Categorified DT-Invariants

• Categorified DT-invariants are graded enhancements of classical DT invariants that utilize derived moduli stacks and shifted symplectic structures to capture refined wall-crossing phenomena.
• They leverage matrix factorization categories and periodic cyclic homology to translate deformation theory and intersection cohomology into robust, invariant counts.
• The framework uses Gauss–Manin connections and derived Lagrangian morphisms in Tyurin degenerations to ensure deformation invariance and link local and global moduli data.

Categorified Donaldson–Thomas (DT) invariants are homological or categorical enhancements of classical DT invariants, which traditionally enumerate stable coherent sheaves or objects on Calabi–Yau threefolds with certain fixed invariants. The program of categorification seeks to replace these integer-valued, often virtual counts with richer structures: graded vector spaces, complexes, or abelian/dg categories, whose invariants (Euler characteristics, graded traces, or K-theory classes) recover the classical DT numbers, but which carry additional geometric, representation-theoretic, or wall-crossing information. Recent developments integrate derived algebraic geometry, shifted symplectic structures, wall-crossing, matrix factorization categories, and even topological field theory, providing a robust framework for understanding not only invariants of spaces and moduli, but also the interplay of deformation, symmetry, and duality at the categorical level.

1. Moduli Problems, Shifted Symplectic Geometry, and Derived Lagrangians

A central geometric input into categorified DT invariants is the use of derived moduli stacks of complexes or sheaves—often governed by (-1)-shifted symplectic structures in the sense of Pantev–Toën–Vaquié–Vezzosi. Given a Calabi–Yau threefold $X$, the moduli stack of (rigidified) perfect complexes or stable sheaves on $X$ is locally modeled as a derived critical locus of a function $f$ on a smooth derived manifold or dg scheme. In the context of Tyurin degenerations $X \rightsquigarrow X_1 \cup_S X_2$, where $X_1$, $X_2$ are quasi-Fano threefolds glued along an anticanonical divisor $S$, the associated moduli spaces $\mathcal{M}_i$ (of perfect complexes on $X_i$) admit derived Lagrangian restriction morphisms to the moduli on $S$. Specifically, for each $i=1,2$, the restriction map

$r_i : \mathcal{M}(X_i) \to \mathcal{M}(S)$

is (–1)-shifted Lagrangian: the tangent-to-cotangent map is a quasi-isomorphism ($$\Theta_{r_i} : \mathbb{T}_{r_i} \to \mathbb{L}_{\mathcal{M}(X_i)}[-1]$$). The derived fiber product

$$\mathcal{M}(X_1) \times^L_{\mathcal{M}(S)} \mathcal{M}(X_2)$$

inherits a natural (–1)-shifted symplectic structure and acts as the primary object for algorithms and invariance theorems in categorified DT theory (Kryczka et al., 23 Oct 2025).

2. Matrix Factorizations, Critical Loci, and Categorified Invariants

Locally, derived moduli problems admit presentations by matrix factorization (MF) categories associated to potential functions derived from deformation theory. The classical case for an object $E$ on a Calabi–Yau threefold is via the cyclic $L_\infty$ algebra governing $E$'s deformations, yielding a dg model of the form $\mathrm{MF}(f)$ where $f$ is the "potential." In Tyurin degenerations, further enhancements allow the potential to be expressed as $\mathbb{W} = f \cdot g$ (incorporating normal directions from degeneration geometry). The associated MF categories $\mathrm{MF}(f_t)$ for the family parameter $t$ describe the categorical structure over each fiber.

The periodic cyclic homology $\operatorname{HP}_\bullet(\mathrm{MF}(f_t))$ serves as the primary receptacle for categorified DT invariants: in established work by Orlov and Efimov, the Euler characteristic (or suitable Hodge characteristic) of $\operatorname{HP}$ recovers the ordinary DT invariant. In the family context, the existence of a flat Gauss–Manin connection on $\operatorname{HP}_\bullet(\mathrm{MF}(f_t))$, constructed via a $u$-connection on the (mixed) Hochschild complex, ensures that the graded dimension of the categorified DT invariant is invariant under deformation, i.e., is constant in the family (Kryczka et al., 23 Oct 2025).

