Skip to main content
Log in

Quantum K-theory and q-difference Equations

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

This is a set of lecture notes for the first author’s lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Adams, C. R.: On the linear ordinary q-difference equation. Ann. of Math. (2), 30(1/4), 195–205 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  2. Adams, C. R.: On the irregular cases of the linear ordinary difference equation. Trans. Amer. Math. Soc., 30(3), 507–541 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aganagic, M., Hori, K., Karch, A., Tong, D.: Mirror symmetry in 2 + 1 and 1 + 1 dimensions. J. High Energy Phys., 07, 022 (2001)

    Article  MathSciNet  Google Scholar 

  4. Carmichael, R. D.: The general theory of linear q-difference equations. Amer. J. Math., 34(2), 147–168 (1912)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chiodo, A., Ruan, Y.: Landau—Ginzburg/Calabi—Yau correspondence for quintic three-folds via symplectic transformations. Invent. Math., 182, 117–165 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dorey, N., Tong, D.: Mirror symmetry and toric geometry in three-dimensional gauge theories. J. High Energy Phys., 5, Paper No. 018, 16 pp. (2000)

  7. Garoufalidis, S., Scheidegger, E.: On the quantum K-theory of the quintic. SIGMA, 18, Paper No. 021, 20 pp. (2022)

  8. Givental, A.: On the WDVV equation in quantum K-theory, Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J., 48, 295–304 (2000)

    MathSciNet  MATH  Google Scholar 

  9. Givental, A.: Permutation-equivariant quantum K-theory V. Toric q-hypergeometric functions, arX-iv:1509.03903 (2015)

  10. Iritani, H., Milanov, T., Tonita, V.: Reconstruction and convergence in quantum K-theory via difference equations. Int. Math. Res. Not. IMRN, 11, 2887–2937 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jockers, H., Mayr, P.: A 3d gauge theory/quantum k-theory correspondence. Adv. Theor. Math. Phys., 24, 327–457 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lee, Y.-P.: Quantum K-theory I: Foundations. Duke Math. J., 121(3), 389–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Roquefeuil, A.: Confluence of quantum K-theory to quantum cohomology for projective spaces, arXiv: 1911.00254 (2019)

  14. Ruan, Y., Wen, Y., Zhou, Z.: Quantum K-theory of toric varieties, level structures, and 3d mirror symmetry, arXiv:2011.07519 (2020)

  15. Ruan, Y., Zhang, M.: The level structure in quantum K-theory and mock theta functions, arXiv:1804.06552 (2018)

  16. Sauloy, J.: Systèmes aux q-différences singuliers réguliers: classification, matrice de connexion et monodromie. Ann. Inst. Fourier (Grenoble), 50(4), 1021–1071 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sauloy, J.: Galois theory of fuchsian q-difference equations. Ann. Sci. École Norm. Sup. (4), 36(6), 925–968 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sauloy, J.: Analytic study of q-difference equations. In: Galois Theories of Linear Difference Equations: An Introduction, Math. Surveys Monogr., Vol. 211, Amer. Math. Soc., Providence, RI, 2016, 103–171

    Chapter  MATH  Google Scholar 

  19. Ueda, K., Yoshida, Y.: 3d \({\cal N} = 2\) Chern—Simons-matter theory, Bethe ansatz, and quantum K-theory of Grassmannians. J. High Energ. Phys., 8, Paper No. 157, 43 pp. (2020)

  20. Wen, Y.: Difference equation for quintic 3-fold. SIGMA, 18, Paper No. 043, 25 pp. (2022)

Download references

Acknowledgements

These lectures were held at the Institute for Advanced Study in Mathematics at Zhejiang University. We express our special thanks to the institute for its wonderful environment and support. The second author would like to thank Prof. Bohan Fang, Prof. Huijun Fan, and Prof. Shuai Guo for their helpful support during the visit, and also wants to thank Prof. Choi for his support.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yong Bin Ruan.

Additional information

Dedicated to Professor Banghe Li on His 80th Birthday

The second author is supported by a KIAS Individual Grant (MG083901) at Korea Institute for Advanced Study and a POSCO Science fellowship

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ruan, Y.B., Wen, Y.X. Quantum K-theory and q-difference Equations. Acta. Math. Sin.-English Ser. 38, 1677–1704 (2022). https://doi.org/10.1007/s10114-022-1616-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-022-1616-2

Keywords

MR(2010) Subject Classification

Navigation