Bipartite Graphs and Microlocal Geometry

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1 Harold Williams University of Texas at Austin joint work in progress with V. Shende, D. Treumann, and E. Zaslow

2 Legendrians and Lagrangians Let S be a surface, T S its cotangent bundle, and T S = T S/R + its cosphere bundle, thought of as the boundary of T S at infinity. Definition: (Generic) Legendrian Knots in T S A knot Λ T S is Legendrian if it is lifted from an oriented immersed curve S by leftward conormal directions: Λ

3 Legendrians and Lagrangians Let S be a surface, T S its cotangent bundle, and T S = T S/R + its cosphere bundle, thought of as the boundary of T S at infinity. Definition: (Generic) Legendrian Knots in T S A knot Λ T S is Legendrian if it is lifted from an oriented immersed curve S by leftward conormal directions: Λ T S has the Liouville form λ = p i dq i and the symplectic form ω = dλ. A surface L S is an exact Lagrangian if λ L is exact. L T S is Lagrangian = L T S is Legendrian

4 Bipartite Surface Graphs To a bipartite graph G S we associate an alternating strand diagram G (following a strand, one meets crossings of alternating orienations). G an reconstruct G from G : components of S G classified into white / black / alternating regions by boundary orientation.

5 Bipartite Surface Graphs To a bipartite graph G S we associate an alternating strand diagram G (following a strand, one meets crossings of alternating orienations). G G an reconstruct G from G : components of S G classified into white / black / alternating regions by boundary orientation.

6 The onjugate Surface, after Goncharov-Kenyon Associate to bipartite G S a surface L G, the conjugate surface, which 1 has a projection L G S with image the white and black regions of S G and which is 1-1 except above crossings. 2 has a canonical isomorphism H 1 (G; Z) = H 1 (L G ; Z).

7 The onjugate Surface, after Goncharov-Kenyon Associate to bipartite G S a surface L G, the conjugate surface, which 1 has a projection L G S with image the white and black regions of S G and which is 1-1 except above crossings. 2 has a canonical isomorphism H 1 (G; Z) = H 1 (L G ; Z). H 1 (L G ; Z) comes with a basis: boundaries of faces of G. Gives canonical coordinates on the algebraic torus H 1 (L G ; ): [H 1 (L G ; )] = [y ±1 1,..., y ±1 n ], where y i measures the holonomy around ith face of G. Taking intersection pairings in H 1 (L G ; Z) defines a quiver whose vertices are the faces of G.

8 Examples of Bipartite Surface Graphs S = D 2, G reduced : positroid varieties (Postnikov,...), soliton graphs (Kodama-L. Williams,...) S arbitrary, G contractible to punctures : cluster algebras from surfaces (Gekhtman-Shapiro-Vainshtein, Fock-Goncharov, Fomin-Shapiro-Thurston,...), (wild) SL n -character varieties (Fock-Goncharov, Goncharov,...) S = T 2, G minimal : toric integrable systems, pentagram maps (Goncharov-Kenyon, Fock-Marshakov, herkis-ward, Glick-Pylyavskyy, Musiker-Lai,...) All appear in diverse physical contexts: brane tilings, bipartite field theories, on-shell diagrams,... (Hanany-Kennaway, Feng-He-Kennaway-Vafa, Yamazaki, Franco-Eager-Schaeffer, Franco, Gaiotto-Moore-Neitzke, Arkani-Hamed-Bourjaily- achazo-goncharov-postnikov-trnka,...)

9 The Square Move and luster Transformations Given a quadrilateral face of G, we can produce a new graph G by performing a square move: G G square move

10 The Square Move and luster Transformations Given a quadrilateral face of G, we can produce a new graph G by performing a square move: G G square move General Pattern For special equivalence classes of graphs there is a space X (positroid variety, character variety, integrable system,...) which contains all H 1 (L G ; ) as open subvarieties: H 1 (L G ; ) X /y-type cluster transformation H 1 (L G ; ) X

11 The onjugate Lagrangian Let Λ G T S be the Legendrian lift of G. Proposition (Shende-Treumann-W.-Zaslow) The projection L G S factors through an exact Lagrangian embedding L G T S, unique up to Hamiltonian isotopy, such that L G T S = Λ G.

12 The onjugate Lagrangian Let Λ G T S be the Legendrian lift of G. Proposition (Shende-Treumann-W.-Zaslow) The projection L G S factors through an exact Lagrangian embedding L G T S, unique up to Hamiltonian isotopy, such that L G T S = Λ G. Not hard to prove, but a ton of information is contained in the embedding L G T S! In particular, it completely determines the general pattern above (the space X, the maps H 1 (L G ; ) X, and hence the cluster transformations H 1 (L G ; ) H 1 (L G ; )), and extends it to bipartite graphs that aren t special.

