Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra

Transcription

1 Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra Department of Mathematics and Systems Analysis, Aalto University joint work with March 5, 8 Columbia University Clément Hongler (EPFL, Lausanne) Fredrik Viklund (KTH, Stockholm)

2 Outline. Introduction: Conformal Field Theory and Virasoro algebra. 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)

3 . INTRODUCTION I. Introduction

4 Intro: Two-dimensional statistical physics (uniform spanning tree) (percolation) etc. etc. (Ising model) I. Introduction

5 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 * is not today s topic 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 I. Introduction

6 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 + n3 n δ n+m,c [C, L n] = Role of Virasoro algebra in CFT? (C a central element) 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 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) I. Introduction

7 II. LOCAL FIELDS IN LATTICE MODELS II. Local fields in lattice models

8 The critical Ising model domain Ω C open, -connected δ > small mesh size lattice approximation Ω δ C δ := δz Ising model: random spin configuration σ = ( σ z )z C δ {+, } C δ σ Cδ \Ω δ + P [ {σ} ] exp ( ) β E(σ) (plus-boundary conditions) (Boltzmann-Gibbs) E(σ) = z w σ z σ w (energy) β = β c = log ( + ) (critical point) II. Local fields in lattice models

9 Celebrated scaling limits of Ising correlations φ: Ω H = {z C Im(z) > } conformal map Thm [Chelkak & Hongler & Izyurov, Ann. Math. 5] = lim δ [ k ] δ E σ k/8 zj j= k φ (z j ) /8 ( C k φ(z ),..., φ(z k ) ) j= z z 4 z z 3 Thm [Hongler & Smirnov, Acta Math. 3] [Hongler, ] = lim δ δ m E [ m j= ( σ zj σ zj +δ + ) ] m φ ( (z j ) E m φ(z ),..., φ(z ) m) j= II. Local fields in lattice models

10 Local fields of the Ising model Local fields F(z) of Ising V Z finite subset P : {+, } V C a function F(z) = P ( (σ z+δx ) x V ) (makes sense when Ω δ is large enough) F space of local fields Null fields: zero inside correlations F(z) null field: [ R > s.t. E F(z) ] n j= σw j whenever z w j > Rδ j N F space of null fields = σ = ( σ z )z Ω δ Ising Examples of local fields: F(z) = σ z (spin) F(z) = σ z σ z+δ (energy) F/N equivalence classes of local fields (same correlations) II. Local fields in lattice models

11 Main result : Virasoro action on Ising local fields Theorem (Hongler & K. & Viklund, 7) The space F/N of correlation equivalence classes of local fields of the Ising model forms a representation of the Virasoro algebra with central charge c =. II. Local fields in lattice models

12 Discrete Gaussian Free Field 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 := δz centered Gaussian field on vertices of discrete domain Ω δ probability density p(φ) exp ( E(φ)) E(φ) = ( ) z w φ(z) φ(w) Dirichlet energy II. Local fields in lattice models

13 Local fields of the dgff Local fields F(z) of dgff V Z finite subset P : R V C polynomial function F(z) = P ( (Φ(z + δx)) x V ) (makes sense when Ω δ is large enough) F space of local fields Null fields: zero inside correlations F(z) null field: [ R > s.t. E F(z) ] n j= σw j whenever z w j > Rδ j N F space of null fields = Φ = ( Φ(z) ) dgff z Ω δ Examples of local fields: F(z) = Φ(z) F(z) = Φ(z+δ) Φ(z δ) F(z) = 36 Φ(z) 3 F/N equivalence classes of local fields (same correlations) II. Local fields in lattice models

14 Main result : Virasoro action on dgff local fields Theorem (Hongler & K. & Viklund, 7) The space F/N of correlation equivalence classes of local fields of the dgff forms a representation of the Virasoro algebra with central charge c =. II. Local fields in lattice models

15 III. AN ALGEBRAIC THEME AND VARIATIONS (SUGAWARA CONSTRUCTION) III. Algebraic theme and variations

16 Bosonic Sugawara construction commutator [A, B] := A B B A Proposition (bosonic Sugawara construction) Suppose: V vector space and a j : V V linear for each j Z v V N Z : j N = a j v = [a i, a j ] = i δ i+j, id V Define: L n := a j a n j + a n j a j j< j Then: L n : V V is well defined [L n, L m] = (n m) L n+m + n3 n δ n+m, id V for n Z V Virasoro representation, central charge c = III. Algebraic theme and variations

