Quantum cluster algebras from geometry

Transcription

1 Based on Chekhov-M.M. arxiv: and Chekhov-M.M.-Rubtsov arxiv:

2 Ptolemy Relation aa + bb = cc a b c c b a

3 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 1 = x 9x 7 + x 8 x 2 x 1 x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

4 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 1 = x 9x 7 + x 8 x 2 x 1 x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

5 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 1 = x 9x 7 + x 8 x 2 x 1 x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

6 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 1 = x 9x 7 + x 8 x 2 x 1 x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

7 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 1 = x 9x 7 + x 8 x 2 x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

8 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

9 Ptolemy Relation (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 9 x 1 x 2 x 2 x 8 x 4 x 7 x 5 x 3 x 6

10 Cluster algebra We call a set of n numbers (x 1,..., x n ) a cluster of rank n. A seed consists of a cluster and an exchange matrix B, i.e. a skew symmetrisable matrix with integer entries. A mutation is a transformation µ i : (x 1, x 2,..., x n ) (x 1, x 2,..., x n), µ i : B B where x i x i = j:b ij >0 x b ij j + j:b ij <0 x b ij j, x j = x j j i. Definition A cluster algebra of rank n is a set of all seeds (x 1,..., x n, B) related to one another by sequences of mutations µ 1,..., µ k. The cluster variables x 1,..., x k are called exchangeable, while x k+1,..., x n are called frozen. [Fomin-Zelevnsky 2002].

11 Example Cluster algebra of rank 9 with 3 exchangeable variables x 1, x 2, x 3 and 6 frozen ones x 4,..., x 9. (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) (x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9 ) x 1 x 1 = x 9x 7 + x 8 x 2 x 1 x 1 x 2 x 9 x 8 x 4 x 7 x 5 x 3 x 6

12 Outline Are all cluster algebras of geometric origin? Introduce bordered cusps Geodesics length functions on a Riemann surface with bordered cusps form a cluster algebra. All Berenstein-Zelevinsky cluster algebras are geometric

13 Teichmüller space For Riemann surfaces with holes: Idea: Hom (π 1 (Σ), PSL 2 (R)) /GL 2 (R). Teichmüller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Develop bordered cusped Teichmüller theory asymptotically. This will provide cluster algebra of geometric type

14 Poincaré uniformsation Σ = H/, where is a Fuchsian group, i.e. a discrete sub-group of PSL 2 (R). Examples γ 2 γ 1

15 Poincaré uniformsation Σ = H/, where is a Fuchsian group, i.e. a discrete sub-group of PSL 2 (R). Examples γ 2 γ 1 Theorem Elements in π 1 (Σ g,s ) are in 1-1 correspondence with conjugacy classes of closed geodesics.

16 Coordinates: geodesic lengths Theorem The geodesic length functions form an algebra with multiplication: G γ G γ = G γ γ + G γ γ 1. γ γ

17 Coordinates: geodesic lengths Theorem The geodesic length functions form an algebra with multiplication: G γ G γ = G γ γ + G γ γ 1. = γ γ

18 Coordinates: geodesic lengths Theorem The geodesic length functions form an algebra with multiplication: G γ G γ = G γ γ + G γ γ 1. = + γ γ γ γ 1 γ γ

19 Poisson structure {G γ, G γ } = 1 2 G γ γ 1 2 G γ γ 1. { } = γ γ γ 1 γ γ γ

20 Two types of chewing-gum moves Connected result: Disconnected result:

21 Chewing gum 1 + ε 1 z 3 z 2 z 1 ( sinh d H(z 1,z 2 ) 2 εl 1 εl 2 ) 2 = z 1 z 2 2 4Iz 1 Iz 2 e d H(z 1,z 2 ) 1 l 1 l 2 + (l 1+l 2 ) 2 ϵ 2 l 1 l 2 + O(ϵ), e d H(z 1,z 3 ) e d H(z 1,z 2 ) + 1 l 1 l 2 + O(ϵ). Rescale all geodesic lengths by e ϵ and take the limit ϵ 0. [Chekhov-M.M. arxiv: ]

22 = skein γa + + the rest γ e γ f γ c γ b γ d γ b γ a γ f γ e G γe G γf = G γa G γc + G γb G γd γ d γ c

23 = skein γa + + the rest γ e γ f γ c γ b γ d γ b γ a γ f γ e G γe G γf = G γa G γc + G γb G γd γ d γ c

24 Poisson bracket Introduce cusped laminations Compute Poisson brackets between arcs in the cusped lamination. Theorem Given a Riemann surface of any genus, any number of holes and at least one cusp on a boundary, there always exists a complete cusped lamination [Chekhov-M.M. ArXiv: ].

