Gukov-Pei-Putrov Model Overview

• The Gukov-Pei-Putrov Model is a theoretical framework assigning q-series invariants (homological blocks) to plumbed 3-manifolds using explicit integral and false theta function formulations.
• It establishes a concrete connection between quantum invariants like the WRT invariant and analytic continuation processes that recover classical topological data.
• The model integrates quantum modular properties, combinatorial structures, and physical interpretations within 3d supersymmetric field theories to unify topology and arithmetic.

The Gukov–Pei–Putrov Model is a theoretical framework assigning q-series invariants—often called homological blocks or “BPS q-series”—to plumbed three-manifolds. It establishes connections between quantum invariants from low-dimensional topology (such as Witten–Reshetikhin–Turaev invariants), modular and quantum modular forms, and physical structures like spectra of BPS states in supersymmetric field theories. Central to this model are the definitions of the q-series invariants, their modular or quantum modular properties, and their deep relationship to classical invariants via specific analytic continuations and arithmetic structures.

1. Definition of Homological Blocks and q-Series Invariants

In the Gukov–Pei–Putrov model, a negative definite plumbed 3-manifold is specified by a plumbing graph or weighted tree Γ. The corresponding q-series invariant, usually denoted $\widehat{Z}_a(q)$ with $a$ a label determined by a (generalized) Spin$^c$ structure, is given by an explicit integral formula: $$\widehat{Z}_a(q) = q^{(-3N + \mathrm{tr} M)/4} \, \mathrm{PV} \int_{\{|w_j|=1\}} \prod_{j=1}^N g(w_j) \prod_{(k, \ell) \in E} f(w_k, w_\ell) \, \Theta_{-M, a}(q;\mathbf{w}) \prod_{j=1}^N \frac{dw_j}{2\pi i w_j}$$ Here:

• $M$ is the plumbing/linking matrix of $\Gamma$,
• $g(w_j)$ and $f(w_k, w_\ell)$ are rational functions encoding local and edge data,
• $\Theta_{-M, a}(q;\mathbf{w})$ is a shifted (multivariate) theta series summing over lattice points, typically

$$\Theta_{-M, a}(q;\mathbf{w}) = \sum_{\mathbf{m}\in M\mathbb{Z}^N+a} q^{\frac14 \mathbf{m}^T M^{-1} \mathbf{m}} \prod_{j=1}^N w_j^{m_j}$$

• $\mathrm{PV}$ denotes Cauchy principal value, imposed to regularize arising divergence.

For positive definite unimodular cases, the integral and spectrum of the operator are translated into constant-term extractions from Laurent expansions, leading to expressions in terms of (generalized) false theta functions, which are central for modularity analysis (Bringmann et al., 2018).

For more intricate graphs, especially those not weakly negative definite, new candidates for the homological block involve indefinite false theta functions—sum over lattice points weighted by sign functions to enforce convergence and modular-like properties—see Section 5 for explicit formulas in the Poincaré homology sphere case (Murakami, 2022).

2. Relationship with WRT Invariants and Analytic Continuation

A centerpiece of the model is the conjecture (now proven in a broad class of cases) that the Witten–Reshetikhin–Turaev invariant $\mathrm{WRT}_k(M)$ of a plumbed manifold $M$ is encoded in the radial limit of the homological block $\widehat{Z}_a(q)$: $$\mathrm{WRT}_k(M) = \frac{1}{2(\zeta_{2k} - \zeta_{2k}^{-1})} \, \lim_{q \to \zeta_k} \widehat{Z}_a(q)$$ where $\zeta_k = e^{2\pi i/k}$ and $q$ approaches the root of unity radially (i.e., $q = \zeta_k\, e^{-t}$, $t\to0^+$) (Murakami, 2022, Mori et al., 2021, Murakami, 2023). The proof requires detailed analysis of the q-series’ asymptotics, comparison of expansions with those of the WRT invariants (often involving weighted Gauss sums), and vanishing of certain “dangerous” negative-degree contributions—guaranteed by holomorphy and confirmed via advanced Euler–Maclaurin expansion techniques.

This analytic transition provides a bridge from non-perturbative topological invariants (WRT) to q-series with arithmetic and modular structure, making the Gukov–Pei–Putrov model foundational in the quantum topology–arithmetic interface.

3. Quantum Modularity and False Theta Functions

The q-series invariants for many plumbed 3-manifolds were found to be false theta functions or generalizations thereof. These functions are q-series whose modularity properties are “defective” in a controlled sense. More precisely:

• For single-leg and star graphs (e.g., Seifert homology spheres), the homological blocks are (derivatives of) unary false theta functions, which are quantum modular forms of depth one.
• For more complicated graphs (e.g., unimodular H-graphs), the homological block is a rank-two (or higher) false theta function and is a quantum modular form of higher depth—in the H-graph, depth two (Bringmann et al., 2019, Mori et al., 2021, Bringmann et al., 2021).

A depth $r$ quantum modular form is a function for which the modular “error” itself decomposes into quantum modular forms of depth $r-1$. For instance, Zhat-invariants for H-graphs: $$Z_0(q) = \sum_{n_1, n_2 \in \mathbb{Z}^2} \mathrm{sgn}^\ast(n_1) \mathrm{sgn}^\ast(n_2) q^{Q(n_1 + a_1, n_2 + a_2)}$$ with quadratic form $Q(\cdot)$ depending on plumbing data—fall under this higher-depth quantum modular paradigm (Bringmann et al., 2019, Bringmann et al., 2021). Modularity completions are constructed via iterated Eichler integrals, clarifying transformation properties under $SL_2(\mathbb{Z})$ and integrating the theory with analytic number theory.

