{"id":"90c2b923-b645-46cc-a289-16185b09e781","arxiv_id":"2510.10867","paper_version":2,"verdict":"UNVERDICTED","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A tutorial on computing partially wrapped Fukaya categories of surfaces via arcs, quivers, and explicit chain complexes.","lead":"This expository paper teaches, through many worked examples, how to compute the Fukaya categories of surfaces with boundary using dissections and quivers. A newcomer gets a step-by-step recipe for an invariant that is normally hard to handle.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Z-graded examples may rely on an unexplained tension: the canonical line field putting all arrows in degree 0 seems to conflict with the Z/2 tip-to-tail degree-1 arrows.","rationale":"The reader's weakest assumption correctly identified the unproven use of the canonical line field as a critical dependency. My concern sharpens this into a specific potential inconsistency: the claim that the line field places the algebra entirely in degree 0 appears to conflict with the Z/2-graded construction, where some intersection points are degree 1. If this is only an expository oversight (e.g., the Z-grading is not meant to reduce to the Z/2-grading described), the paper should state so; if it is a real contradiction, the degree computations in the examples are unreliable. This is load-bearing because the paper's sole contribution is the correctness and clarity of its worked examples. I do not change the reader's UNVERDICTED classification—the paper remains an expository note with no original research claim—but the concern warrants a check before the examples are used as a tutorial.","tokens_in":11966,"tokens_out":6838,"duration_ms":62050,"concrete_test":"For the dissection shown in Figure 16(a), compute the Z-degree of each quiver arrow using the winding-number algorithm described in §1.4 with the claimed canonical line field. Check whether all arrows have degree 0, and whether reducing these degrees modulo 2 reproduces the §1.3 rule (tip-to-tail = degree 1, else degree 0). Then independently recompute Ext^0 and Ext^1 between the blue and orange curves in Figure 16(f) using those Z-degrees and compare with equations (1.1)–(1.3).","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The paper's central claim is that every curve with a local system has an explicit chain complex, computed from a dissection. The Z-graded version in §1.4 depends on the existence of a 'canonical choice of line field associated to a dissection' [8,19,§2] that 'places the algebra entirely in degree 0'. However, §1.3 defines a Z/2-graded category where arrows are degree 1 when the oriented arcs meet tip-to-tail and degree 0 otherwise. If the Z-grading is a lift of the Z/2-grading (the usual relationship), then any tip-to-tail arrow must have odd Z-degree, contradicting 'entirely in degree 0'. The paper does not explain how the two grading conventions are compatible, nor does it prove or even explicitly verify that the cited line field exists and is unique for each depicted dissection. If the canonical line field were misapplied—e.g., if the degrees of arrows in Example 16(f) were shifted by an odd integer—the computed Ext groups in (1.1)–(1.3) would be wrong, invalidating the worked examples that form the paper's pedagogical core.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This expository paper presents a combinatorial method for computing the partially wrapped Fukaya category of a marked surface with boundary. The author explains how to choose a dissection (a full arc system that cuts the surface into polygons), associates a quiver to it, and states rules for reading off compositions and higher A∞-operations from angles and closed polygons. It then explains how to write explicit chain complexes for arbitrary arcs and for closed curves with local systems, working in both a Z/2-graded setting (using orientations) and a Z-graded setting (using line fields). The paper includes several detailed worked examples: the disk (A_n quiver), the annulus (Kronecker quiver), the pair of pants, and a surface with an interior marked point. The author states that nothing is original except the exposition, and the foundational equivalences are cited to [6,19,18].","tokens_in":12223,"tokens_out":20289,"duration_ms":175122,"significance":"If correct, the paper fills a useful gap by providing a single, accessible source with multiple detailed worked computations of Fukaya categories of surfaces. The examples appear internally consistent and are grounded in established theorems; the author explicitly disclaims originality and gives references for every foundational step. The figures and step-by-step chain-complex computations are likely to be genuinely helpful to newcomers. The main weakness is that the Z-graded machinery is introduced only through a citation, and the relation between the Z/2-degree convention and the Z-graded 'degree 0' claim is not explained, which makes the central Z-graded example (16f) impossible for the reader to verify independently.","major_comments":[{"comment":"The paper asserts that the canonical line field of a dissection 'places the algebra entirely in degree 0' (§1.4), but it never reconciles this with the Z/2-degree rule of §1.3, where arrows are degree 1 when the oriented arcs meet tip-to-tail. If the Z-grading is a lift of the Z/2-grading, then a tip-to-tail arrow must have odd Z-degree. The Z-graded example in §1.5 does not specify the integer grading shifts of the generating arcs A,C,R,M,D, so the reader cannot verify that the differential components ba, e, h, g in the complexes before (1.1) have the claimed degrees. A single odd shift would alter the computed Ext groups in (1.1)–(1.3). Please