### 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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