Bordered Legendrian Rational Symplectic Field Theory

Maciej Wlodek

REVIEW 3 major objections 4 minor 12 references

Bordered Legendrian Rational Symplectic Field Theory

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that the commutative Legendrian Rational Symplectic Field Theory algebra of a simple Legendrian knot is the pushout of two bordered differential graded algebras obtained by cutting its front projection along a vertical…

desk verdict The paper builds a genuinely new bordered LSFT and a plausible pushout theorem, but the universal property as stated is not proven: it silently needs the target DGA to be commutative. read the letter →

arxiv 2509.07921 v1 pith:DPIGREOQ submitted 2025-09-09 math.SG

Maciej Wlodek This is my paper · ORCID

classification math.SG MSC 53D4257K10

keywords LegendrianknotsRationalSymplecticFieldTheorybordereddifferentialgradedalgebrapushoutsquareChekanov-EliashbergDGAfrontprojectionholomorphicdiskswithmultiplepositivepuncturesstring

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper builds a bordered version of Legendrian Rational Symplectic Field Theory (LSFT). For a Legendrian knot whose front projection is simple, cutting the front along a generic vertical line produces three commutative differential graded algebras: one for the left half-knot, one for the right half-knot, and one for the dividing line. The main theorem (Theorem 5.1) claims that these three algebras form a pushout square whose apex is the commutative LSFT algebra of the whole knot, so the full invariant is determined by the two halves glued together over the cut. The construction extends the bordered Chekanov-Eliashberg algebra by allowing holomorphic disks with several positive punctures, which is exactly what LSFT adds to ordinary Legendrian contact homology. A two-sided analogue (Theorem 5.55) gives the same gluing statement for adjacent bordered pieces.

What carries the argument

The load-bearing object is a family of commutative DGAs generated by geometric symbols attached to the cut. $A^L_{SFT}$ and $A^R_{SFT}$ are generated by $p_i, q_i$ for crossings and cusps in the half-diagram, by $\alpha_{ij}$ recording half-disks whose boundary interval on the dividing line runs from $i$ to $j$, by $\beta_{ij}$ recording strands that pair dividing-line points, and by $t, t^{-1}$. The middle algebra $A^M_{SFT}$ is generated by $\alpha^L_{ij}, \alpha^R_{ij}, \beta^L_{ij}, \beta^R_{ij}$, with Hamiltonian $h_M = \sum_{i<j} \alpha^L_{ij}\alpha^R_{ij}$. In every case the differential is the sum of an SFT bracket $\{h,\cdot\}$ and a string differential $\delta_{str}$; the bracket satisfies a Jacobi identity, $\delta_{str}$ is a derivation of the bracket, and the two quadratic terms cancel to give $d^2=0$. The maps $\ell, r, L, R$ are defined by counting admissible disks and half-disks with corners at crossings and boundary intervals on $M$; commutativity of the square is checked generator by generator.

What would settle it

Work in the trefoil example of Section 6 and let $Q$ be a noncommutative DGA with maps $f$ and $g$ agreeing on the middle algebra but with two left generators $\alpha$ and $\alpha'$ whose images in the commutative quotient commute, while $f(\alpha)f(\alpha') \neq f(\alpha')f(\alpha)$ in $Q$. The alleged universal map $h$ would assign both orders of the product to the same element of $A^{comm}_{SFT}(\Lambda)$, forcing the two values to be equal; a computation where they differ would contradict the pushout property.

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Extended reading notes

Core claim

The central discovery is a Seifert-van Kampen statement for commutative Legendrian SFT. Cutting a simple Legendrian front $\Lambda$ along a vertical line $M$, one obtains $A^M_{SFT}(M)$, $A^L_{SFT}(\Lambda_L)$ and $A^R_{SFT}(\Lambda_R)$, and the paper proves that $A^{comm}_{SFT}(\Lambda)$ is their pushout: any pair of DGA maps from the left and right bordered algebras that agree on the middle algebra factors through $A^{comm}_{SFT}(\Lambda)$. This means the full LSFT invariant, whose differential counts disks with arbitrarily many positive punctures, can be assembled from counts that live in the half-diagrams and on the dividing line. The proof constructs explicit gluing maps $\ell, r, L, R$ by counting admissible disks and half-disks, and it shows that the bordered differential, written as SFT bracket with a Hamiltonian plus a string differential, squares to zero.