13 The Fukaya ategory Definition Sketch Given a Legendrian Λ T S, the Fukaya category Fuk Λ (T S) has objects exact Lagrangians L with -local systems and such that L T S Λ. Intersection points between Lagrangians define morphisms, modulo analytic relations.

14 The Fukaya ategory Definition Sketch Given a Legendrian Λ T S, the Fukaya category Fuk Λ (T S) has objects exact Lagrangians L with -local systems and such that L T S Λ. Intersection points between Lagrangians define morphisms, modulo analytic relations. Say an object L of Fuk Λ (T S) has microlocal rank one if its local system is rank one and its boundary is equal to Λ. Let M(Λ) be the moduli space of objects in Fuk Λ (T S) of microlocal rank one. Proposition (Shende-Treumann-W.-Zaslow) The canonical inclusion H 1 (L G ; ) M(Λ G ) is open.

15 Isotopy Invariance Shende-Treumann-Zaslow 14: View Fuk Λ (T S) and M(Λ) as Legendrian knot invariants (and study them microlocally). Let Λ t, t [0, 1] be a Legendrian isotopy. Then there is an associated isomorphism M(Λ 0 ) = M(Λ 1 ), compatible with composition.

16 Isotopy Invariance Shende-Treumann-Zaslow 14: View Fuk Λ (T S) and M(Λ) as Legendrian knot invariants (and study them microlocally). Let Λ t, t [0, 1] be a Legendrian isotopy. Then there is an associated isomorphism M(Λ 0 ) = M(Λ 1 ), compatible with composition. The square move defines an isotopy from Λ G to Λ G : Λ 0 = Λ G Λ 1/2 Λ 1 = Λ G The family L t of conjugate Lagrangians develops a singularity at t = 1/2 (L 1 obtained from L 0 by Lagrangian surgery).

17 luster Transformations from onjugate Lagrangians Theorem (Shende-Treumann-W.-Zaslow) Let G, G S be related by a square move. The isotopy isomorphism between M(Λ G ) and M(Λ G ) birationally identifies the open subsets H 1 (L G ; ) and H 1 (L G ; ) by an X /y-type cluster transformation in their face coordinates: X /y-type cluster transformation H 1 (L G ; ) H 1 (L G ; ) M(Λ G ) M(Λ G ) isotopy isomorphism onsistent with general picture, e.g. from Picard-Lefschetz theory, of how Lagrangian surgery affects local systems (Auroux, Seidel, Kontsevich-Soibelman,... ).

18 Microlocal Geometry, after Kashiwara-Schapira Definition: onstructible Sheaves A constructible sheaf on S with singular support on a Legendrian link Λ T S is: 1 A -local system on each component of S π(λ) 2 Possibly noninvertible leftward maps across strands of π(λ) such that S W E N is exact at each crossing. It has microlocal rank one if all maps in (2) are 1-1 with 1-dim l cokernel. W N S E 2 0 microlocal rk one Λ-constructible sheaf = pair of linearly independent vectors in 2

19 Microlocal Geometry, after Kashiwara-Schapira Let Sh Λ (S) be the category of constructible sheaves on S with singular support on Λ. Theorem (Nadler-Zaslow) There is a quasi-equivalence π S : Fuk Λ (T S) = Sh Λ (S), canonical up to a choice of spin structure on S. Rough Idea: for an object L of Fuk Λ (T S), the stalk of π S (L) at x S will be Hom Fuk (T x S, L). In particular, alternate description of M(Λ): the space of microlocal rank one sheaves with singular support on Λ. (Warning: We omit important homological aspects from the present discussion.)

20 Isotopy Isormphisms Revisited Now we can say what the canonical inclusions H 1 (L G ; ) M(Λ G ) have to do with the standard examples from earlier: 1 For special Legendrian knots Λ the space M(Λ), viewed as a space of sheaves, is manifestly isomorphic to some fundamental geometric object X (e.g. positroid variety, character variety,... ). 2 Special bipartite graphs G are those whose associated Legendrian Λ G is isotopic to a special Legendrian Λ, and using isotopy isomorphisms we then have H 1 (L G ; ) M(Λ G ) = M(Λ) = X.

21 Basic Example: X = Open Positroid ell of Gr 2,4 If S has boundary, we can consider a framed moduli space M fr (Λ) of microlocal rank one sheaves with a trivialization along the boundary. 0 0 isotopy 2 0 0

22 Basic Example: X = Open Positroid ell of Gr 2,4 If S has boundary, we can consider a framed moduli space M fr (Λ) of microlocal rank one sheaves with a trivialization along the boundary. 0 0 isotopy M fr (Λ) = { 4 cyclically-ordered vectors in 2 }, /GL neighbors linearly independent 2