Yang-Baxter equation and discrete conformal symmetry Vladimir Bazhanov Department of Theoretical Physics Research School of Physical Sciences and Engineering The Australian National University Discrete Differential Geometry, Berlin, July, 15-19, 2007. [work with V.Mangazeev and S.Sergeev] V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 1 / 19
Connection between statistical mechanics and discrete geometry Exactly solvable models: Yang-Baxter equation, Yang (1967), Baxter (1972), Faddeev-Volkov solution (1992-94) Discrete Riemann mapping theorem. Thurston (1985), Rodin-Sullivan (1987), Stephenson (1987), He-Schramm (1998), ... Discrete analytic functions: Bobenko-& Pinkall (1996), -& Suris (2002), -& Mercat and Suris. Variational principle for circle patterns: Bobenko-Springborn (2002) Topological invariants, braid group, invariants of links, rhombic tilings . . . “Z-invariant” lattices, Baxter (1989) and invariants of links, Jones (1987) Planar embeddings of quad-graphs, Kenyon-Schlenker (2005) Hyperbolic geometry: volumes of polyhedra in the Lobachevskii 3-space Conformal Field Theory, Belavin-Polyakov-Zamolodchikov (1984), A discrete analog? V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 2 / 19
Yang-Baxter equation in statistical mechanics local “spins”: σ i ∈ (set of values) e − E ( σ ) / T , X X Z = E ( { σ } ) = ǫ ( σ i , σ j ) , { spins } ( ij ) ∈ edges W ( σ i , σ j ) = e − ǫ ( σ i ,σ j ) / T X Y W ( σ i , σ j ) = Trace ( T ) m , Z = { spins } ( ij ) ∈ edges Hard to calculate if number of edges, N → ∞ Ising model, dimers, . . . , ⇒ free fermions. “Gaussian models”, reduce to diagonalization of a quadratic form, Pfaffians and determinants. The Boltzmann weights satisfy the Yang-Baxter equation. Commuting transfer-matrices Z = Trace ( T q 1 T q 2 · · · T q m ) , [ T q , T q ′ ] = 0 . V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 3 / 19
θ = p − q , σ, a , b ∈ R W θ ( a − b ) = W π − θ ( a − b ) Yang-Baxter equation Star-triangle relation: Z d σ W q − r ( a − σ ) W p − r ( c − σ ) W p − q ( σ − b ) = W p − q ( c − a ) W p − r ( a − b ) W q − r ( c − b ) . R V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 4 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 5 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 6 / 19
Z Y Y Z = W θ ( ij ) ( σ i − σ j ) d σ i i ( ij ) Partition function Z possesses a remarkable invariance property: it remains unchanged by continuously deforming the rapidity lines (with their boundary position kept fixed) Baxter (1979) factorization theorem. When N → ∞ √ X log Z = f ( θ ( ij ) ) + O ( N ) . ( ij ) V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 7 / 19
Faddeev-Volkov model F θ e 2 ηθ s ϕ ( s + i ηθ/π ) 1 W θ ( s ) = ϕ ( s − i ηθ/π ) , W θ ( s ) = W π − θ ( s ) , (1) „ 1 e − 2i zw Z dw « ϕ ( z ) def = exp , (2) 4 sinh ( wb ) sinh ( w / b ) w R +i0 „ 1 e − 2i zw « Z dw Φ( z ) def = exp , (3) sinh( wb ) sinh( wb − 1 ) cosh( w ( b + b − 1 )) 8 w R +i0 η = ( b + b − 1 ) / 2 . (4) = e i η 2 θ 2 /π +i π (1 − 8 η 2 ) / 24 Φ(2i ηθ/π ) . def F θ (5) With this normalization the edge function f ( θ ) ≡ 0, i.e., √ log Z = O ( N ) , N → ∞ The model related with quantum Liouville and sinh-Gordon equations and the modular double of quantum group q = e i π b 2 , q = e − i π/ b 2 , U q ( sl 2 ) ⊗ U ˜ q ( sl 2 ) , ˜ Parameter b 2 > 0 plays the role of the Planck constant � . V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 8 / 19
