Quadratic differentials of exponential type and stability

Transcription

1 Quadratic differentials of exponential type and stability Fabian Haiden (University of Vienna), joint with L. Katzarkov and M. Kontsevich January 28, 2013

2 Simplest Example Q orientation of A n Dynkin diagram, e.g. }{{} n vertices D b (A n ) bounded derived category of reps of Q, has ) indecomposable objects up to shift ( n+1 2

3 Classification of t-structures A D b (A n ) heart of a bounded t-structure, then A is artinian A has n simple objects These form a tree, embedded in the disc. We also need to record degrees of morphisms between them

4 Classification of Stability Conditions For C = D b (A n ): stability condition = t-structure + n numbers in H One can use results of a 1932 paper of R. Nevanlinna to prove: Stab(C)/Aut(C) = { } / e P (z) dz 2, deg P = n + 1 Aut(C) { = e zn+1 +a n 1 z n a 0 dz 2} / Z/(n + 1) (as sets, also as stacks for n > 1)

5 Bridgeland Stability Conditions Given a triangulated category T, homomorphism cl : K 0 (T ) Γ, Γ finitely generated, a stability condition is Z : Γ C, the central charge A T, the heart of a bounded t-structure satisfying Z is a stability function on A Harder-Narasimhan property support property

6 Bridgeland Stability Conditions Stability function: Z(E) H R <0 for 0 E A so φ(e) = Arg(Z(E)) (0, π] 0 E A is semistable (resp. stable) if 0 F E = φ(f ) φ(e) (resp. φ(f ) < φ(e)) H.-N. property: 0 E A = 0 = E 0 E 1... E n = E with F i = E i /E i 1 semistable, φ(f i 1 ) > φ(f i ), 0 < i n. Support property: C > 0 : E semistable = cl(e) C Z(E)

7 Space of Stability Conditions Stab(T ) set of stability conditions on T Facts: has structure of a complex manifold of dimension rk(γ) Aut(T ) acts holomorphically GL + (2, R) acts smoothly

8 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

9 Quadratic Differentials of Exponential Type Fix S smooth surface (compact, connected) M S set of marked points (non-empty, finite) Positive integer n(p) for each p M Smooth, non-vanishing section of (T M, J) 2 over S \ M for some complex structure J, up to homotopy Consider pairs (C, ϕ) C complex curve with underlying surface S ϕ Holomorphic section of (T C) 2 over C \ M, nonvanishing, such that near p M: ϕ = e z n(p) h(z)dz 2 in some coordinate z, h meromorphic.

10 Geometric Origin Given then C complex curve, M C marked points f meromorphic function on C, holomorphic away from M ( LG potential ) η meromorphic quadratic differential on C, without zeros/poles on C \ M ( CY structure ) e f η is a quadratic differential of exponential type. However, not all are obtained this way, but always locally of this form by definition.

11 Flat Geometry (C, M, ϕ) exponential type flat surface S sm : underlying surface C \ M metric tensor ϕ Incomplete as metric space! Completion S = S sm S sg has a i new points replacing p i.

12 Infinite Angle Singularity Extra points in completion are infinite-angle singularities of S. Local Model: universal cover of R 2 \ {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

13 Finite Geodesics Two types of (maximal) geodesics of finite length on flat surface: 1. Saddle connections endpoints in Ssg rigid 2. Closed loops 1-parameter families foliating cylinder

14 Horizontal Foliation Flat metric horizontal foliation In terms of quadratic differential ϕ: ϕ(v, v) R 0 critical leaves: converge towards S sg (C, M, ϕ) of exponential type, no finite leaves (generic), then critical leaves cut S into rectangular pieces: R (0, h) finite number R (0, ) infinite number

15 Horizontal Foliation

16 Ribbon Graphs Formal definition: triple (H, σ, ι) H set (maybe infinite) σ bijection on H ι involution on H Terminology: H E = H/ι V = H/σ H ι E \ H ι half-edges edges vertices open edges proper edges

17 Mutation There is a notion of left/right mutation along an edge of a ribbon graph Formal definition: Given a proper edge {h 1, h 2 } H, i.e. ι(h 1 ) = h 2, let T be the involution T = (h 1, σ(h 2 ))(h 2, σ(h 1 )) then the left-mutated graph is (H, T 1 σt, ι).

18 Mutation For trivalent ribbon graphs, this is essentially quiver mutation (for special quivers), where Graph Common generalization? On categorical level: Quiver proper edges vertices σ(h 1 ) = h 2 arrow from h 1 to h 2 F L E F Hom 1 (E, F ) E F [1] c.f. Kontsevich Soibelman categorification of cluster mutation

19 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=

20 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).

21 Reconstruction of (C, M, ϕ) Meromorphic Case Before dealing with exponential case, consider simpler correspondence: finite ribbon graph Γ meromorphic differential ϕ Graph Γ Differential ϕ vertex of val. 1 simple pole vertex of val. 2 regular point vertex of val. k 3 zero of order k 2 vertex of Γ pole of order 2 order of pole for vertex v of Γ is: 2 + # of open edges attached to v

22 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 1 + z ) n n n

23 Graded Linear Category for Ribbon Graph Γ = (H, σ, ι) ribbon graph of exp. type Define graded linear category C Γ over C: Objects: set of proper edges (H \ H ι )/ι Morphisms: basis of Hom k (E 1, E 2 ) given by {(h 1, h 2 ) h i E i, σ k (h 1 ) = h 2 } Composition: (h 2, h 3 ) (h 1, h 2 ) = (h 1, h 3 ), (ι(h 2 ), h 3 ) (h 1, h 2 ) = 0 C Γ = augmentation of C Γ (add identities)

24 Classification of Indecomposable Objects Consider (one-sided) twisted complexes over C Γ dg-category Tw(C Γ ), 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.

25 Bondarenko s Matrix Problem Input: X linearly ordered set with involution ι additive category B(X, ι) with Objects: sequence of vector spaces V x, x X, with V x = V ι(x), dim V x < and block matrix B End ( V x ) with B 2 = 0. Morphism from ((V x ) x X, B) to ((W x ) x X, C) is element ( T Hom Vx, ) W x, Ty x Hom(V x, W y ) such that 1. T B = CT 2. x > y implies T x y = 0 (T is lower triangular) 3. T x x = T ι(x) ι(x)

26 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(C Γ ) to that of B(X, ι), where X = (H \ H ι ) Z. Answer in terms of Strings: walks on the graph Γ, without U-turns Bands: closed walks on Γ, not powers, satisfying grading condition ( vanishing Maslov class) Then indecomposables = strings (bands Jordan blocks)

27 Phases Next step: Classification of stability conditions on Tw(C Γ ) Recall: Each stable object E has a phase Z(E) Z(E) S1 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)

28 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(C Γ ) phases are closed, or at least not dense. Then we have Artinian heart Finite number N of simple objects N = rk(k 0 (Tw(C Γ ))) = # of proper edges of Γ

29 Remaining Steps Tw(C Γ ) should depend only on topological data Genus of S Sequence of positive integers n(p) 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.

30 Related Work Gaiotto-Moore-Neitzke (2009) studied wall-crossing for meromorphic quadratic differentials with Simple zeros 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.

31 Possible Future Work Meromorphic differentials: Infinite area case (at least one pole of order > 1): Expect stability conditions correspond, up to tilting, to ribbon graphs. 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? Methods developed here no longer apply, new ideas are needed.

32 The End Thank you for your attention!