Load-bearing premise

The pushout theorem assumes the target algebra is commutative: the universal map is defined on the commutative quotient of the LSFT algebra, and the proof does not show it descends when the target is not commutative.

Editorial extensions

If this is right

For a simple front, $A^{comm}_{SFT}(\Lambda)$ is determined by the left and right half-diagrams together with the pairing data on the dividing line; no count of disks crossing the whole front is needed.
The bordered construction specializes to the bordered Chekanov-Eliashberg algebra: setting all $\beta$ and $\alpha^{R}$ generators to zero and deleting the $t^{\pm1}$ terms recovers the commutative quotient of the earlier pushout theorem.
The two-sided LR algebra satisfies its own pushout theorem (Theorem 5.55), so adjacent bordered pieces can be glued one after another to compute the invariant of a knot assembled from tangle diagrams.
The universal property packages LSFT as a local-to-global invariant: DGA-valued invariants of the halves that agree on the middle algebra factor uniquely through $A^{comm}_{SFT}(\Lambda)$.
The example calculations give a checkable blueprint: the trefoil computation shows how the middle, left, and right algebras and their maps are meant to fit together in practice.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The theorem is best read as a pushout in the category of commutative DGAs; for a noncommutative target the construction would need the extra assumption that the two side images commute.
The same gluing pattern suggests a general local-to-global principle for SFT-type invariants: whenever bordered algebras can be defined for a cut, the commutative invariant is determined by a pushout, making cutting-and-gluing the default computational strategy.
The noncommutative case might be recovered by a homotopy pushout or a curved structure, since the paper notes that the noncommutative maps are not morphisms in general.
Comparing the pushout output to a direct full-front computation of $A^{comm}_{SFT}$ for the trefoil would validate the gluing and reveal any hidden dependence on the choice of dividing line.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper constructs bordered Legendrian Rational SFT: differential graded algebras A^L_SFT, A^R_SFT, A^M_SFT, and a Seifert–van Kampen type pushout recovering Ng's commutative LSFT algebra. That is genuinely new, extending Sivek's bordered CE algebra to the multiple-positive-puncture setting. The definitions are explicit, the trefoil example is fully worked, and a code link is provided. The proofs of (d^M)^2=0 and (d^L)^2=0 are largely written out; the Jacobi and derivation identities are checked on generators. This is a serious piece of work, not a re-coordinatization.

The main soft spot is exactly the one the stress-test identifies. In the proof of Theorem 5.1, the paper takes an arbitrary DGA Q with maps f:A^L_SFT→Q and g:A^R_SFT→Q, defines h on generators of A^comm_SFT by h(x)=f(s) or g(s), and extends by h(xy)=h(x)h(y). But A^comm_SFT is the free commutative algebra on these generators, so h is an algebra map only if the images of f and g commute in Q. The paper never states that Q is commutative, nor proves the commutation. As written, the universal property is not established for general DGA targets. The fix is easy: either state the pushout in the category of commutative DGAs, or add the missing hypothesis. But without that, the main theorem is conditional. This is not a fatal flaw in the construction, just a quantifier error in the statement and proof.

Lesser concern: the proofs that L and R are chain maps (Theorem 5.47) and Lemma 5.41 (h^L→h^L = δ^L_str(h^L)) rely on figure-based cancellation arguments rather than full algebra. These are plausibly correct — the figures are convincing — but they would be stronger with detailed algebra or machine-checked verification. That is a minor-to-moderate concern, fixable by more exposition.

