Legendrian Rational Symplectic Field Theory

• Legendrian rational SFT is a genus-zero sector of symplectic field theory that organizes punctured holomorphic curves into algebraic operations extending the Chekanov–Eliashberg formalism.
• The framework employs commutative L∞ and homotopy Poisson structures along with functorial DGA maps derived from exact Lagrangian cobordisms to capture invariants in R³ and higher dimensions.
• Computable variants—including bordered, convex-surface, and sheaf-theoretic avatars—offer augmentation-like replacements and mutation rules that enhance contact homology analysis.

Legendrian rational symplectic field theory is the genus-zero Legendrian sector of symplectic field theory in which punctured holomorphic curves with boundary on Lagrangian cylinders over Legendrians are organized into algebraic operations that extend the one-positive-puncture Chekanov–Eliashberg formalism by incorporating disks with multiple positive punctures. In the most explicit three-dimensional setting this enlargement appears as a commutative $L_\infty$/homotopy Poisson structure extracted from an LSFT Hamiltonian, while exact Lagrangian cobordisms supply functorial algebra maps; in higher-dimensional or analytically difficult settings, microlocal sheaves, flag moduli, and related algebro-geometric constructions are used as augmentation-like replacements (Ng, 2023, Ekholm et al., 2012, Casals et al., 2020). Across the literature, however, the term does not denote a single completed universal package: some works construct one-positive-puncture truncations, some commutative or bordered shadows, and some sheaf-theoretic or Floer-theoretic avatars (Wlodek, 9 Sep 2025, Eagles et al., 23 Apr 2026, Treumann et al., 2016).

1. Scope and algebraic architecture

A convenient ambient template is the full Eliashberg–Givental–Hofer Legendrian SFT algebra recalled in the convex-surface setting: $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$ with orbit variables $\gamma_i$, chord variables $c_i$, and differentials

$$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$

$$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$

Within this template, the one-positive-puncture Chekanov–Eliashberg differential is a truncation, whereas rational SFT keeps the genus-zero multiple-positive-puncture sector (Eagles et al., 23 Apr 2026).

In the standard contact $\mathbb R^3$ setting, the most explicit algebraic realization is Ng’s commutative LSFT package for pointed Legendrian links. Starting from the graded-commutative algebra

$$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$

one introduces formal dual variables $p_j$, a Hamiltonian

$h=h_1+h_2+h_3+\cdots,$

the SFT bracket generated by $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$0, and the string coproduct $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$1, with the master equation

$$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$2

The ordinary commutative Legendrian contact homology differential is recovered as

$$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$3

while disks with $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$4 positive punctures contribute higher operations $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$5. The resulting structure is a homotopy Poisson, equivalently $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$6, algebra on $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$7, rather than the full noncommutative EGH rational SFT algebra (Ng, 2023).

This algebraic architecture makes clear a persistent distinction in the subject. On one side stand full Hamiltonian or DGA formalisms motivated by punctured holomorphic curves; on the other stand computable shadows obtained by commutativization, linearization, quotienting away orbit variables, or replacing curve counts by sheaf-theoretic moduli. Much of the modern literature on Legendrian rational SFT is organized by precisely these controlled reductions.

2. Three-dimensional theory in standard contact $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$8

For a pointed oriented Legendrian link $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$9, the Reeb vector field is $\gamma_i$0, so Reeb chords are vertical segments. In the commutative LSFT construction, the grading is determined by capping paths and Maslov potentials, with

$\gamma_i$1

The operations are

$\gamma_i$2

$\gamma_i$3

while $\gamma_i$4 contains the essential string-topology correction forced by boundary bubbling. The main theorem states that the $\gamma_i$5 make $\gamma_i$6 into a homotopy Poisson algebra, so

$\gamma_i$7

inherits a degree-zero Poisson bracket from $\gamma_i$8 (Ng, 2023).

The correction in $\gamma_i$9 is not ornamental. In Ng’s formulation, the naive higher brackets extracted from $c_i$0 fail to satisfy the $c_i$1 identities because of the $c_i$2 term in the master equation. The added string term neutralizes precisely that obstruction. This places Legendrian rational SFT in $c_i$3 at the intersection of holomorphic-disk combinatorics, string-topological correction terms, and higher Lie-type algebra.

The theory is also computable. For $c_i$4-closures of admissible positive braids, every immersed disk has at most two positive corners, hence

$c_i$5

In that family the $c_i$6 structure is strict, and $c_i$7 becomes a DG Poisson algebra. For the Legendrian $c_i$8-torus link $c_i$9, the Poisson bracket is written explicitly; for example,

$$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$0

Under the substitution $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$1, $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$2, $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$3, this is identified with the Flaschka–Newell bracket. Invariance is proved through $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$4: if two pointed Legendrian links are isotopic, the induced Poisson algebras on $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$5 are isomorphic.

