Quadratic differentials of exponential type
and stability
Fabian Haiden (University of Vienna),
joint with L. Katzarkov and M. Kontsevich
January 28, 2013
Simplest Example
Q — orientation of An Dynkin diagram, e.g.
•
| ←− • ←−
{z · · · ←− •}
n vertices
Db (An ) —
bounded derived category of reps of Q, has
n+1
indecomposable objects up to shift
2
Classification of t-Structures
A ⊂ Db (An ) heart of a bounded t-structure, then
I
A is artinian
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A has n simple objects
These form a tree, embedded in the disc. We also need to record
degrees of morphisms between them.
6
1
2
4
7
1
Classification of Stability Conditions
For C = Db (An ):
stability condition = t-structure + n numbers in H
One can use results of a 1932 paper of R. Nevanlinna to prove:
n
o.
n
o.
∼ eP (z) dz 2 , deg P = n + 1
Stab(C)/Aut(C) =
∼ ez n+1 +an−1 z n−1 +...+a0 dz 2
=
(as sets, also as stacks for n > 1)
Aut(C)
Z/(n + 1)
Bridgeland Stability Conditions
Given a triangulated category T , homomorphism cl : K0 (T ) → Γ,
Γ finitely generated, a stability condition is
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Z : Γ → C, the central charge
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A ⊂ T , the heart of a bounded t-structure
satisfying
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Z is a stability function on A
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Harder-Narasimhan property
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support property
Bridgeland Stability Conditions
Stability function: Z(E) ∈ H ∪ R<0 for 0 6= E ∈ A
so φ(E) = Arg(Z(E)) ∈ (0, π]
0 6= E ⊂ A is semistable (resp. stable) if
0 ( F ( E =⇒ φ(F ) ≤ φ(E)
(resp. φ(F ) < φ(E))
H.-N. property:
0 6= E ⊂ A =⇒ ∃ 0 = E0 ⊂ E1 ⊂ . . . ⊂ En = E
with Fi = Ei /Ei−1 semistable, φ(Fi−1 ) > φ(Fi ), 0 < i ≤ n.
Support property:
∃C > 0 :
E semistable =⇒ kcl(E)k ≤ C|Z(E)|
Space of Stability Conditions
Stab(T ) — set of stability conditions on T
Facts:
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has structure of a complex manifold of dimension rk(Γ)
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Aut(T ) acts holomorphically
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+
GL^
(2, R) acts smoothly
Dimension One
Consider:
T — Z-graded Fukaya-type category of a Riemann surface
Expectation: Stab(T )/Aut(T ) is related to a space of quadratic
differentials ϕ with prescribed critical points, equivalently flat
surfaces with prescribed singularities, up to equivalence, such that
central charge ←→ contour integrals over
√
ϕ
stable objects ←→ finite length geodesics of |ϕ|
Quadratic Differentials of Exponential Type
Fix
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S — smooth surface (compact, connected)
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M ⊂ S — set of marked points (non-empty, finite)
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Positive integer n(p) for each p ∈ M
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Smooth, non-vanishing section of (T ∗ M, J)⊗2 over S \ M for
some complex structure J, up to homotopy
Consider pairs (C, ϕ)
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C — complex curve with underlying surface S
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ϕ — Holomorphic section of (T ∗ C)⊗2 over C \ M ,
nonvanishing, such that near p ∈ M :
ϕ = ez
−n(p)
h(z)dz 2
in some coordinate z, h meromorphic.
Geometric Origin
Given
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C — complex curve, M ⊂ C marked points
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f — meromorphic function on C, holomorphic away from M
(“LG potential”)
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η — meromorphic quadratic differential on C, without
zeros/poles on C \ M (“CY structure”)
then
ef η
is a quadratic differential of exponential type.
However, not all are obtained this way, but always locally of this
form by definition.
Flat Geometry
(C, M, ϕ) exponential type −→ flat surface Ssm :
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underlying surface C \ M
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metric tensor |ϕ|
Incomplete as metric space!
