Cartan-Fejer Gram Tomography and Flop Covariance for BPS Resummed Gromov-Witten Potentials
Abstract
We introduce a matrix-valued finite difference formalism for the genus zero BPS resummed local Gromov-Witten potential of a threefold flop. Factoring the central difference as Δη=∇η2, ∇η=Thη/2−T−hη/2, we prove that the mixed differences H=(∇ηi∇ηjF)i,j admit an exact rank one signed q-Gram decomposition. Primitive BPS classes are therefore detected by rank one coefficient matrices, and a matrix-valued Möbius inversion reconstructs all multiplicities on a primitive ray. Under a simple threefold flop the Gram forcing is covariant away from the flopped ray and acquires the universal logarithmic anomaly κC(C+⊗ C+)log r, κC=∑d≥1d3ndC. After a natural renormalization it becomes flop covariant, while one classical logarithmic derivative recovers the cubic correction in the crepant transformation formula. We interpret κC as the third moment of the GV width distribution: Toda's noncommutative width is the second moment, and contraction algebra BPS invariants give a finite cohomological width series whose numerical realization contains all these moments. For root supported BPS theories, simple root Gram coefficients reconstruct the signed Cartan matrix. Applied to the standard ADE foldings, the theory distinguishes Bn from Cn, detects the F4 double bond and D4 triality, and yields explicit global q-Gram discriminants.
Related
- Étale spaces and coverings — On a text written by Henri Cartan for Bourbaki
- Einstein-Cartan and Palatini-Cartan Gravity from Graded Poisson Geometry
- The Cartan determinant conjecture for representation-finite Artin algebras
- Isoperimetric and total curvature inequalities in Cartan-Hadamard manifolds with nullity
- simple cartan-free modules of finite rank over the super-virasoro algebras
BibTeX
@misc{li2026cartanfejer,
title = {{Cartan-Fejer Gram Tomography and Flop Covariance for BPS Resummed Gromov-Witten Potentials}},
author = {Xiaobin Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16693},
eprint = {2609.16693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16693},
}
RIS
TY - UNPB
TI - Cartan-Fejer Gram Tomography and Flop Covariance for BPS Resummed Gromov-Witten Potentials
AU - Xiaobin Li
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16693
UR - https://arxiv.org/abs/2609.16693
AB - We introduce a matrix-valued finite difference formalism for the genus zero BPS resummed local Gromov-Witten potential of a threefold flop. Factoring the central difference as Δ_η=∇_η^2, ∇_η=T_hη/2−T_−hη/2, we prove that the mixed differences H=(∇_η_i∇_η_jF)_i,j admit an exact rank one signed q-Gram decomposition. Primitive BPS classes are therefore detected by rank one coefficient matrices, and a matrix-valued Möbius inversion reconstructs all multiplicities on a primitive ray. Under a simple threefold flop the Gram forcing is covariant away from the flopped ray and acquires the universal logarithmic anomaly κ_C(C^+⊗ C^+)log r, κ_C=∑_d≥1d^3n_dC. After a natural renormalization it becomes flop covariant, while one classical logarithmic derivative recovers the cubic correction in the crepant transformation formula. We interpret κ_C as the third moment of the GV width distribution: Toda's noncommutative width is the second moment, and contraction algebra BPS invariants give a finite cohomological width series whose numerical realization contains all these moments. For root supported BPS theories, simple root Gram coefficients reconstruct the signed Cartan matrix. Applied to the standard ADE foldings, the theory distinguishes B_n from C_n, detects the F_4 double bond and D_4 triality, and yields explicit global q-Gram discriminants.
ER -