Floer Theory for Lagrangian Cobordisms

• Floer theory for Lagrangian cobordisms is a framework that defines invariants using quilted Floer homology and geometric composition principles.
• Monotonicity conditions and control of bubbling ensure compact moduli spaces and well-defined Floer complexes in the quilted setting.
• The theory supports modular computation of invariants, enabling practical decomposition, gluing, and applications in homological mirror symmetry.

Floer theory for Lagrangian cobordisms provides the analytic, algebraic, and topological foundation for defining and computing invariants of Lagrangian cobordisms via the machinery of Lagrangian Floer homology, with extensions to Morse–Bott, relative, and family settings; it also offers powerful gluing, decomposition, and functoriality principles that support the robust paper of the topology and dynamics of Lagrangian submanifolds in symplectic geometry.

1. Geometric Composition and Quilt Invariance

A key structural principle of Floer theory for Lagrangian cobordisms is invariance under geometric (transverse, embedded) composition in quilted Floer theory. Given a cyclic sequence of Lagrangian correspondences $L_0, L_1, ..., L_k$ between symplectic manifolds $M_0, ..., M_k$, if adjacent correspondences (say $L_r$ and $L_{r+1}$) compose transversely and the fiber product projection is an embedding, the theory establishes an isomorphism

$$HF(L_0, ..., L_r, L_{r+1}, ..., L_{k-1}) \cong HF(L_0, ..., L_r \circ L_{r+1}, ..., L_{k-1})$$

This geometric composition isomorphism (Lekili et al., 2010) demonstrates that the Floer cohomology is invariant under different quilt decompositions of a Lagrangian cobordism, as long as the composition data remains in the transverse/embedded regime. For cobordisms that can be broken into subpieces, this result implies that the Floer cohomology does not depend on the specific way the cobordism is cut and, crucially, that chain-level operations are compatible with the decomposition and recombination of corresponding patches or correspondences.

2. Monotonicity, Bubbling, and Moduli Space Compactness

Transversality, compactness, and well-definedness of the Floer complex in the quilted setting—and thus in the cobordism context—are established by imposing a monotonicity condition for the surfaces appearing in the construction. The central relation takes the form

$$\int v^* \omega = \tau \cdot I_{\operatorname{Maslov}}(v)$$

where $\tau$ is a monotonicity constant (positive for monotone, negative for strongly negative cases), and $I_{\operatorname{Maslov}}(v)$ is the Maslov index relative to the Lagrangian boundary/seam conditions. In the quilted Floer context, especially at the so-called Y-end (where multiple strips meet), such a relation enforces an energy–index proportionality, which in turn excludes bubbling (formation of nonconstant sphere or disk bubbles) in moduli spaces relevant to the definition of the differential (whose expected dimension is 0 or 1). Absence of bubbling at the Y-end ensures that the boundary strata of moduli spaces correspond only to broken trajectories, so that $d^2=0$ and the Floer theory is rigorously defined (Lekili et al., 2010).

This mechanism is essential when decomposing a cobordism into quilted pieces, as bubbling at composition interfaces would otherwise compromise the invariance results and render the theory ill-defined.

3. Floer Cohomology as a Cobordism Invariant

The invariance property described above elevates Floer cohomology to a robust invariant not only of individual Lagrangian submanifolds but also of Lagrangian cobordisms that interpolate between them. Explicitly, if a cobordism is presented as a sequential composition of correspondences,

$$L_0 \xrightarrow{L_{01}} M_1 \xrightarrow{L_{12}} M_2 \xrightarrow{L_2} \text{(output)} \quad \Longrightarrow \quad L_{02} = L_{01} \circ L_{12}$$

then the isomorphism

$$HF(L_0, L_{01}, L_{12}, L_2) \cong HF(L_0, L_{02}, L_2)$$

shows the well-definedness of the Floer-theoretic invariant for the cobordism as a whole, independent of the precise way the sequence is broken up, provided all compositions are transverse and embedded. This flexibility underlies many gluing and degeneration arguments: Floer cohomology classes persist through neck-stretching, concatenation, and other geometric manipulations as long as the analytic setup (transversality, monotonicity, exclusion of bubbling) is maintained.

4. Analytic and Algebraic Structures of Quilted Moduli

In quilted Floer theory, the domains are partitioned into patches, each mapped to a symplectic manifold, with seams (and possibly seams meeting at a Y-end) where maps are constrained to Lagrangian correspondences. The underlying moduli spaces are constructed by imposing Cauchy–Riemann equations patched together with seam or boundary conditions. The monotonicity property controls the analytic complexity (ensuring Gromov compactness and precluding low-index bubbles), while the combinatorics of patches and seams determine algebraic structures (composition, products, and higher $A_\infty$ maps).

The invariance is not merely at the cohomological level; rather, the entire $A_\infty$ structure (counts of holomorphic polygons and their relations via moduli space boundaries) is respected provided monotonicity and transversality are preserved in each elementary modification.

5. Implications for the Computation and Use of Floer Invariants

The robust nature of Floer cohomology under composition and quilt replacement allows modular computation of invariants for complicated Lagrangian cobordisms: to analyze the effect of a given geometric operation or deformation, one can model it as a quilted composition and, by the invariance property, replace subpieces with their composed image without recalculating the Floer theory from scratch. This enables a flexible approach to practical computations and is the foundation for spectral sequence arguments, decomposition theorems, and computational frameworks that exploit "local to global" methods.

This invariance principle is also essential in the context of homological mirror symmetry, where constructing and comparing $A_\infty$-functors and equivalences between Fukaya categories relies on delicate manipulations of quilted Lagrangians and their compositions.

6. Diagrammatics and Visualization

The structure of the resulting invariants and the role of the monotonicity condition is concisely illustrated by quilt diagrams (see (Lekili et al., 2010)). In a typical schematic,

[---- Patch 1 ----] ↓ seam L_{01} [---- Patch 2 ----] ← Y-end (no bubble allowed) ↓ seam L_{12} [---- Patch 3 ----]

the patches are rectangles or strips mapped to symplectic manifolds, the vertical or horizontal seams impose Lagrangian boundary/correspondence constraints, and the Y-end (the central trivalent point) is the locus where bubbling is analytically dangerous but prevented by the monotonicity hypothesis.

7. Summary and Broader Significance

The foundational results of quilted Floer theory (Lekili et al., 2010) establish that, under appropriate monotonicity and transversality conditions, Floer cohomology for Lagrangian correspondences—and hence Lagrangian cobordisms—is invariant under geometric composition and decomposition. Monotonicity guarantees the analytic control needed to exclude bubble formation, especially at Y-ends, so that the moduli spaces used for defining differentials and higher maps are compact (modulo breaking). Altogether, these features make Floer theory a powerful and computable invariant in both topological and categorical studies of Lagrangian cobordisms, and provide the necessary formalism to support the construction and manipulation of Fukaya categories and related invariants in symplectic topology.