Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Louisa Liles

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.

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

Abstract

Gukov, Pei, Putrov, and Vafa developed a -series invariant of negative definite plumbed -manifolds with spin structures, building on earlier work of Lawrence and Zagier. This was recently generalized to an an infinite family of two-variable -series invariants by Akhmechet, Johnson, and Krushkal (AJK). We calculate one such series for all Seifert manifolds with 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 -series invariants for three infinite families of manifolds, finding mixed modularity properties for one such family.

Paper Structure

This paper contains 16 sections, 22 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction and statement of main results
  2. Background
  3. Negative definite plumbed $3$-Manifolds
  4. Seifert manifolds
  5. Spin$^{c}$ structures
  6. The $(t,q)$-series invariants
  7. Modular and quantum modular forms
  8. Calculation of $\widehat{\newline{\widehat{Z}}\,}\!$ for Seifert manifolds
  9. Quantum modularity of $\widehat{\newline{\widehat{Z}}\,}\!$ for three exceptional fibers
  10. Calculation of $(t,q)$-series invariants
  11. Seifert Manifolds with $3$ Exceptional Fibers
  12. H-shaped graphs
  13. $4$-leg star graphs
  14. New $q$-series invariants
  15. Modularity properties of the $(t,q)$-series for three exceptional fibers
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M(b;(a_{1},b_{1}),\dots,(a_{k},b_{k}))$ be a Seifert manifold, and let $|H|=|H_{1}(M;z)|$. Then:

Figures (2)

  • Figure 1: A plumbing graph and its associated framed link.
  • Figure 2: A plumbing tree for $M(b;(a_{1},b_{1}),\dots,(a_{k},b_{k}))$ obtained from continued fraction decompositions of $\frac{a_{i}}{b_{i}}$.

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 2.2: ajk, Theorem $6.3$ and Remark $6.5$
  • Definition 2.3
  • Lemma 2.4
  • ...and 40 more