $G$ acting transitively on a homogeneous space $s$0, where $s$1 is the isotropy subgroup of a base point $s$2. A random field $s$3 is taken to be almost surely square integrable, and isotropy is formulated in the strict sense by invariance in law of the linear functionals

$s$4

under the left regular action $s$5 (Baldi et al., 2013). For centered second-order isotropic fields, the covariance kernel

$s$6

is $s$7-invariant and is encoded by the positive definite function

$s$8

which is continuous, bi-$s$9-invariant, and positive definite (Baldi et al., 2013).

The scalar construction on homogeneous spaces is based on a convolution square root. Every such $S^2$0 admits a representation

$S^2$1

for some $S^2$2 that is bi-$S^2$3-invariant; if $S^2$4 is real-valued, $S^2$5 may be chosen real-valued (Baldi et al., 2013). Given an isometry $S^2$6 from $S^2$7 into a Gaussian Hilbert space, one defines

$S^2$8

Because $S^2$9 is $s$0-invariant, this is well defined. The resulting field is centered, Gaussian, mean-square integrable, and isotropic, with covariance determined by inner products of left translates of $s$1 (Baldi et al., 2013).

For spin fields on $s$2, the same mechanism is transferred from functions on a homogeneous space to sections of homogeneous line bundles. For $s$3, the spin $s$4 line bundle $s$5 is

$s$6

where $s$7 is the character defining the bundle action (Baldi et al., 2013). A section corresponds uniquely to a function $s$8 of type $s$9, meaning

$SO(3)$0

The induced action on sections,

$SO(3)$1

gives the notion of isotropy for random sections, and the paper proves that isotropy of a section is equivalent to isotropy of the pullback random field on $SO(3)$2 (Baldi et al., 2013).

A key structural point is the distinction between scalar and spin cases. For $SO(3)$3, real isotropic Gaussian fields on homogeneous spaces arise from real $SO(3)$4-invariant kernels. For $SO(3)$5, spin random fields are inherently complex-valued as bundle sections: the covariance under the $SO(3)$6-action prevents a nontrivial spin field from being real. This is precisely why the later literature studies the real part of a complex spin field as a real Gaussian field on $SO(3)$7, rather than as a real spin section on $SO(3)$8 (Baldi et al., 2013, Pistolato et al., 2024).

2. Harmonic analysis on $SO(3)$9 and the spin series model

On $SO(3)$0, with $SO(3)$1 and $SO(3)$2, the irreducible unitary representations are the Wigner matrices $SO(3)$3, $SO(3)$4, with entries $SO(3)$5, and the Peter–Weyl expansion organizes square-integrable functions on the group (Baldi et al., 2013). For type-$SO(3)$6 functions, only the $SO(3)$7-th Wigner column appears, and every type-$SO(3)$8 function admits the decomposition

$SO(3)$9

The spin harmonics are obtained from Wigner matrix coefficients through

$s\neq 0$0

and they form an orthonormal basis of $s\neq 0$1 (Baldi et al., 2013).

The modern probabilistic model used for real left-invariant spin Gaussian fields starts from the complex Gaussian series

$s\neq 0$2

where $s\neq 0$3 are Wigner $s\neq 0$4-matrix coefficients, the $s\neq 0$5 are i.i.d. complex Gaussian variables, and the $s\neq 0$6 specify the angular power spectrum (Pistolato et al., 2024). The field satisfies the right-spin covariance relation

$s\neq 0$7

which expresses the usual spin-weighted transformation law under a change of local tangent frame (Pistolato et al., 2024).

The associated real field is defined by

$s\neq 0$8

This real part is the object of geometric analysis because, in the formulation given in the 2024 work, the real and imaginary parts are completely correlated and therefore carry equivalent statistical information (Pistolato et al., 2024). The 2025 analysis uses the same model and imposes the normalization

$s\neq 0$9

together with the frequency parameter

$SO(3)$0

which determines the local second-order geometry induced by the field (Pistolato et al., 11 Jul 2025).

A useful way to interpret this harmonic expansion is that the complex spin section over $SO(3)$1 and the lifted complex field on $SO(3)$2 encode the same statistical content, while the real left-invariant field $SO(3)$3 is the natural real-valued surrogate for geometric and topological questions on the group manifold. This suggests that the passage from $SO(3)$4 to $SO(3)$5 is not merely a coordinate convenience but a structural re-expression of spin covariance in terms of group actions (Pistolato et al., 11 Jul 2025).

