Radial Conformal Field Theory Joint work with Nikolai G. Makarov - PowerPoint PPT Presentation

SNU Radial Conformal Field Theory Joint work with Nikolai G. Makarov Nam-Gyu Kang Department of Mathematical Science, Seoul National University Conformal maps from probability to physics May 23-28, 2010 Frame: 0/ 24

Outline SNU ◮ Gaussian free field and conformal field theory ◮ Probabilistic setting for CFT. ◮ Calculus of CFT and the source of tensor structures of conformal fields. ◮ Fields = certain types of Fock space fields + tensor nature. ◮ We use “conformal invariance” to denote consistence with conformal structures. ◮ We treat a stress energy tensor in terms of Lie derivatives. ◮ Radial conformal field theory ◮ In radial CFT, several trivial fields are multi-valued. ◮ 2 types of radial CFT and relation to SLE. ◮ Twisted radial CFT. Frame: 1/ 24

Gaussian Free Field Φ and its approximation Φ n SNU ◮ Φ : Gaussian Free Field ∞ � Φ = a n f n . n = 1 ◮ f n : O.N.B. for W 1 , 2 0 ( D ) with Dirichlet inner product. ◮ D : a hyperbolic R.S. ◮ a n : i.i.d. ∼ N ( 0 , 1 ) . ◮ E [Φ( z )] = 0 . ◮ E [Φ( z )Φ( w )] = 2 G ( z , w ) . �� ◮ var (Φ( f )) = 2 G ( z , w ) f ( z ) f ( w ) . √ 2 � n ◮ Φ n ( z ) = j = 1 ( G ( z , λ j ) − G ( z , µ j )) . Frame: 2/ 24

Poles of Φ n Ginibre eigenvalues √ 2 � n ◮ Φ n ( z ) = SNU j = 1 ( G ( z , λ j ) − G ( z , µ j )) . ◮ { λ j } n j = 1 : eigenvalues of the Ginibre ensemble, { µ j } n j = 1 : an independent copy. ◮ Ginibre ensemble is the n × n random matrix ( a j , k ) n j , k = 1 . ◮ a j , k : i.i.d. complex Gaussians with mean zero and variance 1 / n . law ◮ Φ n ( f ) → Φ( f ) . (Y. Ameur, H. Hedenmalm, and N. Makarov) Figure: Ginibre eigenvalues and uniform points ( n = 4096) Frame: 3/ 24

Boundary Conditions Chordal case SNU = + √ H λ ( z ) = 2 λ ( arg ( 1 + z ) − arg ( 1 − z )) Frame: 4/ 24

Level Lines of GFF n Chordal case SNU Figure: Φ n ( z ) + H ( λ = 1 ) ( z ) = 0 . Frame: 5/ 24

Conjecture: Zero Set = Chordal SLE 4 motivated by O. Schramm and S. Sheffield’s SNU Figure: Φ n ( z ) + H ( λ = 1 ) ( z ) = 0 . Frame: 6/ 24

Harmonic Explorer Radial case SNU Frame: 7/ 24

Harmonic Explorer Radial case SNU Frame: 8/ 24

Harmonic Explorer Radial case SNU Frame: 9/ 24

Harmonic Explorer Radial case SNU Figure: As the mesh gets finer, does the HE converge? Frame: 10/ 24

Radial HE and Radial SLE 4 N. Makarov and D. Zhan SNU Figure: As the mesh gets finer, the HE converges to radial SLE 4 . Frame: 11/ 24

Level Lines and Zero Set Radial case (2 covers) SNU n ( z ) := 1 Φ odd 2 (Φ n ( z ) − Φ n ( − z )) . Frame: 12/ 24

Level Lines and Zero Set Radial case (Twisted boundary conditions) SNU √ √ n ( ±√ z ) = ± n ( √ z ) . Φ tw 2 Φ odd 2 Φ odd n ( z ) = Frame: 13/ 24

