[Paper Review] Birational Calabi-Yau 3-folds and BPS state counting

Yukinobu Toda|ArXiv.org|Jul 11, 2007

Algebraic structures and combinatorial models31 references7 citations

TL;DR

This paper introduces motivic Gopakumar-Vafa invariants as counting invariants for D2-branes on Calabi-Yau 3-folds using Bridgeland stability conditions and Joyce's invariants. It proves these invariants are birational invariants, generalizing the invariance of Betti and Hodge numbers under birational maps, and providing a new mathematical framework for Gopakumar-Vafa invariants without relying on virtual classes.

ABSTRACT

This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce's work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar-Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi-Yau 3-folds. The result is similar to the fact that birational Calabi-Yau 3-folds have the same betti numbers or Hodge numbers.

Motivation & Objective

• To define a new mathematical formulation of Gopakumar-Vafa invariants that are invariant under birational transformations of Calabi-Yau 3-folds.
• To extend the invariance of topological invariants—such as Betti and Hodge numbers—beyond cohomological data to include invariants of D2-branes.
• To construct invariants using motivic data from moduli spaces of one-dimensional sheaves, avoiding reliance on virtual fundamental classes.
• To establish a framework where counting invariants of semistable objects under Bridgeland stability conditions yield birational invariants.

Proposed method

• Uses Bridgeland-Douglas stability conditions on the derived category of coherent sheaves on Calabi-Yau 3-folds.
• Applies Joyce’s theory of counting invariants of semistable objects in triangulated categories.
• Defines motivic Gopakumar-Vafa invariants $ n_g^\beta(X) \in \mathbb{Z} $ as refinements of earlier invariants using motivic invariants of moduli spaces.
• Works with the triangulated category $ \mathcal{D}_X \subset D^b(\mathrm{Coh}(X)) $ of complexes with dimension ≤1 support.
• Employs stability conditions parameterized by $ (B+i\omega) \in N^1(X)_{\mathbb{C}} $, and uses convergence and boundedness arguments to establish continuity.
• Proves the invariance of $ n_g^\beta $ under birational maps by analyzing stability conditions and their convergence in the space of stability conditions.

Experimental results

Research questions

• RQ1 Do Gopakumar-Vafa invariants counting D2-branes remain invariant under birational transformations of Calabi-Yau 3-folds?
• RQ2 Can a mathematical formulation of Gopakumar-Vafa invariants be constructed that avoids virtual classes while preserving birational invariance?
• RQ3 Is there a motivic refinement of the invariants defined via intersection cohomology of moduli spaces of one-dimensional sheaves that yields integer invariants under birational maps?
• RQ4 How do Bridgeland stability conditions and Joyce’s counting invariants interact to produce birational invariants in Calabi-Yau 3-folds?
• RQ5 Can the invariance of Betti numbers and Hodge numbers under birational maps be extended to invariants of D2-branes?

Key findings

• The motivic Gopakumar-Vafa invariants $ n_g^\beta(X) \in \mathbb{Z} $ are defined for all $ \beta \in N_1(X) $, including non-effective classes.
• These invariants coincide with the earlier invariants $ \tilde{n}_g^\beta $ when $ M^\beta $ is smooth and $ \beta $ is effective.
• The invariants $ n_g^\beta(W) $ are invariant under birational maps: $ n_g^\beta(W) = n_g^{\phi_*\beta}(X) $ for any birational map $ \phi: W \dashrightarrow X $.
• The construction does not rely on virtual fundamental classes, offering a new approach distinct from Gromov-Witten or Donaldson-Thomas theory.
• The result generalizes the invariance of Betti numbers and Hodge numbers under birational maps to invariants of D2-branes.
• The proof uses convergence of stability conditions and boundedness of central charges in the space of stability conditions, establishing continuity and closure properties.