Quasi-classical expansion, b → 0 Z d ρ i 1 e − 2 π b 2 A [ ρ ] Y Z = 2 π b i „ 1 + e ξ +i θ Z ρ A ( θ | ρ ) = A ( θ | − ρ ) = 1 « log d ξ . e ξ + e i θ i 0 Z x log(1 − x ) “ e ρ − i θ ” “ e ρ +i θ ” A ( θ | ρ ) = i Li 2 − i Li 2 − θρ, Li 2 ( x ) = − dx . x 0 X ` ´ A [ ρ ] = A θ ( ij ) | ρ i − ρ j ( ij ) ∈ E ( G ) e ρ j + e ρ i +i θ ( ij ) ∂ A [ ρ ] ˛ Y ρ = ρ ( cl ) = 0 , ⇒ = 1 , i = ∈ V int ( G ) . ˛ e ρ i + e ρ i +i θ ( ij ) ∂ρ i ˛ ( ij ) ∈ star ( i ) A [ ρ ] precisely coincide with Bobenko-Springborn circle packing action. Our results imply that this action possesses the “ Z -invariance property” and that ∂ 2 A [ ρ ] √ ‚ ‚ 2 π b 2 A [ ρ ( cl ) ] − 1 1 ‚ ‚ log Z = − 2 log det + . . . = O ( N ) , N → ∞ ‚ ‚ ∂ρ i ∂ρ j ‚ ‚ ρ = ρ ( cl ) V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 9 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 10 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 11 / 19
Kenyon-Schlenker: rhombic tilings, Bobenko-Mercat-Suris: isoradial circle patterns, this talk: “rapidity graph” or a braid. Sum rules X θ ( ij ) = 2 π, i ∈ V int ( G ) ( ij ) ∈ star ( i ) V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 12 / 19
Discrete conformal transformations Continuous conformal transformations (i) preserve angles (ii) uniformly rescale all infinitesimal lengths (scale depends on a point) i log r 1 + r 2 e i θ ϕ 1 = 1 r 1 + r 2 e − i θ , (6) Circle flower equations (cross ratio system) X ϕ ( ij ) = 2 π, i ∈ V int ( G ) . (7) ( ij ) ∈ star ( i ) They are identical to the equation of motion in the Faddeev-Volkov model V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 13 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 14 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 15 / 19
Discrete conformal transformations Continuous conformal transformations (i) preserve angles (ii) uniformly rescale all infinitesimal lengths (scale depends on a point) i log r 1 + r 2 e i θ ϕ 1 = 1 r 1 + r 2 e − i θ , (8) Circle flower equations (cross ratio system) X ϕ ( ij ) = 2 π, i ∈ V int ( G ) . (9) ( ij ) ∈ star ( i ) They are identical to the equation of motion in the Faddeev-Volkov model V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 16 / 19
V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 17 / 19
Star-triangle relation and hyperbolic geometry e half-space model { x , y , z ∈ R | z > 0 } , ds 2 = ( dx 2 + dy 2 + dz 2 ) / z 2 . Poincar´ V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 18 / 19
Star-triangle-circle relation V ⋆ = 2 L ( θ 1 2 ) + 2 L ( θ 2 2 ) + 2 L ( θ 3 2 ) + 1 2 A ⋆ [ ρ ( cl ) , ρ 1 , ρ 2 , ρ 3 ] + boundary term (10) 0 V △ = 2 L ( π − θ 1 ) + 2 L ( π − θ 2 ) + 2 L ( π − θ 3 ) + 1 2 A △ [ ρ 1 , ρ 2 , ρ 3 ] + boundary term (11) 2 2 2 V tetrahedron = V ⋆ − V △ = L ( θ 1 ) + L ( θ 2 ) + L ( θ 3 ) , (12) 2 L ( θ 2 ) − 2 L ( π − θ ) = L ( θ ) . (13) 2 V. Bazhanov (ANU) Quantum Circle Patterns Berlin, July 16, 2007 19 / 19
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