In mathematics, and especially symplectic geometry, the Thomas–Yau conjecture asks for the existence of a stability condition, similar to those which appear in algebraic geometry, which guarantees the existence of a solution to the special Lagrangian equation inside a Hamiltonian isotopy class of Lagrangian submanifolds.
The Thomas–Yau conjecture was proposed by Richard Thomas and Shing-Tung Yau in 2001,[1][2] and was motivated by similar theorems in algebraic geometry relating existence of solutions to geometric partial differential equations and stability conditions, especially the Kobayashi–Hitchin correspondence relating slope stable vector bundles to Hermitian Yang–Mills metrics.
The conjecture is intimately related to mirror symmetry, a conjecture in string theory and mathematical physics which predicts that mirror to a symplectic manifold (which is a Calabi–Yau manifold) there should be another Calabi–Yau manifold for which the symplectic structure is interchanged with the complex structure.
[3] In particular mirror symmetry predicts that special Lagrangians, which are the Type IIA string theory model of BPS D-branes, should be interchanged with the same structures in the Type IIB model, which are given either by stable vector bundles or vector bundles admitting Hermitian Yang–Mills or possibly deformed Hermitian Yang–Mills metrics.
Motivated by this, Dominic Joyce rephrased the Thomas–Yau conjecture in 2014, predicting that the stability condition may be understood using the theory of Bridgeland stability conditions defined on the Fukaya category of the Calabi–Yau manifold, which is a triangulated category appearing in Kontsevich's homological mirror symmetry conjecture.
[4] The statement of the Thomas–Yau conjecture is not completely precise, as the particular stability condition is not yet known.
In the work of Thomas and Thomas–Yau, the stability condition was given in terms of the Lagrangian mean curvature flow inside the Hamiltonian isotopy class of the Lagrangian, but Joyce's reinterpretation of the conjecture predicts that this stability condition can be given a categorical or algebraic form in terms of Bridgeland stability conditions.
, when restricted to a Lagrangian submanifold, becomes a top degree differential form.
In principle this phase function is only locally continuous, and its value may jump.
is said to be a special Lagrangian submanifold if the phase angle function
is constant may be written in the following form, which commonly occurs in the literature.
An isotopy is a transformation of a submanifold inside an ambient manifold which is a homotopy by embeddings.
, the mean curvature flow is a differential equation satisfied for a one-parameter family
This is therefore called the Lagrangian mean curvature flow (Lmcf).
Thomas introduced a conjectural stability condition[1] defined in terms of gradings when splitting into Lagrangian connected sums.
is called stable if whenever it may be written as a graded Lagrangian connected sum
In the later language of Joyce using the notion of a Bridgeland stability condition, this was further explained as follows.
An almost-calibrated Lagrangian (which means the lifted phase is taken to lie in the interval
or some integer shift of this interval) which splits as a graded connected sum of almost-calibrated Lagrangians corresponds to a distinguished triangle
is stable if for any such distinguished triangle, the above angle inequality holds.
admits a special Lagrangian representative in its Hamiltonian isotopy class
is stable, then the Lagrangian mean curvature flow exists for all time and converges to a special Lagrangian representative in the Hamiltonian isotopy class
.This conjecture was enhanced by Joyce, who provided a more subtle analysis of what behaviour is expected of the Lagrangian mean curvature flow.
In particular Joyce described the types of finite-time singularity formation which are expected to occur in the Lagrangian mean curvature flow, and proposed expanding the class of Lagrangians studied to include singular or immersed Lagrangian submanifolds, which should appear in the full Fukaya category of the Calabi–Yau.
given by the convergence of the Lagrangian mean curvature flow with surgeries to remove singularities at a sequence of finite times
At these surgery points, the Lagrangian may change its Hamiltonian isotopy class but preserves its class in the Fukaya category.In the language of Joyce's formulation of the conjecture, the decomposition
is a symplectic analogue of the Harder-Narasimhan filtration of a vector bundle, and using Joyce's interpretation of the conjecture in the Fukaya category with respect to a Bridgeland stability condition, the central charge is given by
of the t-structure defining the stability condition is conjectured to be given by those Lagrangians in the Fukaya category with phase
, and the Thomas–Yau–Joyce conjecture predicts that the Lagrangian mean curvature flow produces the Harder–Narasimhan filtration condition which is required to prove that the data
defines a genuine Bridgeland stability condition on the Fukaya category.