Central Manifolds
Central Manifolds Ludmil Katzarkov University of Miami
August 23, 2017
1 / 33
Navier-Stokes For V - velocity, P - pressure: 1 ∂V + (V · ∇)V + ∇P = ν∆V + f (x ) ∂t P
∇V =0 ⇓ L. D. Landau dA =aµ + b|A|2 + h.o.t. dt This produces traveling waves.
2 / 33
Denaturation of DNA double stranded
single stranded denaturation
T Temperature This is described by a one-dimensional lattice and the following lattice ODE. d 2 un + W 0 (un ) = V 0 (un+1 − un ) − V 0 (un − un−1 ) dt 2 W - on-site potential V - interaction potential 3 / 33
PDE Method of Central Manifolds ⇓
ODE 1 d 2y = V 0 (y (x + 1) − y (x )) − V 0 (y (x ) − y (x − 1)) J 2 dx 2 t x =n− 2 un (x ) = y
4 / 33
Recall the method of central manifolds. Central manifolds = of equilibrium points
(
) neither attraction of stable manifold orbits nor repulsion of unstable manifold
Central manifold is given by the linearization of | eigenvalues λi with Reλi = 0 or λi = 0 |. λi = 0 ⇔ slow manifold spanned by eigenvectors
5 / 33
long term dynamics
central manifold
t=∞
Recall If we have a dynamical system dx linearization dx = f (x ) −−−−−−−−→ = Ax dt dt A0 - eigenvectors with λ = 0 A0
tangent to slow manifold
6 / 33
Quiver representations
Q ... finite quiver (directed multigraph) representation of Q: vertex i 7→ vector space Vi arrow α : i → j 7→ linear map φα : Vi → Vj (all over C for now) metrized representation: Vi equipped with Hermitian metric hi
7 / 33
Flow on metrized representations
Potential: S(h) =
X
Tr(hi−1 φ∗α hj φα )
α:i→j
Metric, depending on parameter τi > 0, i ∈ Q0 : < v , w >h =
X
τi Tr(hi−1 vi hi−1 wi )
i∈Q0
Negative gradient flow of S: τi hi−1
X X dhi = hi−1 φ∗α hj φα − φα hj−1 φ∗α hi dt α:i→j α:i→j
8 / 33
Monotonicity
Crucial property of the flow:
Theorem (Monotonicity) If g, h are trajectories of the flow with g(0) 6 h(0), then g(t) 6 h(t),
t > 0.
Consequence: All solutions have same asymptotics up to bounded factors, i.e. log(g(t)/h(t)) = O(1), t → +∞
9 / 33
Asymptotic behavior
Representation Vi , φα is semisimple ⇔ flow converges to fixed point (“harmonic metric”) X α:i→j
hi−1 φ∗α hj φα −
X
φα hj−1 φ∗α hi = 0
α:i→j
Proof: Apply Kempf-Ness theorem to ΠGl(Vi )-action on φα ’s If representation is not semisimple: Metric h grows at different polynomial rates on various subspaces. Limiting filtration (monotonicity ⇒ does not depend on initial metric)
10 / 33
Uniformization and Central Manifolds “Hodge” theory of central manifolds → Vector or Higgs bundles V1 V2
F V3 V4
Example Solutions of
∂H = ΛF + x ∂t ⇓ center manifold ODE 11 / 33
Example (Yang-Mills equation) We start with: V0 V1 polystable
V2
..
.
Here Vi have the same slopes.
12 / 33
The solutions of the ODE equation are connected with the quiver V1 V2 V3 Ji hi−1
P −1 + dhi = hi ϕα hj ϕα − dt α:i→j
−
P α:i→j
ϕα hj−1 ϕ+ α hi
Here, hi are metrics on vector spaces.
