Families DT/GW Correspondence

• The Families DT/GW Correspondence is a framework that unifies GW theory for curves and DT/stable pair theories for sheaves via a universal change of variables and correction matrices.
• It employs degeneration techniques, relative and logarithmic enhancements, and explicit constructions over toric, Calabi–Yau, and Fano threefold families.
• Open issues include the rationality of descendent series, proofs of universal correspondences, and robust degeneration packages for systematic invariant calculations.

The Families DT/GW Correspondence unifies Gromov–Witten (GW) theory of curves and Donaldson–Thomas (DT) or stable pair (P) theories of sheaves, focusing on the behavior across families of algebraic 3-folds and under degenerations. The framework generalizes classical GW/DT correspondences and incorporates descendent insertions, relative geometries, and logarithmic enhancements. The correspondence matches partition functions via a universal change of variables and intricate correction matrices, capturing deep enumerative relationships between moduli spaces of curves and sheaves. This article reviews the core definitions, the universal correspondence structure, key results, degeneration strategies, and major generalizations.

1. Fundamental Framework: Stable Maps, Pairs, and Partition Functions

Consider $\nu: \mathcal{X} \to \mathcal{Y}$, a smooth projective morphism with fibres nonsingular complex 3-folds. For a fibre class $\beta \in H_2(\mathcal{X}_y, \mathbb{Z})$, define $d_\beta = \int_\beta c_1(T_\nu) \in \mathbb{Z}$.

• Gromov–Witten (GW) Theory: For genus $g$, $r$ marked points, and descendent classes $\gamma_i \in H^*(\mathcal{X})$, the moduli space ${}_{g,r}(\nu, \beta)$ parameterizes stable maps $f: C \to \mathcal{X}_y$ representing $\beta$. The disconnected-domain partition function is

$$Z'_{\mathrm{GW}( \nu; u \mid \tau_{k_1,\dots,k_r}(\delta) )_\beta} = \sum_{g \in \mathbb{Z}} \langle \tau_{k_1,\dots,k_r}(\delta) \rangle^{\mathrm{GW}'}_{g,\beta} u^{2g-2} \in H_*(\mathcal{Y})((u)).$$

• Stable Pair/DT Theory: Stable pairs $(F,s)$ consist of pure dimension-1 sheaf $F$, $s: \mathcal{O} \to F$ with 0-dimensional cokernel. The moduli $_n(\nu, \beta)$ carries the virtual class, and the partition function is

$$Z_{\mathrm{P}( \nu; q \mid \tau_{k_1,\dots,k_r}(\delta) )_\beta} = \sum_{n \in \mathbb{Z}} \langle \tau_{k_1,\dots,k_r}(\delta) \rangle^{\mathrm{P}}_{n,\beta} q^n \in H_*(\mathcal{Y})((q)).$$

Both sides extend to relative and logarithmic settings, with boundary conditions and tangency data encoded by partitions and log-cohomological insertions (Pandharipande, 26 Jan 2025, Maulik et al., 2023).

2. Universal Change of Variables and Correction Matrices

The central conjecture (Conjecture I) posits an equality between generating series under the substitution $-q = e^{iu}$, up to explicit dimension-matching powers and a universal insertion-correction:

$$(-q)^{-d_\beta/2} Z_{\mathrm{P}}( \nu; q \mid \prod \tau_{\alpha_i-1}(\gamma_i) )_\beta = (-iu)^{d_\beta} Z'_{\mathrm{GW}}( \nu; u \mid \overline{ \prod \tau_{\alpha_i-1}(\gamma_i) } )_\beta.$$

The “bar” denotes a canonical linear combination of diagonal descendents built via a universal matrix $$\widetilde K_{\alpha,\hat\alpha}(u, c_1, c_2, c_3) \in \mathbb{Q}[i][[u]][c_1,c_2,c_3]$$ indexed by partitions. This insertion-correction is required for general descendent insertions, especially as new diagonal descendents (not factorizable across the base) arise in families (Pandharipande, 26 Jan 2025).