The citation pattern looks fine. The paper builds on Sivek and Ng and credits them properly. The example computations are reproducible in principle given the code, though no commit hash is pinned.

The paper deserves a serious referee. The construction is likely correct and useful for Legendrian contact homology and SFT, but the pushout theorem needs revision before it should be accepted. If the author fixes the category statement and tightens the figure-based proofs, this would be a solid contribution.

Referee Report

3 major / 4 minor

Summary. The paper constructs bordered versions of Legendrian Rational Symplectic Field Theory (LSFT) for a simple Legendrian front cut into left and right halves by a vertical dividing line. It defines a commutative middle DGA A^M_SFT, left and right DGAs A^L_SFT and A^R_SFT, and proves that their differentials square to zero. It then defines DGA maps ℓ, r, L, R among these algebras and claims in Theorem 5.1 that the resulting square is a pushout whose apex is the commutative LSFT algebra A^comm_SFT. A two-sided variant is sketched in Section 5.8, and Section 6 works through the full example of the Legendrian trefoil, including explicit formulas for the algebras, differentials, and gluing maps.

Significance. If the main theorem is correct, it gives a Seifert–van Kampen type decomposition for the commutative LSFT algebra, extending Sivek's bordered Chekanov–Eliashberg theory to the rational/multi-puncture setting. The paper is constructive and explicit: the algebraic definitions are laid out in detail, the proofs of (d^M)^2=0 and (d^L)^2=0 are largely written out, and the trefoil example is fully computed, with code made available. These are real strengths. However, the universal property of the claimed pushout is stated for an arbitrary DGA target without a commutativity assumption, and the chain-map theorem for the maps L and R is justified by figure-based cancellation arguments rather than a complete algebraic pairing. Both issues are load-bearing for the central claim.

major comments (3)

[Section 5.7, proof of Theorem 5.1] The universal property is claimed for 'another DGA Q' with no commutativity hypothesis. But A^comm_SFT is the quotient of the free algebra by all commutators, so a DGA map h:A^comm_SFT→Q forces the images of all generators to commute in Q. In particular, if x=L(s) and y=R(t), the relation xy=yx in A^comm_SFT would force f(s)g(t)=g(t)f(s) in Q, which is not automatic for an arbitrary associative DGA. The theorem is therefore not established as stated. The intended statement is presumably the pushout in the category of commutative DGAs, in which case Q should be required to be commutative and this should be stated explicitly. As written, the quantifier over Q makes the central pushout assertion unsupported.
[Section 5.5.1, Theorem 5.47] The proof that L and R are chain maps is only sketched at the points where it matters most. For x=α_ij the argument says that the relevant terms 'come in pairs which cancel, as depicted in Figure 20', and for x=β_ij it says that exceptional disks 'precisely cancel' (Figures 21, 22). Since d, d^L, and d^R are defined by sums over disks and string insertions, the equality d∘L=L∘d^L requires an explicit, globally defined involution on the set of contributing terms, including all boundary and dividing-line cases. This is not merely a presentational matter: Theorem 5.47 is used directly in the proof of the pushout property, so the central claim depends on this missing detail.
[Section 5.4, Lemma 5.41] Lemma 5.41 asserts h_L→h_L=δ^L_str(h_L), which is used in the proof of Theorem 5.36 to prove (d^L)^2=0. The proof says that 'boundary' terms are 'depicted in Figure 17, and are seen to cancel pairwise', but no precise pairing is given. Because this lemma is another load-bearing point for the left algebra being a DGA, the argument should be expanded into an explicit enumeration of the cancellations, or at least a clearly stated bijection between summands.

minor comments (4)

