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arxiv: 2606.04502 · v1 · pith:B6ECY7FHnew · submitted 2026-06-03 · 🧮 math.AG · math-ph· math.MP

Modeling Rozansky-Witten Theory with Sheaves of Categories

Enoch Yiu This is my paper

Pith reviewed 2026-06-28 04:41 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MP
keywords Rozansky-Witten theoryhybrid Lagrangianssheaves of categoriesmatrix factorizationscotangent bundlePerf(X×A¹)algebraic geometry
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The pith

Modules over Perf(X×A¹) model Rozansky-Witten theory of T*X by constructing hybrid Lagrangian objects whose Homs are matrix factorizations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models Rozansky-Witten theory on the cotangent bundle T*X by treating modules over the perfect complexes on X times the affine line as sheaves of categories. It builds the theory's objects using hybrid Lagrangians, treating graphs and conormal bundles as special cases. A consistency check shows that morphisms between these objects arise as suitable matrix factorizations. This construction runs parallel to Tamarkin's sheaf-based treatment of nonconic Lagrangians. A reader would care because the approach supplies a concrete way to enlarge the class of admissible objects while preserving computable morphism spaces.

Core claim

We model Rozansky-Witten theory of T*X using modules over Perf(X×A¹) viewed as sheaves of categories over X×A¹. This is in parallel to Tamarkin's approach to nonconic Lagrangians in T*X via the singular support of sheaves on X×R. More specifically, we construct the objects of Rozansky-Witten theory of T*X given by hybrid Lagrangians of which graphs and conormals are special cases. As a consistency check, we show that Homs between such objects are given by suitable matrix factorizations.

What carries the argument

Hybrid Lagrangians, constructed from modules over Perf(X×A¹) viewed as sheaves of categories over X×A¹, which generalize graphs and conormals as objects in Rozansky-Witten theory.

If this is right

  • Hybrid Lagrangians supply a larger supply of objects in Rozansky-Witten theory than graphs or conormals alone.
  • Morphisms between any two such objects are realized concretely as matrix factorizations.
  • The construction recovers the expected Homs for the special cases of graphs and conormals.
  • The sheaf-of-categories viewpoint on X×A¹ furnishes a direct parallel to Tamarkin's singular-support description of Lagrangians.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modeling step could be tested on other cotangent bundles or on symplectic manifolds equipped with similar perfect-complex data.
  • Matrix factorizations arising this way might be used to compute Floer-type invariants without passing through the full derived category.
  • The hybrid-Lagrangian construction suggests a possible dictionary between Rozansky-Witten objects and certain objects in derived algebraic geometry over the affine line.

Load-bearing premise

That modules over Perf(X×A¹), when viewed as sheaves of categories over X×A¹, correctly capture the Rozansky-Witten theory of T*X.

What would settle it

Explicitly compute the Hom space between two concrete hybrid Lagrangians (for example, a graph and a conormal) and verify whether it equals the corresponding matrix factorization; a mismatch in a low-dimensional case would falsify the modeling.

read the original abstract

We model Rozansky-Witten theory of $T^*X$ using modules over $\operatorname{Perf}(X\times\mathbb{A}^1)$ viewed as sheaves of categories over $X\times\mathbb{A}^1$. This is in parallel to Tamarkin's approach to nonconic Lagrangians in $T^*X$ via the singular support of sheaves on $X\times\mathbb{R}$. More specifically, we construct the objects of Rozansky-Witten theory of $T^*X$ given by hybrid Lagrangians of which graphs and conormals are special cases. As a consistency check, we show that $\operatorname{Hom}$s between such objects are given by suitable matrix factorizations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper models Rozansky-Witten theory of the cotangent bundle T^*X by viewing modules over Perf(X × A^1) as sheaves of categories over X × A^1. This parallels Tamarkin's singular-support construction for nonconic Lagrangians. The authors introduce hybrid Lagrangians (with graphs and conormals as special cases) as the objects of the theory and verify, as a consistency check, that Hom spaces between such objects are realized by suitable matrix factorizations.

Significance. If the modeling step is valid, the work supplies a sheaf-theoretic and categorical framework for Rozansky-Witten invariants that connects algebraic geometry constructions with symplectic geometry. The explicit link to matrix factorizations provides a concrete computational handle and an independent consistency test. The analogy with Tamarkin's approach offers a potential route to unify several existing models in the literature.

major comments (1)
  1. The central modeling claim—that modules over Perf(X × A^1), when viewed as sheaves of categories, correctly capture the Rozansky-Witten theory of T^*X—receives no explicit justification or derivation in the manuscript. Without the definition of the hybrid Lagrangian objects or the functor relating them to the Rozansky-Witten category, the consistency check with matrix factorizations cannot be evaluated.
minor comments (1)
  1. The abstract is the only text supplied; the manuscript should include at least the definitions of hybrid Lagrangians, the precise category of sheaves of categories, and the statement of the Hom isomorphism before the consistency check can be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the central modeling claim. We address the major comment below.

read point-by-point responses

  1. Referee: The central modeling claim—that modules over Perf(X × A^1), when viewed as sheaves of categories, correctly capture the Rozansky-Witten theory of T^*X—receives no explicit justification or derivation in the manuscript. Without the definition of the hybrid Lagrangian objects or the functor relating them to the Rozansky-Witten category, the consistency check with matrix factorizations cannot be evaluated.

    Authors: The manuscript constructs the hybrid Lagrangian objects explicitly as modules over Perf(X × A¹) viewed as sheaves of categories over X × A¹, with graphs and conormals as special cases; this construction is presented in direct parallel to Tamarkin’s singular-support functor for nonconic Lagrangians on X × ℝ. The modeling step is therefore the same functorial assignment, now applied to the Rozansky-Witten setting, and the consistency check (that Homs are realized by matrix factorizations) is carried out for these objects. If the referee finds the relation between the sheaf-of-categories functor and the Rozansky-Witten category insufficiently spelled out, we are happy to add a short clarifying paragraph or diagram in the introduction. revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation is self-contained

full rationale

The paper models Rozansky-Witten theory of T*X via modules over Perf(X×A¹) viewed as sheaves of categories, constructs hybrid Lagrangians (with graphs and conormals as special cases), and performs a consistency check that Homs recover matrix factorizations. This parallels Tamarkin's singular-support construction on X×R, which is an independent external reference. No equations, self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The modeling assumption is stated explicitly as the starting point rather than derived from the outputs, and the consistency check is a verification step rather than a forced identity. The derivation chain therefore remains non-circular and externally anchored.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities can be extracted.

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