Derived Geometry Approach to DT Theory

• The paper introduces a derived geometry framework that reconstructs DT invariants using (-1)-shifted symplectic structures and a refined obstruction theory.
• It establishes component lattice formalism and stability measures to generalize classical DT constructions and support new wall-crossing theorems.
• The approach leverages perverse sheaves, motivic integration, and categorification to achieve dimensional reduction and refine enumerative invariants.

The derived geometry approach to Donaldson–Thomas (DT) theory reconstructs enumerative invariants for moduli spaces of sheaves or objects on Calabi–Yau threefolds using the intrinsic structures available in derived algebraic geometry, notably (−1)-shifted symplectic derived stacks. This framework unifies and generalizes classical DT constructions, removing the reliance on linear presentation or abelian categories, and replaces them with structural tools from derived symplectic geometry, perverse sheaves, motivic integration, and Hall algebras. Key developments include the introduction of component lattices, stability measures, categorified invariants, and new wall-crossing and reduction theorems.

1. (−1)-Shifted Symplectic Derived Stacks and Obstruction Theory

A derived Artin stack $\mathfrak{X}$ locally of finite presentation over $\mathbb{C}$ is called quasi-smooth if its cotangent complex $$L_\mathfrak{X} \in \operatorname{Perf}(\mathfrak{X})$$ has Tor-amplitude in $[−1,1]$ (Kinjo, 2021). An $n$-shifted symplectic structure is a closed 2-form $$\omega \in \mathcal{A}^{2,\text{cl}}(\mathfrak{X}; n)$$ in the PTVV de Rham complex that induces a quasi-isomorphism $T_\mathfrak{X} \simeq L_\mathfrak{X}[n]$ (Bu et al., 27 Feb 2025, Bu et al., 19 Feb 2025). The canonical example is the (−1)-shifted cotangent stack

$$T^*[-1]\mathfrak{X} := \operatorname{Spec}_{\mathfrak{X}} \operatorname{Sym}(T_\mathfrak{X}[1]),$$

with tautological 1-form and degree (−1) symplectic form. Locally, on a smooth atlas $U \to \mathfrak{X}$, $T^*[-1]U$ is modeled as the critical locus $$\operatorname{Crit}(f) \subset \operatorname{Tot}(E^\vee)$$ for a function $f$ (Kinjo, 2021).

Orientations on (−1)-shifted symplectic stacks are specified by a choice of square root of the virtual canonical bundle, giving rise to a canonical determinant bundle and a corresponding orientation isomorphism (Bu et al., 27 Feb 2025).

The derived enhancement encodes both deformation and obstruction theory: for $X = t_0(\mathfrak{X})$, the truncated cotangent complex yields a two-term perfect complex whose dual gives a symmetric obstruction theory on $X$, abstracting the classical Behrend–Fantechi theory (Bu et al., 27 Feb 2025).

2. Component Lattice Formalism and Stability Measures

The intrinsic DT framework employs the component lattice, $CL(\mathfrak{X})$, which is the set of connected components of the stack of graded points $$\operatorname{Grad}(\mathfrak{X}) = \operatorname{Map}(BG_m, \mathfrak{X})$$ (Bu et al., 19 Feb 2025). This structure generalizes the cocharacter lattice and encodes a wall-and-chamber decomposition reflecting possible filtrations and stratifications.

A face $\alpha$ corresponds to a morphism of formal lattices, and cones represent filtrations. Key results include:

• Constancy theorem: The isomorphism type of graded and filtered components is locally constant within chambers of the cotangent hyperplane arrangement on $CL(\mathfrak{X})$.
• Finiteness theorem: If $\mathfrak{X}$ is quasi-compact with quasi-compact graded points, there are finitely many special faces and cones, hence only finitely many types of components (Bu et al., 19 Feb 2025).

A stability measure $\mu$ assigns weights to special cones in $CL(\mathfrak{X})$, generalizing classical stability conditions and GIT linearizations. $\mu$ provides a combinatorial input for the definition of DT invariants and the formulation of wall-crossing phenomena (Bu et al., 27 Feb 2025).

3. Derived Hall Categories and Associativity

Hall category structures extend classical Hall algebras to the non-linear, intrinsic stack context:

• The morphisms in the Hall category $Hall^+(\mathfrak{X})$ are pairs consisting of inclusions of special faces and cones in the component lattice.
• The associativity theorem states there is a canonical functor from $Hall^+(\mathfrak{X})^\text{op}$ to the category of spans of stacks, recovering classical Hall algebra associativity in the linear case (Bu et al., 19 Feb 2025).

Through the six-functor formalism, operations on stacks—such as motives, cohomology, and constructible functions—can be composed according to this Hall category structure, yielding associative algebra objects that encapsulate DT-theoretic data (Bu et al., 19 Feb 2025, Bu et al., 27 Feb 2025).