[Section 5.8, Theorem 5.55] The two-sided pushout theorem is stated as a theorem but its proof is delegated to the sentence that the earlier proofs 'may be easily adapted'; please include a proof or explicitly label the statement as a sketch.
[Section 5.3.2, Definition 5.23] The set W is described as 'the set of elements of A^L_SFT which are representable as broken closed strings', but it is formally a set of words; this distinction should be made precise.
[Section 4, proof of Proposition 4.8] The proof of well-definedness of the string differential refers to 'Figure 3.8 in [9]' and 'Figure 3.13 in [9]' without reproducing or explaining the relevant arguments; since the numbering is from another paper, a short self-contained statement would help.
[Section 5.7] In the proof of Theorem 5.1, the case in which a generator of A^comm_SFT has representations both as L(s) and R(t) is handled for α and β generators, but the corresponding verification for t and t^{-1} is not written out; this should be made explicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the bordered LSFT construction is a new object compared against the independently defined LSFT algebra.

full rationale

The paper defines new DGAs A^M_SFT, A^L_SFT, and A^R_SFT from tangle data and then proves a comparison theorem relating their pushout to Ng's commutative LSFT algebra. The target A^comm_SFT is not used to define the bordered algebras: the left, right, and middle differentials are specified by explicit generators, SFT brackets, Hamiltonians, and string differentials on the half-diagram, and the chain maps L, R, ℓ, and r are defined by counting admissible disks and half-disks. No parameter is fitted to A^comm_SFT and then renamed as a prediction; the only citations to prior work (Chekanov, Ng, Sivek) are used for the standard Chekanov-Eliashberg DGA, LSFT, and bordered CE constructions rather than as a self-citation chain. The proof of Theorem 5.1 has a real gap for arbitrary (noncommutative) test DGAs Q: extending h to A^comm_SFT by products is only well-defined if the images of A^L_SFT and A^R_SFT commute in Q, which is not assumed or proved; this affects correctness of the universal property as stated, but it is not a circularity because the claimed result is not assumed as an input. Independent example computations in Section 6 test the construction against explicit DGAs, further showing that the derivation chain is not circular. Overall, no load-bearing step reduces by definition to its own inputs, so a score of 0 is appropriate.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on Ng's LSFT setup (string topology, SFT bracket, quantum master equation), Sivek's bordered CE construction, Chekanov's DGA, and the resolution of front projections. These are treated as black boxes. No free parameters or invented entities are introduced; the new DGAs are the content of the theorem.

assumptions (5)

standard math Chekanov-Eliashberg DGA d^2=0 and invariance (Chekanov [3], Thms 2.3/2.6 in paper)
Section 2 reviews and relies on this as background for the bordered construction.
domain assumption Resolution of front projections gives Lagrangian projections (Ng [8], Thm 2.10)
Used in Sections 2.4 and 5 to define front-projection admissible disks and identify them with Lagrangian disks.
domain assumption Properties of Ng's LSFT: quantum master equation, string differential, SFT bracket, commutativity (Ng [9] Props 4.8,4.10, Thms 4.16,4.17)
Section 4 reviews these; Sections 5.2-5.4 cite them for cancellations and Jacobi identities.
domain assumption Sivek's bordered Chekanov-Eliashberg DGA and its pushout theorem (Sivek [11], Thm 3.1)
The new construction is modeled on and generalizes this; comparison is made in Remark 5.18.
domain assumption Simple front projections suffice for any Legendrian knot via Reidemeister moves (Section 2.4.1)
The entire construction assumes a simple front; no invariance of the bordered algebras under the moves is proven.

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Pith. "Pith review of Bordered Legendrian Rational Symplectic Field Theory." pith.science (2026). https://pith.science/paper/DPIGREOQ

@misc{pith2026250907921,
  author       = {Pith},
  title        = {Pith review of: Bordered Legendrian Rational Symplectic Field Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPIGREOQ}},
  note         = {Machine review of arXiv:2509.07921}
}

abstract

Given a Legendrian knot $\Lambda \subset \mathbb{R}^3$ and a vertical line dividing the front projection of $\Lambda$ into two halves, we construct a differential graded algebra associated to each half-knot. We then show that one may obtain the commutative algebra from Legendrian Rational Symplectic Field Theory as a pushout of the two bordered algebras. This construction extends the bordered Chekanov-Eliashberg differential graded algebra by incorporating disks with multiple positive punctures into the differential.

