A Derived Legendrian Category for Shifted Contact Stacks

Efe \.Izbudak; Kadri \.Ilker Berktav

arxiv: 2605.13792 · v2 · pith:J3GHCFGQnew · submitted 2026-05-13 · 🧮 math.AG · math.SG

A Derived Legendrian Category for Shifted Contact Stacks

Efe \.Izbudak , Kadri \.Ilker Berktav This is my paper

Pith reviewed 2026-06-30 21:19 UTC · model grok-4.3

classification 🧮 math.AG math.SG

keywords derived Legendrian categoryshifted contact stackLegendrian correspondenceequivariant descentAKSZ constructionderived algebraic geometryLegendrian surgerymoduli theory

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The pith

The derived Legendrian category F_c(X) is constructed for n-shifted contact derived Artin stacks using equivariant descent on Legendrian morphisms.

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

This paper constructs the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X. Objects in the category are Legendrian morphisms, while morphism spaces and composition are defined using equivariant descent to make the structure work in derived algebraic geometry. The construction also produces an embedding of F_c(X) into an (∞,2)-category of spans coming from the AKSZ construction. It further treats topological cobordisms as Lagrangian correspondences to carry out derived Legendrian surgery, with stated applications to moduli theory.

Core claim

For an n-shifted contact derived Artin stack X the derived Legendrian category F_c(X) is defined by taking Legendrian morphisms as objects and using equivariant descent to supply the morphism spaces and composition laws; this category embeds into the (∞,2)-category of spans furnished by the AKSZ construction, and topological cobordisms may be evaluated as Lagrangian correspondences to define derived Legendrian surgery.

What carries the argument

The derived Legendrian category F_c(X), with objects given by Legendrian morphisms and structure supplied by equivariant descent.

Load-bearing premise

Equivariant descent can be applied to define morphism spaces and composition operations for Legendrian morphisms on n-shifted contact derived Artin stacks in a way that yields a well-defined category.

What would settle it

An explicit calculation on a concrete n-shifted contact stack (for example a simple Artin stack or a point) in which the proposed morphism spaces fail to be associative under composition or the claimed embedding into AKSZ spans does not hold.

read the original abstract

We construct the derived Legendrian category $\mathcal{F}_{c}(X)$ for an $n$-shifted contact derived Artin stack $X$ and the $(\infty,2)$-category $Leg_n$ of Legendrian correspondences in the context of derived algebraic geometry, with several applications to moduli theory. In brief, the objects of the category $\mathcal{F}_{c}(X)$ are Legendrian morphisms; the morphism spaces and composition operations are defined using equivariant descent. We also establish that $\mathcal{F}_{c}(X)$ embeds into an $(\infty, 2)$-category of spans defined by the AKSZ construction. We further evaluate topological cobordisms as Lagrangian correspondences to define derived Legendrian surgery.

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.

Desk Editor's Note private letter to a colleague

The paper constructs a derived Legendrian category F_c(X) for n-shifted contact stacks using equivariant descent and embeds it into an AKSZ span category, but the abstract gives no details to verify the claims.

read the letter

This paper constructs the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X, along with the (∞,2)-category Leg_n of Legendrian correspondences. Objects are Legendrian morphisms, with morphism spaces and composition defined via equivariant descent. It embeds F_c(X) into an AKSZ span category and treats topological cobordisms as Lagrangian correspondences for derived Legendrian surgery, with nods to moduli theory applications.

The construction itself is the main new element. Framing Legendrian correspondences this way in derived algebraic geometry and contact stacks is a concrete step that ties into existing AKSZ machinery.

The soft spots are clear from the abstract alone. No definitions, equations, or descent arguments are shown, so it is impossible to check whether equivariant descent actually yields well-defined morphism spaces and composition without extra assumptions or circularity. The embedding and surgery claims are stated but not evidenced here. The weakest assumption is that descent works cleanly on these stacks; that may hold, but nothing in the text confirms it.

This is for a narrow group working in derived algebraic geometry, higher categories, and contact geometry who already know AKSZ and Legendrian setups. A reader in that subfield could extract a framework to test on moduli problems if the details check out.

It deserves a serious referee to examine the actual definitions and proofs, since the topic is specialized and the construction could be useful if it holds.

Referee Report

0 major / 2 minor

Summary. The manuscript constructs the derived Legendrian category F_c(X) for an n-shifted contact derived Artin stack X, taking Legendrian morphisms as objects and defining morphism spaces and composition via equivariant descent. It introduces the (∞,2)-category Leg_n of Legendrian correspondences, proves that F_c(X) embeds into the (∞,2)-category of spans arising from the AKSZ construction, and defines derived Legendrian surgery by viewing topological cobordisms as Lagrangian correspondences, with stated applications to moduli theory in derived algebraic geometry.