3. Relative, Logarithmic, and Degeneration Formulations

• Relative Extensions: Both GW and P theories admit relative versions (with log-smooth $\nu$, normal-crossing fibers, and weighted boundary conditions), and the correspondence extends, matching boundary insertions and virtual classes.
• Logarithmic Enhancements: The correspondence is formulated for simple normal crossings degenerations, with boundary-tangency profiles $\vec{\mu}$ and incidence conditions realized in the Nakajima basis for Hilb$^n(\partial D)$ (Maulik et al., 2023). Logarithmic degeneration formulas express partition functions of the general fiber in terms of those of the special fiber's strata, with strong cycle-level splitting and compatibility constraints.
• Degeneration Formulas: For degenerations $X \rightarrow X_0 = X_1 \cup_D X_2$, DT and GW invariants decompose via gluing over Hilbert schemes of boundary divisors, with insertion of strict diagonal classes ensuring compatibility of the correspondence (Maulik et al., 2023, Pandharipande, 26 Jan 2025, Oberdieck, 2021).

4. Proven Cases and Explicit Constructions

Five major classes of threefolds/families have been established:

Case	Technique	Reference
Toric 3-folds	$\mathbb{C}^*$-equivariant localization	(Pandharipande, 26 Jan 2025)
Calabi–Yau quintic	Toric degeneration, relative/log theory	(Pandharipande, 26 Jan 2025)
Log–Calabi–Yau (quartic K3)	Toric pair degeneration strategy	(Pandharipande, 26 Jan 2025)
Fano/CY families	Analytic transversality, local-curve	(Pandharipande, 26 Jan 2025)
Universal family over $\overline{M}_{g,n}$	Nodal locus, boundary Reductions	(Pandharipande, 26 Jan 2025)

Marked-relative GW/PT correspondences have also been proved for Fano complete intersections and product K3$\times$C geometries with points up to divisibility 2 (Oberdieck, 2021). In the quiver setting, log GW/DT (via Kronecker quivers) is matched for all genera and refined as Laurent polynomials in $q^{1/2}$ (Bousseau, 2018).

5. Logarithmic Topological Recursion and BPS Index Interpretation

For toric strip geometries, closed-string free energies are derived via Logarithmic Topological Recursion (Log-TR) and $x$–$y$ duality, producing explicit residue formulae for all genera:

$$F_g = \text{Bernoulli/polylogarithm sum over strip parameters},$$

where the infinite product representation of the partition function recovers DT invariants and 5D BPS indices. The polylogarithm terms encode GW/Gopakumar-Vafa invariants, while product exponents match DT (D2–D0) or BPS degeneracies (Banerjee et al., 21 Aug 2025).

6. Degeneration, Flop Invariance, and Compatibility

Degeneration and blow-up formulas allow reduction of general invariants to local or toric cases via strong cycle-level splitting packages, with explicit mapping across Hilbert schemes and log structures (Maulik et al., 2023, Bousseau, 2018). The DT/Flop formulas state that DT partition functions (and, by implication, GW/DT correspondences) are invariant under flops of $(-2)$-curves, matching all BPS numbers and ensuring no wall-crossing in the BPS spectrum for the flopped locus. This is anticipated to extend to arbitrary threefold flops (Ke, 2016).

7. Open Problems and Generalizations

The major current questions include:

• Rationality of general-descendent stable-pair series for families, necessary for rigorous change-of-variable formulation.
• Proofs of universal descendent correspondences (Conjectures II/III) beyond toric and some relative settings.
• Development of logarithmic generalizations to arbitrarily deep normal-crossing degenerations (Pandharipande, 26 Jan 2025, Maulik et al., 2023).
• The universal correspondence over the moduli of K3 surfaces remains open for general descendents.
• Compatibility with exotic incidence insertions and factorization properties in the Hilbert schemes of boundary expansions (Maulik et al., 2023).

A plausible implication is that the establishment of full degeneration packages and rationality results would enable systematic calculation of logarithmic DT invariants from GW side and vice versa.

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The Families DT/GW correspondence thus organizes a rich landscape of enumerative invariants, linking the geometry of curves and sheaves in families, capturing degeneration limits, and encoding physical BPS data, with progress driven primarily by universal correspondence matrices, logarithmic enhancements, and degeneration strategies.