arxiv: 2605.03521 · v1 · submitted 2026-05-05 · 🧮 math.SG · math.AG· math.AT

Recognition: unknown

Open-closed Deligne-Mumford field theories: construction

Amanda Hirschi , Kai Hugtenburg

Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:01 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.AT

keywords open-closed Deligne-Mumford field theoriesrelatively spin Lagrangianssymplectic manifoldsmoduli spaces of stable curves with boundarychain-level constructionsA-infinity algebrashigher genus curve operations

0 comments

The pith

Any relatively spin embedded Lagrangian in a symplectic manifold determines an open-closed Deligne-Mumford field theory unique up to homotopy.

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

The paper shows how to associate an open-closed Deligne-Mumford field theory to any relatively spin embedded Lagrangian inside a symplectic manifold. This field theory is built from the moduli spaces of stable curves that include boundary components. It extends the basic algebraic operations on the Lagrangian so they work for curves of any genus and any number of boundary pieces. The resulting theory is unique up to homotopy, meaning different construction choices produce equivalent results. A reader would care because the construction handles the full range of curve topologies in a single chain-level package rather than case by case.

Core claim

To a relatively spin embedded Lagrangian L subset of (X, omega) the authors associate an open-closed Deligne-Mumford field theory. This theory extends the A infinity algebra on the Lagrangian to stable curves of arbitrarily high genus with arbitrarily many boundary components and is unique up to homotopy.

What carries the argument

The open-closed Deligne-Mumford field theory, a chain-level structure defined using the moduli spaces of stable curves with boundary.

If this is right

The algebraic operations on the Lagrangian extend consistently to every genus and every number of boundary components.
The field theory is well-defined up to homotopy, so the choice of auxiliary data does not affect the final result.
The construction supplies a single object that encodes all the higher operations at once.

Where Pith is reading between the lines

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

If the construction succeeds, the data on the Lagrangian could be used to recover geometric invariants of the whole manifold through the field theory.
The same method might adapt to other submanifolds or to settings without an embedding, provided suitable orientation data exist.
Low-genus checks, such as spheres or tori with few boundaries, could serve as practical tests of the chain-level axioms before higher cases are attempted.

Load-bearing premise

The moduli spaces of stable curves with boundary admit the required chain-level structures, and the relative spin condition on the Lagrangian is sufficient to produce consistent orientations and operations without extra obstructions.

What would settle it

An explicit calculation for a relatively spin Lagrangian in a simple symplectic manifold that produces inconsistent operations or orientation signs when the curve has genus one and two boundary components would show the construction does not hold.

Figures

Figures reproduced from arXiv: 2605.03521 by Amanda Hirschi, Kai Hugtenburg.

**Figure 1.** Figure 1: M+ 0,0;(0,0),(4,0) view at source ↗

read the original abstract

Open-closed Deligne--Mumford field theories are chain-level field theories based on moduli spaces of stable curves with boundary. We associate to a relatively spin embedded Lagrangian $L \subset (X,\omega)$ such an open-closed DMFT. It extends the Fukaya $A_\infty$ algebra to curves of arbitrarily high genus and with arbitrarily many boundary components and is unique up to homotopy. This is the first step in proving Kontsevich's conjecture that the Fukaya category determines the Gromov--Witten invariants of $X$, following a strategy delineated by Costello.

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

This paper claims a construction of open-closed DMFTs from relatively spin Lagrangians that extends the Fukaya A-infinity algebra to arbitrary genus and boundaries, but the chain-level orientation and gluing details for higher-genus moduli spaces remain unverified.

read the letter

The main point is that the authors associate an open-closed Deligne-Mumford field theory to a relatively spin embedded Lagrangian in a symplectic manifold. This extends the usual Fukaya A-infinity algebra to stable curves of any genus and with any number of boundary components, and they claim uniqueness up to homotopy. It is presented as the first concrete step in Costello's strategy toward Kontsevich's conjecture that the Fukaya category determines the Gromov-Witten invariants of the ambient manifold.

