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SLE and CFT Mitsuhiro Kato @ QFT2005 1. Introduction Critical phenomena Conformal Field Theory (CFT) Algebraic approach, field theory, BPZ(1984) Stochastic Loewner Evolution (SLE) Geometrical approach, stochastic process, Schramm(2000)

  1. SLE and CFT Mitsuhiro Kato @ QFT2005

  2. 1. Introduction

  3. Critical phenomena • Conformal Field Theory (CFT) Algebraic approach, field theory, BPZ(1984) • Stochastic Loewner Evolution (SLE) Geometrical approach, stochastic process, Schramm(2000) 1

  4. Percolation Consider triangular lattice whose site is colored black with probability p or white with probabilty 1 − p . Convenient to consider dual lattice whose face (hexagon) is colored accordingly. Study clustering property of colored faces. p → p c : percolation threshold at which mean cluster size diverge 2

  6. Crossing probability C 4 z C z 1 1 4 unit disc C z z C 3 3 2 2 P ( γ 1 , γ 2 ) is a function only of cross-ratio η = ( z 1 − z 2 )( z 3 − z 4 ) ( z 1 − z 3 )( z 2 − z 4 ) Γ( 2 3 ) η 1 / 3 2 F 1 (1 3 , 2 3 , 4 P = 3; η ) Γ( 4 3 )Γ( 1 3 ) 3

  7. SLE treats cluster boundary via stochastic process. 4

  8. Plan of the talk 1. Introduction 2. SLE 3. Critical models 4. Relation to CFT 5. Remarks 5

  9. 2. SLE

  10. Hull A compact subset K in H s.t. H \ K is simply connected, K = K ∩ H is called a hull . g ( z ) K H\K H K 6

  11. Conformal map For any hull K , there exists a unique conformal map g K : H \ K → H z →∞ ( g K ( z ) − z ) = 0 lim This map has an expansion for z → ∞ g K ( z ) = z + a 1 z + · · · + a n z n + · · · a 1 = a 1 ( K ) is called capacity of the hull K . 7

  12. g t (z) Loewner equation H t H K = H \ H t t U t = g t ( γ ( t )) Let γ ( t ) be parametrized s.t. a 1 ( K t ) = 2 t . Then ∂ 2 ∂tg t ( z ) = g 0 ( z ) = z , g t ( z ) − U t 8

  13. example U t = 0 case ∂ 2 ∂tg t ( z ) = g 0 ( z ) = z g t ( z ) , z 2 + 4 t � g t ( z ) = √ γ ( t ) = 2 i t 9

  14. SLE ∂ 2 ∂tg t ( z ) = , g 0 ( z ) = z g t ( z ) − √ κB t where B t is standard Brawnian motion on R , κ is a real parameter. g t ( z ) = g t ( z ) − √ κB t Alternatively, for ˆ g t ( z ) dt − √ κdB t 2 d ˆ g t ( z ) = ˆ 10

  15. Brawnian motion For U t = √ κB t �� U t �� = 0 , �� U t 1 U t 2 �� = κ | t 1 − t 2 | Thus �� dU t dU t �� = κdt 11

  16. Itˆ o formula Suppose X t satisfies stochastic differential eq. dX t = a ( X t , t ) dt + b ( X t , t ) dB t Then for a function f ( X t ) f = ( af ′ + 1 2 b 2 f ′′ ) dt + bf ′ dB t d 12

  17. SLE trace ( z + √ κB t ) z → 0 g − 1 γ ( t ) := lim t 13

  18. Phases of SLE simple curve double points space-filling duality conjecture ∂ K t for κ > 4 SLE trace for ˆ κ = 16 /κ < 4 ⇔ 14

  19. Hausdorff dimensions  1 + κ/ 8 ( κ < 8)  d H = 2 ( κ > 8)  15

  20. Basic properties Denote measure µ ( γ ; D, r 1 , r 2 ) for r 2 D γ r 1 16

  21. Property 1 (Martingale) µ ( γ 2 | γ 1 ; D, r 1 , r 2 ) = µ ( γ 2 ; D \ γ 1 , τ, r 2 ) r 2 D γ 2 τ γ 1 r 1 17

  22. Property 2 Conformal invariance (Φ ∗ µ )( γ ; D, r 1 , r 2 ) = µ (Φ( γ ); D ′ , r ′ 1 , r ′ 2 ) r 2 r’ 2 D D’ γ Φ(γ) r r’ 1 1 18

  23. Example calculation with SLE Schramm’s formula Probability that γ passes to the left of a given point P ( ζ, ¯ ζ ; a 0 ) For infinitesimal dt , g dt : { remainder of γ } → γ ′ 19

  24. By prop. 1 and 2, it has same measure as SLE started from a dt = a 0 + √ κdB t 2 dt ζ → g dt ( ζ ) = ζ + ζ − a 0 γ ′ lies to the left of ζ ′ iff γ does of ζ   ; a 0 + √ κdB t 2 dt 2 dt �� �� P ( ζ, ¯ , ¯ ζ ; a 0 ) =  ζ + ζ + P  ¯ ζ − a 0 ζ − a 0 ↑ over Brownian motion dB t up to time dt 20