Completion S = Ssm ∪ Ssg has ai new points replacing pi .
Infinite Angle Singularity
Extra points in completion are infinite-angle singularities of S.
Local Model: universal cover of R2 \ {0} with additional point S
over the origin
Note: R-torsor of geodesics starting at S, geodesics can meet at S
in any angle ∈ R
Finite Geodesics
Two types of (maximal) geodesics of finite length on flat surface:
1. Saddle connections
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endpoints in Ssg
rigid
2. Closed loops
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1-parameter families foliating cylinder
Horizontal Foliation
Flat metric −→ horizontal foliation
In terms of quadratic differential ϕ:
ϕ(v, v) ∈ R≥0
critical leaves: converge towards Ssg
(C, M, ϕ) of exponential type, no finite leaves (generic), then
critical leaves cut S into rectangular pieces:
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R × (0, h) — finite number
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R × (0, ∞) — infinite number
Horizontal Foliation
Ribbon Graphs
Formal definition: triple (H, σ, ι)
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H — set (maybe infinite)
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σ — bijection on H
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ι — involution on H
Terminology:
H
E = H/ι
V = H/σ
Hι
E \ Hι
half-edges
edges
vertices
open edges
proper edges
Mutation
There is a notion of left/right mutation along an edge of a ribbon
graph.
1
1
1
1
Formal definition: Given a proper edge {h1 , h2 } ⊂ H, i.e.
ι(h1 ) = h2 , let T be the involution
T = (h1 , σ(h2 ))(h2 , σ(h1 ))
then the left-mutated graph is (H, T −1 σT, ι).
Mutation
For trivalent ribbon graphs, this is essentially quiver mutation (for
special quivers), where
Graph
proper edges
σ(h1 ) = h2
Quiver
vertices
arrow from h¯1 to h¯2
Common generalization?
On categorical level:
F −→ LE F −→ Hom1 (E, F ) ⊗ E −→ F [1]
c.f. Kontsevich–Soibelman categorification of cluster mutation
Ribbon Graphs of Exponential Type
Dual to Γ = (H, σ, ι) is Γ∗ = (H, σ ◦ ι, ι)
Valency of vertex v ∈ H/σ = cardinality of v as σ-orbit, so
val(v) ∈ {1, 2, . . . , ∞}
Γ = (H, σ, ι) is of exponential type if
1. finite set of vertices
2. finite set of proper edges
3. all vertices have valency=∞
4. all vertices of dual graph have valency=∞
Correspondence Between Flat Surface and Ribbon Graph
Consider ribbon graphs with “central charge”
Z(E) ∈ H,
E a proper edge of Γ
Then:
Surface
singular points
pieces R × (0, h)
vector between singular points
pieces R × (0, ∞)
gluing along critical leaves
Graph
vertices
proper edges
Z(E)
open edges
σ
Γ can be embedded in S as a deformation retract (follow leaves).
Reconstruction of (C, M, ϕ) — Meromorphic Case
Before dealing with exponential case, consider simpler
correspondence:
finite ribbon graph Γ ←→ meromorphic differential ϕ
Graph Γ
vertex of val. 1
vertex of val. 2
vertex of val. k ≥ 3
vertex of Γ∗
Differential ϕ
simple pole
regular point
zero of order k − 2
pole of order ≥ 2
order of pole for vertex v of Γ∗ is:
2 + # of open edges attached to v
Reconstruction of (C, M, ϕ) — Exponential Case
Idea: ribbon graph of exp. type Γ is limit of finite ribbon graphs
with increasing valency
−→ construct quadratic differential of exp. type as corresponding
limit of meromorphic quadratic differentials
This is a “glued” version of Euler’s approximation
exp(z) = lim
n→∞
1+
z
n
n
Graded Linear Category for Ribbon Graph
Γ = (H, σ, ι) — ribbon graph of exp. type
Define graded linear category C Γ over C:
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Objects: set of proper edges (H \ H ι )/ι
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Morphisms: basis of Homk (E1 , E2 ) given by
{(h1 , h2 ) | hi ∈ Ei , σ k (h1 ) = h2 }
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Composition:
(h2 , h3 ) ◦ (h1 , h2 ) = (h1 , h3 ),
(ι(h2 ), h3 ) ◦ (h1 , h2 ) = 0
CΓ = augmentation of C Γ (add identities)
Classification of Indecomposable Objects
Consider (one-sided) twisted complexes over CΓ
−→ dg-category Tw(CΓ ), its homotopy category, T , is triangulated
Intuition from topology: indecomposable objects of T should
correspond to certain paths on Γ.