3. Derived Intersection Cohomology and Deformation Invariance

A key categorification phenomenon in the degeneration setting is that the global, or "generic fiber," categorified DT invariant (i.e., the periodic cyclic homology of the category over the smooth Calabi–Yau) can be expressed in terms of the derived intersection cohomology of the Fano moduli spaces on the special (degenerated) fiber. Explicitly, the derived fiber product

$$\mathcal{M}(X_1) \times^L_{\mathcal{M}(S)} \mathcal{M}(X_2)$$

as a (–1)-shifted symplectic derived stack, is locally given by the derived critical locus of a composite potential, with associated MF category whose $\operatorname{HP}$ computes the invariant. The calculation proceeds via a spectral sequence in which the first page involves the Tor-groups of the ideal sheaves of the Lagrangians (the Fano moduli spaces) in $\mathcal{M}(S)$; the degeneration at $E_2$ then yields the periodic cyclic homology, capturing the categorified DT invariant as a derived intersection cohomology (Kryczka et al., 23 Oct 2025). This structure intertwines the geometry of the degeneration with the wall-crossing and deformation invariance properties expected in DT theory.

4. Gauss–Manin Connections and Flatness

The existence of a flat Gauss–Manin connection on periodic cyclic homology $\operatorname{HP}_\bullet(\mathrm{MF}(f_t))$ as $t$ varies in the base of the Tyurin degeneration is a central technique for proving deformation invariance of the categorified DT invariant. The technical construction involves the $u$-connection (a generalization of the connection in the Deligne–Illusie theory for crystalline cohomology) acting on the $u$-completed mixed Hochschild complex, whose cohomology defines $\operatorname{HP}$. The explicit calculation of the twisted de Rham complex and the $u$-connection shows that the dimension (over suitable ground fields) is constant, and thus the categorified DT invariants are invariant under the degeneration, integrating the global geometry of the total family.

5. Implications and Derived Geometric Framework

The derived Lagrangian approach to Tyurin degenerations offers a model in which the categorified DT invariants—the periodic cyclic homologies of the relevant MF categories—satisfy strong deformation invariance and can be expressed in terms of local (special fiber) data, providing a powerful method for calculating or comparing categorified invariants. The categorical approach generalizes earlier methods using intersection cohomology, perverse sheaves, and motivic vanishing cycles, but now connects directly to the geometry of derived moduli, shifted symplectic structures, and aligns with the rich deformation theory of Calabi–Yau threefolds and their degenerations.

6. Summary of Structural Consequences

• Tyurin degenerations $(X \rightsquigarrow X_1 \cup_S X_2)$ enable control over the global-to-local transition in moduli space geometry, allowing one to relate DT invariants across families.
• Derived Lagrangian morphisms and the (–1)-shifted symplectic structure ensure that intersection theory, as formalized via derived critical loci, governs the computation and invariance of the categorified DT invariants.
• The Gauss–Manin connection on periodic cyclic homology provides a mechanism for tracking and proving deformation invariance and facilitating comparisons of invariants across singular and smooth fibers.
• Matrix factorization categories encode the categorified DT theory, with their periodic cyclic homology matching intersection theory computations, and are directly linked to the critical locus geometry arising from the deformation theory of the moduli problem.
• The final identification of the categorified DT invariant of the generic (smooth) fiber with the derived intersection cohomology of Fano moduli spaces on the special fiber provides a precise computational and conceptual bridge between DT theory, derived algebraic geometry, and symplectic topology.

Overall, this framework situates categorified Donaldson–Thomas invariants within the rigorous paradigm of derived algebraic geometry, yielding powerful invariance, comparison, and computation results crucial for further advances in enumerative and homological aspects of Calabi–Yau geometry and its applications in mathematical physics (Kryczka et al., 23 Oct 2025).