Curve counting via stable pairs in the derived category

Abstract: For a nonsingular projective 3-fold $X$, we define integer invariants virtually enumerating pairs $(C,D)$ where $C\subset X$ is an embedded curve and $D\subset C$ is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of $X$. The resulting invariants are conjecturally equivalent, after universal transformations, to both the Gromov-Witten and DT theories of $X$. For Calabi-Yau 3-folds, the latter equivalence should be viewed as a wall-crossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric Calabi-Yau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We prove that our integrality predictions for Gromov-Witten invariants agree with the BPS integrality. Conversely, the BPS geometry imposes strong conditions on the enumeration of pairs.

• The paper defines new curve-counting invariants using stable pairs in the derived category and constructs a moduli space with a virtual fundamental class for calculating these invariants.
• The authors conjecture that this stable pairs theory is equivalent to both Gromov-Witten and Donaldson-Thomas theories, potentially providing a unified framework via derived category transformations.
• The research predicts partition functions with rationality and integrality properties, supported by explicit calculations for local curves and extensions to toric geometry.

Curve Counting via Stable Pairs in the Derived Category

The paper by Pandharipande and Thomas focuses on the definition and study of new curve-counting invariants on nonsingular projective 3-folds via stable pairs in the derived category. This research builds upon the cornerstone theories of Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants, proposing an alternative approach for counting curves that potentially unifies these established theories and provides new insights into their integrality and equivalence.

Stable pairs are defined as pairs $(F, s)$ where $F$ is a sheaf on the threefold $X$ supported in dimension one, and $s$ is a section with specified properties to ensure stability. The main contribution lies in constructing a moduli space $P(X)$ of these pairs, endowed with a virtual fundamental class suitable for defining integer-valued invariants. This construction presents a distinguished component within the broader, less tractable, moduli space of complexes in the derived category $D^b(X)$.

Significant Findings and Predictions

• Virtual Class Construction: The authors construct a virtual class for the moduli space of stable pairs based on a perfectly obfuscating theory upon realizing $P(X)$ as a component of fixed determinant complexes. This is aligned with the classical DT setup but offers a new perspective by anchoring it in derived geometric constructs.
• Equivalence with GW and DT Theories: A conjectural relationship is postulated, suggesting that the stable pairs theory is equivalent to both Gromov-Witten and Donaldson-Thomas theories after appropriate transformations. Particularly, for Calabi-Yau 3-folds, this work conjectures an equality of invariants:

$$Z_{P,\beta}(q) = Z'_{DT,\beta}(q) = Z_{GW,\beta}(u)$$

postulating a transformation relationship akin to wall-crossing in the derived category.

• Rationality and Integrality: The authors predict that the partition functions of the stable pairs theory are Laurent expansions of rational functions invariant under $q \leftrightarrow q^{-1}$. Further, these functions adhere to an integrality constraint akin to the BPS invariants suggested by Gopakumar-Vafa, and conjecture that negative-genus Gopakumar-Vafa invariants vanish, which supports a fundamental rationality structure.

Practical Implications and Future Directions

• Local Curve Contributions: For local curve geometries, explicit calculations support the conjectures, including those for isolated contributions of curves as poles of the partition function, harmonizing results with established GW theory expectations.
• Vertex Decomposition in Toric Geometry: The study extends to toric Calabi-Yau 3-folds, providing the notion of a stable pairs vertex, potentially offering a refined view of partition functions that coincide with DT counts while adopting a new algebraic combinatorial structure based on $\mathbb{P}^3$.

The implications of this research extend towards an enriched understanding of integer-valued curve counting theories, proposing a unified framework that uses the derived category as a novel foundational aspect. Future explorations may solidify the conjectural GW/DT/stable pairs correspondence and address the computational techniques in both theoretical and empirical contexts. The extension of the theory to cover broader categories, including non-Calabi-Yau 3-folds and relative curve counting, represent robust arenas for further exploration and verification.