Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space

Abstract

We consider the equivariant quantum differential equation for the projective space \(P^{n-1}\) and introduce a compatible system of difference equations. We prove an equivariant gamma theorem for \(P^{n-1}\), which describes the asymptotics of the differential equation at its regular singular point in terms of the equivariant characteristic gamma class of the tangent bundle of \(P^{n-1}\). We describe the Stokes bases of the differential equation at its irregular singular point in terms of the exceptional bases of the equivariant K-theory algebra of \(P^{n-1}\) and a suitable braid group action on the set of exceptional bases. Our results are an equivariant version of the well-known results of Dubrovin and Guzzetti..

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Notes

  1. 1.

    These formulas were explained to us by Givental.

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Acknowledgements

The authors thank Giordano Cotti, Alexander Givental, and Richard Rimányi for many helpful discussions.

Funding

Funding was provided by Division of Mathematical Sciences (Grant Nos. 1665239, 1954266) and Simons Foundation (Grant No. 430235).

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Correspondence to Vitaly Tarasov.

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To Boris Dubrovin with admiration

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Vitaly Tarasov was supported in part by Simons Foundation grant 430235. Alexander Varchenko was supported in part by NSF grants DMS-1665239, DMS-1954266.

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Tarasov, V., Varchenko, A. Equivariant quantum differential equation, Stokes bases, and K-theory for a projective space. European Journal of Mathematics (2021). https://doi.org/10.1007/s40879-021-00455-y

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Keywords

  • Equivariant quantum differential equation
  • Equivariant K-theory
  • q-hypergeometric solutions
  • Braid group action

Mathematics Subject Classification

  • 14N35
  • 34M40
  • 17B80