Open-closed Deligne-Mumford field theories: geometric foundations

Amanda Hirschi; Kai Hugtenburg

arxiv: 2501.04687 · v2 · submitted 2025-01-08 · 🧮 math.SG

Open-closed Deligne-Mumford field theories: geometric foundations

Amanda Hirschi , Kai Hugtenburg This is my paper

Pith reviewed 2026-05-23 06:13 UTC · model grok-4.3

classification 🧮 math.SG

keywords Kuranishi chartspseudo-holomorphic mapsLagrangian submanifoldsmoduli spacesopen-closed field theoryDeligne-Mumford field theorychain-level operationssymplectic geometry

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The pith

Global Kuranishi charts are constructed for moduli spaces of pseudo-holomorphic maps with boundary on embedded Lagrangians of arbitrary genus.

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

The paper constructs global Kuranishi charts for moduli spaces of pseudo-holomorphic maps from surfaces of arbitrary genus whose boundaries map to an embedded Lagrangian submanifold. These charts are required to be compatible with the boundary conditions and with changes in genus so that consistent chain-level operations can be defined. The resulting geometric setup supplies the foundations for defining those operations. The operations are then used in follow-up work to construct an open-closed Deligne-Mumford field theory. A sympathetic reader would care because the charts turn geometric moduli spaces into objects on which algebraic structures can be built directly.

Core claim

We construct global Kuranishi charts for moduli spaces of pseudo-holomorphic maps of arbitrary genus with boundary on an embedded Lagrangian submanifold. We then build the geometric foundations required for obtaining compatible chain-level operations, which are employed in follow-up work to construct an open-closed Deligne-Mumford field theory.

What carries the argument

Global Kuranishi charts for the moduli spaces of pseudo-holomorphic maps with boundary, which furnish local models compatible across genera and boundary conditions so that chain-level operations can be defined consistently.

Load-bearing premise

The moduli spaces of pseudo-holomorphic maps with boundary admit global Kuranishi charts that are compatible with the boundary conditions and genus variations in a way that permits consistent chain-level operations.

What would settle it

A concrete example of an embedded Lagrangian submanifold, a fixed genus, and a nonempty moduli space of maps for which no global Kuranishi chart exists that respects both the boundary condition and the required compatibilities would show the construction fails.

read the original abstract

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 constructs global Kuranishi charts for moduli spaces of pseudo-holomorphic maps with Lagrangian boundary conditions across arbitrary genus.

read the letter

The main advance is the construction of global Kuranishi charts on these moduli spaces, which the authors position as the geometric base for compatible chain-level operations in open-closed Deligne-Mumford field theories. They handle the boundary conditions on an embedded Lagrangian and the variation over genus in one framework, which extends earlier chart constructions that were often limited to lower genus or closed cases. The abstract is direct about the goal and the intended use in follow-up work, so the paper delivers a clear statement of what the charts are meant to achieve. The technical step of making the charts global while preserving compatibility with the Lagrangian looks like the part that could be useful to others building virtual cycles in the open-closed setting. A soft spot is that the abstract gives no indication of how transversality or stabilization is managed when genus changes or when boundary components interact, so the actual strength depends on those sections in the body. If those steps contain hidden choices or fail to produce charts that glue cleanly, the claim would need revision. This is aimed at researchers already working on symplectic moduli spaces and field theory constructions. A reader who needs the foundations for chain-level open-closed operations would find the setup relevant, while someone outside that niche would not. The work deserves a serious referee because the claim is concrete and the area has been waiting for such charts; the review can check whether the construction actually closes the gap without new gaps of its own.

Referee Report

0 major / 1 minor

Summary. The paper constructs global Kuranishi charts for moduli spaces of pseudo-holomorphic maps of arbitrary genus with boundary on an embedded Lagrangian submanifold. It then builds the geometric foundations required for obtaining compatible chain-level operations, which are employed in follow-up work to construct an open-closed Deligne-Mumford field theory.