Figures

Figures reproduced from arXiv: 2509.07921 by the authors.

**Figure 1.** Left: a Legendrian trefoil in the front projection. Right: a Legendrian trefoil in the Lagrangian projection. Consider the projection Π : N → R 2 xy, (x, y, z, t) 7→ (x, y). This projects L down to an immersed Lagrangian Π(L) ⊂ R 2 , and Reeb chords ci to self-intersections of Π(L). Elements of M(c1, . . . , ck) are projected to immersed or branched disks in R 2 with boundary on Π(L), and in particular elements of z… view at source ↗

**Figure 2.** Reeb signs at a crossing. (1) u is an immersion apart from a finite set of boundary points, where it maps to self-intersections of Πxy(Λ). These points are called corners of u. (2) At each corner c ∈ ∂D2 of u, a neighborhood of c is mapped to precisely one quadrant of q = u(c). Label a corner as positive or negative in accordance with the Reeb sign of this quadrant. (3) u has precisely one positive corner (and any n… view at source ↗

**Figure 3.** A disk with a non-convex corner may be cut in two ways [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (21 more)

**Figure 5.** Figure 5: A left half-diagram. M Λ R R 2 R [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: A right half-diagram. As in Section 2, let Λ ⊂ R 3 be a Legendrian knot, and let Πxy and Πxz denote the Lagrangian and front projections, respectively. The bordered Chekanov-Eliashberg DGA is formulated in terms of the front projection. We assume that the front project…

**Figure 7.** Figure 7: x contains a p factor and y contains a corresponding q factor; we may glue as shown to get a summand of {x, y}. Let γ and γ ′ be two broken closed strings, and suppose that we can write w(γ) = w1piw2 and w(γ ′ ) = w ′ 1 qiw ′ 2 (in other words, γ and γ ′ have corners a…

**Figure 8.** Figure 8: A diagram associated to a middle algebra. The thick line is the dividing line M, and the dotted curves represent the left and right pairings β L = {(1, 3),(2, 4)} and β R = {(1, 2),(3, 4)}, respectively. The generators of this algebra are {α L 12, αL 13, αL 14, αL 23,…

**Figure 9.** Figure 9: A left half-diagram. The dotted curves represent β = {(1, 2),(3, 4)}. The generators of AL SF T in this case are: {p1, q1, α12, α13, α14, α23, α24, α34, β12, β34, t, t−1}. • A generator αij for every 1 ⩽ i < j ⩽ n. (representing a right half-disk) • A generator βij for…

**Figure 10.** Figure 10: A broken closed string representing the element α24 (left; drawn red and slightly offset), and a pq insertion resulting in the element pqα24 (right; drawn red and slightly offset. At the crossing, the new broken closed string jumps across the Reeb chord from the lower…

**Figure 11.** Figure 11: A broken closed string representing the element α24 (left; drawn red and slightly offset), and an α insertion resulting in the element α23α34. p p q q [PITH_FULL_IMAGE:figures/full_fig_p031_11.png]

**Figure 12.** Figure 12: A broken closed string representing the element q (red, drawn slightly offset). Because the broken closed string has a jump from strand 3 to 4, we can perform a β insertion resulting in the element qβ34 (not pictured because qβ34 ̸∈ W). Then define δ L str(βij ) = β 2…

**Figure 13.** Figure 13: Local moves of broken closed strings. (1) δ L str is well defined, i.e. when w ∈ W, δ L str(w) is independent of the choice of γ. (2) δ L str satisfies the Leibniz rule δ L str(xy) = δ L str(x)y + xδL str(y) Proof. (1) Suffices to check that δ L str(w) doesn’t change …

