Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 043, 25 pages      arXiv:2011.07527      https://doi.org/10.3842/SIGMA.2022.043

Difference Equation for Quintic 3-Fold

Yaoxinog Wen
Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea

Received September 28, 2021, in final form June 04, 2022; Published online June 14, 2022

Abstract
In this paper, we use the Mellin-Barnes-Watson method to relate solutions of a certain type of $q$-difference equations at $Q=0$ and $Q=\infty$. We consider two special cases; the first is the $q$-difference equation of $K$-theoretic $I$-function of the quintic, which is degree 25; we use Adams' method to find the extra 20 solutions at $Q=0$. The second special case is a fuchsian case, which is confluent to the differential equation of the cohomological $I$-function of the quintic. We compute the connection matrix and study the confluence of the $q$-difference structure.

Key words: $q$-difference equation; quantum $K$-theory; Fermat quintic.

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References

1. Adams C.R., On the irregular cases of the linear ordinary difference equation, Trans. Amer. Math. Soc. 30 (1928), 507-541.
2. Adams C.R., Linear $q$-difference equations, Bull. Amer. Math. Soc. 37 (1931), 361-400.
3. Candelas P., de la Ossa X.C., Green P.S., Parkes L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nuclear Phys. B 359 (1991), 21-74.
4. Chiodo A., Ruan Y., Landau-Ginzburg/Calabi-Yau correspondence for quintic three-folds via symplectic transformations, Invent. Math. 182 (2010), 117-165, arXiv:0812.4660.
5. Garoufalidis S., Scheidegger E., On the quantum K-theory of the quintic, SIGMA 18 (2022), 021, 20 pages, arXiv:2101.07490.
6. Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and its Applications, Vol. 96, Cambridge University Press, Cambridge, 2004.
7. Givental A., A mirror theorem for toric complete intersections, in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996), Progr. Math., Vol. 160, Birkhäuser Boston, Boston, MA, 1998, 141-175.
8. Givental A., On the WDVV equation in quantum $K$-theory, Michigan Math. J. 48 (2000), 295-304, arXiv:math.AG/0003158.
9. Givental A., Permutation-equivariant quantum $K$-theory V. Toric $q$-hypergeometric functions, arXiv:1509.03903.
10. Gu W., Pei D., Zhang M., On phases of 3d $\mathcal{N} = 2$ Chern-Simons-matter theories, Nuclear Phys. B 973 (2021), 115604, 20 pages, arXiv:2105.02247.
11. Lee Y.-P., Quantum $K$-theory. I. Foundations, Duke Math. J. 121 (2004), 389-424, arXiv:math.AG/0105014.
12. Milanov T., Roquefeuil A., Confluence in quantum $K$-theory of weak Fano manifolds and $q$-oscillatory integrals for toric manifolds, arXiv:2108.08620.
13. Roquefeuil A., Confluence of quantum $K$-theory to quantum cohomology for projective spaces, arXiv:1911.00254.
14. Ruan Y., Wen Y., Quantum $K$-theory and $q$-difference equations, arXiv:2109.02218.
15. Sauloy J., Systèmes aux $q$-différences singuliers réguliers: classification, matrice de connexion et monodromie, Ann. Inst. Fourier (Grenoble) 50 (2000), 1021-1071.
16. 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.