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Stable pairs, flat connections and Gopakumar–Vafa invariants

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Abstract

We offer a quick introduction to certain recent aspects of Donaldson–Thomas theory in the context of curve-counting invariants. We present a new computation which recasts the correspondence between the Gromov–Witten and Donaldson–Thomas theories of a Calabi–Yau threefold in the language of flat connections, their monodromy, and their flat sections.

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Notes

  1. In the apprach of [1, 6], fixing a generic Z, one introduces a formal parameter s and an inverse system of principal bundles \(P^j = {\text {Aut}}({\mathbb {C}}[\Gamma ][s]/(s)^j) \times {\mathbb {C}}^*\), with connections \(\nabla ^j\), whose generalised monodromies converge to a formal power series version of \(\{{\mathbb {S}}_{\ell }(Z), \ell \subset {\mathbb {C}}^*\}\), and are constant in a possibly decreasing sequence of open neighbourhoods \(U_j\) of Z in \({\text {Hom}}(\Gamma , {\mathbb {C}})\). But in general, without extra finiteness assumptions, we have \(\cap _{j} U_j= \emptyset \).

  2. In the approach of [1, 6], Y(t, Z) takes values in \({\text {Aut}}({\mathbb {C}}[\Gamma ][[s]])\), and its reductions modulo \((s)^j\) give canonical sections for the inverse system of connections \(\nabla ^j\).

  3. The appearence of branch cuts is expected: around a double pole, the “canonical solutions” of a holomorphic differential equation are only defined in a sector.

  4. We assume primitivity and the vanishing in order to simplify the statement. A similar result holds in general.

  5. A relation between Theorem 1 and classical ODEs is discussed in [19]. A different type of relation between the connection \(\nabla \) and curve-counting in described in [6, 7].

  6. In general, as we explained, one should work with the inverse system of bundles and connections \((P^i, \nabla ^i)\), see [1, 6].

  7. Rays of this type could be dense in a region, and in general the statement only makes sense for each \((P^i, \nabla ^i)\).

  8. The operator \([q^n]\) applied to Laurent polynomials extracts the coefficient of \(q^n\).

References

  1. Barbieri, A., Stoppa, J.: Frobenius type and CV-structures for Donaldson–Thomas theory and a convergence property. Comm. Anal. Geom. 27(2), 287–327 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bridgeland, T.: Riemann–Hilbert problems from Donaldson–Thomas theory. Invent. Math. 216(1), 69–124 (2019)

    Article  MathSciNet  Google Scholar 

  3. Bridgeland, T., Toledano-Laredo, V.: Stability conditions and Stokes factors. Invent. Math. 187(1), 61–98 (2012)

    Article  MathSciNet  Google Scholar 

  4. Donaldson, S.K., Thomas, R.P.: Gauge Theory in Higher Dimensions, The Geometric Universe (Oxford, 1996), pp. 31–47. Oxford University Press, Oxford (1998)

    Google Scholar 

  5. Felder, G., Kazhdan, D.: Regularization of divergent integrals. Sel. Math. (N.S.) 24(1), 157–186 (2018)

    Article  MathSciNet  Google Scholar 

  6. Filippini, S., Garcia Fernandez, M., Stoppa, J.: Stability data, irregular connections and tropical curves. Sel. Math. (N.S.) 23(2), 1355–1418 (2017)

    Article  MathSciNet  Google Scholar 

  7. Filippini, S., Stoppa, J.: TBA type equations and tropical curves. Int. J. Math. 27(7), 1640005 (2016)

    Article  MathSciNet  Google Scholar 

  8. Gaiotto, D.: Opers and TBA. arXiv:1403.6137 [hep-th]

  9. Gaiotto, D., Moore, G., Neitzke, A.: Four dimensional wall-crossing via three-dimensional field theory. Comm. Math. Phys. 299(1), 163–224 (2010)