3. Left invariance, right-spin covariance, and non-isotropy

The defining symmetry of the complex spin model is right covariance under the circle subgroup generated by rotations about the vertical axis: $SO(3)$6 Taking real parts yields

$SO(3)$7

In particular, for $SO(3)$8,

$SO(3)$9

(Pistolato et al., 11 Jul 2025).

At the same time, the field is left-invariant in law: $SO(3)$0 and is also invariant in law under the right $SO(3)$1-action generated by $SO(3)$2: $SO(3)$3 (Pistolato et al., 11 Jul 2025). In covariance form, this becomes

$SO(3)$4

for the 2024 formulation (Pistolato et al., 2024), and more generally

$SO(3)$5

in the 2025 formulation (Pistolato et al., 11 Jul 2025).

A recurring misconception is to identify left invariance with isotropy on $SO(3)$6. The later papers explicitly reject this equivalence. The field is not isotropic on $SO(3)$7 in the usual scalar sense, because it is not right-invariant under the full group and its covariance is not a function of distance alone (Pistolato et al., 11 Jul 2025). Right invariance would be a much stronger symmetry and occurs only in the special homothetic case $SO(3)$8 (Pistolato et al., 2024). Thus the term “real left-invariant spin Gaussian field” refers to a real field whose law is invariant under left translations, while its residual right-structure retains the spin information.

This asymmetry is essential rather than accidental. The physical polarization signal is not scalar, and the right $SO(3)$9-action corresponds to frame rotations around the line of sight. A plausible implication is that the non-isotropic character of $G$0 on $G$1 is the geometric trace of the underlying bundle-valued nature of the spin field on $G$2 (Pistolato et al., 11 Jul 2025).

4. Geometry on $G$3 and the Adler–Taylor metric

The geometric analysis is carried out on $G$4 equipped with its standard Riemannian metric $G$5. In Euler-angle coordinates $G$6, the Gram matrix of $G$7 is

$G$8

(Pistolato et al., 2024). The 2025 paper describes the same geometry through the embedding

$G$9

and the projection

$s$00

which is a Riemannian submersion whose fibers are circles of length $s$01 (Pistolato et al., 11 Jul 2025).

For a nondegenerate smooth Gaussian field $s$02, the field-induced metric is the Adler–Taylor metric

$s$03

In the spin setting, this is another left-invariant metric $s$04, with local Gram matrix

$s$05

which at $s$06 reduces to

$s$07

(Pistolato et al., 2024, Pistolato et al., 11 Jul 2025). Relative to the standard metric, the eigenvalues are constant: $s$08 (Pistolato et al., 2024).

The metric is therefore non-homothetic in general. The classical Adler–Taylor Gaussian kinematic formula applies directly when

$s$09

but spin fields on $s$10 are not homothetic except in the special case $s$11 (Pistolato et al., 2024). This distinction is central to later developments, because the Lipschitz–Killing curvatures are computed with respect to the background geometry $s$12, not merely the field-induced geometry $s$13.

The 2025 paper further decomposes the gradient into horizontal and vertical components. Writing

$s$14

one has

$s$15

with orthogonality for both $s$16 and $s$17 (Pistolato et al., 11 Jul 2025). Moreover,

$s$18

and

$s$19

(Pistolato et al., 11 Jul 2025). This makes the spin contribution geometrically explicit: $s$20 controls the horizontal directions, while $s$21 controls the fiber direction.

The curvature of $s$22 can also be computed explicitly. Its scalar curvature is constant: $s$23 and the corresponding Lipschitz–Killing curvatures of $s$24 include

$s$25

with $s$26 and $s$27 (Pistolato et al., 2024).

5. Excursion geometry and expected Lipschitz–Killing curvatures

The excursion set of the real field is

$s$28

For unit-variance fields, the 2024 paper states that $s$29 is almost surely a smooth manifold with boundary, and derives explicit, non-asymptotic formulas for the expected Lipschitz–Killing curvatures measured with respect to the background metric $s$30 (Pistolato et al., 2024).

The theorem gives

$s$31

$s$32

$s$33

and

$s$34

(Pistolato et al., 2024). The $s$35 and $s$36 terms are expressed via the auxiliary expectations

$s$37

with $s$38 independent, specialized in the spin case to $s$39 (Pistolato et al., 2024).