add an explicit explanation of how the Z/2 and Z conventions are related, and state the gradings used in the example.","section":"§1.4 and §1.5, Example 16(f), Eqs. (1.1)–(1.3)"},{"comment":"The existence and uniqueness of the canonical line field for a dissection is cited to [8] and [19, §2], but the paper does not verify it for the dissections actually depicted, nor does it show how the laminate construction yields the Z-degrees of the arrows used in the pair-of-pants example. The reader is asked to accept that 'after choosing Z-gradings' the blue and orange curves have the stated complexes and that (1.1)–(1.3) are correct. For an expository paper whose goal is to enable independent computation, a short verification for Figure 16(f) — even just the degree of one or two arrows — would make the example reproducible and would also help resolve the compatibility issue raised above.","section":"§1.4, Figures 1.14 and 1.16"}],"minor_comments":[{"comment":"Typo: 'the higher the higher A∞-operations vanish' should read 'the higher A∞-operations vanish'.","section":"§1.1"},{"comment":"The formula for μ^n is not clearly written: 'μ^n(γ α1, α2, ..., αn) = γ' appears to have n+1 arguments. Please clarify the intended notation.","section":"§1.3, after Figure 1.7"},{"comment":"The basis elements for Hom^0 and Hom^1 are written as matrices with shorthand like (id_A 0; 0 0). A short sentence explaining the ordering of the basis and the degrees of the generators a,b,c,d would help the reader follow the matrix differential.","section":"§1.5, Example 16"},{"comment":"The author states that the computations were verified with a computer, but no code or ancillary files are provided. While not required for an expository paper, a link to the scripts would strengthen reproducibility.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"This is an expository paper whose value depends on the accuracy of the worked examples. The main technical concern is the compatibility of the Z/2-grading rule with the Z-graded 'degree 0' claim for the canonical line field. If the author can clarify this with a concrete explanation (and ideally a verification in the pair-of-pants example), the paper would be a useful contribution to the pedagogical literature. The scope is modest but honest, and the examples are otherwise carefully presented."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know upfront: this is an expository paper, not a research paper. The author says so in the first pages, and that's true. There are no new theorems or methods. What it does well is exactly what it promises: it works through many concrete examples of computing partially wrapped Fukaya categories of surfaces from dissections, including chain complexes, Ext groups, and cones. That is genuinely rare and useful. I expect it to become a standard reading for graduate students entering this area. The writing is clear, the examples are detailed, and the author is honest about the sources.\n\nThe main thing I'd want a referee to look at is Section 1.4. The paper defines a Z/2-graded category where tip-to-tail oriented arcs give degree-1 maps. It then says the canonical line field associated to a dissection 'places the algebra entirely in degree 0.' If the Z-grading is supposed to be a lift of the Z/2 grading, those two statements cannot both be correct: tip-to-tail arrows would have to have odd Z-degree. The examples in Section 1.5 do show degree-1 maps in the Z-graded category, so the 'entirely in degree 0' comment is at least confusing and at worst indicates a misstatement. The author cites [8] and [19] for the line field construction, so it's likely the examples are right and the text is sloppy, but an expert referee should check.\n\nA second, smaller issue is reproducibility: the author says the matrices were verified by computer, but no code or scripts are included. That's fine for an expository note, but it means readers have to trust the computations.\n\nBottom line: this paper deserves a serious referee. It fills a real pedagogical gap, the main ideas are correct as far as I can tell, and the grading issue is a fixable exposition problem rather than a fundamental one. The right outcome is probably minor revision before publication.\n\nBest,\n[You]","headline":"Useful, clearly-written expository note on computing Fukaya categories of surfaces; the main thing to check is the Z-grading claim, which seems to conflict with the Z/2 tip-to-tail rule.","tokens_in":12681,"tokens_out":4986,"would_cite":false,"duration_ms":43342,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53D37","16G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper provides an explicit combinatorial recipe — a dissection into polygons — for computing the partially wrapped Fukaya category of a marked surface, with detailed worked examples.","keywords":["Fukaya categories","surfaces with boundary","marked surfaces","dissections","gentle algebras","quiver representations","A∞-categories","local systems"],"falsifier":"Take the once-punctured torus (or any surface with an interior marked point) and compute Ext^• between two arcs ending at the puncture using the dissection recipe; if the countably many generators do not match the geometric wrapping count, or if the differential fails to square to zero for one of the paper's worked matrices, the claimed correspondence fails at the computational level.","tokens_in":11846,"feed_emoji":"✂️","tokens_out":4957,"duration_ms":46274,"temperature":0.7,"pith_summary":"The paper's aim is to make the partially wrapped Fukaya category of a marked surface with boundary concretely computable. It argues that once a surface is cut into polygons by a dissection of generating arcs, every curve with a local