3. Exact Lagrangian cobordisms, fillings, and functoriality

Exact Lagrangian cobordisms supply the functorial backbone for the one-positive-puncture sector of Legendrian SFT. An exact Lagrangian cobordism from $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$6 to $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$7 is a pair $$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$8 with cylindrical ends

$$\widehat{\partial}(\gamma) = \sum \#\bigl(\mathcal M(\gamma;\{\gamma_1,\dots,\gamma_k\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k,$$9

and with $$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$0 constant on each end. Such a cobordism induces a DGA morphism

$$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$1

defined on a Reeb chord $$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$2 by

$$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$3

This map is a chain map,

$$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$4

and is functorial under composition: $$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$5 The construction is explicitly identified as the one-positive-puncture truncation of rational SFT relevant to Legendrian contact homology (Ekholm et al., 2012).

A major technical advance is the replacement of holomorphic disks by gradient flow trees for Morse cobordisms in completions of cotangent bundles. For suitable $$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$6,

$$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$7

This makes cobordism maps combinatorial for elementary moves. Triple point, birth, death, minimum, and saddle cobordisms recover the familiar Chekanov DGA maps as actual exact Lagrangian cobordism maps. For a simple contractible chord $$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$8, the saddle map takes the form

$$\widehat{\partial}(c) = \sum \#\bigl(\mathcal M(c;\{\gamma_1,\dots,\gamma_k\},\{c_1,\dots,c_l\})/\mathbb R\bigr)\, \gamma_1\cdots\gamma_k\,c_1\cdots c_l.$$9

Fillings are the special case $\mathbb R^3$0. A filling induces an augmentation, and different fillings can yield different augmentations. One application is the existence of non-isotopic exact Lagrangian fillings of the same Legendrian link; for the right-handed trefoil, there are $\mathbb R^3$1 exact Lagrangian fillings corresponding to the $\mathbb R^3$2 Catalan resolution orders. This part of the subject is not yet full rational SFT in the multiple-positive-puncture sense, but it supplies the geometric morphisms and augmentation-theoretic functoriality from which Legendrian rational SFT draws much of its structure.

4. Bordered and convex-surface variants

A bordered version of commutative Legendrian rational SFT is obtained by cutting a simple front projection by a vertical line and assigning DGAs to the left half, right half, and interface. The middle algebra $\mathbb R^3$3, and the left and right algebras $\mathbb R^3$4, $\mathbb R^3$5, are commutative $\mathbb R^3$6-DGAs equipped with an SFT bracket, a Hamiltonian counting disks with arbitrarily many positive corners, and a string differential. Their total differentials have the form

$\mathbb R^3$7

and the global commutative LSFT algebra is recovered by a pushout square

$\mathbb R^3$8

This extends bordered Chekanov–Eliashberg theory by incorporating multiple-positive-puncture disks and string-topology corrections, but the pushout theorem is established only in the commutative quotient, not for the full curved noncommutative LSFT algebra (Wlodek, 9 Sep 2025).

A different extension appears for Legendrian knots in thickened convex surfaces $\mathbb R^3$9, where the dividing set $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$0 is built into the algebra. Here

$$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$1

with $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$2 countably generated because decomposable $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$3-chords can wrap around components of $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$4 arbitrarily many times. The $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$5-sector differential is not a direct full-SFT disk count; rather, after a degree shift,

$$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$6

where $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$7 splits a $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$8-chord by concatenation and $$A=\Bbbk[q_1,\dots,q_n,t_1^{\pm1},\dots,t_s^{\pm1}],$$9 count polygons in $p_j$0. The differential squares to zero, and the stable tame isomorphism type is invariant under Legendrian isotopy. The authors explicitly position this as a reduced or contact-homology-type model inspired by rational SFT, not as the actual full SFT differential, since curves with several negative $p_j$1-ends are not included (Eagles et al., 23 Apr 2026).

These variants show two complementary tendencies. Bordered LSFT seeks locality and gluing in the commutative multiple-positive-puncture theory; convex-surface DGAs enlarge the ambient geometry while retaining polygonal combinatorics. In both cases the theme is the same: extract a computable algebra from genus-zero holomorphic-curve behavior, even if the full noncommutative relative SFT package is not yet available.

5. Higher-dimensional Legendrians and sheaf-theoretic avatars

For Legendrian surfaces in contact five-manifolds, direct holomorphic-curve calculations are substantially harder, and the literature often replaces analytic rational SFT by sheaf-theoretic or algebro-geometric models. In the theory of Legendrian weaves, the basic combinatorial input is an $p_j$2-graph

$p_j$3

on a surface $p_j$4, encoding crossings of adjacent sheets, trivalent branch points, and hexagonal triple interactions. From $p_j$5 one constructs a Legendrian surface $p_j$6, and a local calculus of graph moves realizes isotopies, surgeries, mutations, connected sums, and stabilization. The associated flag moduli space

$p_j$7

is identified with microlocal rank-one constructible sheaves: $p_j$8 This moduli space is a Legendrian isotopy invariant and is repeatedly interpreted as an augmentation-type avatar. Its local coordinates include cross ratios

$p_j$9

and triple ratios, and under mutation these coordinates satisfy cluster-type transformations such as

$h=h_1+h_2+h_3+\cdots,$0

The surgery and cobordism calculus is particularly suggestive from an SFT perspective, because exact Lagrangian cobordisms are encoded combinatorially and expected to induce maps on these moduli analogous to augmentation-variety maps (Casals et al., 2020).