Conceptually, these quantities are the Minkowski functionals in dimension three: volume, half the boundary area, Euler characteristic, and a curvature-sensitive functional involving mean and scalar curvature (Pistolato et al., 2024). Their role in cosmology is to detect geometric and topological deviations from Gaussianity and isotropy not visible in the power spectrum alone, particularly for the spin-2 polarization field relevant to LiteBIRD and to tests of inflationary physics and primordial gravitational waves (Pistolato et al., 2024).

The comparison with previous asymptotic work is also explicit. The exact formulas are presented as coherent with the asymptotic high-frequency results for spin-2 fields, and after identifying metric conventions they recover the same leading behavior (Pistolato et al., 2024). This suggests that real left-invariant spin Gaussian fields admit both an exact finite-frequency geometric theory and a high-frequency asymptotic regime, with the former sharpening the constants left implicit in earlier analyses.

6. Chaos decomposition of level-set area and the spin-sensitive fluctuation regime

The 2025 paper studies the level-set area

$s$40

especially for sets $s$41 that are unions of fibers over $s$42 (Pistolato et al., 11 Jul 2025). This is the second Lipschitz–Killing curvature of the excursion boundary in the geometric setting adapted to polarization.

The area measure admits the Wiener–Itô decomposition

$s$43

with convergence in $s$44, and only even chaos orders appear (Pistolato et al., 11 Jul 2025). A central ingredient is the fiberwise spin norm

$s$45

which is well defined because $s$46 (Pistolato et al., 11 Jul 2025). The resulting chaos expansion is

$s$47

for $s$48, where $s$49 is given explicitly in terms of hypergeometric and Beta factors (Pistolato et al., 11 Jul 2025).

The coefficient is written as

$s$50

with $s$51, while the derivation starts from a general nodal-chaos identity involving the Hermite terms $s$52 and a vertical-gradient chaos polynomial $s$53 (Pistolato et al., 11 Jul 2025).

The zero-spin case collapses to the scalar spherical model. When $s$54,

$s$55

for an isotropic Gaussian field $s$56 on $s$57, and

$s$58

so the area problem on $s$59 becomes a nodal-length problem on the sphere (Pistolato et al., 11 Jul 2025). The corresponding chaos expansion then replaces the spin-dependent fiber terms by scalar Hermite terms on $s$60.

A principal conclusion of the 2025 work is the separation between first-order and higher-order behavior. The leading expectation becomes asymptotically spin-blind in the high-frequency regime: $s$61 so the expected geometry has the same leading asymptotics for all spins (Pistolato et al., 11 Jul 2025). By contrast, higher chaos orders remain spin-sensitive because of the terms

$s$62

and the paper records the inequality

$s$63

(Pistolato et al., 11 Jul 2025). The stated implication is that expectations may lose sensitivity to spin at high frequency, whereas variances and higher-order fluctuations do not.

7. Relation to classical spin bundles and applications to CMB polarization

The representation-theoretic formulation is compatible with the classical spin-bundle formalism. The 2013 paper identifies the homogeneous line bundle $s$64 with the usual spin-$s$65 bundle on $s$66 by showing that the transition functions in local charts are

$s$67

which coincide with the classical spin transition factors $s$68 (Baldi et al., 2013). This establishes that the global $s$69-based formulation and the local coordinate-based Newman–Penrose and Geller–Marinucci descriptions are equivalent, while the group-theoretic version is cleaner for representation-theoretic purposes (Baldi et al., 2013).

The cosmological motivation is explicit in both papers. Spherical spin random fields are used to model Cosmic Microwave Background polarization, and polarization is a spin-$s$70 object rather than a scalar field (Pistolato et al., 2024, Pistolato et al., 11 Jul 2025). The lift to $s$71 is natural because the right $s$72-action encodes rotation of the local polarization frame around the line of sight (Pistolato et al., 11 Jul 2025). In this setting, real left-invariant spin Gaussian fields provide a real-valued geometric observable whose excursion sets and level sets can be analyzed with exact differential-geometric and stochastic tools.

Two complementary lines of investigation therefore meet in this topic. The first is representational: every isotropic complex Gaussian spin section arises from a bi-$s$73-associated $s$74-kernel via convolution square roots on $s$75 (Baldi et al., 2013). The second is geometric-probabilistic: the real part of the lifted field admits explicit formulas for expected Lipschitz–Killing curvatures and for Wiener–Itô chaos decompositions of level-set area (Pistolato et al., 2024, Pistolato et al., 11 Jul 2025). Taken together, these results place real left-invariant spin Gaussian fields at the intersection of harmonic analysis on compact groups, Gaussian random geometry, and the statistical theory of polarized cosmological data.