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Characters of tangent spaces at torus fixed points and 3d-mirror symmetry

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Abstract

Let X be a Nakajima quiver variety and \(X'\) its 3d-mirror. We consider the action of the Picard torus \({\mathsf {K}}=\mathrm {Pic}(X)\otimes {\mathbb {C}}^{\times }\) on \(X'\). Assuming that \((X')^{{\mathsf {K}}}\) is finite, we propose a procedure for obtaining the \({\mathsf {K}}\)-character of the tangent spaces at the fixed points in terms of certain enumerative invariants of X known as vertex functions.

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Notes

  1. We believe that our main conjecture holds in full generality. We restrict the exposition to the quiver varieties for the sake of simplicity of the exposition.

  2. The choice of the cone \(C_{\text {eff}}(X)\) corresponds to the choice of the stability parameter for the Nakajima variety X.

  3. The term “symplectic duality”, describing the symmetry of this type is more common in mathematical literature. In this paper we call it 3d-mirror symmetry following the terminology of [2, 26, 27].

  4. See also the talk by A.Okounkov “Enumerative symplectic duality” at the MSRI workshop Structures in Enumerative Geometry in April 2018 for the first discussion of these ideas (available online).

  5. We may assume \(\hbar ^{1/2}\) exists by passing to the double cover of \({\mathsf {T}}\) if needed.

  6. To the best of the authors’ knowledge this is not proved.

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Acknowledgements

This work is supported by the Russian Science Foundation under Grant 19-11-00062.

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  1. University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3250, USA

    Andrey Smirnov & Hunter Dinkins

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  1. Andrey Smirnov
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Correspondence to Hunter Dinkins.

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Smirnov, A., Dinkins, H. Characters of tangent spaces at torus fixed points and 3d-mirror symmetry. Lett Math Phys 110, 2337–2352 (2020). https://doi.org/10.1007/s11005-020-01292-y

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