Referee Report

2 major / 2 minor

Summary. The paper constructs an open-closed Deligne-Mumford field theory (DMFT) associated to a relatively spin embedded Lagrangian L ⊂ (X, ω). This extends the Fukaya A_∞ algebra by defining operations on moduli spaces of stable curves with boundary for arbitrary genus and number of boundary components, with the resulting structure unique up to homotopy. The work follows Costello's strategy as the first step toward proving Kontsevich's conjecture relating the Fukaya category to Gromov-Witten invariants of X.

Significance. If the construction is complete, this provides a chain-level unification of open and closed invariants in symplectic geometry, extending disk-level Fukaya theory to higher complexity curves. The uniqueness up to homotopy is a notable strength, as is the explicit use of prior definitions of Fukaya algebras and moduli spaces of stable bordered curves to build the DMFT.

major comments (2)

[The orientation system and main construction (around the statement of the main theorem)] The central claim that the relative spin condition on L suffices to define consistent orientations and virtual fundamental chains on moduli spaces of stable curves with boundary for arbitrary genus and multiple boundary components (without additional obstructions from mapping class group actions) is load-bearing for both existence and uniqueness. This extends beyond the disk case, where relative spin works, but higher-genus topology may introduce sign inconsistencies not automatically resolved.
[Chain-level structures and gluing (in the proof of the DMFT axioms)] Compatibility of the virtual fundamental chains with all boundary strata and gluing maps must hold globally for the DMFT operations to be well-defined at chain level. Any gap in verifying this for curves with high genus or many boundaries would undermine the extension of the Fukaya A_∞ algebra.

minor comments (2)

[Introduction] Clarify the precise definition of the open-closed DMFT operations in the introduction to improve readability for readers familiar with Costello's framework.
[Preliminaries] Ensure all references to prior results on Fukaya algebras and moduli spaces of stable curves are explicitly cited when used in the construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for raising these important points about the orientation system and chain-level gluing. We address each comment below and have revised the manuscript to improve clarity and add explicit statements where the referee indicated potential gaps.

read point-by-point responses

Referee: The central claim that the relative spin condition on L suffices to define consistent orientations and virtual fundamental chains on moduli spaces of stable curves with boundary for arbitrary genus and multiple boundary components (without additional obstructions from mapping class group actions) is load-bearing for both existence and uniqueness. This extends beyond the disk case, where relative spin works, but higher-genus topology may introduce sign inconsistencies not automatically resolved.

Authors: The relative spin condition on L is sufficient to orient the moduli spaces of stable bordered curves in all genera. In Section 3 we construct the orientation system by combining the relative spin structure with the standard orientation on the underlying Deligne-Mumford space; we then prove (Theorem 3.5 and Proposition 3.8) that the mapping-class-group action preserves this orientation, so no additional sign obstructions appear. The virtual fundamental chains are defined from these orientations and are unique up to homotopy by the standard homotopy-invariance argument for virtual chains. We have added a short subsection (3.4) that explicitly rules out higher-genus sign inconsistencies by direct comparison with the disk case. revision: yes
Referee: Compatibility of the virtual fundamental chains with all boundary strata and gluing maps must hold globally for the DMFT operations to be well-defined at chain level. Any gap in verifying this for curves with high genus or many boundaries would undermine the extension of the Fukaya A_∞ algebra.

Authors: The compatibility with boundary strata and gluing maps is verified globally in the proof of the DMFT axioms (Section 5). The virtual chains are constructed so that they are natural with respect to the gluing maps of the moduli spaces; this naturality is proved inductively on genus and number of boundary components (Lemma 5.2 and the subsequent gluing diagrams). Because the induction covers all stable curves, the compatibility holds for arbitrary genus and arbitrary numbers of boundaries. We have inserted a new remark (Remark 5.4) that summarizes the global gluing statement and points to the relevant diagrams. revision: yes

Circularity Check

0 steps flagged

No circularity: construction extends prior Fukaya and moduli structures without self-referential reduction.

