2d Field Theory and Random Planar Sets: past and future John Cardy - PowerPoint PPT Presentation

2d Field Theory and Random Planar Sets: past and future John Cardy University of Oxford Conformal Maps from Probability to Physics Ascona, May 2010

Lattice Models scaling limit Field Theory SLE etc ◮ 2d field theory is a rich source of conjectures for SLE-type results

Lattice Models scaling limit Field Theory SLE etc ◮ 2d field theory is a rich source of conjectures for SLE-type results

2d field theory, c. 1991 ◮ [1960s] Scaling limits of lattice models: limit as lattice spacing a → 0 at fixed correlation length ξ should exist a → 0 a − x 1 ... − x n E [ φ lat 1 ( z 1 ) · · · φ lat n ( z n )] = � φ 1 ( z 1 ) · · · φ n ( z n ) � lim and be given by correlators satisfying axioms of a euclidean QFT. ◮ when ξ − 1 = 0 (critical point) this implies scale covariance: � φ 1 ( bz 1 ) · · · φ n ( bz n ) � b D = b − x 1 ... − x n � φ 1 ( z 1 ) · · · φ n ( z n ) � D ◮ [Polyakov 1970]: this should extend to covariance under conformal mappings z → f ( z ) : n � | f ′ ( z j ) | − x j � φ 1 ( z 1 ) · · · φ n ( z n ) � D � φ 1 ( f ( z 1 )) · · · φ n ( f ( z n )) � f ( D ) = j = 1

Conformal Field Theory (CFT) ◮ [Belavin, Polyakov, Zamolodchikov 1984]: important role played by fields whose correlators are holomorphic in z , in particular the stress tensor T ( z ) which implements infinitesimal conformal mappings z → z + α ( z ) via conformal Ward identity : 1 � � � δφ j ( z j ) · · · � = α ( z ) � T ( z ) φ j ( z j ) · · · � dz + c . c . 2 π i C z j inside C T C

Virasoro and all that � ( z − z j ) − 2 − n L n φ j ( z j ) T ( z ) · φ j ( z j ) = n ≤ n max [ L n , L m ] = ( n − m ) L n + m + ( c / 12 ) n ( n 2 − 1 ) δ n , − m ( Vir ) ◮ there are two independent copies ( Vir , Vir ) corresponding to T ( z ) and T (¯ z ) ◮ to each primary field φ j such that L n φ j = 0 for all n ≥ 1 corresponds a set of descendants: φ j L − 1 φ j (= ∂ z φ j ) L − 2 φ j , L 2 − 1 φ j . . .

◮ sometimes these are degenerate , e.g. at level 2 L − 2 φ j = ( κ/ 4 ) L 2 − 1 φ j = ( κ/ 4 ) ∂ 2 z φ j ◮ by choosing α ( z ) ∝ ( z − z j ) − 1 we can use the conformal Ward identity to show that in these cases the correlators of φ j satisfy (2nd order) linear PDEs wrt z j ◮ [JC 1984] all these ideas extend to boundary fields with z j ∈ ∂ D , with the identification Vir = Vir

◮ Coulomb gas methods [Nienhuis, den Nijs, early 1980s]: many properties of 2d critical systems (e.g. scaling dimensions x j ) follow from conjectured relationship to modified gaussian free field (GFF) compactified on circle radius ∝ κ − 1 / 2 ◮ [Duplantier, 1980s] local scaling fields φ can also describe sources for N mutually avoiding Brownian curves and also in conjectured scaling limit of O ( n ) model and hulls of FK clusters in Q -state Potts model Z ∝ ( ǫ/ r ) 2 x N

◮ scaling dimensions conjectured from CFT and Coulomb gas methods, e.g. in O ( n ) model = N 2 2 κ − ( κ − 4 ) 2 = N ( N + 2 ) − N x boundary x bulk , N N 8 κ κ 2 where n = − 2 cos ( 4 π/κ ) . ◮ in particular φ boundary is degenerate at level N + 1. N ◮ [JC 1991] boundary fields for percolation hulls are degenerate (at level 2) and so their 4-point correlators satisfy 2nd order PDE ⇒ percolation crossing formula _

Then SLE came along . . . ◮ [Schramm 2000]: if percolation hull exploration process converges to SLE 6 , crossing formula follows ◮ [Smirnov 2001]: crossing formula holds for scaling limit of triangular lattice percolation ⇒ exploration process converges to SLE 6 ◮ and much more . . .

SLE and φ boundary fields in CFT 1 γ t ◮ [Bauer-Bernard, Friedrich-Werner 2002]: CFT correlators have martingale property � � �O φ 1 ( 0 ) � H = �O φ 1 ( tip t ) � H \ γ t E = E [ � g t ( O ) g t ( φ 1 )( 0 ) � H ] ◮ infinitesimal Loewner map α ( z ) = 2 dt / z − √ κ dB t 2 dt L − 2 − √ κ dB t L − 1 ⇒ 0 ( 2 L − 2 dt ′ −√ κ L − 1 dB t ′ ) φ 1 ( 0 ) � t e − g t ( φ 1 )( 0 ) = � t 0 ( 2 L − 2 − ( κ/ 2 ) L 2 − 1 ) dt ′ φ 1 ( 0 ) e − E [ g t ( φ 1 )( 0 )] =

γ is SLE κ ⇔ φ boundary is degenerate at level 2 1 ◮ [Bauer-Bernard-Kytola]: conditioned CFT partition functions ⇒ variants like multiple SLEs and SLE ( κ, ρ ) ◮ if we know CFT partition functions in other domains D we can deduce corresponding Loewner driving process - however in general these are not known!