25 Cluster algebras Teichmu ller Theory Geodesic lengths Quatisation Decorated character variety Poisson structure Theorem The Poisson algebra of the λ-lengths of a complete cusped lamination is a cluster algebra [Chekhov-M.M. ArXiv: ]. {gsi,tj, gpr,ql } = gsi,tj gpr,ql Isi,tj,pr,ql Isi,tj,pr,ql = ϵi r δs,p +ϵj r δt,p +ϵi l δs,q +ϵj l δt,q 4

26 Quantisation For standard geodesic lengths G γ G ħ γ [Chekhov-Fock 99]: [ ] = q q 1 2 G ħ γ G ħ γ G ħ γ 1 γ G ħ γ γ For arcs g si,t j g ħ s i,t j : [G ħ γ, G ħ γ ] = q 1 2 G ħ γ 1 γ + q 1 2 G ħ γ γ q Is i,t j,p r,q l g ħ s i,t j g ħ p r,q l = g ħ p r,q l g ħ s i,t j q Ip r,q l,s i,t j This identifies the geometric basis of the quantum cluster algebras introduced by Berenstein and Zelevinsky.

27 Decorated character variety What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom (π 1 (Σ), PSL 2 (C)) /GL 2 (C). For Riemann surfaces with bordered cusps: Decorated character variety [Chekhov-M.M.-Rubtsov arxiv: ] Replace π 1 (Σ) with the groupoid of all paths γ ij from cusp i to cusp j modulo homotopy. Replace tr by two characters: tr and tr K.

28 Shear coordinates in the Teichmüller space LEONID CHEKHOV, MARTA MAZZOCCO, VLADIMIR RUBTSOV Fatgraph: p 1 s 1 s 3 p 3 s 2 p 2 Decompose each hyperbolic element in Right, Left and Edge Figure 4. The fat graph of the 4 holed Riemann sphere. The red dashed geodesic is x matrices 1. Fock, Thurston ( 1 1 ) ( 0 1 tained by decomposing each hyperbolic matrix γ g,s into a product of called right, left and edge matrices: ( ) 1 1 R := ( ) 0 1, L ( := 0 exp ( ) ) si, L :=, X si := 2 1 exp ( ). 1 si s setting our x 1,x 2,x( 3 are the geodesic lengths of thee geodesics which go d two holes without self intersections. 0 exp ( y ) ) are now going X to produce a similar shear coordinate description of each of the Painlevé cubics. y := 2 We will provide exp ( a geometric y ). description of the corresponding nn surface and its fat-graph. Our geometric 2 0 description agrees with the one ed in [32], which was obtained by building a Strebel differential from the nodromic problems associated to each of the Painlevé equations. ),

