Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra Kalle Kytölä ❦❛❧❧❡✳❦②t♦❧❛❅❛❛❧t♦✳❢✐ Department of Mathematics and Systems Analysis, Aalto University joint work with Clément Hongler (EPFL, Lausanne) Fredrik Viklund (KTH, Stockholm) June 22, 2018 — KIAS, "Random Conformal Geometry and Related Fields"
Outline 1. Introduction: Conformal Field Theory and Virasoro algebra 2. Main results: local fields of probabilistic lattice models form Virasoro representations ◮ discrete Gaussian free field ◮ Ising model 3. An algebraic theme and variations (Sugawara construction) 4. Proof steps (discrete complex analysis) Conformal Field Theory on the lattice Kalle Kytölä
1. I NTRODUCTION Conformal Field Theory on the lattice I. Introduction Kalle Kytölä
Intro: Two-dimensional statistical physics (uniform spanning tree) (Ising model) (percolation) etc. etc. Conformal Field Theory on the lattice I. Introduction Kalle Kytölä
Intro: Conformally invariant scaling limits Conventional wisdom: Any interesting scaling limit of any two-dimensional random lattice model is conformally invariant: ◮ interfaces − → SLE-type random curves ◮ correlations − → CFT correlation functions Remarks: ◮ SLE: Schramm-Loewner Evolution * [cf. the other talks] ◮ CFT: Conformal Field Theory * powerful algebraic structures (Virasoro algebra, modular invariance, quantum groups, . . . ) * exact solvability (critical exponents, PDEs for correlation fns, . . . ) * mysteries — what is CFT, really? ◮ This talk: concrete probabilistic role for Virasoro algebra Conformal Field Theory on the lattice I. Introduction Kalle Kytölä
Intro: The role of Virasoro algebra Virasoro algebra: ∞ -dim. Lie algebra, basis L n ( n ∈ Z ) and C [ L n , L m ] = ( n − m ) L n + m + n 3 − n 12 δ n + m , 0 C [ C , L n ] = 0 ( C a central element ) Role of Virasoro algebra in CFT? ◮ stress tensor T : first order response to variation of metric (in particular “infinitesimal conformal transformations”) ◮ Laurent modes of stress tensor T ( z ) = � n ∈ Z L n z − 2 − n ◮ C acts as c × id, with c ∈ R the “central charge” of the CFT ◮ action on local fields (effect of variation of metric on correlations) ◮ local fields form a Virasoro representation ◮ highest weights of the representation � critical exponents ◮ degenerate representations � PDEs for correlations (exact solvability & classification) Conformal Field Theory on the lattice I. Introduction Kalle Kytölä
II. L OCAL FIELDS IN LATTICE MODELS Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
The critical Ising model on Z 2 ◮ domain Ω � C open, 1-connected ◮ δ > 0 small mesh size ◮ lattice approximation Ω δ ⊂ C δ := δ Z 2 Ising model : random spin configuration � � z ∈ C δ ∈ { + 1 , − 1 } C δ σ = σ z � C δ \ Ω δ ≡ + 1 (plus-boundary conditions) σ � � � � � P { σ } ∝ exp − β E ( σ ) (Boltzmann-Gibbs) � E ( σ ) = − σ z σ w (energy) � z − w � = δ � √ β = β c = 1 � 2 log 2 + 1 (critical point) Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
Celebrated scaling limits of Ising correlations � � ℑ m ( z ) > 0 } conformal map φ : Ω → H = { z ∈ C Thm [Chelkak & Hongler & Izyurov, Ann. Math. 2015] k 1 � � � lim δ k / 8 E σ z j δ → 0 j = 1 k | φ ′ ( z j ) | 1 / 8 × C k � � � = φ ( z 1 ) , . . . , φ ( z k ) j = 1 ↓ Thm [Hongler & Smirnov, Acta Math. 2013] [Hongler, 2011] � m �� 1 − σ z j σ z j + δ + 1 � � lim δ m E √ z 1 δ → 0 2 z 2 j = 1 m z 3 � | φ ′ ( z j ) | × E m � � = φ ( z 1 ) , . . . , φ ( z m ) j = 1 z 4 + [Gheissari & Hongler & Park, 2013 — Sung Chul’s talk] + [Chelkak & Hongler & Izyurov, 2018+ — Kostya’s talk] Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
Local fields of the Ising model � � σ = σ z Ising Local fields F ( z ) of Ising z ∈ Ω δ ◮ V ⊂ Z 2 finite subset ◮ P : { + 1 , − 1 } V → C a function ◮ F ( z ) = P � ( σ z + δ x ) x ∈ V � � F space of local fields Null fields: “zero inside correlations” ◮ F ( z ) null field: � � F ( z ) � n ∃ R > 0 s.t. E j = 1 σ w j = 0 Examples of local fields: whenever � z − w j � 1 > R δ ∀ j * F ( z ) = σ z (spin) � N ⊂ F space of null fields * F ( z ) = − σ z σ z + δ (energy) F / N — equivalence classes of local fields, “same correlations” Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
