Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds

Abstract

Let X be a compact Calabi-Yau 3-fold, and write M, M for the moduli stacks of objects in coh(X), D b coh(X). There are natural line bundles KM → M, K M → M, analogues of canonical bundles. Orientation data on M, M is an isomorphism class of square root line bundles KM , satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman [36, §5] in their theory of motivic Donaldson-Thomas invariants, and is also important in categorifying Donaldson-Thomas theory using perverse sheaves.We show that natural orientation data can be constructed for all compact Calabi-Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi-Yau 3-folds X that admit a spin smooth projective compactification X ֒→ Y . This proves a long-standing conjecture in Donaldson-Thomas theory.These are special cases of a more general result. Let X be a spin smooth projective 3-fold. Using the spin structure we construct line bundles KM → M, K M → M. We define spin structures on M, M to be isomorphism classes of square roots KM . We prove that natural spin structures exist on M, M. They are equivalent to orientation data when X is a Calabi-Yau 3-fold with the trivial spin structure.We prove this using our previous paper [34], which constructs 'spin structures' (square roots of a certain complex line bundle K E• P → BP ) on differential-geometric moduli stacks BP of connections on a principal U(m)-bundle P → X over a compact spin 6-manifold X. 0 (X). For Calabi-Yau 3-folds, such strong spin structures are equivalent to strong orientation data, which is important in the categorification of Donaldson-Thomas theory using perverse sheaves.We begin in §2 by summarizing the results of [34] on spin structures on connection moduli spaces B P . Section 3 studies spin structures on sheaf moduli spaces M, M for a spin smooth projective m-fold X with m odd. Section 4 restricts to X Calabi-Yau, and relates spin structures and orientation data on M, M. The proofs of Theorem 3.6 and 3.10 are deferred to §5- §6.

Batalin--Vilkovisky quantization and supersymmetric twists

“…[Pan+13]). It is shown in [JU21b] that RBun GC (X) carries a canonical orientation data for G = SU(n).…”

Dimensional reduction in cohomological Donaldson-Thomas theory

“…It studies the vanishing cycle cohomologies of the moduli stacks of representations over Jacobi algebras associated with quivers with potentials. Later [BBBBJ15, BBDJS15,JU20] opened the door to CoDT theory for CY 3-folds by defining a natural perverse sheaf ϕ M H-ss X (resp. ϕ M H-st X ) on the moduli stack M H-ss X of compactly supported H-semistable sheaves (resp.…”

“…In[JU20], natural orientation data for a wide class of CY 3-folds including all projective ones and local surfaces are constructed using gauge theoretic techniques. In [JU20, Remark 4.12] it is conjectured that our choice coincides with theirs for local surfaces.…”