Legendrian Lift in Contact Geometry

• Legendrian lift is the process of transforming a Lagrangian submanifold into a Legendrian one within a contact manifold, preserving key geometric and algebraic invariants.
• This construction enables the utilization of contact invariants, such as Legendrian contact homology and functorial DGA mappings, to study symplectic and contact topology.
• Various lifting techniques, including prequantization and conical approaches, offer practical tools to analyze rigidity, non-squeezing phenomena, and spectral constructions.

A Legendrian lift is a canonical construction that associates to a Lagrangian submanifold—often with additional structure, such as a generating family or primitive—a Legendrian submanifold in an appropriate contact manifold. This procedure occurs naturally in the study of symplectic and contact topology, particularly in the context of 1-jet spaces, cotangent bundles, and prequantization spaces. Legendrian lifts encode geometric and algebraic data critical for invariants in contact homology, generating family theory, and quantized sheaf categories. The lift transforms problems about Lagrangians into the Legendrian setting, thereby enabling the use of contact invariants and functorial frameworks. Variant lift constructions—such as conical Legendrian lifts of immersed Lagrangian cobordisms or prequantization lifts—enable additional functorialities, spectral constructions, and rigidity phenomena.

1. Legendrian Lifts: Foundational Constructions

Several Legendrian lift mechanisms are employed depending on the source manifold and target contact structure:

a. Lagrangian Lifts to Contactizations.

Given an exact Lagrangian $L$ in an exact symplectic manifold $(V,d\lambda)$, equipped with a primitive $f$ such that $\lambda|_L=df$, the Legendrian lift $\tilde L\subset V\times\mathbb{R}_z$ is $\tilde L = \{(x,z)\mid x\in L,\,z=-f(x)\},$ where the contact form is $\alpha=dz+\lambda$. This construction ensures that $\alpha|_{T\tilde L}=0$, so $\tilde L$ is Legendrian (Nakamura, 2023).

b. Prequantization Lifts in $\mathbb{C}^n$.

For a closed Lagrangian $L\subset\mathbb{C}^n$ with $\lambda|_L$ integral, a “primitive” $$\theta(z)=\int_{p_0}^{z}\lambda\quad\in\mathbb{R}/\mathbb{Z}$$ gives the Legendrian lift $$\widetilde L = \{(t=\theta(z), z)\mid z\in L\} \subset S^1\times\mathbb{C}^n,$$ in the contact manifold $(S^1\times\mathbb{C}^n, dt+\lambda)$. The lift is unique up to Reeb translation in $t$ (Kilgore, 2024).

c. Lifts of Cooriented Wavefronts.

A cooriented wavefront $\omega:S^1\to \Sigma$ on an oriented surface $\Sigma$ lifts canonically to a Legendrian curve $\Lambda\subset U^*\Sigma$ by associating, at each regular point, the unique unit covector $p(t)$ annihilating the tangent and positive on the cooriented half-plane. In coordinates: $$\Lambda(x) = (x,f(x); -f'(x),1)/\sqrt{1+(f'(x))^2}\in U^*\Sigma,$$ with the contact 1-form $\alpha=p_xdx+p_ydy$ (Cahn et al., 2013).

2. Conical Legendrian Lifts of Immersed Lagrangian Cobordisms

For 1-dimensional Legendrians in 1-jet spaces $J^1M$, the conical Legendrian lift of an immersed exact Lagrangian cobordism $L$ is constructed in the symplectization $\mathbb{R}_t\times J^1M$ as $L \subset \text{Symp}(J^1M)$ with cylindrical ends. Equipped with a primitive $\rho:L\to\mathbb{R}$, the cobordism admits a Legendrian lift $$\Sigma^{\text{Symp}} \subset \text{Symp}(J^1M)\times\mathbb{R}_w,\quad w=-\rho,$$ which under a specific contactomorphism $\Phi$ becomes an embedded Legendrian submanifold $\Sigma\subset J^1(\mathbb{R}_{>0}\times M)$, agreeing with scaled fronts of the ends for large $|s|$ (Pan et al., 2019).

These lifts are critical for functorial constructions in Legendrian contact homology, as they allow one to transcribe algebraic structures—such as DGAs—to cobordisms beyond the embedded case.

3. Algebraic Structures via Legendrian Lifts

The main algebraic object arising from Legendrian lifts is the Legendrian contact homology DGA $(A(\Lambda),\partial)$, whose generators are Reeb chords and whose differential counts rigid holomorphic disks or polygons with specified boundary conditions. For conical Legendrian cobordisms, the DGA functoriality proceeds via mapping-cylinders:

• For cobordism $\Sigma$ from $\Lambda_-$ to $\Lambda_+$, two DGA maps arise:
  • $i:A(\Lambda_-)\hookrightarrow A(\Sigma)$ (inclusion);
  • $f:A(\Lambda_+)\rightarrow A(\Sigma)$, defined by counts of rigid gradient flow trees computed in Morse-theoretic neighborhoods.