29 LEONID CHEKHOV, MARTA MAZZOCCO, VLADIMIR RUBTSOV p 1 s 1 s 3 p 3 s 2 p 2 The three geodesic lengths: x i = Tr(γ jk ) Figure 4. The fat graph of the 4 holed Riemann sphere. The red dashed geodesic is x 1. x 1 = e s 2+s 3 +e s 2 s 3 +e s 2+s 3 +(e p 2 2 +e p 2 2 )e s 3 +(e p 3 2 +e p 3 2 )e s 2 x 2 = e s 3+s 1 +e s 3 s 1 +e s 3+s 1 +(e p 3 2 +e p 3 2 )e s 1 +(e p 1 2 +e p 1 2 )e s 3 x 3 = e s 1+s 2 +e s 1 s 2 +e s 1+s 2 +(e p 1 2 +e p 1 2 )e s 2 +(e p 2 2 +e p 2 2 )e s 1 tained by decomposing each hyperbolic matrix γ g,s into a product of called right, left and edge matrices: ( ) ( ) ( exp ( ) ) si, L :=, X si := 2 1 exp ( ). si 2 0 s setting our x 1,x 2,x 3 are the geodesic lengths of thee geodesics which go d two holes without self intersections. are now going to produce a similar shear coordinate description of each of the Painlevé cubics. We will provide a geometric description of the corresponding nn surface and its fat-graph. Our geometric description agrees with the one ed in [32], which was obtained by building a Strebel differential from the nodromic problems associated to each of the Painlevé equations. {x 1, x 2 } = 2x 3 + ω 3, {x 2, x 3 } = 2x 1 + ω 1, {x 3, x 1 } = 2x 2 + ω 2. hear coordinates for PV. The confluence from the cubic associated to o the one associated to PV is realised by p 3 p 3 2 log[ϵ], limit ϵ 0. We obtain the following shear coordinate description for Quantum the cluster algebras from geometry

30 Cluster algebras Teichmu ller Theory Geodesic lengths Quatisation Decorated character variety PV γb = X (k1 )RX (s3 )RX (s2 )RX (p2 )RX (s2 )LX (s3( )LX (k1 ) ) 0 0 BUT its length is b = trk (γb ) = tr(bk ), K = 1 0 PV flipped:

31 Cluster algebras Teichmu ller Theory Geodesic lengths Quatisation Decorated character variety PV {gsi,tj, gpr,ql } = gsi,tj gpr,ql ϵi r δs,p +ϵj r δt,p +ϵi l δs,q +ϵj l δt,q 4 {b, d} = PV{g 13,14, g21,18 } flipped: ϵ3 1 δ1,2 + ϵ4 1 δ1,2 + ϵ3 8 δ1,1 + ϵ4 8 δ1,1 = g13,14 g21, = bd 2

32 Mutations Example 28 LEONID CHEKHOV, MARTA MAZZOCCO, VLADIMIR RUBTSOV any other branch of that solution will corresponds to points (y1,y2,y3) on the same cubic such that each yi is a Laurent polynomial of the initial (y 0 1,y 0 2,y 0 3) Generalised cluster algebra structure for PV and PVdeg. In this case, we have a Riemann surface Σ0,3,2 with two bordered cusps on one hole. The only nontrivial Dehn twist is around the closed geodesic γ encircling these two holes (this Riemann sphere with three holes, and two cusps on one of the geodesic is unique). Its geodesic function Gγ is the Hamiltonian MCG invariant. We now consider the effect of this MCG transformation on the system of arcs in holes. Frozen variables: c, d, e. Exchangeable variables: a, b. Fig. 6. c c c e d Ma e d Mb e d ω2 ω1 ω2 a ω1 ω2 a ω1 b γ a b b (5.61) Gγ = c ω2 b + c ω1 a + a b + b a + c2 ab and this function Marta is the Mazzocco (Hamiltonian) MCG Quantum invariant: cluster it generates algebras the corre- from geometry Here the generalized mutations Ma and Mb are given by the formulas a = g 15,1 a a = b 2 + c 2 + ω1bc; b b =(a ) 2 + c 2 + ω2a c, 6, b = g 13,1 4, d = g 18,2 2, {a, b} = ab, {a, d} = ad 2. or, explicitly, [ ] b 2 + c 2 + ω1bc a (5.60) a b (b 2 + c 2 + ω1bc) 2 a 2 + ω2c b2 + c 2 + ω1bc. Chekhov-M.M.-Rubtsov arxiv: : + c2 b ba b The geodesic function of γ is Sub-algebra of functions that commute with the frozen variables {x 1, x 2 } = 2x 3 + ω 3, {x 2, x 3 } = 2x 1 + ω 1, {x 3, x 1 } = 2x 2 + ω 2.

33 Conclusion A Riemann surface of genus g, n holes and k cusps on the boundary admits a complete cusped lamination of 6g 6 + 2n + 2k arcs which triangulate it. Any other cusped lamination is obtained by the cluster algebra mutations. By quantisation: quantum cluster algebra of geometric type. New notion of decorated character variety Many thanks for your attention!!!