Significance. If the equivariant descent construction yields a well-defined (∞,2)-category with the claimed embedding and surgery operation, the work supplies a new categorical tool linking shifted contact geometry, Lagrangian correspondences, and AKSZ spans, which may organize moduli problems involving Legendrian data in derived settings.

minor comments (2)

[Abstract] Abstract and introduction: the precise statement of the embedding F_c(X) → AKSZ span category (including the functor on objects and 1-morphisms) is not visible in the provided summary and would benefit from an explicit theorem number or displayed statement early in the text.
The applications to moduli theory are announced but not illustrated with even a single concrete example (e.g., a specific moduli stack or a computation of morphism spaces); adding one short illustrative computation would strengthen readability without altering the central construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a direct construction of the derived Legendrian category F_c(X) whose objects are Legendrian morphisms and whose morphism spaces and composition are defined via equivariant descent on n-shifted contact derived Artin stacks. This is a definitional construction in derived algebraic geometry rather than a deduction or prediction that reduces to fitted parameters or prior self-citations by construction. The embedding into the AKSZ span category and the evaluation of topological cobordisms are stated as consequences of the construction itself. No equations, self-referential definitions, or load-bearing self-citations appear in the provided abstract or description that would make the central result equivalent to its inputs. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, background axioms, or new postulated entities.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Equivariant Contact Darboux Quotients and Perversely Categorified Legendrian Correspondences
math.AG 2026-06 unverdicted novelty 5.0

Establishes equivariant contact Darboux quotients for -1-shifted derived Artin stacks and constructs categorified Legendrian 2-categories via ell-adic perverse sheaves and Fourier-Mukai functors.

Reference graph

Works this paper leans on

8 extracted references · 3 canonical work pages · cited by 1 Pith paper · 2 internal anchors

[1]

Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

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work page internal anchor Pith review Pith/arXiv arXiv 2026
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T. Pantev, B. To¨en, M. Vaqui´e, G. Vezzosi,Shifted Symplectic Structures, Publ. Math. Inst. Hautes ´Etudes Sci. 117 (2013), 271-328

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Calaque, R

D. Calaque, R. Haugseng, C. Scheimbauer,The AKSZ Construction in Derived Algebraic Geometry as an Extended T opological Field Theory, arXiv:2108.02473, 2022

work page arXiv 2022
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K. ˙I. Berktav,Shifted Contact Structures and Their Local Theory, Ann. Fac. Sci. Toulouse, Math., Serie 6, Vol. 33(4): 1019-1057, 2024

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K. ˙I. Berktav,On shifted contact derived Artin stacks, Higher Structures 9(2):103–135, 2025

2025
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Guillermou and P

S. Guillermou and P . Schapira,Microlocal Theory of Sheaves and T amarkin’s Non Displaceability Theorem, in:Homological Mirror Symmetry and Tropical Geometry, Springer International Publishing, Cham, 2014, pp. 43–85

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The global nilpotent variety is Lagrangian

V . Ginzburg,The global nilpotent variety is Lagrangian, arXiv:alg-geom/9704005, 2000. DEPARTMENT OFMATHEMATICS, METU, 06800, ANKARA, T ¨URKIYE Email address:efe.izbudak@metu.edu.tr DEPARTMENT OFMATHEMATICS, METU, 06800, ANKARA, T ¨URKIYE Email address:berktav@metu.edu.tr

work page internal anchor Pith review Pith/arXiv arXiv 2000

[1] [1]

Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem

E. ˙Izbudak and K. ˙I. Berktav,Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersec- tion Theorem, 2026. arXiv:2605.08394

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Pantev, B

T. Pantev, B. To¨en, M. Vaqui´e, G. Vezzosi,Shifted Symplectic Structures, Publ. Math. Inst. Hautes ´Etudes Sci. 117 (2013), 271-328

2013

[3] [3]

Calaque,Shifted cotangent stacks are shifted symplectic, Ann

D. Calaque,Shifted cotangent stacks are shifted symplectic, Ann. Fac. Sci. Toulouse Math. 28 (2019), no. 1, 67-90

2019

[4] [4]

Calaque, R

D. Calaque, R. Haugseng, C. Scheimbauer,The AKSZ Construction in Derived Algebraic Geometry as an Extended T opological Field Theory, arXiv:2108.02473, 2022

work page arXiv 2022

[5] [5]

K. ˙I. Berktav,Shifted Contact Structures and Their Local Theory, Ann. Fac. Sci. Toulouse, Math., Serie 6, Vol. 33(4): 1019-1057, 2024

2024

[6] [6]

K. ˙I. Berktav,On shifted contact derived Artin stacks, Higher Structures 9(2):103–135, 2025

2025

[7] [7]

Guillermou and P

S. Guillermou and P . Schapira,Microlocal Theory of Sheaves and T amarkin’s Non Displaceability Theorem, in:Homological Mirror Symmetry and Tropical Geometry, Springer International Publishing, Cham, 2014, pp. 43–85

2014

[8] [8]

The global nilpotent variety is Lagrangian

V . Ginzburg,The global nilpotent variety is Lagrangian, arXiv:alg-geom/9704005, 2000. DEPARTMENT OFMATHEMATICS, METU, 06800, ANKARA, T ¨URKIYE Email address:efe.izbudak@metu.edu.tr DEPARTMENT OFMATHEMATICS, METU, 06800, ANKARA, T ¨URKIYE Email address:berktav@metu.edu.tr

work page internal anchor Pith review Pith/arXiv arXiv 2000