Mock Modular Forms as Z-Invariants Bridging Quantum 3-Manifolds and Modularity — E8 Intelligence Research

FINDING: Mock modular forms serve as Z-invariants for quantum 3-manifold invariants, bridging categorification, lattice theta functions, and quantum modularity. | MATH: Key objects: mock theta functions (Ramanujan), non-holomorphic modular forms (Zwegers: \( \hat{f}(\tau) = f(\tau) + \frac{i}{2\pi}\int_{-\bar{\tau}}^{i\infty} \frac{\Theta(w)}{\sqrt{-i(w+\tau)}} dw \)), quantum modular forms (Zagier: functions \( f: \mathbb{Q} \to \mathbb{C} \) with modular transformation errors that are "nice" — e.g., \( f(x) - f(x+1) \), \( f(-1/x) - x^k f(x) \) are real-analytic on \( \mathbb{R}\setminus\mathbb{Q} \)). Z-invariants: partition functions of \( \text{SU}(2) \) Chern–Simons theory on 3-manifolds, \( Z(M) = \sum_{j} \frac{S_{0j}^2}{S_{0j}} \dots \) (WRT invariant), categorified to homological invariants. Even unimodular lattice theta: \( \Theta_\Lambda(\tau) = \sum_{v\in\Lambda} q^{v\cdot v/2} \), with \( \Lambda \) even unimodular (e.g., \( E_8 \), Leech), modular of weight \( k = \text{

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com