Annulus SLE partition functions and martingale-observables Joint - PowerPoint PPT Presentation

Annulus SLE partition functions and martingale-observables Joint work with Nam-Gyu Kang, Hee-Joon Tak Sung-Soo Byun Seoul National University Random Conformal Geometry and Related Fields June 18, 2018

Outline 1 Annulus SLE partition functions Annulus SLE ( κ, Λ) Null-vector equation CFT of GFF in a doubly connected domain 2 GFF with Dirichlet and Excursion-Reflected boundary conditions Eguchi-Ooguri and Ward’s equations Coulomb gas formalism Connection to SLE theory 3 One-leg operator and Insertion Martingale-observables for annulus SLE Screening Work in progress: Multiple SLEs in the annulus 4

Chordal type annulus SLE( κ, Λ ) r ˜ g t r − t ξ t = ˜ g t ( γ ( t )) Loewner Flow: ∂ t ˜ g t ( z ) = H ( r − t , ˜ g t ( z ) − ξ t ) , H ( r , z ) := 2 ∂ z log Θ( r , z ) d ξ t = √ κ dB t + Λ( r − t , ξ t − ˜ Driving process: For κ > 0, g t ( q )) dt . Annulus SLE partition function Z ( r , x ) : Λ( r , x ) = κ∂ x log Z ( r , x ) Null-vector equation (Zhan): � 3 � ∂ r Z = κ κ − 1 2 Z ′′ + H Z ′ + H ′ Z 2

The null-vector equation Null-vector equation (Zhan): � 3 � ∂ r Z = κ κ − 1 2 Z ′′ + H Z ′ + H ′ Z 2 Lawler used Brownian loop measures to define annulus SLE( κ, Λ ) and proved that the SLE partition function (total mass) satisfies the null-vector equation. (B.-Kang-Tak) For each κ > 0, � 2 Θ( r , x − ζ ) − 4 κ Θ( r , ζ ) − 4 κ d ζ Z ( r , x ) := Θ( r , x ) κ γ solves the null-vector equation. Examples κ Z ( r , · ) Θ − 1 / 2 4 Θ − 1 H 2 3 H 2 − 2 H ′ + 4 ζ r ( π ) Θ − 3 / 2 � � 4 / 3 π 4 H 3 − 6 HH ′ + H ′′ + 12 ζ r ( π ) Θ − 2 � � 1 H π

Outline 1 Annulus SLE partition functions Annulus SLE ( κ, Λ) Null-vector equation CFT of GFF in a doubly connected domain 2 GFF with Dirichlet and Excursion-Reflected boundary conditions Eguchi-Ooguri and Ward’s equations Coulomb gas formalism Connection to SLE theory 3 One-leg operator and Insertion Martingale-observables for annulus SLE Screening Work in progress: Multiple SLEs in the annulus 4

Green’s functions in a doubly connected domain Dirichlet BM and ERBM Figure: ERBM (Drenning, Lawler) Figure: BM • In the cylinder C r := { z : 0 < Im z < r } / � z �→ z + 2 π � , the Green’s function G r is represented as � �  � � Θ( r , ζ − z ) � − Im ζ · Im z  � �  log for Dirichlet b.c.  �  Θ( r , ζ − z ) r  G r ( ζ, z ) = � �   � �  Θ( r , ζ − z )  � �  log for ER b.c. � � Θ( r , ζ − z )

GFF in a doubly connected domain Dirichlet and ER boundary conditions Φ ( 0 ) : GFF − π + ir π + ir e − r 1 D r − π π w E [Φ( ζ )Φ( z )] = 2 G D r ( ζ, z ) = 2 G r ( w ( ζ ) , w ( z )) � �  � � Θ( r , ζ − z ) � − Im ζ · Im z  � �  log for Dirichlet b.c.  �  Θ( r , ζ − z ) r  G r ( ζ, z ) = � �   � �  Θ( r , ζ − z )  � �  log for ER b.c. � � Θ( r , ζ − z )

Central charge modification of GFF Φ ( 0 ) : GFF − π + ir π + ir e − r 1 D r π − π w � � κ/ 8 − 2 /κ ) and define Fix a real parameter b (= Φ ( b ) ( z ) := Φ ( 0 ) ( z ) − 2 b arg w ′ ( z ) . c = 1 − 12 b 2 = ( 6 − κ )( 3 κ − 8 ) / 2 κ The central charge is given as The Fock space fields are obtained from the GFF by applying basic operations: 1. derivatives; 2. Wick’s product ⊙ ; 3. multiplying by scalar functions and taking linear combinations.