13 / 33
V1
ϕα
V2
•
•
h1
h2
Mean curvature flow ∂t f = ρ(x , f )∂x f ⇓ f (x ) = a + b(x − xi ) + cϕ (x 2 + 1)ϕ = 1 ϕ(0) = ϕ0 (0) = 0 ⇓ •
•
xi
•
14 / 33
Solutions are filtered by the speed of approaching. ϕ(x ) =
π (x ) − log |x | 2
•
•
x1
x2
An category
15 / 33
We propose an approach to the theory of moduli spaces of Higgs Bundles using the theory of Central Manifolds. We consider a semistable Higgs bundle of the form: E1 φ 1 E2 φ 2 ..
.
E En φ n
16 / 33
We consider the spectral covering C1 ∪ C2 ∪ · · · ∪ Cn .
∇∗ C
C1 C2 Cn
• •
These are all Lagrangians. We consider the Lagrangians: Li = Ci ∇∗ C
and the mean curvature flow. As before, we have a flow on Lagrangians.
17 / 33
↓
necks Conjecture 1. The Yang-Mills Higgs flow on the semistable bundle E converges to solutions of ODE associated with the quiver. •
L1
•
L2
• •
Ln
18 / 33
The filtrations are given by the asymptotic behavior of the area of the necks. In the next conjecture we propose a connection of the technique of central manifolds with the theory of spectral networks. We consider a limit of Φ so that the spectral covering becomes several non-intersecting components.
We will put the additional ramification points using the Fredrickson equation.
t = time
R r•
R = surf r = radius 19 / 33
We have a space of fiducial solutions of ∂2 1 ∂ + 2 ∂r r ∂r
!
u = t 2 r sinh(ut ).
So we have with R → ∞ the following conjecture. Conjecture 2. We have the following correspondences. (
YMH metrics
)
(
)
Central manifolds of fiducial solutions
(
)
Spectral networks
20 / 33
Conjecture 1 Consider (E , Φ) on S and H ⊂ S a curve. Let (E , Φ)|H be the restriction of (E , Φ) on H. Then we have a map Central(E , Φ)|H → Central(E , Φ) compatible with the filtration.
21 / 33
Conjecture 2 Consider a family (Et , Φt ) with a monodromy •
Then we have weight filtration Wk on Central(E , Φ) compatible with the filtration on Central(E , Φ).
22 / 33
Let V be a semistable bundle w.r.t. L.
V1
..
. V2
Then P(0 + V1 ⊕ V2 ) has c.s.c. metric on components M1 ...... M2 We obtain a quiver.
23 / 33
Similarly we obtain a quiver for a toric variety. • •
...
• •
• •
•
···
•
We get a quiver of k-stable surface.
24 / 33
The same is true for CY. E ×E •
•
•
•
Thurston surface
Kronecker quiver
25 / 33
New Invariants
Ji hi−1
P −1 + dhi = hi ϕα hj ϕα − dt α:i→j
−
P α:i→j
•
•
F1 , F2
ϕα hj−1 ϕ+ α hi
•
•
•
Functors h stable
h stable
26 / 33
X1 #X2
Functors
(MX1 , MX2 )
(Q1 ) → C
Example Cn = {F : K (n) → C} β· F = F 0 · α α ∈ Aut(K (n)) B = Serre C
27 / 33
Example - Ck for P2 - n → 3(M), M = Markov number This formula can be changed so that they depend on the stability condition. Cnδ = {F : K (n) → C so that • → • 7→ semi-stable}
28 / 33
Example •
F : K (2)
C2δ
•
•
2 1
1 0
Wallcrossings
29 / 33
Example (A side case)
•
•
exceptional
spherical CY (n) 0
0
How to compute Cn = {F : CY (n) → Fuk}
30 / 33
Example (elliptic curve case) in how many ways we compute Cn
= gcd(k, n) = 1 T 2 with (1, 0) ∪ (k, n) primitive
Question Are Cn birational invariants?