Significance. If the construction holds, this supplies the geometric prerequisite for defining consistent chain-level operations on moduli spaces with Lagrangian boundary conditions and arbitrary genus. Such global charts would enable well-defined virtual chains and gluing maps, directly supporting the algebraic structures of open-closed field theories in symplectic geometry.

minor comments (1)

The abstract states the main result but provides no indication of the stabilization or compatibility conditions used to ensure the charts are global and compatible across genus and boundary strata.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the paper and for recognizing the significance of constructing global Kuranishi charts for moduli spaces of pseudo-holomorphic maps with Lagrangian boundary conditions of arbitrary genus. No major comments were listed in the report, so we have no specific points to address. We remain available to provide additional details or clarifications on the geometric foundations if requested.

Circularity Check

0 steps flagged

No significant circularity; direct geometric construction

full rationale

The paper's central claim is a direct construction of global Kuranishi charts on moduli spaces of pseudo-holomorphic maps with Lagrangian boundary conditions, followed by building geometric foundations for chain-level operations. No equations, fitted parameters, or self-citations are invoked in a load-bearing way that reduces the result to its inputs by definition. The abstract and described structure present an independent geometric prerequisite rather than a renaming, prediction from fit, or uniqueness imported via self-citation. The derivation chain is self-contained against external benchmarks of virtual cycle constructions in symplectic geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

pith-pipeline@v0.9.0 · 5564 in / 1064 out tokens · 18548 ms · 2026-05-23T06:13:30.148188+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

The topology of Lagrangian submanifolds via open-closed string topology
math.SG 2026-04 unverdicted novelty 5.0

A curved deformation of the dg algebra on the based loop space of L is built from holomorphic disc moduli spaces via string topology, implying non-vanishing Maslov class when pi2(L) vanishes.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper

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[AK10] Gregory Arone and Marja Kankaanrinta, On the functoriality of the blow-up construction , Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 5, 821–832. MR 2777772 82 [AMS21] M. Abouzaid, M. McLean, and I. Smith, Complex cobordism, Hamiltonian loops and global Kuranishi charts, arXiv:2110.14320 (2021). 5, 15, 66 [AMS24] , Gromov-Witten invariants in...

work page arXiv 2010

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MR 4647902 71 [BB12] J. E. Borzellino and V. Brunsden, Elementary orbifold differential topology , Topology Appl. 159 (2012), no. 17, 3583–3589. MR 2973378 46 AN OPEN-CLOSED DELIGNE–MUMFORD FIELD THEORY 87 [BCT09] Andrew J. Blumberg, Ralph L. Cohen, and Constantin Teleman, Open-closed field theories, string topology, and Hochschild homology, Alpine perspe...

work page arXiv 2012

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MR 4704417 5, 6, 18, 25, 27, 28, 29, 30, 31, 40, 68, 69, 70 [Des22] Yash Deshmukh, A homotopical description of Deligne-Mumford compactifications , 2022, arXiv:2211.05168

©2024. MR 4704417 5, 6, 18, 25, 27, 28, 29, 30, 31, 40, 68, 69, 70 [Des22] Yash Deshmukh, A homotopical description of Deligne-Mumford compactifications , 2022, arXiv:2211.05168. 3 [ES24] Tobias Ekholm and Vivek Shende, Skeins on branes ,

work page arXiv 2024

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Differential Geom

3 [Flo88] Andreas Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), no. 3, 513–547. MR 965228 2 [FO99] Kenji Fukaya and Kaoru Ono, Arnold conjecture and Gromov-Witten invariant , Topology 38 (1999), no. 5, 933–1048. MR 1688434 2 [FOO16] K. Fukaya, H. Oh, Y.-G.and Ohta, and K. Ono, Exponential decay estimates and smoothness...

work page arXiv 1988

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MR 4406774 39 [Fuk10] Kenji Fukaya, Cyclic symmetry and adic convergence in Lagrangian Floer theory , Kyoto J. Math. 50 (2010), no. 3, 521–590. MR 2723862 2, 3 [Fuk11] , Counting pseudo-holomorphic discs in Calabi-Yau 3-folds , Tohoku Math. J. (2) 63 (2011), no. 4, 697–727. MR 2872962 2 [Geo13] Penka Georgieva, The orientability problem in open Gromov-Wit...