OPE family of GFF � � Fix a real parameter b (= κ/ 8 − 2 /κ ) and define Φ ( b ) ( z ) := Φ ( 0 ) ( z ) − 2 b arg w ′ ( z ) . c = 1 − 12 b 2 = ( 6 − κ )( 3 κ − 8 ) / 2 κ The central charge is given as The Fock space fields are obtained from the GFF by applying basic operations: 1. derivatives; 2. Wick’s product ⊙ ; 3. multiplying by scalar functions and taking linear combinations. Operator product expansion (OPE) of two (holomorphic) fields X ( ζ ) and Y ( z ) are given as � C n ( z )( ζ − z ) n , X ( ζ ) Y ( z ) = ζ → z . In particular, the OPE multiplication X ∗ Y := C 0 . OPE family F ( b ) of the Φ ( b ) : the algebra (over C ) spanned by the generators 1 , mixed derivatives of Φ ( b ) , those of OPE exponentials e ∗ α Φ ( b ) ( α ∈ C )

Ward’s equation in doubly connected domain Stress energy tensor A ( b ) : � � A ( b ) := − 1 2 J ( 0 ) ⊙ J ( 0 ) + ib ∂ − E [ J ( b ) ] J ( 0 ) , J ( b ) = ∂ Φ ( b ) . Theorem (B.-Kang-Tak) For any string X of fields in the OPE family F ( b ) , we have � � � � L + v ζ + L − 2 E A ( b ) ( ζ ) X = E [ X ] + ∂ r E [ X ] , v ¯ ζ where all fields are evaluated in the identity chart of C r and the Loewner vector field v ζ is given by C r )( z ) = H ( r , ζ − z ) = 2 Θ ′ ( r , ζ − z ) ( v ζ � id ¯ Θ( r , ζ − z ) . Cf. On a complex torus of genus one, similar form of Ward’s equation holds. Eguchi-Ooguri: Conformal and current algebras on a general Riemann surface , Nuclear Phys. B, 282(2):308-328, 1987. Kang-Makarov: Calculus of conformal fields on a compact Riemann surface, arXiv:1708.07361, 86 pp.

Eguchi-Ooguri’s type equation in a doubly connected domain Lemma For any string X of fields in the OPE family F ( b ) , in C r , � 1 E [ A ( ζ ) X ] d ζ = ∂ r E [ X ] . π [ − π + ir ,π + ir ] Ingredients of proof 1 Heat equation of Jacobi theta function: ∂ r Θ( r , z ) = Θ( r , z ) ′′ . 2 Frobenius-Stickelberger’s pseudo-addition theorem for Weierstrass ζ -function: � � 2 + ζ ′ ( z 1 )+ ζ ′ ( z 2 )+ ζ ′ ( z 3 ) = 0 , ζ ( z 1 )+ ζ ( z 2 )+ ζ ( z 3 ) ( z 1 + z 2 + z 3 = 0 ) .

Neutrality condition and multi-vertex field Given divisors σ = � n j = 1 σ j · z j , σ ∗ = � n j = 1 σ j ∗ · z j , we set n � σ j Φ + ( b ) ( z j ) − σ ∗ j Φ − Φ ( b ) [ σ , σ ∗ ] := ( b ) ( z j ) , j = 1 where Φ ( b ) = Φ + ( b ) + Φ − ( b ) , Φ − ( b ) = Φ + ( b ) . Then Φ ( b ) [ σ , σ ∗ ] is a well-defined Fock space field if and only if the following neutrality condition (NC 0 ) holds: � n ( σ j + σ ∗ j ) = 0 j = 1 We define the multi-vertex field O [ σ , σ ∗ ] ≡ O ( b ) [ σ , σ ∗ ] by O ( b ) [ σ , σ ∗ ] = C ( b ) [ σ , σ ∗ ] e ⊙ i Φ ( 0 ) [ σ , σ ∗ ] where C ( b ) [ σ , σ ∗ ] is Coulomb gas correlation function .

