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HIGHER GENUS GROMOV–WITTEN THEORY OF AND ASSOCIATED TO LOCAL CURVES
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| Zeitschriftentitel: | Forum of Mathematics, Pi |
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| Personen und Körperschaften: | , |
| In: | Forum of Mathematics, Pi, 7, 2019 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
Cambridge University Press (CUP)
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| Schlagwörter: |
| Zusammenfassung: | <jats:p>We study the higher genus equivariant Gromov–Witten theory of the Hilbert scheme of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline5" /> <jats:tex-math>$n$</jats:tex-math> </jats:alternatives> </jats:inline-formula> points of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline6" /> <jats:tex-math>$\mathbb{C}^{2}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [<jats:italic>Invent. Math.</jats:italic> <jats:bold>179</jats:bold> (2010), 523–557], is semisimple, the higher genus theory is determined by an <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline7" /> <jats:tex-math>$\mathsf{R}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-matrix via the Givental–Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline8" /> <jats:tex-math>$\mathsf{R}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-matrix by explicit data in degree <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline9" /> <jats:tex-math>$0$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline10" /> <jats:tex-math>$\mathsf{Hilb}^{n}(\mathbb{C}^{2})$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and the Gromov–Witten/Donaldson–Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [<jats:italic>Transform. Groups</jats:italic> <jats:bold>15</jats:bold> (2010), 965–982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov–Witten theory of the symmetric product <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S2050508619000040_inline11" /> <jats:tex-math>$\mathsf{Sym}^{n}(\mathbb{C}^{2})$</jats:tex-math> </jats:alternatives> </jats:inline-formula> is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, <jats:italic>Algebraic Geometry–Seattle 2005, Part 1</jats:italic>, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23–42; Coates <jats:italic>et al.</jats:italic>, <jats:italic>Geom. Topol.</jats:italic> <jats:bold>13</jats:bold> (2009), 2675–2744; Coates & Ruan, <jats:italic>Ann. Inst. Fourier (Grenoble)</jats:italic> <jats:bold>63</jats:bold> (2013), 431–478].</jats:p> |
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| ISSN: |
2050-5086
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| DOI: | 10.1017/fmp.2019.4 |