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,[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.
Consider a Calabi–Yau manifold
(X,\omega,\Omega)
n
2n
n
L\subsetX
L
\left.\omega\right|L=0
\Omega\in\Omegan,0(X)
dVL
L
\left.\Omega\right|L=fdVL
f:L\toC
X
f
f=ei\Theta
\Theta:L\to[0,2\pi)
f
\vartheta:L\toR
R
\Theta=\vartheta\mod2\pi
L
An oriented, graded Lagrangian
L
\vartheta
L
\theta
\theta=\arg\intL\Omega,
and only depends on the Hamiltonian isotopy class of
L
\Theta
Im(e-i\theta\left.\Omega\right|L)=0.
The condition of being a special Lagrangian is not satisfied for all Lagrangians, but the geometric and especially physical properties of Lagrangian submanifolds in string theory are predicted to only depend on the Hamiltonian isotopy class of the Lagrangian submanifold. An isotopy is a transformation of a submanifold inside an ambient manifold which is a homotopy by embeddings. On a symplectic manifold, a symplectic isotopy requires that these embeddings are by symplectomorphisms, and a Hamiltonian isotopy is a symplectic isotopy for which the symplectomorphisms are generated by Hamiltonian functions. Given a Lagrangian submanifold
L
L
[L]
L
M
\iota:N\hookrightarrowM
\iotat
t
[0,T)
Nt
N0=N
where
d\iotat dt =
H \iotat
H | |
\iotat |
Nt\subsetM
M
N
On a Calabi–Yau manifold, if
L
L
(L,\vartheta)
Lt\in[L]
t\in[0,T)
Thomas introduced a conjectural stability condition defined in terms of gradings when splitting into Lagrangian connected sums. Namely a graded Lagrangian
(L,\vartheta)
the average phases satisfy the inequality(L,\vartheta)=(L1,\vartheta1)\#(L2,\vartheta2)
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\theta1<\theta2.
(-\pi/2,\pi/2)
in the Fukaya category. The LagrangianL1\toL1\#L2\toL2\toL1[1]
(L,\vartheta)
The conjecture as originally proposed by Thomas is as follows:
Conjecture: An oriented, graded, almost-calibrated LagrangianFollowing this, in the work of Thomas–Yau, the behaviour of the Lagrangian mean curvature flow was also predicted.admits a special Lagrangian representative in its Hamiltonian isotopy classL
if and only if it is stable in the above sense.[L]
Conjecture (Thomas–Yau): If an oriented, graded, almost-calibrated LagrangianThis 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.is stable, then the Lagrangian mean curvature flow exists for all time and converges to a special Lagrangian representative in the Hamiltonian isotopy classL
.[L]
Conjecture (Thomas–Yau–Joyce): An oriented, graded, almost-calibrated LagrangianIn the language of Joyce's formulation of the conjecture, the decompositionsplits as a graded Lagrangian connected sumL
of special Lagrangian submanifoldsL=L1\# … \#Lk
with phase anglesLi
given by the convergence of the Lagrangian mean curvature flow with surgeries to remove singularities at a sequence of finite times\theta1> … >\thetak
. At these surgery points, the Lagrangian may change its Hamiltonian isotopy class but preserves its class in the Fukaya category.0<T1< … <Tk
L=L1\# … \#Lk
the heart,Z(L)=\intL\Omega
l{A}
\theta\in(-\pi/2,\pi/2)
(Z,l{A})