Conjecture The specialization of families of derived categories of rational varieties is a non commutative deformation of a rational variety. 31 / 33
The End
32 / 33
August 23, 2017
1 / 33
Navier-Stokes For V - velocity, P - pressure: 1 ∂V + (V · ∇)V + ∇P = ν∆V + f (x ) ∂t P
∇V =0 ⇓ L. D. Landau dA =aµ + b|A|2 + h.o.t. dt This produces traveling waves.
2 / 33
Denaturation of DNA double stranded
single stranded denaturation
T Temperature This is described by a one-dimensional lattice and the following lattice ODE. d 2 un + W 0 (un ) = V 0 (un+1 − un ) − V 0 (un − un−1 ) dt 2 W - on-site potential V - interaction potential 3 / 33
PDE Method of Central Manifolds ⇓
ODE 1 d 2y = V 0 (y (x + 1) − y (x )) − V 0 (y (x ) − y (x − 1)) J 2 dx 2 t x =n− 2 un (x ) = y
4 / 33
Recall the method of central manifolds. Central manifolds = of equilibrium points
(
) neither attraction of stable manifold orbits nor repulsion of unstable manifold
Central manifold is given by the linearization of | eigenvalues λi with Reλi = 0 or λi = 0 |. λi = 0 ⇔ slow manifold spanned by eigenvectors
5 / 33
long term dynamics
central manifold
t=∞
Recall If we have a dynamical system dx linearization dx = f (x ) −−−−−−−−→ = Ax dt dt A0 - eigenvectors with λ = 0 A0
tangent to slow manifold
6 / 33
Quiver representations
Q ... finite quiver (directed multigraph) representation of Q: vertex i 7→ vector space Vi arrow α : i → j 7→ linear map φα : Vi → Vj (all over C for now) metrized representation: Vi equipped with Hermitian metric hi
7 / 33
Flow on metrized representations
Potential: S(h) =
X
Tr(hi−1 φ∗α hj φα )
α:i→j
Metric, depending on parameter τi > 0, i ∈ Q0 : < v , w >h =
X
τi Tr(hi−1 vi hi−1 wi )
i∈Q0
Negative gradient flow of S: τi hi−1
X X dhi = hi−1 φ∗α hj φα − φα hj−1 φ∗α hi dt α:i→j α:i→j
8 / 33
Monotonicity
Crucial property of the flow:
Theorem (Monotonicity) If g, h are trajectories of the flow with g(0) 6 h(0), then g(t) 6 h(t),
t > 0.
Consequence: All solutions have same asymptotics up to bounded factors, i.e. log(g(t)/h(t)) = O(1), t → +∞
9 / 33
Asymptotic behavior
Representation Vi , φα is semisimple ⇔ flow converges to fixed point (“harmonic metric”) X α:i→j
hi−1 φ∗α hj φα −
X
φα hj−1 φ∗α hi = 0
α:i→j
Proof: Apply Kempf-Ness theorem to ΠGl(Vi )-action on φα ’s If representation is not semisimple: Metric h grows at different polynomial rates on various subspaces. Limiting filtration (monotonicity ⇒ does not depend on initial metric)
10 / 33
Uniformization and Central Manifolds “Hodge” theory of central manifolds → Vector or Higgs bundles V1 V2
F V3 V4
Example Solutions of
∂H = ΛF + x ∂t ⇓ center manifold ODE 11 / 33
Example (Yang-Mills equation) We start with: V0 V1 polystable
V2
..
.
Here Vi have the same slopes.
12 / 33
The solutions of the ODE equation are connected with the quiver V1 V2 V3 Ji hi−1
P −1 + dhi = hi ϕα hj ϕα − dt α:i→j
−
P α:i→j
ϕα hj−1 ϕ+ α hi
Here, hi are metrics on vector spaces.
