SLE and CFT. Mitsuhiro QFT2005

Transcription

1 SLE and CFT Mitsuhiro 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

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6 Crossing probability C 1 C 4 z 1 z 4 unit disc C 2 C 3 z z 2 3 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 ) P = Γ(2 3 ) Γ( 4 3 )Γ(1 3 ) η1/3 2F 1 ( 1 3, 2 3, 4 3 ; η) 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, is called a hull. g (z) K K = K H H\K H K 6

11 Conformal map For any hull K, there exists a unique conformal map g K : H \ K H lim z (g K(z) z) = 0 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 Loewner equation g t (z) H t H K t = H \ H t U t = g t (γ(t)) Let γ(t) be parametrized s.t. a 1 (K t ) = 2t. Then t g t(z) = 2 g t (z) U t, g 0 (z) = z 8

13 example U t = 0 case t g t(z) = 2 g t (z), g 0(z) = z g t (z) = z 2 + 4t γ(t) = 2i t 9

14 SLE t g t(z) = 2 g t (z) κb t, g 0 (z) = z where B t is standard Brawnian motion on R, κ is a real parameter. Alternatively, for ĝ t (z) = g t (z) κb t dĝ t (z) = 2 ĝ t (z) dt κdb t 10

15 Brawnian motion For U t = κb t U t = 0, U t1 U t2 = κ t 1 t 2 Thus du t du t = κdt 11

16 Itô 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 ) df = (af b2 f )dt + bf db t 12

17 SLE trace γ(t) := lim z 0 g 1 t (z + κb t ) 13

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

19 Hausdorff dimensions d H = 1 + κ/8 (κ < 8) 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 γ 1 τ 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 1 r 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 ζ g dt (ζ) = ζ + 2dt ζ a 0 γ lies to the left of ζ iff γ does of ζ P (ζ, ζ; a 0 ) = P ζ + 2dt ζ a 0, ζ + 2dt ; a 0 + κdb t 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 ζ a 0 ζ + 2 ζ a 0 ζ + κ 2 2 a 2 0 P (ζ, ζ; a 0 ) = 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 = Γ(2/3) πγ(1/6) (cot θ) 2 F 1 ( 1 2, 2 3, 3 2 ; cot2 θ ) 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

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30 q-states Potts model Z = {s} exp = {s} j,k = β δ sj,s k j,k (1 + (e β 1)δ sj,s k ) (e β 1) b q c graphs q = cos(8π/κ) 23

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32 4. Relation to CFT

33 BCFT Hilbert space of BCFT = {ψ Γ } on Γ C Γ 0 = [dψ Γ ] ψ Γ =ψ Γ [dψ]e S[ψ] ψ Γ φ = [dψ Γ ] ψ Γ =ψ Γ [dψ]φ(0)e S[ψ] ψ Γ L n φ = [dψ Γ ] ψ Γ =ψ Γ [dψ] C dz 2πi zn+1 T (z)φ(0)e S[ψ] ψ Γ 25

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

35 Measure is also determined by SLE dĝ t = 2dt ĝ t κ db t This is an infinitesimal conformal mapping which corresponds to the insertion (1/2πi) (2dt/z κ db t )T (z). Thus for t 1 < t g t1 (γ t ) = T exp ( t1 0 (2L 2dt L 1 κ dbt ) ) γ t 27

36 (measure on γ t ) = (measure on γ t \ γ t1, conditioned on γ t1 ) (measure on γ t1 ) = (measure on g t1 (γ t )) (measure on γ t1 ) 28

37 h t = dµ(g t1 (γ t )) dµ( κb t [0,t 1 ] )Te 0 t1 (2L 2 dt L 1 κdbt ) gt1 (γ t ) h t = exp ( (2L 2 κ ) 2 L2 1 )t 1 h t t1 However, h t is independent of t. Thus (2L 2 κ 2 L2 1) h = 0 29

38 h = h 2,1 = 6 κ 2κ c = 13 6( κ 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) dw t = κ db t n dx (j) t = 2dt X (j) t This is a special case of dw t j=1 ρ j dt X (j) t dw t = κdb t J x t (0)dt 31

41 (2L 2 κ 2 L2 1 J 1 L 1 ) h = 0 (J 1 = J x 0(0)) J µ ɛ 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, jose/papers.html, W.Kager, B.Nienhuis; A guide to stochastic Löwner evolution and its applications, math-ph/ J.Cardy; SLE for theoretical physicists, cond-mat/