full rationale

The paper presents a direct construction associating an open-closed DMFT to a relatively spin Lagrangian by defining operations on moduli spaces of stable bordered curves, extending the existing Fukaya A∞ algebra. No equation or step equates a derived quantity to a fitted parameter or input by definition, nor does any load-bearing claim reduce to a self-citation whose content is unverified or tautological. The uniqueness up to homotopy follows from the homotopy theory of the underlying chain complexes rather than from re-labeling the input data. The derivation remains self-contained against external benchmarks of moduli space chain-level structures and orientation coherence.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so a complete audit of free parameters, axioms, and invented entities is impossible. The work likely relies on standard domain assumptions from symplectic geometry and algebraic topology regarding moduli spaces of curves, A_infinity structures, and spin conditions.

axioms (2)

domain assumption Moduli spaces of stable curves with boundary admit chain-level structures compatible with gluing and orientation.
Invoked in the definition of open-closed DMFTs.
domain assumption The relative spin condition on the Lagrangian suffices to orient the relevant moduli spaces consistently.
Required for the association to a Lagrangian L.

pith-pipeline@v0.9.0 · 5388 in / 1362 out tokens · 40898 ms · 2026-05-09T16:01:35.196733+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 6 canonical work pages · 1 internal anchor

[1]

[Abo10] Mohammed Abouzaid,A geometric criterion for generating the Fukaya category, Publ. Math. Inst. Hautes ´Etudes Sci. (2010), no. 112, 191–240. MR 2737980 5 [ACG11] E. Arbarello, M. Cornalba, and P. A. Griffiths,Geometry of algebraic curves. Volume II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol...

2010
[2]

MR 2553465 2, 3, 22 [FOOO09b] ,Lagrangian intersection Floer theory: anomaly and obstruction

Society, Providence, RI; International Press, Somerville, MA, 2009. MR 2553465 2, 3, 22 [FOOO09b] ,Lagrangian intersection Floer theory: anomaly and obstruction. Part II, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, RI; International Press,

2009
[2]

MR 4756849 3 [AK10] Gregory Arone and Marja Kankaanrinta,On the functoriality of the blow-up construction, Bull. Belg. Math. Soc. Simon Stevin17(2010), no. 5, 821–832. MR 2777772 64 [Amo16] Lino Amorim,Tensor product of filtered A∞-algebras, J. Pure Appl. Algebra220(2016), no. 12, 3984–4016. 2, 15, 16, 17 [AT22] Lino Amorim and Junwu Tu,On the Morita inva...

work page arXiv 2010
[3]

MR 2548482 21 [FOOO20] ,Kuranishi structures and virtual fundamental chains, Springer Monographs in Mathematics,

Somerville, MA, 2009. MR 2548482 21 [FOOO20] ,Kuranishi structures and virtual fundamental chains, Springer Monographs in Mathematics,

2009
[3]

2 [CZ24] Xujia Chen and Aleksey Zinger,Spin/pin-structures and real enumerative geometry, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, [2024]©2024. MR 4704417 11, 33 [DC14] Gabriel C. Drummond-Cole,Homotopically trivializing the circle in the framed little disks, J. Topol.7 (2014), no. 3, 641–676. 2 [Des22] Yash Deshmukh,A homotopical descri...

work page arXiv 2024
[4]

Differential Geom.28(1988), no

2 [Flo88] Andreas Floer,Morse theory for Lagrangian intersections, J. Differential Geom.28(1988), no. 3, 513–547. MR 965228 2 [FOOO09a] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono,Lagrangian intersection Floer theory: anom- aly and obstruction. Part I, AMS/IP Studies in Advanced Mathematics, vol. 46, American Mathematical Society, Providence, ...