γ is SLE κ ⇔ φ boundary is degenerate at level 2 1 ◮ [Bauer-Bernard-Kytola]: conditioned CFT partition functions ⇒ variants like multiple SLEs and SLE ( κ, ρ ) ◮ if we know CFT partition functions in other domains D we can deduce corresponding Loewner driving process - however in general these are not known!

γ is SLE κ ⇔ φ boundary is degenerate at level 2 1 ◮ [Bauer-Bernard-Kytola]: conditioned CFT partition functions ⇒ variants like multiple SLEs and SLE ( κ, ρ ) ◮ if we know CFT partition functions in other domains D we can deduce corresponding Loewner driving process - however in general these are not known!

Can we get the whole of CFT from SLE (or CLE)? ◮ [Friedrich-Werner 2002, Doyon-Riva-JC 2005]: identification of stress tensor T in SLE setting ◮ when conformal restriction on curves γ holds � d θ e − 2 i θ 1 γ separates ( z ± ǫ e i θ ) ǫ → 0 ǫ − 2 T ( z ) ∝ lim ◮ this T satisfies conformal Ward identities (with c = 0) ◮ more generally for c � = 0, T can be defined by the notion of conformal derivative [Doyon 2010]

Can we get the whole of CFT from SLE (or CLE)? ◮ [Friedrich-Werner 2002, Doyon-Riva-JC 2005]: identification of stress tensor T in SLE setting ◮ when conformal restriction on curves γ holds � d θ e − 2 i θ 1 γ separates ( z ± ǫ e i θ ) ǫ → 0 ǫ − 2 T ( z ) ∝ lim ◮ this T satisfies conformal Ward identities (with c = 0) ◮ more generally for c � = 0, T can be defined by the notion of conformal derivative [Doyon 2010]

Holomorphic fields ◮ [Smirnov, Riva-JC, Rajabpour-JC, Ikhlef-JC]: in many lattice models, local observables of curves γ can be identified which are discretely holomorphic , e.g. � d θ e − i σθ 1 γ ends at z with winding angle θ ψ σ ( z ) ∝ z ◮ in the cases where convergence of � ψ σ ( z ) � to a continuous holomorphic function can be proved with suitable boundary conditions this implies convergence of γ to SLE κ with σ = ( 6 − κ ) / 2 κ (e.g. Ising [Chelkak-Smirnov]) ◮ the existence of discretely holomorphic observables appears to be linked to integrability of lattice models

Holomorphic fields ◮ [Smirnov, Riva-JC, Rajabpour-JC, Ikhlef-JC]: in many lattice models, local observables of curves γ can be identified which are discretely holomorphic , e.g. � d θ e − i σθ 1 γ ends at z with winding angle θ ψ σ ( z ) ∝ z ◮ in the cases where convergence of � ψ σ ( z ) � to a continuous holomorphic function can be proved with suitable boundary conditions this implies convergence of γ to SLE κ with σ = ( 6 − κ ) / 2 κ (e.g. Ising [Chelkak-Smirnov]) ◮ the existence of discretely holomorphic observables appears to be linked to integrability of lattice models

Other correlators of holomorphic fields z z 1 2 ◮ 2-point function in R 2 � ψ σ ( z 1 ) ψ σ ( z 2 ) � ∼ ( z 1 − z 2 ) − 2 σ + + ◮ 4-point function: Ising case � � 1 � ψ 1 2 ( z 1 ) ψ 1 2 ( z 2 ) ψ 1 2 ( z 3 ) ψ 1 2 ( z 4 ) � R 2 ∝ Pf z j − z k

◮ for general κ , conjectured scaling limit of these Smirnov observables corresponds to holomorphic CFT fields which are degenerate at level 2 and so we know their higher-order correlators ◮ in general the solution space has dimension > 1 and they have non-trivial monodromy, e.g. � 2 σ � z 13 z 24 � ψ σ ( z 1 ) · · · ψ σ ( z 4 ) � = ( A 1 F 1 ( η ) + A 2 F 2 ( η )) z 12 z 23 z 34 z 41 where η = z 12 z 34 / z 13 z 24 and F j ( η ) are hypergeometric functions ◮ these correlators can be considered as multi-particle wave functions of a quantum system in 2+1 dimensions ◮ non-Abelian fractional statistics, can be used in principle to make a quantum computer!

◮ for general κ , conjectured scaling limit of these Smirnov observables corresponds to holomorphic CFT fields which are degenerate at level 2 and so we know their higher-order correlators ◮ in general the solution space has dimension > 1 and they have non-trivial monodromy, e.g. � 2 σ � z 13 z 24 � ψ σ ( z 1 ) · · · ψ σ ( z 4 ) � = ( A 1 F 1 ( η ) + A 2 F 2 ( η )) z 12 z 23 z 34 z 41 where η = z 12 z 34 / z 13 z 24 and F j ( η ) are hypergeometric functions ◮ these correlators can be considered as multi-particle wave functions of a quantum system in 2+1 dimensions ◮ non-Abelian fractional statistics, can be used in principle to make a quantum computer!

Other degenerate bulk fields ◮ [Gamsa-JC, Simmons-JC]: in the conjectured CFT description of the O ( n ) model ’twist’ fields are also degenerate at level 2 and so their correlators satisfy 2nd order PDEs z ) ∝ ( − 1 ) number of curves separating z and z 0 φ twist ( z , ¯ ◮ gives 2-point information about SLE 8 / 3 ◮ in all these examples of bulk level 2 degenerate fields, is there a stochastic calculus interpretation?