For indefinite plumbing cases, the “homological blocks” are given by “indefinite false theta functions,” which still reproduce WRT invariants via radial limits and coincide with original blocks in classical examples (Murakami, 2022).

4. Algebraic and Combinatorial Structures: Splice Diagrams and Root Lattices

The invariants $\widehat{Z}_a(q)$ are intimately connected to combinatorial and algebraic structures used in singularity and 3-manifold theory:

• Splice diagrams: By interpreting plumbing graphs via normal surface singularity theory, it is shown that the normalized sum of all $\widehat{Z}_b$ is determined by the splice diagram—topological structures encoding the “universal” abelian cover of the link of the singularity. This demonstrates that the q-series invariant depends only on the splice diagram class of the 3-manifold, not on the specific plumbing presentation (Gukov et al., 2023).
• Root lattices and Weyl assignments: The q-series formalism extends by twisting with arbitrary root lattices $Q$ (not just $A_1$). The unique, Neumann-move-invariant series $\widehat{Z}_a(q)$ is recovered as a symmetrized average over series labeled by Weyl group assignments (on graph vertices) and input “Kostant” collections. This decomposition reveals that, for Seifert and Brieskorn spheres, individual building blocks coincide, while more complex graphs require averaging over distinct, locally invariant series (Moore et al., 23 May 2024).

Graph Type	False Theta Structure	Quantum Modularity Depth	Average/Decomposition Feature
3-leg star	unary false theta (1-dim)	1	single series
H-graph (6 verts)	rank-two false theta (2-dim)	2	single/averaged over Weyl assignments
Non-Seifert	higher-rank/indefinite	$r$ (vertices $\geq 3$)	nontrivial average of distinct building blocks

This algebraic and combinatorial underpinning assures that the q-series invariants are genuine topological invariants, i.e., indistinguishable under the standard set of plumbing/Neumann moves for the corresponding 3-manifold.

5. Indefinite Plumbing and the Poincaré Homology Sphere

Not all 3-manifolds arising from plumbings possess weakly negative definite linking matrices. In such cases (e.g., certain H-graphs realizing the Poincaré homology sphere), the classical definition of homological blocks as false theta functions does not directly apply. The methodology is generalized by defining “indefinite false theta functions”: $$\widehat{Z}_\Gamma(q) = q^{[-3/2]} \sum_{n=-1}^{\infty} \chi_{60}(n) q^{(n^2 - 1)/120}$$ where $\chi_{60}(n)$ is a character determined by explicit congruence data (Murakami, 2022). The radial limit of the resulting q-series recovers the WRT invariant up to an explicit normalization, and coincides with previous constructions (including the formulation of Lawrence–Zagier and Gukov–Pei–Putrov–Vafa).

This construction demonstrates that even when the underlying plumbing is indefinite, the analytic and arithmetic properties of the q-series—now involving more sophisticated regularizations and asymptotic expansions—are still robust enough to recover classical 3-manifold invariants.

6. Modular, Quantum Modular, and Physical Implications

The Gukov–Pei–Putrov model solidifies the profound connection between topology, modular forms, and quantum field theory:

• The (quantum) modularity of homological blocks implies that the quantum invariants of 3-manifolds are arithmetic objects, with transformation properties closely paralleling mock modular forms and Eichler integrals (Bringmann et al., 2018, Bringmann et al., 2019, Bringmann et al., 2021).
• Via the 3d–3d correspondence, homological blocks are interpreted as partition functions for effective 3d $\mathcal{N}=2$ field theories on suitable backgrounds, where the Spin$^c$ or homological labels index BPS sectors (Chauhan et al., 2023, Chauhan, 14 Dec 2024).
• The model’s flexibility, including averaging over Weyl group data and twisting by different root systems, points to applications in categorification, knot homologies, and representation theory.

7. Summary Table of Structural Features and Results

Aspect	Key Result/Property	Reference(s)
Definition	q-series via plumbing/linking data, integral over torus	(Bringmann et al., 2018, Moore et al., 23 May 2024)
Modular/Quantum modular property	(Higher-depth) quantum modular form, rank matches # of branching vertices	(Bringmann et al., 2019, Bringmann et al., 2021)
WRT recovery by analytic continuation	Radial limit at roots of unity recovers WRT invariants	(Murakami, 2022, Mori et al., 2021)
Algebraic/topological invariance	Invariance under all Neumann moves justified via root lattice symmetries	(Gukov et al., 2023, Moore et al., 23 May 2024)
Indefinite case/Poincaré sphere	Indefinite false theta function construction coincides with homological block	(Murakami, 2022)
Extension to general gauge/Lie algebras	Invariant depends only on Lie algebra, not global group structure	(Chauhan et al., 2023)

The Gukov–Pei–Putrov model thus provides a comprehensive and mathematically robust framework for lifting quantum invariants of three-manifolds to q-series with rich modular behavior and deep connections to both topology and arithmetic. Its structure has proved resilient under generalizations (to higher rank, more general plumbings, and beyond), and its techniques—integral formulations, modularity analysis, and algebraic decompositions—now form a cornerstone for quantum topology, representation theory, and modern number theory research.