system can be written as an iterated cone of arcs, with arrows given by angles between arcs and higher operations given by closed polygons. The author is explicit that the underlying equivalence theorems come from earlier work; the contribution is a detailed, example-driven exposition with explicit matrices for chain complexes, Ext groups, and cones. A sympathetic reader should come away able to perform such computations by hand on new surfaces.","feed_headline":"Cut a surface into polygons to compute its Fukaya category","feed_subtitle":"A combinatorial recipe replaces hard symplectic analysis with quivers, angles, and matrices, with worked examples.","key_machinery":"The load-bearing object is the dissection — a family of nonintersecting arcs cutting the surface into polygons — together with its dual 'laminate': perpendicular arcs whose endpoints lie on the boundary stops. The laminate's line field gives the Z-grading and makes all arrows degree 0 when using the dissection's canonical line field; the quiver of arcs and angles then directly encodes the A-infinite structure, with cones of degree-1 maps performing geometric gluing.","core_discovery":"The central claim is that the Z/2- and Z-graded partially wrapped Fukaya category of a marked surface is presented by the A-infinite category of a dissection: choose nonintersecting arcs including all boundary arcs that cut the surface into polygons with at most one boundary edge (or one interior marked point). Then vertices of a quiver are the arcs, arrows are the angles between them (surface on the right), degree 0 or 1 by tip-to-tail orientation, composition is concatenating adjacent angles, and higher multiplications are nonzero exactly for closed polygons. For a dissection the higher operations vanish, so ordinary quiver algebra computes the category. Every arc corresponds to a projecti","pith_inferences":["The explicit matrices suggest a direct algorithm: choose a dissection, build the quiver by scanning angles, and mechanically compute cones; this could be automated in a computer algebra system, potentially making Fukaya-category computations routine for surfaces.","The sensitivity of Z-grading to line-field choice implies that mirror-symmetry statements for surfaces must specify the grading datum; the Z/2-graded category is a safer target because it requires no such choices.","The closed-curve construction with monodromy matrices offers a concrete laboratory for Hall-algebra and skein-relation computations, since cones of degree-1 maps between bands are written out explicitly.","To teach the subject, one can sequence the material as the paper does: first Z/2-graded examples to learn quiver combinatorics, then line fields for integer gradings, since the latter build on the former without new geometric input."],"forward_implications":["Any curve with a rank-1 local system on a dissected surface has an explicit chain complex of projectives; the paper writes these matrices for the disk, annulus, pair of pants, and a once-punctured surface.","Ext groups between two curves can be read off from their intersection points: each interior intersection contributes one generator in each direction, with degrees summing to 1 (or 0 in the Z/2 case if orientations match).","Cones of degree-1 maps are computed by matrix block manipulation, so geometric resolutions can be produced by elementary row and column operations, as shown in the pair-of-pants example.","For surfaces with interior marked points, wrapping produces countably many generators and infinite projective resolutions; the combinatorial model reproduces these infinite structures exactly.","The resulting algebra is finite dimensional exactly when the surface has no interior marked points, and classical (no higher A-infinite operations) exactly when the chosen arc system is a dissection."],"fun_headline_variants":["Combinatorial recipe computes surface Fukaya categories","Cut surfaces into polygons to compute Fukaya categories","Polygon dissections yield explicit Fukaya categories","Quiver algebra from angles computes Fukaya categories","Surface Fukaya categories via combinatorial quivers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper assumes, citing foundational work without proof, that the A-infinite category computed from any dissection is independent of the choice of arcs and equivalent to the actual partially wrapped Fukaya category; if this equivalence fails for non-simply-connected or multiply-punctured surfaces, the explicit complexes and degrees in the examples would be wrong.","fun_headline_variants_meta":{"raw":{"variants":["Combinatorial recipe computes surface Fukaya categories","Cut surfaces into polygons to compute Fukaya categories","Polygon dissections yield explicit Fukaya categories","Quiver algebra from angles computes Fukaya categories","Surface Fukaya categories via combinatorial quivers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1321,"prompt_tokens":580,"completion_tokens":741,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":324,"completion_tokens_details":{"reasoning_tokens":671}},"tokens_in":324,"tokens_out":741,"duration_ms":6229,"temperature":1.0,"reasoning_tokens":671,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T10:13:24.547463+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the once-punctured torus (or any surface with an interior marked point) and compute Ext^• between two arcs ending at the puncture using the dissection recipe; if the countably many generators do not match the geometric wrapping count, or if the differential fails to square to zero for one of the paper's worked matrices, the claimed correspondence fails at the computational level.","supporting_citations":[],"review_version":1}