13 / 33
V1
ϕα
V2
•
•
h1
h2
Mean curvature flow ∂t f = ρ(x , f )∂x f ⇓ f (x ) = a + b(x − xi ) + cϕ (x 2 + 1)ϕ = 1 ϕ(0) = ϕ0 (0) = 0 ⇓ •
•
xi
•
14 / 33
Solutions are filtered by the speed of approaching. ϕ(x ) =
π (x ) − log |x | 2
•
•
x1
x2
An category
15 / 33
We propose an approach to the theory of moduli spaces of Higgs Bundles using the theory of Central Manifolds. We consider a semistable Higgs bundle of the form: E1 φ 1 E2 φ 2 ..
.
E En φ n
16 / 33
We consider the spectral covering C1 ∪ C2 ∪ · · · ∪ Cn .
∇∗ C
C1 C2 Cn
• •
These are all Lagrangians. We consider the Lagrangians: Li = Ci ∇∗ C
and the mean curvature flow. As before, we have a flow on Lagrangians.
17 / 33
↓
necks Conjecture 1. The Yang-Mills Higgs flow on the semistable bundle E converges to solutions of ODE associated with the quiver. •
L1
•
L2
• •
Ln
18 / 33
The filtrations are given by the asymptotic behavior of the area of the necks. In the next conjecture we propose a connection of the technique of central manifolds with the theory of spectral networks. We consider a limit of Φ so that the spectral covering becomes several non-intersecting components.
We will put the additional ramification points using the Fredrickson equation.
t = time
R r•
R = surf r = radius 19 / 33
We have a space of fiducial solutions of ∂2 1 ∂ + 2 ∂r r ∂r
!
u = t 2 r sinh(ut ).
So we have with R → ∞ the following conjecture. Conjecture 2. We have the following correspondences. (
YMH metrics
)
(
)
Central manifolds of fiducial solutions
(
)
Spectral networks
20 / 33
Conjecture 1 Consider (E , Φ) on S and H ⊂ S a curve. Let (E , Φ)|H be the restriction of (E , Φ) on H. Then we have a map Central(E , Φ)|H → Central(E , Φ) compatible with the filtration.
21 / 33
Conjecture 2 Consider a family (Et , Φt ) with a monodromy •
Then we have weight filtration Wk on Central(E , Φ) compatible with the filtration on Central(E , Φ).
22 / 33
Let V be a semistable bundle w.r.t. L.
V1
..
. V2
Then P(0 + V1 ⊕ V2 ) has c.s.c. metric on components M1 ...... M2 We obtain a quiver.
23 / 33
Similarly we obtain a quiver for a toric variety. • •
...
• •
• •
•
···
•
We get a quiver of k-stable surface.
24 / 33
The same is true for CY. E ×E •
•
•
•
Thurston surface
Kronecker quiver
25 / 33
New Invariants
Ji hi−1
P −1 + dhi = hi ϕα hj ϕα − dt α:i→j
−
P α:i→j
•
•
F1 , F2
ϕα hj−1 ϕ+ α hi
•
•
•
Functors h stable
h stable
26 / 33
X1 #X2
Functors
(MX1 , MX2 )
(Q1 ) → C
Example Cn = {F : K (n) → C} β· F = F 0 · α α ∈ Aut(K (n)) B = Serre C
27 / 33
Example - Ck for P2 - n → 3(M), M = Markov number This formula can be changed so that they depend on the stability condition. Cnδ = {F : K (n) → C so that • → • 7→ semi-stable}
28 / 33
Example •
F : K (2)
C2δ
•
•
2 1
1 0
Wallcrossings
29 / 33
Example (A side case)
•
•
exceptional
spherical CY (n) 0
0
How to compute Cn = {F : CY (n) → Fuk}
30 / 33
Example (elliptic curve case) in how many ways we compute Cn
= gcd(k, n) = 1 T 2 with (1, 0) ∪ (k, n) primitive
Question Are Cn birational invariants?
Conjecture The specialization of families of derived categories of rational varieties is a non commutative deformation of a rational variety. 31 / 33
The End
32 / 33
