q-Holonomic Systems and Quantum Invariants
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q-Holonomic Systems and Quantum Invariants


The topics of this dissertation fall under the purview of quantum topology, which seeks tobuild connections between the insights and constructions of quantum physics and classical topol- ogy. A pivotal theme will be the appearance of topologically interesting q-holonomic systems in quantum invariants. These manifest in the quasiperiodic behavior of Witten-Reshetikhin-Turaev (WRT) invariants, and as certain modules associated to lagrangians in quantized character vari- eties. This work was motivated by the AJ conjecture [Gar04, Guk05], which predicts that these two manifestations are the two sides of a single coin. The main result of this dissertation is that the ADO invariant is q-holonomic, meaning it exhibits strong recursive behavior. Some subtlety is involved in the definition of q-holonomicity in this setting, as the ADO invariant exhibits a topologically uninteresting quasi-periodicity because of the appearance of roots of unity. This invariant is closely related to the colored Jones polynomial of the AJ conjecture, and acts as its analytic continuation.

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