Outline 1 Annulus SLE partition functions Annulus SLE ( κ, Λ) Null-vector equation CFT of GFF in a doubly connected domain 2 GFF with Dirichlet and Excursion-Reflected boundary conditions Eguchi-Ooguri and Ward’s equations Coulomb gas formalism Connection to SLE theory 3 One-leg operator and Insertion Martingale-observables for annulus SLE Screening Work in progress: Multiple SLEs in the annulus 4

One-leg operator We choose real parameters a and b in terms of SLE parameter κ as � � � a = 2 /κ, b = κ/ 8 − 2 /κ. Given a divisor β = � N j = 1 β j · q j define the one-leg operator Ψ ≡ Ψ β by Ψ β ( p , q ) := O [ a · p + β , 0 ] . Now we consider SLE ( κ, Λ) , where Λ is given by � � Λ( r , p , q ) := κ ∂ ξ ξ = p log E [Ψ( ξ, q )] d ξ t = √ κ dB t + Λ( r − t , ξ t − ˜ g t ( q 1 ) , · · · , ξ t − ˜ g t ( q N )) dt . i.e., r ˜ g t r − t

Insertion Using Ψ as an insertion field, set � � ( 0 ) ( p )+ i � β j Φ + E [ X ] := E [Ψ( p , q ) X ] ⊙ ia Φ + ( 0 ) ( q j ) X � E [Ψ( p , q )] = E . e Example. Suppose that q j ’s are on the outer boundary component. In the cylinder C r , E [Φ ( b ) ]( z ) = 0

Insertion Using Ψ as an insertion field, set � � ( 0 ) ( p )+ i � β j Φ + E [ X ] := E [Ψ( p , q ) X ] ⊙ ia Φ + ( 0 ) ( q j ) X � E [Ψ( p , q )] = E . e Example. Suppose that q j ’s are on the outer boundary component. In the cylinder C r , � E [Φ ( b ) ]( z ) = 2 a arg Θ( r , p − z ) � + 2 β j arg Θ( r , q j − z ) . � E [Φ ( b ) ] has piecewise Dirichlet boundary condition with jump 2 a π at p , 2 πβ j at q j and by NC 0 all jumps add up to 0. Izyurov-Kyt¨ ol¨ a: Hadamard’s formula and couplings of SLEs with free field . Probab. Theory Related Fields, 155(1-2):35-69, 2013.

Martingale-observables for annulus SLE By definition, a non-random field M is a martingale-observable for annulus SLE ( κ, Λ) if for any z 1 , · · · , z n , the process � � w − 1 M t ( z 1 , · · · , z n ) := M � ˜ ( z 1 , · · · , z n ) , w t = ˜ ˜ g t − ξ t t is a local martingale on the SLE probability space. Theorem (B-Kang-Tak) For any string X of fields in the OPE family F ( b ) of Φ ( b ) , the non-random fields M = � E X are martingale-observables for SLE ( κ, Λ) . Idea of proof Ward’s equation + level 2 degeneracy equation for Ψ ⇒ BPZ-Cardy equation ⇔ M t is driftless.

Screening Goal To find explicit solutions of Zhan’s PDE for Z ✄ � Idea : Z ⇐ ⇒ E Ψ ✂ ✁ Consider an chordal type annulus SLE ( κ, Λ) in cylinder from p to q . p q

Screening Goal To find explicit solutions of Zhan’s PDE for Z ✄ � Idea : Z ⇐ ⇒ E Ψ ✂ ✁ Ψ( p , q ) = O [? · p +? · q ] The conformal dim λ z at z with charge σ is given as λ z = σ 2 / 2 − σ b . Candidate 1 Candidate 2 To satisfy level-2 degeneracy eq., 2 b − a λ p = a 2 / 2 − ab . p a