Stable pairs, flat connections and Gopakumar–Vafa invariants

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. 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. 2.

    In the approach of [1, 6], Y(tZ) 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. 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. 4.

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

  5. 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. 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. 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. 8.

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

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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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Correspondence to Jacopo Stoppa.

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Stoppa, J. Stable pairs, flat connections and Gopakumar–Vafa invariants. Boll Unione Mat Ital (2020). https://doi.org/10.1007/s40574-020-00243-8

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Mathematics Subject Classification

  • 14N35
  • 34M50