{"title":"希格斯束准 BPS 类别的拓扑 K 理论","authors":"Tudor Pădurariu, Yukinobu Toda","doi":"arxiv-2409.10800","DOIUrl":null,"url":null,"abstract":"In a previous paper, we introduced quasi-BPS categories for moduli stacks of\nsemistable Higgs bundles. Under a certain condition on the rank, Euler\ncharacteristic, and weight, the quasi-BPS categories (called BPS in this case)\nare non-commutative analogues of Hitchin integrable systems. We proposed a\nconjectural equivalence between BPS categories which swaps Euler\ncharacteristics and weights. The conjecture is inspired by the Dolbeault\nGeometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus\nmirror symmetry, and by the $\\chi$-independence phenomenon for BPS invariants\nof curves on Calabi-Yau threefolds. In this paper, we show that the above conjecture holds at the level of\ntopological K-theories. When the rank and the Euler characteristic are coprime,\nsuch an isomorphism was proved by Groechenig--Shen. Along the way, we show that\nthe topological K-theory of BPS categories is isomorphic to the BPS cohomology\nof the moduli of semistable Higgs bundles.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Topological K-theory of quasi-BPS categories for Higgs bundles\",\"authors\":\"Tudor Pădurariu, Yukinobu Toda\",\"doi\":\"arxiv-2409.10800\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In a previous paper, we introduced quasi-BPS categories for moduli stacks of\\nsemistable Higgs bundles. Under a certain condition on the rank, Euler\\ncharacteristic, and weight, the quasi-BPS categories (called BPS in this case)\\nare non-commutative analogues of Hitchin integrable systems. We proposed a\\nconjectural equivalence between BPS categories which swaps Euler\\ncharacteristics and weights. The conjecture is inspired by the Dolbeault\\nGeometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus\\nmirror symmetry, and by the $\\\\chi$-independence phenomenon for BPS invariants\\nof curves on Calabi-Yau threefolds. In this paper, we show that the above conjecture holds at the level of\\ntopological K-theories. When the rank and the Euler characteristic are coprime,\\nsuch an isomorphism was proved by Groechenig--Shen. Along the way, we show that\\nthe topological K-theory of BPS categories is isomorphic to the BPS cohomology\\nof the moduli of semistable Higgs bundles.\",\"PeriodicalId\":501038,\"journal\":{\"name\":\"arXiv - MATH - Representation Theory\",\"volume\":\"63 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-17\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Representation Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.10800\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Representation Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.10800","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
在上一篇论文中,我们介绍了可迷惑希格斯束的模叠的准BPS范畴。在秩,欧拉特性和权重的特定条件下,准 BPS 范畴(这里称为 BPS)是希金可积分系统的非交换类似物。我们提出了 BPS 范畴之间的等价猜想,即交换欧拉特征和权重。这一猜想受到了多纳吉--潘特夫(Donagi--Pantev)的多尔博几何朗兰兹等价、豪塞尔--塔德斯镜像对称性以及卡拉比--尤三折上曲线的BPS不变量的$\chi$-independence现象的启发。在本文中,我们证明了上述猜想在拓扑 K 理论层面上成立。当秩和欧拉特征为共素时,这种同构由格罗切尼-申证明。同时,我们还证明了BPS范畴的拓扑K理论与半稳希格斯束模态的BPS同调同构。
Topological K-theory of quasi-BPS categories for Higgs bundles
In a previous paper, we introduced quasi-BPS categories for moduli stacks of
semistable Higgs bundles. Under a certain condition on the rank, Euler
characteristic, and weight, the quasi-BPS categories (called BPS in this case)
are non-commutative analogues of Hitchin integrable systems. We proposed a
conjectural equivalence between BPS categories which swaps Euler
characteristics and weights. The conjecture is inspired by the Dolbeault
Geometric Langlands equivalence of Donagi--Pantev, by the Hausel--Thaddeus
mirror symmetry, and by the $\chi$-independence phenomenon for BPS invariants
of curves on Calabi-Yau threefolds. In this paper, we show that the above conjecture holds at the level of
topological K-theories. When the rank and the Euler characteristic are coprime,
such an isomorphism was proved by Groechenig--Shen. Along the way, we show that
the topological K-theory of BPS categories is isomorphic to the BPS cohomology
of the moduli of semistable Higgs bundles.