The mapping cylinder DGA $C := A* \widehat{A} * B$ encodes these morphisms with the differential on $\widehat{A}$ $$\partial(\widehat{a}) = f(a) + a + \Gamma(\partial_A a),$$ where $\Gamma$ is a derivation tied to $(f,i)$. Isomorphisms and homotopies in this context provide well-defined functorial correspondences between categories of Legendrians (with metrics and potentials) and categories of DGAs (with immersed maps) (Pan et al., 2019).

4. Moduli Spaces and Holomorphic Curve Lifting

Legendrian lifts play a pivotal role in moduli-theoretic correspondences:

Given a pseudo-holomorphic polygon $$u: (\Sigma,\partial \Sigma)\to (P,\Pi_{\text{Lag}}(\Lambda))$$ in an exact symplectic manifold $P$, lifting procedures yield $$\bar{u}(s,u,z): \Sigma \to \mathbb{R}_s\times P \times \mathbb{R}_z$$ where $\bar{u}$ is $J_{\tilde{P}}$-holomorphic and its boundary satisfies $$\bar{u}(\partial\Sigma)\subset \mathbb{R}_s\times \Lambda$$. The vanishing of the $\bar{\partial}$ operator yields a system relating the holomorphicity of $u$, the $s$-coordinate via the primitive $\theta$, and the $z$-coordinate via harmonic conjugacy. This lifts counting problems of polygons in $P$ to discs in $\mathbb{R}_s\times P\times \mathbb{R}_z$, enabling equivalences $$A(\Lambda)_{\text{polygon}} \cong A(\Lambda)_{\text{symplectization}}$$ on contact DGA invariants and all derived augmentations (Rizell, 2013).

5. Legendrian Operations and Generating Functions

Legendrian lifts facilitate algebraic operations—sum, convolution, Fourier—in jet spaces $J^1(\mathbb{R}^n,\mathbb{R})$:

For $L_1$, $L_2$ Legendrians, their sum and convolution

$$L_1+L_2 = \{(u_1+u_2, q, p_1+p_2)\mid (u_i,q,p_i)\in L_i\},$$

$$L_1\star L_2 = \{(u_1+u_2, q_1+q_2, p)\mid (u_i,q_i,p)\in L_i\},$$

produce immersed Legendrians generically.

Generating functions $F: \mathbb{R}^n\times \mathbb{R}^k\to \mathbb{R}$ define Legendrians via critical set conditions. Under operations, generating functions correspond to sum and infimal-convolution at the function level—concretely:

$(j^1f_1) + (j^1f_2) = j^1(f_1+f_2),\quad (j^1f_1)\star(j^1f_2) = j^1(f_1 \infconv f_2),$

with the Fourier-type involution $\mathbf{T}$ realizing Legendre transforms at the level of generating functions (Limouzineau, 2016).

6. Stable Homotopy and Generating Family Invariants

Legendrian lifts equipped with linear-at-infinity generating families $F$ produce stable homotopy spectra $\mathcal{G}(F)$. The difference-function construction

$\delta_F(x,\eta,\zeta) = F(x,\eta) - F(x,\zeta)$

gives a mapping-cone prespectrum whose suspension stabilizations produce an honest spectrum $\mathcal{G}(F)$. The homology $\pi_\ast(\mathcal{G}(F))$ lifts generating family homology $GFH_\ast(\Lambda, F)$. When the generating family extends over a filling, the spectrum-level Seidel isomorphism

$\mathcal{G}(F) \simeq \Sigma^\infty(L/\partial L)$

recovers invariants of the Lagrangian filling and yields constraints on fiber-dimensions, existence of fillings, and stable homotopy invariants of $L$ (Tanaka et al., 2024).

7. Rigidity, Metrics, and Non-Squeezing via the Legendrian Lift

Legendrian lifts encode analytic and combinatorial invariants leading to quantitative rigidity:

a. Vanishing of Shelukhin–Chekanov–Hofer Metric.

For an exact, displaceable Lagrangian $L$, its Legendrian lift in $(V\times\mathbb{R}, dz+\lambda)$ has vanishing Shelukhin–Chekanov–Hofer metric: $d_\alpha(\tilde{L},\tilde{L}')=0$ for all Legendrian isotopic $\tilde{L}'$. This establishes counterexamples to conjectures about non-degeneracy of this metric (Nakamura, 2023).

b. Non-Squeezing Phenomena.

Prequantization Legendrian lifts of Lagrangians in $\mathbb{C}^n$ cannot be isotoped into arbitrarily small prequantized cylinders when certain microsheaf category invariants (Nadler–Shende $\Sh_\Lambda$ and its rank-one subcategory $\Sh^1_\Lambda$) exhibit rank inequalities not realized inside the cylinder. This non-squeezing result is derived by Legendrian isotopy invariance and explicit combinatorics for Clifford, Chekanov, and Whitney tori (Kilgore, 2024).

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This constellation of constructions and results demonstrates that the Legendrian lift is an indispensable tool for translating symplectic, Morse-theoretic, and homological data into the contact-geometric setting, with broad implications for functorial invariants, rigidity, and the interface between algebraic and geometric topology.