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©2023, pp. 2530–2550. MR 4680329 2 [GMT86] Francesco Guaraldo, Patrizia Macr` ı, and Alessandro Tancredi,Topics on real analytic spaces , Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig,

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MR 1013362 39 [Gro85] M. Gromov, Pseudo holomorphic curves in symplectic manifolds , Invent. Math. 82 (1985), no. 2, 307–347. MR 809718 2 [GZ16] Penka Georgieva and Aleksey Zinger, On the topology of real bundle pairs over nodal symmetric surfaces , Topology Appl. 214 (2016), 109–126. MR 3571041 27, 68 [Han24] Sebastian Haney, Infinity inner products and ...

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6 [Hir23] Amanda Hirschi, Properties of Gromov-Witten invariants defined via global Kuranishi charts , 2023, arXiv:2312.03625

2 AN OPEN-CLOSED DELIGNE–MUMFORD FIELD THEORY 88 [HH] Amanda Hirschi and Kai Hugtenburg, Open Gromov-Witten invariants and Lagrangian cobordisms , In preparation. 6 [Hir23] Amanda Hirschi, Properties of Gromov-Witten invariants defined via global Kuranishi charts , 2023, arXiv:2312.03625. 4, 17, 18 [HPV00] John Hubbard, Peter Papadopol, and Vladimir Vesel...

work page arXiv 2023

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83 [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 79 [KL06] Sheldon Katz and Chiu-Chu Melissa Liu, Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc , The interaction of finit...

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Math., vol

MR 2289519 82 [Kon95] Maxim Kontsevich, Enumeration of rational curves via torus actions , The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkh¨ auser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062 6 [Kot] C. Kottke, Extension off the boundary of a manifold with corners , https://ckottke.ncf.edu/docs/ extension.pdf, Accessed: ...

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16, 2001, pp

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MR 2954391 15, 16, 58, 69 [MW17] Dusa McDuff and Katrin Wehrheim, Smooth Kuranishi atlases with isotropy , Geom. Topol. 21 (2017), no. 5, 2725–2809. MR 3687107 2 [Pal61] Richard S. Palais, On the existence of slices for actions of non-compact Lie groups , Ann. of Math. (2) 73 (1961), 295–323. MR 126506 11 [Pan00] R. Pandharipande, The Toda equations and t...

work page 2017

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Robbin and Dietmar A

5 [RS06] Joel W. Robbin and Dietmar A. Salamon, A construction of the Deligne-Mumford orbifold , J. Eur. Math. Soc. (JEMS) 8 (2006), no. 4, 611–699. MR 2262197 74 AN OPEN-CLOSED DELIGNE–MUMFORD FIELD THEORY 89 [RS18] Joel W Robbin and Dietmar A Salamon, Introduction to differential topology. 75 [RT97a] Yongbin Ruan and Gang Tian, Higher genus symplectic i...

work page 2006

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Solomon and Sara B

MR 2441780 2 [ST21] Jake P. Solomon and Sara B. Tukachinsky, Point-like bounding chains in open Gromov-Witten theory , Geom. Funct. Anal. 31 (2021), no. 5, 1245–1320. MR 4356703 2, 6 [ST22] , Differential forms, Fukaya A∞ algebras, and Gromov-Witten axioms , J. Symplectic Geom. 20 (2022), no. 4, 927–994. MR 4573438 2, 5, 6, 22, 28, 31, 48, 55, 74, 75, 77 ...

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Ye, Gromov’s compactness theorem for pseudo holomorphic curves , Trans

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work page 1994