Flat endomorphisms for mod p equivariant quantum connections from quantum Steenrod operations

• Jae Hee Lee ^[1]
  1. [1] Stanford, CA, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 1, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-024-01015-4
• Enlaces
  • Texto completo
• Resumen

  • We present a method for constructing covariantly constant endomorphisms for the mod p equivariant quantum connection, using the quantum Steenrod power operations of Fukaya and Wilkins. The example of the cotangent bundle of the projective line is fully computed, and we discuss the relationship with the mod p solutions of trigonometric KZ equation recently constructed by Varchenko. As a byproduct, we compute the first examples of quantum Steenrod operations that are not a priori determined by ordinary Gromov–Witten theory and classical Steenrod operations, which may be of independent interest.

• Referencias bibliográficas
  • 1. Aspinwall, P.S., Morrison, D.R.: Topological field theory and rational curves. Commun. Math. Phys. 151(2), 245–262 (1993)
  • 2. Allday, C., Puppe, V.: Cohomological Methods in Transformation Groups, Cambridge Studies in Advanced Mathematics, vol. 32. Cambridge University...
  • 3. Atiyah, M.F.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A 247, 237–244 (1958)
  • 4. Bryan, J., Gholampour, A.: Root systems and the quantum cohomology of ADE resolutions. Algebra Number Theory 2(4), 369–390 (2008)
  • 5. Braverman, A., Maulik, D., Okounkov, A.: Quantum cohomology of the Springer resolution. Adv. Math. 227(1), 421–458 (2011)
  • 6. Çineli, E., Ginzburg, V.L., Gürel, B.Z.: From pseudo-rotations to holomorphic curves via quantum Steenrod squares. Int. Math. Res. Not....
  • 7. Cox, D.A., Sheldon, K.: Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs, vol. 68. American Mathematical Society,...
  • 8. Fukaya, K.: Morse homotopy and its quantization, Geometric topology (Athens, GA: AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc. Providence,...
  • 9. Givental, A.B.: Homological geometry. I. Projective hypersurfaces. Selecta Math. (N.S.) 1(2), 325–345 (1995)
  • 10. Katz, S., Morrison, D.R.: Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups. J. Algeb. Geom....
  • 11. Manin, J.I.: The Hasse–Witt matrix of an algebraic curve. Izv. Akad. Nauk SSSR Ser. Mat. 25, 153–172 (1961)
  • 12. Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. Astérisque 408, ix+209 (2019)
  • 13. McDuff, D., Salamon, D.: J -holomorphic Curves and Symplectic Topology, 2nd edn. American Mathematical Society Colloquium Publications,...
  • 14. Rezchikov, S.: Rational quantum cohomology of Steenrod uniruled manifolds (2021). arxiv:2111.09157
  • 15. Ritter, A.F.: Floer theory for negative line bundles via Gromov–Witten invariants. Adv. Math. 262, 1035–1106 (2014)
  • 16. Seidel, P.: Formal groups and quantum cohomology. Geom. Topol. (2019) (to appear)
  • 17. Shelukhin, E.: Pseudo-rotations and Steenrod squares revisited. Math. Res. Lett. 28(4), 1255–1261 (2021)
  • 18. Seidel, P., Smith, I.: Localization for involutions in Floer cohomology. Geom. Funct. Anal. 20(6), 1464–1501 (2010)
  • 19. Schechtman, V., Varchenko, A.: Solutions of KZ differential equations modulo p. Ramanujan J. 48(3), 655–683 (2019)
  • 20. Smirnov, A., Varchenko, A.: The p-adic approximations of vertex functions via 3d-mirror symmetry (2023). arxiv:2302.03092
  • 21. Seidel, P., Wilkins, N.: Covariant constancy of quantum Steenrod operations. J. Fixed Point Theory Appl. 24(2), Paper No. 52, 38 (2022)
  • 22. Tarasov, V., Varchenko, A.: Hypergeometric solutions of the quantum differential equation of the cotangent bundle of a partial flag variety....
  • 23. Varchenko, A.: An invariant subbundle of the KZ connection mod p and reducibility of sl82 Verma modules mod p. Math. Notes 109(3–4), 386–397...
  • 24. Varchenko, A.: Dynamical and qKZ equations modulo ps, an example (2022). arxiv:2205.03980
  • 25. Varchenko, A.: Notes on solutions of KZ equations modulo ps and p-adic limit s → ∞, Hypergeometry, integrability and Lie theory. Contemp....
  • 26. Voisin, C.: A mathematical proof of a formula of Aspinwall and Morrison. Compos. Math. 104(2), 135–151 (1996)
  • 27. Wendl, C.: Lectures on holomorphic curves in symplectic and contact geometry. arxiv:1011.1690
  • 28. Wilkins, N.: A construction of the quantum Steenrod squares and their algebraic relations. Geom. Topol. 24(2), 885–970 (2020)
  • 29. Wilkins, N.: A survey of equivariant operations on quantum cohomology for symplectic manifolds (2024). arXiv:2409.01743
  • 30. Zinger, A.: Pseudocycles and integral homology. Trans. Am. Math. Soc. 360(5), 2741–2765 (2008)