从 p-二次积分看 BPS 不变量

IF 1.3 1区 数学 Q1 MATHEMATICS Compositio Mathematica Pub Date : 2024-05-30 DOI:10.1112/s0010437x24007176
Francesca Carocci, Giulio Orecchia, Dimitri Wyss
{"title":"从 p-二次积分看 BPS 不变量","authors":"Francesca Carocci, Giulio Orecchia, Dimitri Wyss","doi":"10.1112/s0010437x24007176","DOIUrl":null,"url":null,"abstract":"<p>We define <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$p$</span></span></img></span></span>-adic <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathrm {BPS}$</span></span></img></span></span> or <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></img></span></span> invariants for moduli spaces <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$\\operatorname {M}_{\\beta,\\chi }$</span></span></img></span></span> of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span>. Our definition relies on a canonical measure <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$\\mu _{\\rm can}$</span></span></img></span></span> on the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$F$</span></span></img></span></span>-analytic manifold associated to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$\\operatorname {M}_{\\beta,\\chi }$</span></span></img></span></span> and the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></img></span></span> invariants are integrals of natural <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline12.png\"><span data-mathjax-type=\"texmath\"><span>${\\mathbb {G}}_m$</span></span></img></span></span> gerbes with respect to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline14.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mu _{\\rm can}$</span></span></span></span>. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline15.png\"/><span data-mathjax-type=\"texmath\"><span>$\\chi$</span></span></span></span>-independence result for these <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline16.png\"/><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></span></span> invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline17.png\"/><span data-mathjax-type=\"texmath\"><span>$p\\mathrm {BPS}$</span></span></span></span> with usual <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline18.png\"/><span data-mathjax-type=\"texmath\"><span>$\\mathrm {BPS}$</span></span></span></span> invariants through a result of Maulik and Shen [<span>Cohomological</span> <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline19.png\"/><span data-mathjax-type=\"texmath\"><span>$\\chi$</span></span></span></span><span>-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles</span>, Geom. Topol. <span>27</span> (2023), 1539–1586].</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"BPS invariants from p-adic integrals\",\"authors\":\"Francesca Carocci, Giulio Orecchia, Dimitri Wyss\",\"doi\":\"10.1112/s0010437x24007176\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We define <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p$</span></span></img></span></span>-adic <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm {BPS}$</span></span></img></span></span> or <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></img></span></span> invariants for moduli spaces <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\operatorname {M}_{\\\\beta,\\\\chi }$</span></span></img></span></span> of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$F$</span></span></img></span></span>. Our definition relies on a canonical measure <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline8.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mu _{\\\\rm can}$</span></span></img></span></span> on the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline9.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$F$</span></span></img></span></span>-analytic manifold associated to <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline10.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\operatorname {M}_{\\\\beta,\\\\chi }$</span></span></img></span></span> and the <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline11.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></img></span></span> invariants are integrals of natural <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline12.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>${\\\\mathbb {G}}_m$</span></span></img></span></span> gerbes with respect to <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline14.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mu _{\\\\rm can}$</span></span></span></span>. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline15.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\chi$</span></span></span></span>-independence result for these <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline16.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></span></span> invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline17.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$p\\\\mathrm {BPS}$</span></span></span></span> with usual <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline18.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\mathrm {BPS}$</span></span></span></span> invariants through a result of Maulik and Shen [<span>Cohomological</span> <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240529183756228-0512:S0010437X24007176:S0010437X24007176_inline19.png\\\"/><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\chi$</span></span></span></span><span>-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles</span>, Geom. Topol. <span>27</span> (2023), 1539–1586].</p>\",\"PeriodicalId\":55232,\"journal\":{\"name\":\"Compositio Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Compositio Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1112/s0010437x24007176\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x24007176","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

我们通过对非阿基米德局部场 $F$ 的积分,定义了 del Pezzo 和 K3 曲面上一维剪切的模空间 $operatorname {M}_{\beta,\chi }$ 的 $p$-adic $\mathrm {BPS}$ 或 $p\mathrm {BPS}$ 不变量。我们的定义依赖于与 $\operatorname {M}_{\beta,\chi }$ 相关联的 $F$-analytic 流形上的规范度量 $\mu _{\rm can}$ ,而 $p\mathrm {BPS}$ 不变式是自然 ${mathbb {G}}_m$ gerbes 关于 $\mu _{\rm can}$ 的积分。类似的构造也可以用于曲线上的全形希格斯束和通常希格斯束。我们的主要定理是这些 $p\mathrm {BPS}$ 不变量的 $\chi$-independence 结果。对于del Pezzo曲面上的一维剪切和全形希格斯束,我们通过Maulik和Shen的一个结果[Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom.Topol.27 (2023), 1539-1586].
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BPS invariants from p-adic integrals

We define $p$-adic $\mathrm {BPS}$ or $p\mathrm {BPS}$ invariants for moduli spaces $\operatorname {M}_{\beta,\chi }$ of one-dimensional sheaves on del Pezzo and K3 surfaces by means of integration over a non-archimedean local field $F$. Our definition relies on a canonical measure $\mu _{\rm can}$ on the $F$-analytic manifold associated to $\operatorname {M}_{\beta,\chi }$ and the $p\mathrm {BPS}$ invariants are integrals of natural ${\mathbb {G}}_m$ gerbes with respect to $\mu _{\rm can}$. A similar construction can be done for meromorphic and usual Higgs bundles on a curve. Our main theorem is a $\chi$-independence result for these $p\mathrm {BPS}$ invariants. For one-dimensional sheaves on del Pezzo surfaces and meromorphic Higgs bundles, we obtain as a corollary the agreement of $p\mathrm {BPS}$ with usual $\mathrm {BPS}$ invariants through a result of Maulik and Shen [Cohomological $\chi$-independence for moduli of one-dimensional sheaves and moduli of Higgs bundles, Geom. Topol. 27 (2023), 1539–1586].

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来源期刊
Compositio Mathematica
Compositio Mathematica 数学-数学
CiteScore
2.10
自引率
0.00%
发文量
62
审稿时长
6-12 weeks
期刊介绍: Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.
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