Fock space fields SNU Fock space fields ( F-fields ) are obtained from GFF by applying the following basic operations: i. derivatives; ii. Wick’s products; iii. multiplying by scalar functions and taking linear combinations. Examples J ⊙ ¯ J = ∂ Φ , Φ ⊙ Φ , J ⊙ Φ , J ⊙ J , J . Correlations (at distinct points) are defined for any finite collections of Fock fields: (i) by differentiation; (ii) by Wick’s yoga; (iii) by linearity. Examples ◮ E [ J ( ζ )Φ( z )] = 2 ∂ ζ G ( ζ, z ) . ◮ E [Φ ⊙ 2 ( ζ )Φ( z 1 )Φ( z 2 )] = 2 E [Φ( ζ )Φ( z 1 )] E [Φ( ζ )Φ( z 2 )] . ◮ E [ e ⊙ Φ( ζ ) � � ∞ α n Φ ⊙ n ( z ) � ] = e | α | 2 E [Φ( ζ )Φ( z )] = e 2 | α | 2 G ( ζ, z ) n = 0 n ! Frame: 14/ 24

Conformal geometry and conformal invariance SNU Definition We consider a non-random field f on a Riemann surface. We say that a. f is a differential of degree ( d , d # ) if for two overlapping charts φ and ˜ φ, we have f = ( h ′ ) d (¯ h ′ ) d # � f ◦ h , where f is the notation for ( f � φ ) , ˜ f for ( f � ˜ φ ) , and h is the transition map. b. f is a PS-form (pre-Schwarzian form) or 1-from of order µ if ( N h = h ′′ / h ′ = ( log h ′ ) ′ ); f = ( h ′ ) 1 � f ◦ h + µ N h c. f is an S-form (Schwarzian form) or 2-from of order µ if f = ( h ′ ) 2 � ( S h = N ′ h − N 2 f ◦ h + µ S h h / 2 ) . A field X is invariant wrt to some conformal automorphism τ of M if E [( X � φ )Φ( p 1 ) ⊙ · · · ⊙ Φ( p n )] = E [( X � φ ◦ τ − 1 )Φ( τ p 1 ) ⊙ · · · ⊙ Φ( τ p n )] . Conformal invariance allows to define fields in conformally equivalent situations. Frame: 15/ 24

Lie derivative SNU ψ t M t M φ ◦ ψ t φ Suppose M is a Riemann surface. Consider a (local) flow of a vector field v on M ˙ ψ t : M → M , ψ 0 ( z ) = v ( z ) . Suppose X is a field in M and v is holomorphic in U = U v ⊂ M . By definition, the field X t in U is ( X t ( z ) � φ ) = ( X ( ψ t z ) � φ ◦ ψ − t ) . Definition � L v X = d � t = 0 X t . � dt Frame: 16/ 24

Stress tensor Abstract theory SNU ◮ A pair of quadratic differentials W = ( A + , A − ) is called a stress tensor for X if “residue form of Ward’s identity” holds: � � 1 1 L v X ( z ) = vA + X ( z ) − ¯ vA − X ( z ) . 2 π i 2 π i ( z ) ( z ) Notation: F ( W ) is the family of fields with stress tensor W = ( A + , A − ) . 2 J ⊙ J , W = ( A , ¯ ◮ For A = − 1 A ) is a stress tensor for GFF Φ . ◮ If X , Y ∈ F ( W ) , then ∂ X , X ∗ Y ∈ F ( W ) . Frame: 17/ 24

Virasoro field Abstract theory SNU Definition Let W = ( A , ¯ A ) be a stress tensor. A Fock space field T is the Virasoro field for the family F ( W ) if ◮ T − A is a non-random holomorphic Schwarzian form, ◮ T ∈ F ( W ) . Example Twisted Radial CFT � � 1 − √ ζ ¯ � � 1 + ζ/ z z � � ◮ G ( ζ, z ) = log 1 + √ ζ ¯ � � z 1 − ζ/ z ◮ T = − 1 2 J ∗ J = A + S , where A = − 1 2 J ⊙ J . �� � w ′ ( ζ ) w ′ ( z ) � ◮ E [ J ( ζ ) J ( z )] = − 1 w ( ζ ) / w ( z ) + w ( z ) / w ( ζ ) . ( w ( ζ ) − w ( z )) 2 2 w ′ 2 ( ζ − z ) 2 − 1 1 S = T − A = 3 ◮ E [ J ( ζ ) J ( z )] = − 6 S ( ζ, z ) , w 2 + S w . 4 Frame: 18/ 24

