Mock Modular Forms as Z-Invariants Bridging Quantum 3-Manifolds and Modularity — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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: f̂(τ) = f(τ) + i/2π∫−τ̅i∞ Θ(w)/(√(−i(w+τ))) dw), quantum modular forms (Zagier: functions f: ℚ → ℂ with modular transformation errors that are "nice" — e.g., f(x) − f(x+1), f(−1/x) − xk f(x) are real-analytic on ℝ∖ℚ). Z-invariants: partition functions of SU(2) Chern–Simons theory on 3-manifolds, Z(M) = ∑j S0j2/S0j … (WRT invariant), categorified to homological invariants. Even unimodular lattice theta: ΘΛ(τ) = ∑v∈Λ qv· v/2, with Λ even unimodular (e.g., E8, Leech), modular of weight \( k = \text{
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026mock,
title = {{Mock Modular Forms as Z-Invariants Bridging Quantum 3-Manifolds and Modularity — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23490690},
url = {https://zenodo.org/records/23490690},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mock Modular Forms as Z-Invariants Bridging Quantum 3-Manifolds and Modularity — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/20
PB - Zenodo
DO - 10.5281/zenodo.23490690
UR - https://zenodo.org/records/23490690
AB - 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: f̂(τ) = f(τ) + i/2π∫_−τ̅^i∞ Θ(w)/(√(−i(w+τ))) dw), quantum modular forms (Zagier: functions f: ℚ → 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 ℝ∖ℚ). Z-invariants: partition functions of SU(2) Chern–Simons theory on 3-manifolds, Z(M) = ∑_j S_0j^2/S_0j … (WRT invariant), categorified to homological invariants. Even unimodular lattice theta: Θ_Λ(τ) = ∑_v∈Λ q^v· v/2, with Λ 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
ER -