Difference Equation for Quintic 3-Fold

TL;DR

This paper investigates the solutions and connection matrices of specific q-difference equations related to the quintic 3-fold, revealing new solutions and the confluence to differential equations using advanced mathematical methods.

Contribution

It introduces a novel application of the Mellin-Barnes-Watson and Adams' methods to analyze q-difference equations of the quintic, including solutions at different points and their confluence to differential equations.

Findings

Found 20 additional solutions at Q=0 for the q-difference equation.

Computed the connection matrix relating solutions at Q=0 and Q=∞.

Analyzed the confluence of q-difference equations to differential equations.

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.