Ward equation in D SNU Proposition (Ward equation) In D , for a string X of differentials in F ( W ) , we obtain ζ 2 + 2 ζ z j − z 2 � � � 1 z j ζ + z j j E [ T ( ζ ) X ] = E [ T ( ζ )] E [ X ] + ζ − z j ∂ j + d j E [ X ] 2 ζ 2 ( ζ − z j ) 2 j ζ ∗ 2 + 2 ¯ � ζ ∗ + ¯ � � ¯ ¯ ζ ∗ ¯ z 2 z j − ¯ 1 z j ∂ j + d # ¯ j + z j ¯ E [ X ] . ζ ∗ − ¯ ¯ j (¯ ζ ∗ − ¯ 2 ζ 2 z j ) 2 z j j ◮ Consider a vector field v ζ ( z ) = z ζ + z ζ − z . ◮ The reflection of a vector field in ∂ D is defined by v # ( z ) = − v ( 1 / ¯ z ) z 2 . ζ = v ζ ∗ and ζ ∗ := 1 / ¯ ◮ v # ζ is the symmetric point of ζ with respect to ∂ D . ◮ ¯ ∂ v ζ = − 2 πζ 2 δ ζ . Frame: 19/ 24

SLE numerology SNU ◮ SLE κ map g t ( z ) : D t → D ∂ g t ( z ) = g t ( z ) 1 + k ( t ) g t ( z ) 1 − k ( t ) g t ( z ) , g 0 ( z ) = z , where k ( t ) = e − i √ κ B t . Set w t ( z ) = k ( t ) g t ( z ) . ◮ B t : a 1- D standard Brownian motion on R , B 0 = 0 . ◮ SLE hulls: K t := { z ∈ D : τ ( z ) ≤ t } .   ( e i √ κ B t ) . � g t ◮ SLE path: γ t = γ [ 0 , t ] , where γ ( t ) = g − 1 t ◮ When κ = 4 , we consider Makarov-Zhan’s martingale observable � ϕ t ( z ) = 2 a arg 1 + w t ( z ) � , 1 − w t ( z ) √ where a = ± 1 / 2 . Frame: 20/ 24

Boundary conditions Insertion of a chiral vertex SNU E p [ X ] = E [ e ⊙ ia Φ † ( q , p ) X ] and let � Denote � X p denote the string X of F-fields under the boundary condition with u = − 2 a arg 1 + √ w 1 − √ w w : ( D , q , p ) → ( D , 0 , 1 ) . w q 0 1 p � E p [ X ] = E [ � Lemma X p ] . � � � 1 − √ ζ ¯ � � 1 + ζ/ z z � � Main idea. Recall that E [Φ( ζ )Φ( z )] = 2 log 1 + √ ζ ¯ � � in D . Thus � � � z 1 − ζ/ z � tw ( q , z ) = − 2 i arg 1 + w ( z ) E [Φ † ( q , p )Φ( z )] = G † tw ( p , z ) − G † � . 1 − w ( z ) Frame: 21/ 24

Wick meets Itˆ o and Schramm SNU Suppose A j ’s are conformally invariant (holomorphic differentials) in F ( W ) . For every ( D , q ) consider R p ( z 1 , · · · , z n ) = � E p [ A 1 ( z 1 ) · · · A n ( z n )] , z j ∈ D , p ∈ ∂ D . Denote M t : = � E γ ( t ) [ A 1 D t ( z 1 ) · · · ]   t ( z 1 ) d 1 · · · � ( = w ′ E 1 [ A D ( w t ( z 1 )) , · · · ]) � w t Then M t is a (local) martingale, or the CFT F ( W ) does not change under SLE κ evolution. Main idea: conformal invariance + degeneracy at level two θ R e i θ = − 1 ∂ 2 2 L v R e i θ , v = v e i θ . Frame: 22/ 24

Wick meets Itˆ o and Schramm Hadamard’s formula SNU ◮ A 1-pt function ϕ ( z ) := � � E [Φ( z )] is a martingale-observable. ◮ A 2-pt function  � E [Φ( z 1 )Φ( z 2 )] = � ϕ ( z 1 ) � ϕ ( z 2 ) + 2 G ( z 1 , z 2 )  � w t is a martingale-observable. ◮ Equating the drifts, 2 dG t ( z 1 , z 2 ) = − d � ϕ ( z 1 ) , ϕ ( z 2 ) � t � � w t ( z 1 ) w t ( z 2 ) = − 8 ℜ 1 − w t ( z 1 ) ℜ 1 − w t ( z 2 ) dt . Frame: 23/ 24