Lagrangian Liftings: Derived and Microlocal Frameworks

• Lagrangian liftings are methods that represent data from classical, derived, and algebraic structures as Lagrangian objects in enriched ambient settings.
• They encompass a range of constructions including shifted symplectic intersections, microlocal characteristic cycles, and lifts to Legendrian submanifolds or Lagrangian cones.
• These techniques impact quantum homology, categorical symplectic relations, and Lagrangian field theories, enabling effective geometric and topological computations.

Searching arXiv for the cited papers and closely related work on Lagrangian liftings. “Lagrangian liftings” denotes a family of constructions in which data on one geometric or algebraic object is represented by a Lagrangian, Legendrian, Lagrangian fibration, Lagrangian cone, or Lagrangian relation on another space. In the literature considered here, the phrase includes the production of Lagrangians in shifted symplectic derived stacks by multiple intersections (Ben-Bassat, 2013), the natural Lagrangian fibration on the derived critical locus (Grataloup, 2020), the assignment of conic Lagrangian cycles to constructible invariants through MacPherson’s graph construction (Liao, 2018), the lifting of Lagrangian immersions in $\mathbb{C}P^{n-1}$ to Legendrian submanifolds of $S^{2n-1}$ and hence to Lagrangian cones in $\mathbb{C}^n$ (Baldridge et al., 2017), and several algebraic and homotopical liftings in quantum homology, categorical symplectic linear algebra, and field theory [(Biran et al., 2014); (Comfort et al., 2021); (Jurčo et al., 2024); (Schiavina et al., 31 Jul 2025)].

1. Scope of the term

In these works, “lifting” does not have a single fixed meaning. It may refer to the passage from a classical object to a derived one, as in the derived critical locus and multiple derived intersections; to the replacement of singularity-theoretic or constructible-function data by conic Lagrangian cycles; to the elevation of a Lagrangian immersion to a Legendrian embedding in a prequantization space; to the iteration of bi-Lagrangian structures to higher-dimensional bundles; or to the replacement of strict morphisms by Lagrangian relations and generalized Lagrangians in symplectic categories (Grataloup, 2020, Liao, 2018, Baldridge et al., 2017, Ndawa, 2021, Jurčo et al., 2024).

A common misconception is that a Lagrangian lifting must always be the lift of a submanifold to a larger symplectic manifold. The cited literature shows a broader usage. In one direction, the lift of a constructible function is a characteristic cycle in a cotangent bundle (Liao, 2018). In another, the lift of a homology class is its quantum and Floer-theoretic incarnation (Biran et al., 2014). In yet another, the lift is a categorical morphism represented by a Lagrangian subspace of $\overline{V}\times W$ or by a distributional half-density supported on a Lagrangian relation (Comfort et al., 2021, Jurčo et al., 2024). This suggests that “lifting” functions less as a single operation than as a recurrent structural pattern: geometric, microlocal, derived, or algebraic data is encoded by a Lagrangian object in a richer ambient setting.

2. Derived and shifted-symplectic liftings

The shifted-symplectic framework of Pantev–Toën–Vaquié–Vezzosi underlies several derived versions of Lagrangian lifting. An $n$-shifted symplectic derived stack $(S,\omega)$ is a derived stack equipped with a closed, non-degenerate $2$-form $\omega$ of degree $n$, and a Lagrangian is a morphism $f:X\to S$ with an isotropic structure satisfying a nondegeneracy condition. Oren Ben-Bassat’s “Multiple Derived Lagrangian Intersections” proves that an $S^{2n-1}$0-fold fiber product of Lagrangians in a shifted symplectic derived stack is itself Lagrangian in a certain cyclic product of pairwise homotopy fiber products of the Lagrangians (Ben-Bassat, 2013). The construction produces new Lagrangians directly from multiple intersections rather than from a single pairwise intersection.

The paper “A Derived Lagrangian Fibration on the Derived Critical Locus” places this idea in a more explicit local model. For a functional $S^{2n-1}$1, the derived critical locus is

$S^{2n-1}$2

realized as the derived intersection of the graph of $S^{2n-1}$3 and the zero section in $S^{2n-1}$4 (Grataloup, 2020). Since the derived intersection of two Lagrangian morphisms in an $S^{2n-1}$5-shifted symplectic stack has a canonical $S^{2n-1}$6-shifted symplectic structure, $S^{2n-1}$7 carries a $S^{2n-1}$8-shifted symplectic structure in this case. The main additional statement is that the projection

$S^{2n-1}$9

has a natural Lagrangian fibration structure.

When the strict critical locus $\mathbb{C}^n$0 is smooth and $\mathbb{C}^n$1 is non-degenerate, the local geometry of the Lagrangian fibration is controlled by the Hessian quadratic form

$\mathbb{C}^n$2

The non-degeneracy of the induced map

$\mathbb{C}^n$3

is equivalent to non-degeneracy of $\mathbb{C}^n$4 in the normal directions (Grataloup, 2020). In this context, “Lagrangian lifting” designates a derived-geometric refinement of the classical critical locus: the intersection is enlarged by homotopical information, and the resulting derived stack is equipped with lower-shift symplectic or Lagrangian-fibration data.

