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2020

Difference Equation for Quintic 3-Fold

Wen, Yaoxiong

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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.

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Adams C.R., On the irregular cases of the linear ordinary difference equation,
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Candelas P., de la Ossa X.C., Green P.S., Parkes L., A pair of Calabi–Yau manifolds as an exactly soluble superconformal theory,
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Givental A., A mirror theorem for toric complete intersections, in Topological Field Theory, Primitive Forms and Related Topics (Kyoto, 1996),
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Chiodo A., Ruan Y., Landau–Ginzburg/Calabi–Yau correspondence for quintic three-folds via symplectic transformations,
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Garoufalidis S., Scheidegger E., On the quantum K-theory of the quintic,
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