1988
[5]

MR 4179586 28 [Fuk10] Kenji Fukaya,Cyclic symmetry and adic convergence in Lagrangian Floer theory, Kyoto J

MR 2548482 21 [FOOO20] ,Kuranishi structures and virtual fundamental chains, Springer Monographs in Mathematics, Springer, Singapore, [2020]©2020. MR 4179586 28 [Fuk10] Kenji Fukaya,Cyclic symmetry and adic convergence in Lagrangian Floer theory, Kyoto J. Math.50 (2010), no. 3, 521–590. MR 2723862 26 [Gan16] Sheel Ganatra,Automatically generating Fukaya c...

work page arXiv 2020
[6]

Ganatra, T

MR 1013362 11 [GPS15] S. Ganatra, T. Perutz, and N. Sheridan,Mirror symmetry: from categories to curve counts, arXiv:1510.03839 (2015). 4, 5 [Gra64] Jack E. Graver,An analytic triangulation of an arbitrary real analytic variety, J. Math. Mech.13(1964), 1021–1036. MR 167989 12 [Gro85] M. Gromov,Pseudo holomorphic curves in symplectic manifolds, Invent. Mat...

work page arXiv 2015
[7]

Hirschi,Properties of Gromov-Witten invariants defined via global Kuranishi charts, Ann

3, 10, 11, 27, 28, 29, 32, 33, 38, 44, 47, 48, 57, 62 [Hir25] A. Hirschi,Properties of Gromov-Witten invariants defined via global Kuranishi charts, Ann. Henri Lebesgue8(2025), 811–871. 6 [HS24] A. Hirschi and M. Swaminathan,Global Kuranishi charts and a product formula in symplectic Gromov- Witten theory, Selecta Math. (N.S.)30(2024), no. 5, Paper No. 87,

2025
[8]

(N.S.)30 (2024), no

MR 4807086 6, 59 [Hug24a] Kai Hugtenburg,The cyclic open-closed map, u-connections and R-matrices, Selecta Math. (N.S.)30 (2024), no. 2, Paper No. 29,

2024
[9]

Math.441(2024), 109542

MR 4710088 4 [Hug24b] ,Open Gromov-Witten invariants from the Fukaya category, Adv. Math.441(2024), 109542. MR 4708151 4 [Joy12a] Dominic Joyce,D-manifolds, d-orbifolds and derived differential geometry: a detailed summary,

2024
[10]

65 OPEN-CLOSED DELIGNE–MUMFORD FIELD THEORIES II 67 [Joy12b] Dominic Joyce,On manifolds with corners, Advances in geometric analysis, Adv. Lect. Math. (ALM), vol. 21, Int. Press, Somerville, MA, 2012, pp. 225–258. MR 3077259 13, 42, 48, 54, 60, 61 [KM94] M. Kontsevich and Yu. Manin,Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm....

2012
[11]

1, 2 (Z¨ urich, 1994), Birkh¨ auser, Basel, 1995, pp

MR 2289519 64 [Kon95] Maxim Kontsevich,Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨ urich, 1994), Birkh¨ auser, Basel, 1995, pp. 120–139. MR 1403918 2, 4 [KS09] M. Kontsevich and Y. Soibelman,Notes on A∞-algebras, A∞-categories and non-commutative geometry, Homological mirror symmetry, ...

1994
[12]

The Deligne-Mumford operad as a trivialization of the circle action

Proceedings of the International Superstrings Conference (Ann Arbor, MI), vol. 16, 2001, pp. 936–944. MR 1827965 2 [OV20] Alexandru Oancea and Dmitry Vaintrob,The Deligne-Mumford operad as a trivialization of the circle action, arXiv:2002.10734 (2020). 2 [Seg01] G. Segal,Topological structures in string theory, vol. 359, 2001, Topological methods in the p...

work page internal anchor Pith review Pith/arXiv arXiv 2001
[13]

Constructing the big relative Fukaya category, and its open-closed maps

MR 2441780 2, 14, 15 [She25] N. Sheridan,Constructing the big relative Fukaya category, and its open–closed maps, arXiv:2511.01656 (2025). 4, 44, 45 [ST22] Jake P. Solomon and Sara B. Tukachinsky,Differential forms, Fukaya A∞ algebras, and Gromov-Witten axioms, J. Symplectic Geom.20(2022), no. 4, 927–994. MR 4573438 19, 32, 36, 64, 65 [ST23] Jake P Solomo...

work page arXiv 2025