Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions Discrete Holomorphicity and Quantum Affine Algebras Robert Weston Heriot-Watt University, Edinburgh MSP, Kyoto, Aug 3rd, 2013 Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 1 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions Plan Introduction 1 Non-local quantum group currents in vertex models 2 From vertex models to loop models 3 Interacting boundaries 4 The continuum limit 5 Conclusions & Comments 6 Ref: Y. Ikhlef, R.W., M. Wheeler, P. Zinn-Justin: Discrete Holomorphicity and Quantized Affine Algebras, J. Phys.A 46 (2013) 265205, arxiv:1302.4649 Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 2 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions What is 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 Cauchy-Riemann reln z 2 − z 1 z 1 − z 0 Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 3 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions What is use of DH in SM/CFT? For review see [S. Smirnov, Proc. ICM 2006, 2010] DH observables used in proof of long-standing conjectures on conformal invariance of scaling limit, e.g., planar Ising model [S. Smirnov, C. Hongler, D. Chelkak . . . , 2001-] percolation on honeycomb lattice - Cardy’s crossing formula and reln to SLE(6) [S. Smirnov: 2001] Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 4 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions Relation to Integrability DH seems also to be related to integrability [Riva & Cardy 07, Cardy & Ikhlef 09, Ikhlef 12, Alam & Batchelor 12, de Gier et al13] e.g. parafermions of dilute O ( n ) loop model are DH precisely in the case when loop weights obey a linear relation whose solution corresponds to a solution of Yang-Baxter relation. How to interprete linear relation for R implying YB? Natural to assume that R ∆( x ) = ∆( x ) R for a quantum group is behind this. i.e. DH observables should be understood in terms of quantum group generators [Bernard & Fendley have publicly made this point]. Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 5 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions Our Key Results Dense/dilute 0( n ) PFs are essentially non-local quantum group currents for U q ( A (1) 1 )/ U q ( A (2) 2 ) DH of these currents just comes from R ∆( x ) = ∆( x ) R Currents of boundary (co-ideal) subalgebra gives rise to observables that have discrete boundary conditions of form � � Re Ψ( z 01 )( z 1 − z 0 ) + Ψ( z 12 )( z 2 − z 1 ) = 0 z 2 z 1 z 0 Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 6 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum 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) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 7 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum 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 and obvious extensions to ∆ ( N ) ( x ). With R : V 1 ⊗ V 2 → V 2 ⊗ V 1 1 2 Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 8 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions R ∆( x ) = ∆( x ) R becomes: a a a a + = + R (Θ ab ⊗ J b ) (Θ ab ⊗ J b ) R R ( J a ⊗ 1) + ( J a ⊗ 1) R + = a a b b = , = a a b b R (Θ ac ⊗ Θ cb ) (Θ ac ⊗ Θ cb ) R , R ( � Θ bc ⊗ � ( � Θ bc ⊗ � = Θ ca ) Θ ca ) R = Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 9 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions For monodromy matrix, 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) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 10 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions Quantum Affine Algebras Consider algebra U gen. by e i , f i , t ± 1 with standard relns and i ∆( e i ) = e i ⊗ 1 + t i ⊗ e i , ∆( t i ) = t i ⊗ t i Hence can consider currents: i e i ( x , t + 1 2) ∼ i e i ( x + 1 2 , t ) ∼ Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 11 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions We consider two cases with i ∈ { 0 , 1 } with irreps: U q ( A (1) 1 ): 6-Vertex Model � 0 � � q − 1 � 0 0 e 0 = z , t 0 = 1 0 0 q U q ( A (2) 2 ): 19-Vertex Izergin-Korepin Model q − 2 0 0 0 0 0 e 0 = z 1 − ℓ , t 0 = 1 0 0 0 1 0 q 2 0 0 0 0 q Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 12 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions From vertex models to loop models - the A (1) dense case 1 A ( z ) 0 0 0 0 B ( z ) C ( z ) 0 6-vertex model R ( z = z h / z v ) = can be 0 C ( z ) B ( z ) 0 0 0 0 A ( z ) written in dressed-loop picture as A ( z ) = , B ( z ) = , C ( z ) = + plus reversed arrow cases. Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 13 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions These can be rewriiten as appropriate loop weights a ( z ) = qz − q − 1 z − 1 , b ( z ) = z − z − 1 : a ( z ) b ( z ) δ 2 π from directed line turning through times additional factor ( − q ) angle δ . Acute angle α given by z = ( − q ) − α π . Thus A ( z ) = a ( z ), B ( z ) = b ( z ), π − 1 = q − q − 1 . α α π + b ( z )( − q ) C ( z ) = a ( z )( − q ) Partition fn becomes: Z = � a N a b N b ( − q − q − 1 ) N loops Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 14 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions e 0 ( x , t ) in the loop picture - the A (1) dense case 1 � 0 � 0 For U q ( A (1) 1 ), we have e 0 = z , so sends up arrow to down, or 1 0 right arrow to left: Simple boundary conditions consistent with � e 0 ( a , b ) � � = 0 are below, with a free line passing through ( a , b ) and attached to boundaries as shown: � � e 0 ( x , t + 1 2 ) � = 1 Z The tail can be moved through loops on boundary. Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 15 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions To express purely in terms of loop configuration C , consider angle turns of and and effects of Both and have same angle turn θ ( C ) = π k ( C ), where k ( C ) ∈ Z , equals 2 in example. Weight = ( − q ) k ( C ) . also k ( C ). Weight = q k ( C ) . No. down - no. up crossing of � Hence � e 0 ( x , t + 1 2 ) � = z v W ( C )( − q 2 ) θ ( C ) /π . Z C | ( x + 1 2 , t ) ∈ γ Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 16 / 27
Introduction Currents in vertex models Vertex to loops Boundaries Continuum Conclusions Similarly � � e 0 ( x + 1 2 , t ) � = 1 Z Z q α/π � Z e − i α � = z h W ( C )( − q 2 ) θ ( C ) /π = z v W ( C )( − q 2 ) θ ( C ) C | ( x , t + 1 C | ( x , t + 1 2 ) ∈ γ 2 ) ∈ γ Robert Weston (Heriot-Watt) Disc. Hol. & QAA MSP, Kyoto, Aug 3rd, 2013 17 / 27
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