Exceptional supersphere integration and logarithmic Pizzetti kernels
Abstract
We study orthosymplectically invariant supersphere integration at the exceptional superdimensions M=−2u, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function 𝒢μ(ρ;x,y) =Γ(μ/2)/2πμ/2 (1+ρ{x,y}+ρ2x2y2)−μ/2. At μ=−2u, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees k≤ u, the residue inverts a canonical renormalized pairing on 𝒫k. In the collision range u<k≤2u, the ordinary pairing has radical (x2)k−u𝒫2u−k; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For k≥2u+1, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head--socle pairing on the generalized harmonic modules. As an application, we derive covariant right--left radial q-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided q-Fischer projection.
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BibTeX
@misc{boryreyes2026exceptional,
title = {{Exceptional supersphere integration and logarithmic Pizzetti kernels}},
author = {Juan Bory-Reyes and Baruch Schneider and Diana Schneiderova and Yifan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.21241},
eprint = {2607.21241},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21241},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Exceptional supersphere integration and logarithmic Pizzetti kernels
AU - Juan Bory-Reyes
AU - Baruch Schneider
AU - Diana Schneiderova
AU - Yifan Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2607.21241
UR - https://arxiv.org/abs/2607.21241
AB - We study orthosymplectically invariant supersphere integration at the exceptional superdimensions M=−2u, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function 𝒢_μ(ρ;x,y) =Γ(μ/2)/2π^μ/2 (1+ρ{x,y}+ρ^2x^2y^2)^−μ/2. At μ=−2u, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees k≤ u, the residue inverts a canonical renormalized pairing on 𝒫_k. In the collision range u<k≤2u, the ordinary pairing has radical (x^2)^k−u𝒫_2u−k; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For k≥2u+1, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head--socle pairing on the generalized harmonic modules. As an application, we derive covariant right--left radial q-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided q-Fischer projection.
ER -