$(t,q)$-Series Invariants of Seifert Manifolds

TL;DR

This paper computes and analyzes two-variable $(t,q)$-series invariants for all Seifert manifolds with $b_1=0$, extending previous theorems and exploring modularity properties across multiple manifold families.

Contribution

It extends the calculation of $(t,q)$-series invariants to all Seifert manifolds with $b_1=0$, generalizing prior results and analyzing their modularity properties.

Findings

Calculated invariants for all Seifert manifolds with $b_1=0$

Extended previous theorems to the two-variable case

Identified mixed modularity properties in specific manifold families

Abstract

Gukov, Pei, Putrov, and Vafa developed a $q$-series invariant of negative definite plumbed $3$-manifolds with spin$^{c}$ structures, building on earlier work of Lawrence and Zagier. This was recently generalized to an an infinite family of two-variable $(t,q)$-series invariants by Akhmechet, Johnson, and Krushkal (AJK). We calculate one such series for all Seifert manifolds with $b_{1}=0.$ These results extend a previous theorem of Liles and McSpirit to any number of exceptional fibers and the Reduction Theorem of Gukov, Svoboda, and Katzarkov to the two-variable case. As a consequence, a previous result of Liles and McSpirit on modularity properties and radial limits is enhanced to a larger family of manifolds. We also calculate the infinite collection of $(t,q)$-series invariants for three infinite families of manifolds, finding mixed modularity properties for one such family.