Consequence: T has tame representation type, i.e.
indecomposable objects form at most 1-dimensional families.
Fukaya category proof?
We take a more algebraic approach: matrix problems.
Bondarenko’s Matrix Problem
Input: X — linearly ordered set with involution ι
−→ additive category B(X, ι) with
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Objects: sequence of vector spaces Vx , x ∈ X, with
Vx = Vι(x) ,
dim
and “block matrix” B ∈ End (
L
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M
Vx < ∞
Vx ) with B 2 = 0.
Morphism from ((Vx )x∈X , B) to ((Wx )x∈X , C) is element
T ∈ Hom
M
Vx ,
M
Wx ,
Tyx ∈ Hom(Vx , Wy )
such that
1. T B = CT
2. x > y implies Tyx = 0 (T is lower triangular)
ι(x)
3. Txx = Tι(x)
Solution to Classification Problem
Bondarenko classifies objects in B(X, ι) (in terms of
k[x, x−1 ]-modules) in a 1975 paper, based on methods of
Nazarova-Roiter.
We can reduce classification of objects in Tw(CΓ ) to that of
B(X, ι), where X = (H \ H ι ) × Z.
Answer in terms of
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Strings: walks on the graph Γ, without U-turns
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Bands: closed walks on Γ, not powers, satisfying grading
condition (↔ vanishing Maslov class)
Then
indecomposables = strings t (bands × Jordan blocks)
Phases
Next step: Classification of stability conditions on Tw(CΓ )
Recall: Each stable object E has a phase
Z(E)
∈ S1
|Z(E)|
Fact: Given a stability condition on any category,
phases of stable objects not dense in S 1
=⇒ heart of t-structure is artinian (after tilting)
Closedness of Phases
For stability conditions from quadratic differentials of exp. type,
the set of phases of all stable objects is closed in S 1
In geometric terms: Slopes of finite geodesics form closed set
Need to show: For arbitrary stability condition on Tw(CΓ ) phases
are closed, or at least not dense. Then we have
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Artinian heart
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Finite number N of simple objects
N = rk(K0 (Tw(CΓ ))) = # of proper edges of Γ
Remaining Steps
Tw(CΓ ) should depend only on topological data
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Genus of S
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Sequence of positive integers n(p)
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Maslov map, up to homotopy (element of
H 1 (S \ M, Z)-torsor)
The most basic version of the result would identify stability
conditions with quadratic differentials, both up to equivalence.
Would also like to understand autoequivalences of the category,
and their relation to the mapping class group of the surface with
marked points.
Related Work
Gaiotto-Moore-Neitzke (2009) studied wall-crossing for
meromorphic quadratic differentials with
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Simple zeros
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Poles of order ≥ 2, at least one
These correspond to trivalent ribbon graphs, which are in turn dual
to ideal triangulations.
Bridgeland-Smith announced results relating this to CY-3
categories and stability conditions.
Possible Future Work
Meromorphic differentials:
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Infinite area case (at least one pole of order > 1): Expect
stability conditions correspond, up to tilting, to ribbon graphs.
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Finite area case (no poles of order > 1): Qualitatively
different, no longer expect heart of t-structure to be artinian.
Simplest example: elliptic curve with constant differential
Higher dimensions?
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Methods developed here no longer apply, new ideas are
needed.
The End
Thank you for your attention!