High-rank motivic degree-zero Donaldson--Thomas theory on singular curves, and q-series

Abstract: My main point is that high-rank motivic degree-zero DT invariants on singular curves appear to give infinite products of Rogers--Ramanujan type. This is based on explicit computation of certain Quot schemes, which is where the new ideas and results lie, but this seems to be a new phenomenon that I cannot explain from physics or other conceptual connection. For context, the rank-1 case has been observed to relate to knot theory and Catalan combinatorics in the last decade (keyword: Oblomkov--Rasmussen--Shende conjecture).

A down-to-earth statement that captures all the essence is the following (stated for the singular curve $y^2=x^3$): For a random $n\times n$ matrix $A$ over a finite field $\mathbb{F}_q$, what is the expected number of matrices $B$ such that $AB=BA$ and $A^3=B^2$? It turns out that as $n\to \infty$, the limiting answer is $\prod (1-q^{-i})$ over all positive $i$ congruent to $1,4$ mod $5$, the famous Rogers--Ramanujan infinite product.

The reported results contain joint work with Ruofan Jiang (on the $y^2=x^n$ case) and joint work in progress with RJ and Alexei Oblomkov (on the $y^m=x^n$ case with $m,n$ coprime).

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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