4. Perverse Sheaves, Cohomological DT Invariants, and Dimensional Reduction

On an oriented (−1)-shifted symplectic derived Artin stack $(\mathfrak{X},\omega,\mathcal{O})$, a perverse sheaf $\mathcal{P}_\mathfrak{X}$ is constructed via local vanishing cycle complexes, glued globally using d-critical stack formalism (Kinjo, 2021). When $\mathfrak{X}$ is a derived moduli stack of sheaves on a Calabi–Yau threefold, the hypercohomology $H^*(t_0(\mathfrak{X}); \mathcal{P}_\mathfrak{X})$ categorifies the DT invariant and recovers the Behrend-weighted DT invariant upon taking the Euler characteristic.

Dimensional reduction theorem: For the (−1)-shifted cotangent stack $T^*[-1]\mathfrak{X}$ of virtual dimension $\operatorname{vdim}\mathfrak{X}$, there is a canonical isomorphism:

$$H^*\big(T^*[-1]\mathfrak{X};\mathcal{P}_{T^*[-1]\mathfrak{X}}\big) \cong H_*^{\mathrm{BM}}\big(t_0(\mathfrak{X})\big)[+\operatorname{vdim}\mathfrak{X}],$$

where $H_*^{\mathrm{BM}}$ denotes Borel–Moore homology. This globalizes Davison’s local reduction and has applications to the paper of moduli for local surfaces and the sheaf-theoretic construction of virtual fundamental classes, providing a derived-geometric Thom isomorphism for dual obstruction cones (Kinjo, 2021).

5. Rings of Motives, Motivic Integration, and DT Invariants

Derived geometry allows for the definition of motivic and cohomological refinements of DT invariants. The motivic ring $\mathcal{M}(\mathfrak{X};A)$ is generated by classes of representable quasicoherent substacks, and integration employs the motivic Behrend function $\nu_\mathfrak{X}^{\text{mot}}$ constructed using vanishing cycles (Bu et al., 27 Feb 2025). The DT invariants are defined as:

• Numerical DT invariants:

$$DT_\mathfrak{X}^{(k)}(\mu) = \int_\mathfrak{X} (1-\mathbb{L})^k \epsilon_\mathfrak{X}^{(k)}(\mu)\cdot \nu_\mathfrak{X}$$

• Motivic DT invariants:

$$DT_\mathfrak{X}^{(k),\mathrm{mot}}(\mu) = \int_\mathfrak{X} (\mathbb{L}^{1/2} - \mathbb{L}^{-1/2})^k \epsilon_\mathfrak{X}^{(k)}(\mu) \nu_\mathfrak{X}^{\mathrm{mot}}$$

where $\epsilon_\mathfrak{X}^{(k)}(\mu)$ is constructed via Möbius inversion in the Hall category, and $\nu_\mathfrak{X}$ is the Behrend function (Bu et al., 27 Feb 2025).

Wall-crossing formulae arise from the transformation behavior of invariants under changes in the stability measure $\mu$, now governed by the action of the Hall algebra on motivic data. The generalized no-pole theorem ensures that the motivic integration is finite and well-defined for all (−1)-shifted symplectic stacks (Bu et al., 27 Feb 2025).

6. Derived Category and Stability Conditions Perspective

The moduli stacks of Bridgeland semistable objects in the derived category $\mathcal{D} = D^b(\operatorname{Coh}(X))$ underpin classical DT theory for Calabi–Yau threefolds, but the derived approach allows extension to more general, possibly nonlinear, stacks (Toda, 2014). Bridgeland stability conditions and their spaces encode wall-crossing structure, with associated wall-crossing formulas expressed in quantum torus algebras. The component lattice and stability measure frameworks generalize these moduli-theoretic perspectives, supporting DT invariants independent of linear or abelian embeddings (Bu et al., 27 Feb 2025, Bu et al., 19 Feb 2025).

7. Categorification, Geometry of Stability Spaces, and Joyce Structures

Further categorification is achieved by associating geometric structures, such as Joyce structures, to the space of stability conditions $M = \operatorname{Stab}(\mathcal{C})$ for a chosen Calabi–Yau 3-category. Derived DT invariants control the isomonodromic data of a pencil of flat connections $\mathcal{A}$ on $TM \times \mathbb{P}^1$, reconstructing DT/BPS wall-crossing as Stokes phenomena (Bridgeland, 2019).

Joyce structures parallel Frobenius manifold structures: they consist of a flat connection, a constant lattice, a skew form, a Joyce function $J$, and an Euler field, with axioms encoding DT wall-crossing and associativity. Explicit calculations for quivers, geometric categories, and local Calabi–Yau varieties realize DT invariants as Stokes multipliers and encode the enumerative data in holomorphic generating functions (Bridgeland, 2019).

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The derived geometry approach, by replacing abelian-category or ambient-critical-locus dependence with (−1)-shifted symplectic structures, stability measures, component lattices, and sheaf-theoretic or motivic tools, enables a uniform and general construction of DT invariants for Artin stacks, offering a robust framework for motivic, cohomological, and categorified enumerative invariants and their wall-crossing behavior (Kinjo, 2021, Bu et al., 27 Feb 2025, Bu et al., 19 Feb 2025, Bridgeland, 2019, Toda, 2014).