3. Microlocal and characteristic-cycle liftings

A second major meaning of Lagrangian lifting appears in microlocal geometry. For a holomorphic map $\mathbb{C}^n$5 between complex manifolds, assumed flat and sans éclatement en codimension $\mathbb{C}^n$6, MacPherson’s graph construction is applied to the vector bundle map

$\mathbb{C}^n$7

The graph family

$\mathbb{C}^n$8

degenerates as $\mathbb{C}^n$9, and the limit cycle decomposes as

$\overline{V}\times W$0

Here $\overline{V}\times W$1 is the Nash modification of $\overline{V}\times W$2 relative to $\overline{V}\times W$3, while $\overline{V}\times W$4 is supported over the critical locus $\overline{V}\times W$5 (Liao, 2018).

The paper identifies the Milnor number constructible function

$\overline{V}\times W$6

with a Lagrangian cycle supported on the “vertical” part of the graph limit. Its Lagrangian lift is expressed by

$\overline{V}\times W$7

and the characteristic cycle of $\overline{V}\times W$8 is the projectivised Lagrangian cycle $\overline{V}\times W$9 (Liao, 2018). When $n$0 has finite contact type, the Chern class transformation is computed by

$n$1

In this microlocal usage, lifting is not a passage to a covering space or a cone. It is the assignment of a conic Lagrangian cycle to a constructible function or singularity invariant. The underlying theme is that local analytic invariants, such as the Milnor number and the local Euler obstruction, are encoded by explicit geometric cycles in cotangent bundles (Liao, 2018). This suggests a close relation between “lifting” and microlocalization: the invariant is represented by a Lagrangian object whose support records the singular geometry.

4. Lifts to Legendrian submanifolds and Lagrangian cones

A more classical geometric form of Lagrangian lifting is the production of actual Lagrangian submanifolds or cones in a larger symplectic or contact manifold. In “Lifting Lagrangian immersions in $n$2 to Lagrangian cones in $n$3,” the Hopf fibration

$n$4

is used to lift a Lagrangian immersion $n$5 to a Legendrian immersion $n$6, and hence to a Lagrangian cone in $n$7 (Baldridge et al., 2017). The lifting theorem imposes two conditions: a monodromy condition

$n$8

and a double-point separation condition requiring distinct nonzero phase values mod $n$9 for paths joining points with the same image in $(S,\omega)$0. Under these hypotheses, the lift is an embedding, its image is Legendrian, and its cone is Lagrangian.

This framework produces explicit families of cones, including cones isotopic to the Harvey–Lawson and trivial cones, and examples whose projections to $(S,\omega)$1 have only few transverse double points (Baldridge et al., 2017). The same paper treats special Lagrangian cones by imposing additional phase constraints. The geometric meaning is direct: a Lagrangian immersion in the symplectic quotient is promoted to a Legendrian link in the prequantization bundle, and then extended radially to a cone in $(S,\omega)$2.

A complementary construction appears in Matessi’s work on tropical geometry. A smooth tropical hypersurface in $(S,\omega)$3 can be lifted to a smooth embedded Lagrangian submanifold in $(S,\omega)$4 by first constructing a piecewise-linear Lagrangian lift $(S,\omega)$5, then replacing vertex neighborhoods by Lagrangian pairs of pants, extending along edges by cylinders, and gluing the local models with trimming and partitions of unity (Matessi, 2018). The local model is the graph

$(S,\omega)$6

of an exact $(S,\omega)$7-form on the blown-up coamoeba, with

$(S,\omega)$8

The result is a smooth embedded Lagrangian $(S,\omega)$9 converging to the PL model as the scaling parameter $2$0 (Matessi, 2018).

Together, these constructions show that “lifting” can mean either horizontal lifting through a contact prequantization bundle or smoothing a combinatorial tropical object into an exact Lagrangian submanifold. In both cases, combinatorial or symplectic data in a lower-dimensional quotient determines a higher-dimensional Lagrangian geometry.

5. Lifts of geometric structures and dynamical systems

Some works use Lagrangian lifting for ambient structures rather than individual submanifolds. A bi-Lagrangian structure $2$1 on a $2$2-manifold $2$3 consists of a symplectic form and two transversal Lagrangian foliations. Such a structure admits a unique torsion-free Hess connection preserving $2$4 and both foliations. “Infinite Lifting of an Action of Symplectomorphism Group on the set of Bi-Lagrangian Structures” proves that a bi-Lagrangian structure on $2$5 lifts to a bi-Lagrangian structure on $2$6 with

$2$7

and that this process can be iterated indefinitely (Ndawa, 2021). Affineness is preserved, the symplectomorphism group action is compatible with Hess connections, and for parallelizable manifolds analogous lifted structures can be transferred to $2$8 and $2$9.