A closely related surface theory begins from cubic planar graphs $h=h_1+h_2+h_3+\cdots,$1. The associated hyperelliptic wavefront determines a Legendrian surface $h=h_1+h_2+h_3+\cdots,$2, and the microlocal rank-one sheaf moduli has the explicit description

$h=h_1+h_2+h_3+\cdots,$3

where $h=h_1+h_2+h_3+\cdots,$4 is the open set of face colorings assigning distinct points to adjacent faces. Over finite fields,

$h=h_1+h_2+h_3+\cdots,$5

so distinct chromatic polynomials of the dual graphs distinguish Legendrian surfaces with the same classical invariants. At the same time, these Legendrians have no smooth oriented graded exact Lagrangian fillings in $h=h_1+h_2+h_3+\cdots,$6. Nevertheless, the period map embeds $h=h_1+h_2+h_3+\cdots,$7 as a holomorphic Lagrangian in

$h=h_1+h_2+h_3+\cdots,$8

and after choosing a phase and an OGW framing one obtains a generating function $h=h_1+h_2+h_3+\cdots,$9 on a filling torus, conjecturally of dilogarithmic form

$$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$00

This is not rational SFT in the direct holomorphic-curve/DGA sense, but it functions as an augmentation-like and potential-theoretic replacement in a setting where direct Legendrian SFT is presently inaccessible (Treumann et al., 2016).

6. Broader rational-SFT context and present limits

The Legendrian story sits inside a broader genus-zero SFT landscape. On the closed-contact side, the $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$01 formalism packages rational SFT operations $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$02 into an operator $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$03 satisfying $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$04, and exact symplectic cobordisms induce $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$05-morphisms. This yields order-valued invariants such as algebraic planar torsion, semi-dilation, and planarity. The construction is not Legendrian, but it provides a model for how genus-zero curve counts, augmentations, linearizations, and cobordism functoriality can be organized algebraically (Moreno et al., 2020).

A related closed-string application appears in the computation of the pair-of-pants product and BV-operator on symplectic homology via Legendrian surgery. There the ordinary Legendrian homology algebra determines the BV-operator, but the product requires the first genuinely rational-SFT layer beyond one-positive-puncture contact homology, namely a two-positive-puncture term $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$06. The chain-level product is written as

$$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$07

This shows, in a concrete Weinstein-handlebody setting, that operations on symplectic homology detect precisely the extra structure expected from Legendrian rational SFT (Bourgeois et al., 2010).

Other nearby theories are explicitly described as adjacent rather than identical to Legendrian rational SFT. Rational SFT of Seifert fibrations over orbifold projective lines is expressed through genus-zero orbifold Gromov–Witten theory of the base, making the Hamiltonian formalism concrete in a closed-string circle-bundle setting (Rossi, 2008). Rabinowitz Floer homology for Legendrian lifts in prequantization bundles defines a Reeb-chord Floer theory with strip and triangle operations and compares it to $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$08, but it is not the Legendrian rational SFT algebra (Bae et al., 30 Jun 2026). The derived contactification of homogeneous shifted symplectic stacks and the derived Legendrian intersection theorem develop a contact/Legendrian moduli formalism with shifted contact structures on intersections, again without holomorphic-curve counts, Reeb chords, or SFT Hamiltonians (İzbudak et al., 8 May 2026).

The main limitations are therefore structural, not merely technical. Ng’s $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$09 package proves isotopy invariance through $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$10, while full $$\mathcal A(Y,\Lambda)=S(C(Y,\alpha))\otimes T(C(\Lambda)),$$11 invariance remains conjectural (Ng, 2023). The bordered pushout theorem is currently a theorem for the commutative quotient of LSFT, not for the full noncommutative curved theory (Wlodek, 9 Sep 2025). The convex-surface DGA is presented as a reduced contact-homology-type model and is conjectured to be quasi-isomorphic to a reduced sutured Legendrian contact homology DGA rather than claimed to realize full SFT (Eagles et al., 23 Apr 2026). In higher dimensions, Legendrian weaves and cubic-planar-graph surfaces provide explicit moduli, mutation rules, and filling constructions, but not direct rational SFT Hamiltonians or Chekanov–Eliashberg DGAs (Casals et al., 2020, Treumann et al., 2016). This suggests that the present state of Legendrian rational symplectic field theory is best understood as a constellation of compatible formalisms: full genus-zero curve-counting structures in a few very explicit three-dimensional settings, functorial one-positive-puncture cobordism maps, commutative and bordered shadows of LSFT, and sheaf-theoretic or Floer-theoretic avatars that preserve much of the expected augmentation, wall-crossing, and filling behavior when direct analytic SFT remains out of reach.