  25. Using �� dB t �� = 0 and �� ( dB t ) 2 �� = dt , one obtains ∂ 2   2 ∂ 2 ∂ ζ + κ  P ( ζ, ¯ ∂ζ + ζ ; a 0 ) = 0  ∂a 2 ¯ ∂ ¯ ζ − a 0 ζ − a 0 2 0 By scale inv., P depends only on θ = arg( ζ − a 0 ) → linear 2nd-order ordinary diff. eq. (hypergeometric) With b.c. P ( θ = π ) = 0, P ( θ = 0) = 1 → P = 1 Γ(2 / 3) � 1 2 , 2 3 , 3 � 2; − cot 2 θ 2 + √ π Γ(1 / 6)(cot θ ) 2 F 1 21

  26. 3. Critical Models

  27. κ = 2 loop-erased random walk κ = 8 / 3 self-avoiding walk ∗ κ = 3 cluster boundary in Ising model ∗ κ = 4 BCSOS model of roughening transition ∗ (4- state Potts), harmonic explorer, dual to the KT transition in XY model ∗ κ = 6 cluster boundary in critical percolation κ = 8 Peano curve associated with uniform spanning tree 22

  30. q-states Potts model   � � Z = exp  β δ s j ,s k    { s } � j,k � 1 + ( e β − 1) δ s j ,s k � � � � = { s } � j,k � ( e β − 1) b q c = � graphs q = 2 + 2 cos(8 π/κ ) 23

  32. 4. Relation to CFT

  33. Γ BCFT C Hilbert space of BCFT = { ψ Γ } on Γ � � [ dψ ] e − S [ ψ ] | ψ ′ [ dψ ′ | 0 � = Γ ] Γ � ψ Γ = ψ ′ Γ � � [ dψ ′ [ dψ ] φ (0) e − S [ ψ ] | ψ ′ | φ � = Γ ] Γ � ψ Γ = ψ ′ Γ dz � � � [ dψ ′ 2 πiz n +1 T ( z ) φ (0) e − S [ ψ ] | ψ ′ L n | φ � = Γ ] [ dψ ] Γ � ψ Γ = ψ ′ C Γ 25

  34. Insertion of a boundary condition changing operator Γ � | h � = | h t � = dµ ( γ t ) | γ t � � � Γ ; γ t [ dψ ] e − S [ ψ ] | ψ ′ [ dψ ′ | γ t � = Γ ] Γ � ψ Γ = ψ ′ dµ ( γ t ) is given by the path-integral in H . | h � is independent of t . 26

  35. Measure is also determined by SLE − √ κ dB t g t = 2 dt d ˆ ˆ g t This is an infinitesimal conformal mapping which cor- � (2 dt/z −√ κ dB t ) T ( z ). responds to the insertion (1 / 2 πi ) Thus for t 1 < t �� t 1 √ κ dB t ′ ) � 0 (2 L − 2 dt ′ − L − 1 | g t 1 ( γ t ) � = T exp | γ t � 27

  36. (measure on γ t ) = (measure on γ t \ γ t 1 , conditioned on γ t 1 ) × (measure on γ t 1 ) = (measure on g t 1 ( γ t )) × (measure on γ t 1 ) 28

  37. | h t � = √ κdB t ′ ) | g t 1 ( γ t ) � � 0 dµ ( √ κB t ′ ∈ [0 ,t 1 ] )T e t 1 (2 L − 2 dt ′ − L − 1 � � dµ ( g t 1 ( γ t )) ↓ − (2 L − 2 − κ � � 2 L 2 | h t � = exp − 1 ) t 1 | h t − t 1 � However, | h t � is independent of t . Thus (2 L − 2 − κ 2 L 2 − 1 ) | h � = 0 29

  38. ⇓ h = h 2 , 1 = 6 − κ 2 κ c = 13 − 6( κ 2 + 2 κ ) P ( ζ ; a 0 ) = � φ 2 , 1 ( a 0 ) O ( ζ ) φ 2 , 1 ( ∞ ) � � φ 2 , 1 ( a 0 ) φ 2 , 1 ( ∞ ) � 30

  39. 5. Remarks

  40. A generalization SLE( κ , � ρ ) : a minimal generalization of SLE which retains self-similarity σ − 1 g σ 2 t ( σz ) n dW t = √ κ dB t − ρ j dt � X ( j ) j =1 t 2 dt dX ( j ) = − dW t t X ( j ) t This is a special case of dW t = √ κdB t − J x t (0) dt 31

  41. (2 L − 2 − κ 2 L 2 ( J − 1 = J x − 1 − J − 1 L − 1 ) | h � = 0 0 (0)) J µ ∝ ǫ µν ∂ ν φ → κ = 4 case free field with piecewise constant Dirichlet b.c. κ � = 4 case Coulomb gas representation. 32

  42. Review articles G.F.Lawler ; An introduction to the stochastic Loewner evolution, http://www.math.duke.edu/˜jose/papers.html, 2001. W.Kager, B.Nienhuis ; A guide to stochastic L¨ owner evolution and its ap- plications, math-ph/0312056. J.Cardy ; SLE for theoretical physicists, cond-mat/0503313. 34

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