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Euler characteristic of stable envelopes

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Abstract

In this paper we prove a formula relating the equivariant Euler characteristic of K-theoretic stable envelopes to an object known as the index vertex for the cotangent bundle of the full flag variety. Our formula demonstrates that the index vertex is the power series expansion of a rational function. This result is a consequence of the 3d mirror self-symmetry of the variety considered here. In general, one expects an analogous result to hold for any two varieties related by 3d mirror symmetry.

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Notes

  1. In other words, if A and B are \({\mathsf {T}}^{!}_q\)-modules, then \(\text {rk}(A-B)=\text {rank}(A)-\text {rank}(B)\).

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Acknowledgements

This work was partially supported by NSF grant DMS-2054527.

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Authors and Affiliations

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

    Hunter Dinkins & Andrey Smirnov

  2. Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina str. 8, Moscow, Russia, 119991

    Andrey Smirnov

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  1. Hunter Dinkins
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  2. Andrey Smirnov
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Correspondence to Hunter Dinkins.

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Dinkins, H., Smirnov, A. Euler characteristic of stable envelopes. Sel. Math. New Ser. 28, 72 (2022). https://doi.org/10.1007/s00029-022-00788-w

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