Equivariant Contact Darboux Quotients and Perversely Categorified Legendrian Correspondences

Technical Summary: Equivariant Contact Darboux Quotients and Perverse Categorified Legendrian Correspondences

Problem Statement
The paper addresses the challenge of defining enumerative invariants and linearizing categorical structures within derived contact geometry. While prior work established that shifted contact derived Artin stacks admit smooth Darboux atlases, these local models are insufficient for calculating equivariant étale vanishing cycles, which are necessary for enumerative theories (such as Donaldson-Thomas invariants). Furthermore, the global non-linear 2-categories of Legendrian correspondences, previously constructed by the author, lack a linearized framework that connects derived contact moduli spaces to microlocal sheaf theory. The core difficulty lies in the "generic topological acyclicity" of contact spaces due to the structural $\mathbb{G}_m$-action (Reeb flow), which causes naive topological Euler characteristics to vanish.

Methodology
The author employs a combination of derived algebraic geometry, equivariant geometry, and $\ell$-adic sheaf theory. The methodology proceeds through the following steps:

1. Equivariant Local Models: The paper establishes an Equivariant Contact Darboux Theorem. By utilizing the derived symplectification of a $-1$-shifted contact stack and applying Kresch's theorem on stratification by quotient stacks, the author refines the local smooth atlas to a geometric quotient stack $[\Delta_{\text{loc}}(s)/G]$. Here, $\Delta_{\text{loc}}(s)$ is the derived discriminant locus associated with a relative section $s$, and $G$ is a reductive stabilizer group.
2. Descent of Perverse Sheaves: Using the BBDJS (Beilinson-Bernstein-Deligne-Gabber-Joyce-Schürmann) minimal model for vanishing cycles on the derived symplectification, the author descends these structures to the contact stack. This descent relies on Verdier's specialization equivalence for monodromic sheaves. The $\mathbb{G}_m$-action on the symplectification induces a tame geometric monodromy automorphism $T$ on the resulting $\ell$-adic perverse sheaf.
3. Linearization of Categories: The paper formulates a contact analogue of Joyce's conjecture to linearize the non-linear Legendrian categories. By utilizing the six-functor formalism for $\ell$-adic sheaves and Fourier-Mukai pull-push functors, the author constructs linearized 2-categories where 2-morphisms are defined by the étale hypercohomology of monodromic perverse sheaves.
4. Invariants via Trace Formulas: To resolve the issue of topological acyclicity, the paper defines enumerative invariants using the $\ell$-adic Grothendieck-Lefschetz trace of the twisted monodromy operator $\tau$ acting on the perverse sheaf of vanishing cycles. This leads to the definition of a contact Behrend function.

Key Contributions and Results

• Theorem 1.1 (Equivariant Contact Darboux Theorem): Proves that every $-1$-shifted contact derived Artin stack over an algebraically closed field $K$ of characteristic zero admits a smooth atlas by a contact Darboux scheme $\text{Spec}(A) \simeq \Delta_{\text{loc}}(s)$. In the presence of reductive stabilizers $G$, the stack is locally equivalent in the smooth topology to the geometric quotient $[\Delta_{\text{loc}}(s)/G]$.
• Theorem 1.2 (Monodromic $\ell$-adic Perverse Sheaves): Establishes that every oriented $-1$-shifted contact derived Artin stack admits an $\ell$-adic perverse sheaf $P_Z$ equipped with a tame geometric monodromy automorphism $T$. This structure is derived by descending the BBDJS vanishing cycles from the symplectification.
• Theorem 1.3 (Categorified Legendrian Category): Constructs a linearized 2-category $\mathcal{LF}_c(X)$ for a $1$-shifted contact stack $X$. The 1-morphisms are proper oriented Legendrian correspondences, and the 2-morphisms are given by the étale hypercohomology of monodromic perverse sheaves on derived intersections. A global categorified linear weak 2-category $\mathcal{LLeg}_0$ is also constructed for $0$-shifted contact stacks.
• Theorem 7.2 ($\ell$-adic Contact Topological Integration): Provides a formula for the contact Donaldson-Thomas (DT) invariant as the $\ell$-adic topological integration of the contact Behrend function $\nu^{\text{contact}}_Z$. This resolves the vanishing of topological invariants by weighting strata with the trace of the monodromy operator.
• Computational Examples: The paper computes the contact Behrend function for $A_k$-singularities and non-degenerate quadratic singularities. In both cases, the local contact multiplicity is found to be 1, contrasting with the classical Behrend function values (e.g., $k$ for $A_k$), demonstrating how the alternating trace of cyclic monodromy cancels contributions from multi-sheeted branches.

Significance and Claims
The paper claims to resolve the obstruction to defining enumerative invariants in derived contact geometry by introducing equivariant local models and a monodromic perverse sheaf formalism. By linearizing the Legendrian categories, the work connects derived contact moduli spaces to microlocal sheaf theory, providing a "Contact analogue of Joyce's conjecture."

The author asserts that this framework allows for the extraction of derived enumerative invariants via the $\ell$-adic Grothendieck-Lefschetz trace, effectively bypassing the generic topological acyclicity inherent in contact spaces. The results lay the groundwork for future developments in motivic contact DT invariants, Cohomological Hall Algebras (CoHAs), and a potential "Contact Homological Mirror Symmetry" via a Contact Nadler-Zaslow correspondence. The paper positions these results as the necessary geometric and cohomological foundations for a derived Symplectic Field Theory (SFT) formulated algebraically.