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. In particular the conjecture contains two difficulties: first it asks what a suitable stability condition might be, and secondly if one can prove stability of an isotopy class if and only if it contains a special Lagrangian representative. The Thomas–Yau conjecture was proposed by Richard Thomas and Shing-Tung Yau in 2001, 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. 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.
Statement 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.
Special Lagrangian submanifolds Consider a Calabi–Yau manifold ( X , ω , Ω ) {\displaystyle (X,\omega ,\Omega )} of complex dimension n {\displaystyle n} , which is in particular a real symplectic manifold of dimension 2 n {\displaystyle 2n} . Then a Lagrangian submanifold is a real n {\displaystyle n} -dimensional submanifold L ⊂ X {\displaystyle L\subset X} such that the symplectic form is identically zero when restricted to L {\displaystyle L} , that is ω | L = 0 {\displaystyle \left.\omega \right|_{L}=0} . The holomorphic volume form Ω ∈ Ω n , 0 ( X ) {\displaystyle \Omega \in \Omega ^{n,0}(X)} , when restricted to a Lagrangian submanifold, becomes a top degree differential form. If the Lagrangian is oriented, then there exists a volume form d V L {\displaystyle dV_{L}} on L {\displaystyle L} and one may compare this volume form to the restriction of the holomorphic volume form: Ω | L = f d V L {\displaystyle \left.\Omega \right|_{L}=fdV_{L}} for some complex-valued function f : L → C {\displaystyle f:L\to \mathbb {C} } . The condition that X {\displaystyle X} is a Calabi–Yau manifold implies that the function f {\displaystyle f} has norm 1, so we have f = e i Θ {\displaystyle f=e^{i\Theta }} where Θ : L → [ 0 , 2 π ) {\displaystyle \Theta :L\to [0,2\pi )} is the phase angle of the function f {\displaystyle f} . In principle this phase function is only locally continuous, and its value may jump. A graded Lagrangian is a Lagrangian together with a lifting ϑ : L → R {\displaystyle \vartheta :L\to \mathbb {R} } of the phase angle to R {\displaystyle \mathbb {R} } , which satisfies Θ = ϑ mod 2 π {\displaystyle \Theta =\vartheta \mod 2\pi } everywhere on L {\displaystyle L} .
An oriented, graded Lagrangian L {\displaystyle L} is said to be a special Lagrangian submanifold if the phase angle function ϑ {\displaystyle \vartheta } is constant on L {\displaystyle L} . The average value of this function, denoted θ {\displaystyle \theta } , may be computed using the volume form as
… excerpt ends here. Continue reading the full article.
