1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Discrete Holomorphicity in the Chiral Potts Model Robert Weston Heriot-Watt University, Edinburgh GGI, 19/05/15 Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 1 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Plan The Talk in 1 Slide 1 Discrete Holomorphicity 2 Non-Local Quantum Group Currents 3 The Z ( N ) Chiral Potts Model 4 The CP Model via Representation Theory 5 DH relations and perturbed CFT 6 Conclusions 7 [ Ref: Y. Ikhlef, RW, M. Wheeler and P. Zinn-Justin, J. Phys.A 46 (2013) 265205, arxiv:1302.4649; Y. Ikhlef and RW, (2015) arxiv:1502.04944] Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 2 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Talk in 1 Slide DH means a lattice analog of Cauchy-Riemann relations We use underlying quantum group to construct DH operators for stat-mech models DH follows from fact that lattice model weights are QG R-matrices DH relns in massless case are discrete version of ∂ ¯ z Ψ( z , ¯ z ) = 0 z ) = � DH relns in massive case are of form ∂ ¯ z Ψ( z , ¯ χ i ( z , ¯ z ) i where in CFT w ) = · · · + χ i ( w , ¯ w ) Ψ( z )Φ pert ( w , ¯ + · · · i z − w Can thus identify the CFT perturbing fields DH operators hopefully useful in rigorous proof of scaling to CFT Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 3 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Discrete Holomorphicity Λ a planar graph in R 2 , embedded in complex plane. Let f be a complex-valued fn defined at midpoint of edges Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 4 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Discrete Holomorphicity Λ a planar graph in R 2 , embedded in complex plane. Let f be a complex-valued fn defined at midpoint of edges � f said to be DH if it obeys lattice version of f ( z ) dz = 0 around any cycle. Around elementary plaquette, we use: f ( z 01 )( z 1 − z 0 )+ f ( z 12 )( z 2 − z 1 )+ f ( z 23 )( z 3 − z 2 )+ f ( z 30 )( z 0 − z 3 ) = 0 z 3 z 2 z ij = ( z i + z j ) / 2 z 0 z 1 Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 4 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Discrete Holomorphicity Λ a planar graph in R 2 , embedded in complex plane. Let f be a complex-valued fn defined at midpoint of edges � f said to be DH if it obeys lattice version of f ( z ) dz = 0 around any cycle. Around elementary plaquette, we use: f ( z 01 )( z 1 − z 0 )+ f ( z 12 )( z 2 − z 1 )+ f ( z 23 )( z 3 − z 2 )+ f ( z 30 )( z 0 − z 3 ) = 0 z 3 z 2 z ij = ( z i + z j ) / 2 z 0 z 1 Can be written for this cycle as f ( z 23 ) − f ( z 01 ) = f ( z 12 ) − f ( z 30 ) a discrete C-R reln ¯ , ∂ f = 0 z 2 − z 1 z 1 − z 0 Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 4 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions What is use of DH in SM/CFT? For review see [S. Smirnov, Proc. ICM 2006, 2010] DH of observables has been as a key tool in rigorous proof of existence and uniqueness of scaling limit to particular conformal field theories, e.g., planar Ising model [S. Smirnov, C. Hongler D. Chelkak . . . , 2001-] - convergence of interfaces to SLE(3) site percolation on triangular lattice - Cardy’s crossing formula and reln to SLE(6) [S. Smirnov: 2001] We find DH condition also useful in identifying the particular integrable CFT perturbation to which SM lattice model corresponds Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 5 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions DH and Integrability Observed by [Ikhlef, Cardy (09); de Gier, Lee, Rasmussen (09); Alam, Batchelor (12,14)] that candidate operators in various lattice models obey DH in the case when R-matrix obeys Yang-Baxter Our construction explains this by showing how DH operators arise naturally from Quantum Groups Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 6 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Non-local quantum group currents in vertex models Following Bernard and Felder [1991] we consider a set of elements { J a , Θ ab , � Θ ab } , a , b = 1 , 2 , . . . , n , of a Hopf algebra U . Θ ab � Θ ba Θ bc = δ a , c � Θ cb = δ a , c Properties: and Co-product ∆ and antipode S are (with summation convention): ∆( J a ) = J a ⊗ 1 + Θ ab ⊗ J b S ( J a ) = − � Θ ba J b ∆(Θ ab ) = Θ ac ⊗ Θ cb S (Θ ab ) = � Θ ba ∆( � Θ ab ) = � Θ ac ⊗ � S ( � Θ cb Θ ab ) = Θ ba . Acting on rep of U , we represent as Θ ab = a � Θ ab = J a = a , b , a b Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 7 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions Coproducts pictures are: ∆( J a ) = + a a Θ ab ⊗ J b J a ⊗ 1 ∆( � ∆(Θ ab ) Θ ab ) = = b , a a b Θ ac ⊗ Θ cb Θ ac ⊗ � � Θ cb with obvious extensions to ∆ ( N ) ( x ): i � ∆ ( N ) ( J a ) = a i Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 8 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions With ˇ R ∆( x ) = ∆( x ) ˇ ˇ R : V 1 ⊗ V 2 → V 2 ⊗ V 1 1 , R is 2 a a a a + = + R (Θ ab ⊗ J b ) (Θ ab ⊗ J b ) ˇ ˇ ˇ ( J a ⊗ 1) ˇ R ( J a ⊗ 1) + = R + R a a b b = , = a a b b R (Θ ac ⊗ Θ cb ) (Θ ac ⊗ Θ cb ) ˇ ˇ R ( � ˇ Θ bc ⊗ � ( � Θ bc ⊗ � Θ ca ) ˇ = R Θ ca ) = R Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 9 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions So we have non-local currents a a + a a = + Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 10 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions So we have non-local currents a a + a a = + Gives j a ( x − 1 2 , t ) − j a ( x + 1 2 , t ) + j a ( x , t − 1 2) − j a ( x , t + 1 2) = 0 when inserted into a correlation function Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 10 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions So we have non-local currents a a + a a = + Gives j a ( x − 1 2 , t ) − j a ( x + 1 2 , t ) + j a ( x , t − 1 2) − j a ( x , t + 1 2) = 0 when inserted into a correlation function Idea: Construct DH operators in terms of such currents: - Dense ( U q ( � sl 2 ) ) and dilute loop models ( U q ( A (2) 2 )): [Ikhlef, RW, Wheeler, Zinn-Justin (13)] - Chiral Potts ( U q ( � sl 2 ) ) : [Iklef, RW (15)] Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 10 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Integrable Z ( N ) Chiral Potts Model Introduced by [Howes, Kadonoff, den Nijs (83); Au-Yang, Perk, McCoy, Tang, Yan, Sah (87); Baxter, Perk, Au-Yang (88)]. See [B. McCoy, Advanced Statistical Mech, OUP, 2010] Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 11 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Integrable Z ( N ) Chiral Potts Model Introduced by [Howes, Kadonoff, den Nijs (83); Au-Yang, Perk, McCoy, Tang, Yan, Sah (87); Baxter, Perk, Au-Yang (88)]. See [B. McCoy, Advanced Statistical Mech, OUP, 2010] r s r s r s r s Heights a ∈ { 0 , 1 , · · · , N − 1 } on vertices: Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 11 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Integrable Z ( N ) Chiral Potts Model Introduced by [Howes, Kadonoff, den Nijs (83); Au-Yang, Perk, McCoy, Tang, Yan, Sah (87); Baxter, Perk, Au-Yang (88)]. See [B. McCoy, Advanced Statistical Mech, OUP, 2010] r s r s r s r s Heights a ∈ { 0 , 1 , · · · , N − 1 } on vertices: Boltzmann weights are a r s r s W rs ( a − b ) = a b , W rs ( a − b ) = . b Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 11 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Integrable Z ( N ) Chiral Potts Model . . . Rapidities r , s in W rs ( a − b ) are points on algebraic curve C k Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 12 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Integrable Z ( N ) Chiral Potts Model . . . Rapidities r , s in W rs ( a − b ) are points on algebraic curve C k C k given by ( x , y , µ ) with 1 − kx N = 1 − ky N k ′ x N + y N = k (1 + x N y N ) , µ N = , k ′ genus ( N − 1) 2 Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 12 / 28
1 slide DH QG Currents The CP model CP via rep th pert CFT Conclusions The Integrable Z ( N ) Chiral Potts Model . . . Rapidities r , s in W rs ( a − b ) are points on algebraic curve C k C k given by ( x , y , µ ) with 1 − kx N = 1 − ky N k ′ x N + y N = k (1 + x N y N ) , µ N = , k ′ genus ( N − 1) 2 Obeys star-triangle N − 1 � W rs ( a − d ) W rt ( d − b ) W st ( d − c ) d =0 = ρ rst × W rs ( c − b ) W rt ( a − c ) W st ( a − b ) but no difference property Robert Weston (Heriot-Watt) DH in the CP model GGI, 19/05/15 12 / 28
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