## 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.
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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*
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