**Figure 14.** Figure 14: Top: x contains a p factor and y contains a corresponding q factor; we may glue as shown to get a summand of {x, y}. Bottom: x contains an αij factor and y contains an adjacent αjk factor; we may glue as shown to get a summand of {x, y}. Definition 5.33 (Hamiltonian…

**Figure 15.** Figure 15: Top: an exceptional term in δ{x, y}. Bottom: an exceptional term in {x, δy}. The top and bottom terms give the same element in AL SF T and thus cancel. Lemma 5.38. (δ L str) 2 = 0. Proof. (1) Let x = βij . Then (δ L str) 2 (x) = 2β 3 ij = 0. (2) Let x ∈ W. Then each t…

**Figure 16.** Figure 16: The boundary exceptional terms cancel pairwise as shown. Proof. Consider the summands in (h L → h L) + δ L str(h L). Some ‘interior’ terms cancel pairwise in the same way as in the proof of Theorem 4.16 (see [PITH_FULL_IMAGE:figures/full_fig_p037_16.png]

**Figure 17.** Figure 17: The pairs of boundary canceling terms in (h L → h L) + δ(h L). left half-disks and strands, respectively. The differential once again has an SFT and string component, both defined in an analogous way to d L SF T and δ L str. 5.5. L, R morphisms [PITH_FULL_IMAGE:figur…

**Figure 18.** Figure 18: Left: a right admissible disk contributing to L(α34). Right: a left admissible disk contributing to R(α34). Note that the left admissible disk may ‘loop back’ across the dividing line, but the same is not possible for right admissible disks, since we have restricted c…

**Figure 19.** Figure 19: A right disk with two boundary components on the dividing line must have a non-convex corner (marked with a dot). It is clear that δ(pi) = L(δ L str(pi)). So it suffices to show {h, pi} = L({h L, pi}). Let w = {w ′ , pi} be a summand of {h, pi}. Write w ′ as ∂D for an…

**Figure 20.** Figure 20: Canceling pairs in {h, L(αij )} + L{h L, αij} + δ(L(αij )) + L(δ L str(αij )). M Λ i j p1 q1 p2 q2 h [PITH_FULL_IMAGE:figures/full_fig_p043_20.png]

**Figure 21.** Figure 21: Here, L(βij ) = p1q1 + p2q2 + . . . . The disk labeled h contributes to both {h, p1q1} and {h, p2q2} [PITH_FULL_IMAGE:figures/full_fig_p043_21.png]

**Figure 22.** Figure 22: Here, L(βij ) = pq + . . . . The disk labeled h contributes to both {h, pq} and L({h L, βij}). Given any u ∈ HL ij , let ∂ ∗ (u) denote the boundary of u, excluding the distinguished dividing line interval, read off as a monomial in AL SF T . Define a right admissib…

**Figure 23.** Figure 23: A two-sided diagram. • x = R(s) for s ∈ AR SF T . d Q(h(x)) = d Q(g(s)) = g(d R(s)) = h(R(d R(s))) = h(d(R(s))) = h(d(x)) □ 5.8. LR algebra. Consider now a “two-sided diagram”, i.e., one with a dividing line on both the left and right, such as in [PITH_FULL_IMAGE:fig…

**Figure 24.** Figure 24: Adjacent two-sided diagrams. • α L ij for all 1 ⩽ i < j ⩽ m. • β R ij for 1 ⩽ i < j ⩽ n such that {i, j} ∈ β R. • β L ij for 1 ⩽ i < j ⩽ m such that {i, j} ∈ β L. The string differential on ALR is defined similarly to the previously discussed cases (see Theorem 5.10 a…

**Figure 25.** Figure 25: Legendrian trefoil. 6. Examples In this section, we go over some example calculations of Bordered LSFT. 6.1. Example 1. Let Λ be the Legendrian trefoil depicted in [PITH_FULL_IMAGE:figures/full_fig_p049_25.png]

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Works this paper leans on

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