    Article  MathSciNet  Google Scholar 

  10. Gopakumar, R., Vafa, C.: M-theory and topological strings. II. arXiv:hep-th/9812127 (1998)

  11. Joyce, D.: Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds. Geom. Topol. 11, 667–725 (2007)

    Article  MathSciNet  Google Scholar 

  12. Joyce, D., Song, Y.: A theory of generalized Donaldson–Thomas invariants. Mem. Am. Math. Soc. 217(1020), iv+199 (2012). ISBN: 978-0-8218-5279-8

    MathSciNet  MATH  Google Scholar 

  13. Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson–Thomas invariants and cluster transformations. arXiv:0811.2435 [math.AG]

  14. Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R.: Gromov–Witten theory and Donaldson–Thomas theory. I. Compos. Math. 142(5), 1263–1285 (2006)

    Article  MathSciNet  Google Scholar 

  15. Maulik, D., Toda, Y.: Gopakumar–Vafa invariants via vanishing cycles. Invent. Math. 213(3), 1017–1097 (2018)

    Article  MathSciNet  Google Scholar 

  16. Pandharipande, R., Pixton, A.: GW/P correspondence for the quintic 3-fold. J. Am. Math. Soc. 30, 389–449 (2017)

    Article  Google Scholar 

  17. Pandharipande, R., Thomas, R.: Curve counting via stable pairs in the derived category. Invent. Math. 178(2), 407–447 (2009)

    Article  MathSciNet  Google Scholar 

  18. Pandharipande, R., Thomas, R.: Stable pairs and BPS invariants. J. Am. Math. Soc. 23(1), 267–297 (2010)

    Article  MathSciNet  Google Scholar 

  19. Scalise, J., Stoppa, J.: Variations of BPS structure and a large rank limit. J. Inst. Math. Jussieu. (2017). https://doi.org/10.1017/S1474748019000136

    Article  Google Scholar 

  20. Stoppa, J.: Joyce–Song wall-crossing as an asymptotic expansion. Kyoto J. Math. 54(1), 103–156 (2014)

    Article  MathSciNet  Google Scholar 

  21. Stoppa, J.: A note on BPS structures and Gopakumar–Vafa invariants. Commun. Number Theory Phys. 13(3), 627–645 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Thomas, R.P.: A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations. J. Differ. Geom. 54(2), 367–438 (2000)

    Article  MathSciNet  Google Scholar 

  23. Toda, Y.: Curve counting theories via stable objects I. DT/PT correspondence. J. Am. Math. Soc. 23(4), 1119–1157 (2010)

    Article  MathSciNet  Google Scholar 

  24. Toda, Y.: Curve counting invariants around the conifold point. J. Differ. Geom. 89(1), 133–184 (2011)

    Article  MathSciNet  Google Scholar 

  25. Toda, Y.: Stability conditions and curve counting invariants on Calabi-Yau 3-folds. Kyoto J. Math. 52(1), 1–50 (2012)

    Article  MathSciNet  Google Scholar 

  26. Toda, Y.: Hall algebras in the derived category and higher rank DT invariants. arXiv:1601.07519v2 [math.AG]

  27. Toda, Y.: Gopakumar–Vafa invariants and wall-crossing. arXiv:1710.01843 [math.AG]

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Acknowledgements

I thank A. Barbieri, T. Bridgeland, J. Scalise, and R. Thomas for important comments and suggestions about this work. I am grateful to the organisers and participants in the workshop D-modules, quantum geometry, and related topics, Kyoto, December 2018, and in the XXI Congresso UMI, Pavia, September 2019, for very helpful feedback. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 307119.

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  1. SISSA, via Bonomea 265, 34136, Trieste, Italy

    Jacopo Stoppa

  2. Institute for Geometry and Physics (IGAP), Via Beirut 2-4, 34014, Trieste, Italy

    Jacopo Stoppa

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Stoppa, J. Stable pairs, flat connections and Gopakumar–Vafa invariants. Boll Unione Mat Ital 14, 117–136 (2021). https://doi.org/10.1007/s40574-020-00243-8

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