Yang-Baxter equation and discrete conformal symmetry Vladimir - - PowerPoint PPT Presentation

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Yang-Baxter equation and discrete conformal symmetry Vladimir - - PowerPoint PPT Presentation

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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,

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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

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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

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Yang-Baxter equation in statistical mechanics

local “spins”: σi ∈ (set of values) Z = X

{spins}

e−E(σ)/T , E({σ}) = X

(ij)∈edges

ǫ(σi, σj), W (σi, σj) = e−ǫ(σi ,σj )/T Z = X

{spins}

Y

(ij)∈edges

W (σi, σj) = Trace (T)m, 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 (Tq1 Tq2 · · · Tqm), [Tq, Tq′] = 0.

  • V. Bazhanov (ANU)

Quantum Circle Patterns Berlin, July 16, 2007 3 / 19

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θ = p − q, σ, a, b ∈ R W θ(a − b) = Wπ−θ(a − b)

Yang-Baxter equation

Star-triangle relation: Z

R

dσ W q−r (a − σ) Wp−r (c − σ) W p−q (σ − b) = Wp−q (c − a)W p−r (a − b) Wq−r (c − b) .

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Z = Z Y

(ij)

Wθ(ij)(σi − σj) Y

i

dσi 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 → ∞ log Z = X

(ij)

f (θ(ij)) + O( √ N).

  • V. Bazhanov (ANU)

Quantum Circle Patterns Berlin, July 16, 2007 7 / 19

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Faddeev-Volkov model

Wθ (s) = 1 Fθ e2ηθs ϕ(s + iηθ/π) ϕ(s − iηθ/π) , W θ (s) = Wπ−θ(s) , (1) ϕ(z) def = exp „ 1 4 Z

R+i0

e−2izw

sinh(wb)sinh(w/b) dw w « , (2) Φ(z) def = exp „ 1 8 Z

R+i0

e−2izw

sinh(wb) sinh(wb−1) cosh(w(b + b−1)) dw w « , (3) η = (b + b−1)/2 . (4) Fθ

def

= eiη2θ2/π+iπ(1−8η2)/24 Φ(2iηθ/π) . (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

  • f quantum group

Uq(sl2) ⊗ U˜

q(sl2),

q = eiπb2, ˜ q = e−iπ/b2, Parameter b2 > 0 plays the role of the Planck constant .

  • V. Bazhanov (ANU)

Quantum Circle Patterns Berlin, July 16, 2007 8 / 19

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Quasi-classical expansion, b → 0

Z = Z e−

1 2πb2 A[ρ] Y

i

dρi 2πb A(θ|ρ) = A(θ| − ρ) = 1 i Z ρ log „ 1 + eξ+iθ

eξ + eiθ

« dξ . A(θ|ρ) = i Li2 “eρ−iθ” − i Li2 “eρ+iθ” − θρ, Li2(x) = − Z x log(1 − x) x dx . A[ρ ] = X

(ij)∈E(G)

A ` θ(ij) | ρi − ρj ´ ∂A[ρ ] ∂ρi ˛ ˛ ˛

ρ=ρ(cl) = 0, ⇒

Y

(ij)∈star(i)

eρj + eρi +iθ(ij) eρi + eρi +iθ(ij)

= 1 , i =∈ Vint(G) . A[ρ] precisely coincide with Bobenko-Springborn circle packing action. Our results imply that this action possesses the “Z-invariance property” and that log Z = − 1 2πb2 A[ρ(cl)] − 1 2 log det ‚ ‚ ‚ ‚ ∂2A[ρ] ∂ρi∂ρj ‚ ‚ ‚ ‚

ρ=ρ(cl)

+ . . . = O( √ N), N → ∞

  • V. Bazhanov (ANU)

Quantum Circle Patterns Berlin, July 16, 2007 9 / 19

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Kenyon-Schlenker: rhombic tilings, Bobenko-Mercat-Suris: isoradial circle patterns, this talk: “rapidity graph” or a braid. Sum rules X

(ij)∈star(i)

θ(ij) = 2π, i ∈ Vint(G)

  • V. Bazhanov (ANU)

Quantum Circle Patterns Berlin, July 16, 2007 12 / 19

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Discrete conformal transformations

Continuous conformal transformations

(i) preserve angles (ii) uniformly rescale all infinitesimal lengths (scale depends on a point)

ϕ1 = 1 i log r1 + r2eiθ r1 + r2e−iθ , (6) Circle flower equations (cross ratio system) X

(ij)∈star(i)

ϕ(ij) = 2π, i ∈ Vint(G) . (7) 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

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Discrete conformal transformations

Continuous conformal transformations

(i) preserve angles (ii) uniformly rescale all infinitesimal lengths (scale depends on a point)

ϕ1 = 1 i log r1 + r2eiθ r1 + r2e−iθ , (8) Circle flower equations (cross ratio system) X

(ij)∈star(i)

ϕ(ij) = 2π, i ∈ Vint(G) . (9) 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

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Star-triangle relation and hyperbolic geometry

Poincar´ e half-space model {x, y, z ∈ R|z > 0} , ds2 = (dx2 + dy2 + dz2)/z2.

  • V. Bazhanov (ANU)

Quantum Circle Patterns Berlin, July 16, 2007 18 / 19

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Star-triangle-circle relation

V⋆ = 2L( θ1 2 ) + 2L( θ2 2 ) + 2L( θ3 2 ) + 1 2 A⋆[ρ(cl) , ρ1, ρ2, ρ3] + boundary term (10) V△ = 2L( π − θ1 2 ) + 2L( π − θ2 2 ) + 2L( π − θ3 2 ) + 1 2 A△[ρ1, ρ2, ρ3] + boundary term (11) Vtetrahedron = V⋆ − V△ = L(θ1) + L(θ2) + L(θ3) , (12) 2L( θ 2 ) − 2L( π − θ 2 ) = L(θ) . (13)

  • V. Bazhanov (ANU)

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