Uniform localization and boundary corrections for two-node quiver Coulomb branches
Abstract
We construct simple coherent classes and compute their normalized difference operators for the two-node quiver Coulomb branch with gauge group GL1× GL2 and m−1 framing columns at the second node. The first magnetic charge is fixed at zero. For every m≥1, every support (n,−1) with n≥1, and every integral electric twist, we prove that termwise saturation of the matter Koszul complex gives the simple intermediate extension. A graded normalization formula expresses its localization on every stratum as a finite binomial shuffle sum. The resulting operators satisfy bar symmetry, positive bubbling order, paired root-residue cancellation, and integral Laurent polynomial action. For m=4, we construct the derived boundary correction on (2,−2), retaining the nonsemisimple odd-electric boundary object, and compute all electric twists. Finally, matter Euler divisibility proves triangular uniqueness on arbitrary fixed rank-two magnetic supports inside the actual integral Coulomb image. A pole-free counterexample shows why residue conditions alone do not give this uniqueness. Existence on general deeper mixed-sign supports remains open.
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BibTeX
@misc{tang2026uniform,
title = {{Uniform localization and boundary corrections for two-node quiver Coulomb branches}},
author = {Xiaomin Tang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06509},
eprint = {2611.06509},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06509},
note = {CC0},
}
RIS
TY - UNPB
TI - Uniform localization and boundary corrections for two-node quiver Coulomb branches
AU - Xiaomin Tang
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06509
UR - https://arxiv.org/abs/2611.06509
AB - We construct simple coherent classes and compute their normalized difference operators for the two-node quiver Coulomb branch with gauge group GL_1× GL_2 and m−1 framing columns at the second node. The first magnetic charge is fixed at zero. For every m≥1, every support (n,−1) with n≥1, and every integral electric twist, we prove that termwise saturation of the matter Koszul complex gives the simple intermediate extension. A graded normalization formula expresses its localization on every stratum as a finite binomial shuffle sum. The resulting operators satisfy bar symmetry, positive bubbling order, paired root-residue cancellation, and integral Laurent polynomial action. For m=4, we construct the derived boundary correction on (2,−2), retaining the nonsemisimple odd-electric boundary object, and compute all electric twists. Finally, matter Euler divisibility proves triangular uniqueness on arbitrary fixed rank-two magnetic supports inside the actual integral Coulomb image. A pole-free counterexample shows why residue conditions alone do not give this uniqueness. Existence on general deeper mixed-sign supports remains open.
ER -