A different but related infrastructure for lifting appears in the study of the dual of the first jet bundle. For a bundle $\omega$0, the paper “Lifting geometric objects to the dual of the first jet bundle of a bundle fibred over $\omega$1” defines vertical, horizontal, and complete lifts from $\omega$2 to $\omega$3. If $\omega$4 is a $\omega$5-tensor field on $\omega$6 with $\omega$7, its complete lift $\omega$8 is characterized by

$\omega$9

and satisfies

$n$0

The pair $n$1 is a Poisson–Nijenhuis structure on $n$2 if and only if $n$3 (Sarlet et al., 2013). The paper states that this framework is crucial for the theory of Lagrangian liftings and their applications in time-dependent geometry.

The Bohlin variant of the Eisenhart lift extends the vocabulary of lifting to conservative Lagrangian mechanics. A Lagrangian conservative dynamical system with $n$4 degrees of freedom is embedded into timelike geodesics of the conformally flat Lorentzian metric

$n$5

on a $n$6-dimensional space-time (Galajinsky, 24 Feb 2026). The lifted geodesic equations

$n$7

recover the original Newtonian system after identifying $n$8. Here the “lift” is not itself Lagrangian in the symplectic-topological sense, but it arises from a Lagrangian dynamical system and geometrizes its trajectories as geodesics in a higher-dimensional ambient space.

6. Algebraic, categorical, and field-theoretic liftings

Algebraic symplectic topology uses “lifting” to describe the passage from classical invariants to quantum and homotopical ones. In “The Lagrangian Cubic Equation,” the class $n$9 of a Lagrangian submanifold in quantum homology satisfies

$f:X\to S$0

under the stated monotonicity and Floer-theoretic hypotheses (Biran et al., 2014). The paper explicitly describes the passage from the classical homology class to the quantum homology class $f:X\to S$1, and further to its image in $f:X\to S$2 via the Piunikhin–Salamon–Schwarz isomorphism, as a “lifting.” The discriminant

$f:X\to S$3

packages the quadratic-algebra structure on $f:X\to S$4.

Categorical approaches replace maps by Lagrangian relations. In “A Graphical Calculus for Lagrangian Relations,” the category of Lagrangian relations over a field is presented as a doubled category of linear relations by the assignment

$f:X\to S$5

with the orthogonal complement functor playing the role of conjugation in a CPM-style construction (Comfort et al., 2021). By adjoining a single affine shift operator one obtains affine Lagrangian relations, and for odd prime $f:X\to S$6 the resulting prop is equivalent to qudit stabilizer theory modulo invertible scalars. In “Lagrangian Relations and Quantum $f:X\to S$7 Algebras,” a category $f:X\to S$8 is formed from $f:X\to S$9-shifted symplectic vector spaces and Lagrangian relations $S^{2n-1}$00; in the quantum extension, morphisms are generalized Lagrangians or distributional half-densities, and composition recovers homotopy transfer of quantum $S^{2n-1}$01 algebras (Jurčo et al., 2024). A quantum $S^{2n-1}$02 algebra is encoded by a BV-closed morphism

$S^{2n-1}$03

and composition with a Lagrangian relation realizes the effective-action construction.

Functorial Legendrian contact homology provides another instance of lifting. A good immersed exact Lagrangian cobordism with primitive $S^{2n-1}$04 is lifted to a conical Legendrian cobordism by taking the graph of $S^{2n-1}$05 and then applying a contactomorphism to a jet space, producing an immersed DGA map

$S^{2n-1}$06

(Pan et al., 2019). The lift eliminates self-intersection issues at the Legendrian level and extends functoriality beyond the embedded case.

Field theory supplies a higher-homotopical version of the same pattern. For local functionals on the variational bicomplex, a $S^{2n-1}$07-symplectic local form $S^{2n-1}$08 induces a Lie$S^{2n-1}$09 algebra on Hamiltonian local functionals, and for any compatible cohomological vector field $S^{2n-1}$10 the paper “Homotopies for Lagrangian field theory” constructs an explicit $S^{2n-1}$11 algebra on a resolution of $S^{2n-1}$12 (Schiavina et al., 31 Jul 2025). For $S^{2n-1}$13, the construction provides an explicit lift of the standard Batalin–Vilkovisky framework to local forms enriched by the $S^{2n-1}$14 structure. The associated multisymplectic datum is summarized by

$S^{2n-1}$15

Across these algebraic constructions, Lagrangian lifting means that geometric or homological information is reformulated in a category whose morphisms, relations, or observables are intrinsically Lagrangian. A plausible implication is that the term identifies a methodological principle rather than a single theorem: one passes to a larger symplectic, contact, derived, or homotopical environment in which the original data is expressed by a Lagrangian object and thereby becomes composable, functorial, or computable.