Main result 1: Virasoro action on Ising local fields Theorem (Hongler & K. & Viklund, 2017) The space F / N of correlation equivalence classes of local fields of the critical Ising model on Z 2 forms a representation of the Virasoro algebra with central charge c = 1 2 . Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
Discrete Gaussian Free Field on Z 2 Discrete Gaussian Free Field (dGFF): � � Φ = Φ( z ) z ∈ Ω δ Domain and discretization: ◮ Ω � C open, simply connected ◮ lattice approximation: Ω δ ⊂ C δ := δ Z 2 ◮ centered Gaussian field on vertices of discrete domain Ω δ 1 � � p ( φ ) ∝ exp − 16 π E ( φ ) probability density � � 2 � φ ( z ) − φ ( w ) E ( φ ) = “Dirichlet energy” � z − w � = δ Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
Local fields of the dGFF � � Local fields F ( z ) of dGFF Φ = Φ( z ) dGFF z ∈ Ω δ ◮ V ⊂ Z 2 finite subset ◮ P : R V → C polynomial function Examples of local fields: * F ( z ) = Φ( z ) ◮ F ( z ) = P � (Φ( z + δ x )) x ∈ V � * F ( z ) = 1 2 Φ( z + δ ) − 1 2 Φ( z − δ ) � F space of local fields * F ( z ) = 361 Φ( z ) 3 Null fields: “zero inside correlations” ◮ F ( z ) null field: � � F ( z ) � n ∃ R > 0 s.t. E j = 1 Φ( w j ) = 0 whenever � z − w j � 1 > R δ ∀ j � N ⊂ F space of null fields F / N — equivalence classes of local fields, “same correlations” Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
Main result 2: Virasoro action on dGFF local fields Theorem (Hongler & K. & Viklund, 2017) The space F / N of correlation equivalence classes of local fields of the discrete Gaussian free field on Z 2 forms a representation of the Virasoro algebra with central charge c = 1. Conformal Field Theory on the lattice II. Local fields in lattice models Kalle Kytölä
III. A N ALGEBRAIC THEME AND VARIATIONS (S UGAWARA CONSTRUCTION ) Conformal Field Theory on the lattice III. Algebraic theme and variations Kalle Kytölä
Bosonic Sugawara construction commutator [ A , B ] := A ◦ B − B ◦ A Proposition (bosonic Sugawara construction) ◮ V vector space and a j : V → V linear for each j ∈ Z Suppose: ◮ ∀ v ∈ V ∃ N ∈ Z : j ≥ N = ⇒ a j v = 0 ◮ [ a i , a j ] = i δ i + j , 0 id V � � L n := 1 a j ◦ a n − j + 1 a n − j ◦ a j for n ∈ Z Define: 2 2 j ≥ 0 j < 0 ◮ L n : V → V is well defined Then: ◮ [ L n , L m ] = ( n − m ) L n + m + n 3 − n δ n + m , 0 id V 12 ∴ V Virasoro representation, central charge c = 1 Conformal Field Theory on the lattice III. Algebraic theme and variations Kalle Kytölä
Fermionic Sugawara construction 1 commutator [ A , B ] := A ◦ B − B ◦ A anticommutator [ A , B ] + := A ◦ B + B ◦ A Proposition (fermionic Sugawara, Neveu-Schwarz sector) ◮ V vector space, b k : V → V linear for each k ∈ Z + 1 2 Suppose: ◮ ∀ v ∈ V ∃ N ∈ Z k ≥ N = ⇒ b k v = 0 : ◮ [ b k , b ℓ ] + = δ k + ℓ, 0 id V � 1 � � 1 � � � L n := 1 b n − k b k − 1 2 + k 2 + k ( n ∈ Z ) b k b n − k Def.: 2 2 k > 0 k < 0 ◮ L n : V → V is well defined Then: ◮ [ L n , L m ] = ( n − m ) L n + m + n 3 − n δ n + m , 0 id V 24 V Virasoro representation, central charge c = 1 ∴ 2 Conformal Field Theory on the lattice III. Algebraic theme and variations Kalle Kytölä
Fermionic Sugawara construction 2 commutator [ A , B ] := A ◦ B − B ◦ A anticommutator [ A , B ] + := A ◦ B + B ◦ A Proposition (fermionic Sugawara, Ramond sector) ◮ V vector space, b j : V → V linear for each j ∈ Z Suppose: ◮ ∀ v ∈ V ∃ N ∈ Z : j ≥ N = ⇒ b j v = 0 ◮ [ b i , b j ] + = δ i + j , 0 id V � 1 � 1 L n := 1 � b n − j b j − 1 � � � ( n ∈ Z \ { 0 } ) 2 + j 2 + j b j b n − j 2 2 j ≥ 0 j < 0 Def.: L 0 := 1 j b − j b j + 1 � 16 id V 2 j > 0 ◮ L n : V → V is well defined Then: ◮ [ L n , L m ] = ( n − m ) L n + m + n 3 − n δ n + m , 0 id V 24 V Virasoro representation, central charge c = 1 ∴ 2 Conformal Field Theory on the lattice III. Algebraic theme and variations Kalle Kytölä
IV. P ROOF STEPS ( DISCRETE COMPLEX ANALYSIS ) Conformal Field Theory on the lattice IV. Proof steps: discrete complex analysis Kalle Kytölä
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