Bounding CochainEdit
Bounding cochain is a central construction in modern symplectic geometry, playing a key role in the deformation theory of Lagrangian Floer theory and the structure of the Fukaya category. Roughly speaking, it is an algebraic gadget attached to a Lagrangian submanifold that records how obstructions arising from disc counting can be canceled, allowing one to define a well-behaved Floer theory even when a naive set-up would fail. In practical terms, a bounding cochain b encodes a choice of deformation that makes the differential square to zero and turns an obstructed object into a workable one for defining morphisms and higher operations.
This article surveys what a bounding cochain is, how it enters the A-infinity framework that underpins much of the modern approach to Lagrangian intersections, and why it matters both in pure mathematics and in the broader landscape of ideas connected to mirror symmetry. It also sketches the main fronts of foundational discussion, where different technical approaches aim to secure invariance and computability in the presence of obstructions.
Definition and basic ideas
Setup
Let (M, ω) be a symplectic manifold and L ⊂ M a compact, oriented Lagrangian submanifold with suitable extra structure (for example, a spin or Pin structure) to ensure well-defined signs in counts of holomorphic discs. The Floer cochain complex CF^(L, L) carries higher products m_k for k ≥ 0, forming an A-infinity algebra. The map m_k takes k inputs from CF^(L, L) and yields an output in CF^*(L, L). Among these, m_0 encodes disc contributions with no input, and m_1 is the original Floer differential when no deformation is present.
The bounding cochain
A bounding cochain b is an element of CF^1(L, L) (i.e., of degree 1 in the Floer cochain complex) that satisfies the Maurer–Cartan equation m_0 + m_1(b) + m_2(b, b) + m_3(b, b, b) + ··· = 0. Equivalently, the deformed higher operations m_k^b(a_1, ..., a_k) = ∑ m_{k+j}(b, ..., b, a_1, ..., a_k) define a consistent A-infinity structure in which the deformed differential m_1^b satisfies (m_1^b)^2 = 0. In that sense, b “kills” the obstruction encoded by m_0 and curates a well-defined Floer theory for the pair (L, b).
Unobstructedness and gauge equivalence
Not every Lagrangian admits a bounding cochain; those for which such b exists are called unobstructed (or weakly unobstructed in a precise sense when the framework allows certain controlled obstructions). If b and b′ are related by a gauge equivalence (an A-infinity version of a change of coordinates in the deformation problem), they define equivalent deformations of the Floer theory. The space of bounding cochains up to gauge equivalence forms a moduli space that parametrizes deformations of the Lagrangian brane (the pair consisting of L and the extra data provided by b).
Role in the Fukaya category
In the modern formulation, objects of the Fukaya category are Lagrangian submanifolds equipped with bounding cochains (often called branes). Morphisms between two such objects (L_0, b_0) and (L_1, b_1) are given by the Floer cochains with the deformed differential, and the higher A-infinity structure maps m_k^b encode counts of holomorphic polygons with boundary on the chosen branes. Bounding cochains thus enrich the category by allowing a broader and more robust class of objects, essential for the categorical formulation of mirror symmetry.
The Maurer–Cartan equation and deformation theory
The core equation for b is a manifestation of the Maurer–Cartan equation in the A-infinity setting. It encodes consistency conditions for disc counts with boundary on L in the ambient symplectic geometry. Solving the equation is tantamount to choosing a deformation that cancels obstructions and yields a meaningful, computable Floer theory. The equation can be viewed as a nonlinear algebraic constraint that ties together all higher products m_k, reflecting the intricate geometry of holomorphic discs with boundary on L.
Connections to the mirror side
In many instances arising from mirror symmetry, bounding cochains on the symplectic side correspond to choices of complex structure or superpotential data on the mirror side. The disc potential, assembled from counts of holomorphic discs weighted by the bounding cochain, plays the role of a holomorphic function W whose critical points reflect stability and unobstructedness of the brane from the mirror perspective. This bridge illuminates why bounding cochains are a natural and essential ingredient in the open-string sector of mirror symmetry, linking geometry to physics-inspired deformation theory.
Significance in theory and practice
Objects and morphisms with brane data
The introduction of bounding cochains broadens the set of objects in the Fukaya category and refines morphisms between them. The deformed differential and higher products capture more refined interactions between Lagrangians, including information from disc counts that would otherwise obstruct a clean Floer theory. This refinement is crucial for constructing categories that behave well under symplectic operations and under mirror symmetry.
Computations and examples
In concrete settings, bounding cochains can often be described explicitly for certain Lagrangians (for example, monotone or torus fibers in toric manifolds) and lead to computable invariants. In many classical examples, the trivial cochain b = 0 suffices (no obstruction), while in others one must select a nonzero b to obtain well-defined Floer cohomology. The space of such b's can itself have interesting geometry, sometimes forming a torus or more complicated moduli that reflect the ambient symplectic topology.
Foundations and debates
Foundational questions around bounding cochains center on issues of transversality and the analytic underpinnings of disc counts. Defining and stabilizing the counts of holomorphic discs in the presence of bubbling requires auxiliary technology, such as Kuranishi structures or polyfold theory, to produce robust, invariant structures. Different schools favor different technical frameworks, leading to ongoing discussions about the most transparent, flexible, and computable foundations. These debates are typical of a field that blends deep geometry with delicate analysis and aims to connect to physics-inspired predictions from mirror symmetry.
Perspectives and critiques
From a mathematical standpoint, bounding cochains are valued for making Floer theory workable in broad settings and for enabling a categorical approach that mirrors the structure seen in complex geometry. Critics, when present, often point to the technical overhead and the reliance on sophisticated perturbation schemes. Proponents argue that the payoff — a coherent, deformation-theoretic picture compatible with mirror symmetry and moduli problems — justifies the machinery. In the broader dialogue around open-string phenomena, the bounding cochain framework aligns with the idea that physical and geometric notions of stability are encoded through algebraic deformations.
Examples and applications
Exact Lagrangians in exact symplectic manifolds: the unobstructed case where the trivially zero bounding cochain b = 0 suffices, recovering the standard Floer cohomology.
Monotone or toric cases: bounding cochains parameterize deformations of branes whose associated disk potentials can be computed explicitly, yielding a landscape of objects that reflect mirror-symmetric data on the dual side.
General Lagrangians in Calabi–Yau or non-Calabi–Yau settings: the existence and selection of bounding cochains interact with the global geometry and with the open-closed string sector.
Relations to bulk deformations: in some frameworks, one may also deform the A-infinity structure using bulk parameters in addition to the boundary bounding cochains, enriching the deformation theory further.