Quantum cluster algebras from geometry Marta Mazzocco Based on - PowerPoint PPT Presentation

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Quantum cluster algebras from geometry Marta Mazzocco Based on Chekhov-M.M. arXiv:1509.07044 and Chekhov-M.M.-Rubtsov arXiv:1511.03851 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Ptolemy Relation aa ′ + bb ′ = cc ′ a c ′ c b ′ b a ′ Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 9 x 7 + x 8 x 2 • x 9 x ′ • 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 8 • x ′ x 2 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 n ), µ i : B → B ′ where µ i : ( x 1 , x 2 , . . . , x n ) → ( x ′ 1 , x ′ 2 , . . . , x ′ x b ij x − b ij x i x ′ ∏ ∏ x ′ i = + , j = x j ∀ j ̸ = i . j j j : b ij > 0 j : b ij < 0 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]. Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Teichm¨ uller space For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( R )) / GL 2 ( R ) . Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Develop bordered cusped Teichm¨ uller theory asymptotically. This will provide cluster algebra of geometric type Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Poincar´ e uniformsation Σ = H / ∆ , where ∆ is a Fuchsian group, i.e. a discrete sub-group of P SL 2 ( R ). Examples γ 2 γ 1 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Poincar´ e uniformsation Σ = H / ∆ , where ∆ is a Fuchsian group, i.e. a discrete sub-group of P SL 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. Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety 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 − 1 γ ˜ γ γ ˜ γ Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Poisson structure γ } = 1 γ − 1 { G γ , G ˜ γ − 1 . 2 G γ ˜ 2 G γ ˜ { } = 1 − 1 2 2 ˜ γ γ − 1 ˜ γ ˜ γ γ γ Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Two types of chewing-gum moves Connected result: Disconnected result: Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety Chewing gum 1 + ε 1 z 3 z 1 z 2 εℓ 1 εℓ 2 ) 2 ( = | z 1 − z 2 | 2 sinh d H ( z 1 , z 2 ) 2 4 ℑ z 1 ℑ z 2 l 1 l 2 ϵ 2 + ( l 1 + l 2 ) 2 e d H ( z 1 , z 2 ) ∼ 1 + O ( ϵ ), l 1 l 2 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:1509.07044] Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety = + + the rest skein γ a γ e γ f γ c γ b γ d ˜ γ a γ f = G ˜ γ c + G ˜ ˜ γ f ˜ γ e G ˜ γ e G ˜ γ a G ˜ γ b G ˜ γ d ˜ γ b ˜ γ d ˜ γ c Marta Mazzocco Quantum cluster algebras from geometry

Cluster algebras Teichm¨ uller Theory Geodesic lengths Quatisation Decorated character variety = + + the rest skein γ a γ e γ f γ c γ b γ d ˜ γ a γ f = G ˜ γ c + G ˜ ˜ γ f ˜ γ e G ˜ γ e G ˜ γ a G ˜ γ b G ˜ γ d ˜ γ b ˜ γ d ˜ γ c Marta Mazzocco Quantum cluster algebras from geometry