17 Fermionic Sugawara construction commutator [A, B] := A B B A anticommutator [A, B] + := A B + B A Proposition (fermionic Sugawara, Neveu-Schwarz sector) Suppose: V vector space, b k : V V linear for each k Z + v V N Z : k N = b k v = [b k, b l ] + = δ k+l, id V Def.: L n := ( ) + k b n k b k ( ) + k b k b n k (n Z) k> k< Then: L n : V V is well defined [L n, L m] = (n m) L n+m + n3 n 4 δ n+m, id V V Virasoro representation, central charge c = III. Algebraic theme and variations

18 Fermionic Sugawara construction commutator [A, B] := A B B A anticommutator [A, B] + Proposition (fermionic Sugawara, Ramond sector) := A B + B A Suppose: Def.: Then: V vector space, b j : V V linear for each j Z v V N Z : j N = b j v = [b i, b j ] + = δ i+j, id V L n := ( ) + j b n j b j ( ) + j b j b n j (n Z \ {}) j j< L := j b j b j + 6 id V j> L n : V V is well defined [L n, L m] = (n m) L n+m + n3 n 4 δ n+m, id V V Virasoro representation, central charge c = III. Algebraic theme and variations

19 IV. PROOF STEPS (DISCRETE COMPLEX ANALYSIS)

20 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

21 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

22 Lattices (square lattice and related lattices) fix small mesh size δ > C δ C δ C m δ square lattice C δ medial lattice C m δ dual lattice C δ diamond lattice C δ C δ = δz corner lattice C c δ

23 Lattices (discretization of differential operators) : i +i + : +i i + : Discrete and : δ f (z) = δ f (z) = ( f ( z + δ ) ( δ f z ( f ( z + δ ) ) i ) ( δ ) ) f z + i ( f ( z + iδ ( f ( z + iδ ) ( iδ ) ) f z ) ( iδ ) ) f z f : C m δ C δf, δ f : C δ C f : C δ C δf, δ f : C m δ C Discrete Laplacian: ( δ = δ δ = δ δ ) δ f (z) = f (z) + 4 f ( z + δ ) + 4 f ( z δ ) + 4 f ( z + iδ ) + 4 f ( z iδ )

24 Discrete residue calculus (contour integral) two functions f : C m δ C and g : C δ C γ path on the corner lattice C c δ oriented edge of C c δ f defined on C m δ g defined on C δ [γ] f ( z m ) g ( z ) [dz]δ := e γ f ( e m ) g ( e ) e

25 Discrete residue calculus (properties) f ( ) ( ) z m g z [dz]δ := f ( ) ( ) e m g e e [γ] e γ Proposition (properties of discrete contour integral) Green s formula (sum over w m C m δ int(γ) and w C δ int(γ)) f ( ) ( ) z m g z [dz]δ = i f ( ) ( w m δ g )( ) ( w m + i δ f )( ) ( ) w g w [γ] w m w contour deformation γ, γ two counterclockwise closed contours on C c δ δ f and δ g on symm. diff. int ( ) ( ) γ int γ f ( ) ( ) z m g z [dz]δ = f ( ) ( ) z m g z [dz]δ [γ ] integration by parts γ counterclockwise closed contour on C c δ δ f and δ g on neighbors of γ ( δ f )( ) ( ) z m g z [dz]δ = f ( ) ( z m δ g )( ) z [dz]δ [γ ] [γ] [γ]

26 Discrete monomial functions (defining properties) Proposition (discrete monomial functions) functions z z [p], p Z, defined on C δ Cm δ, such that δ z [p] = whenever... discrete holomorphicity p and z C δ Cm δ p < and z C δ Cm δ, z > R p δ z [] for all z C δ Cm δ δ z [ ] = π δ z, + π δ z [p] = p z [p ] x {± δ,±i δ } δz,x z [p] has the same 9 rotation symmetry as z p for p < we have z [p] as z for any z there exists D z such that z [p] = for p D z constant function Green s function For γ large enough counterclockwise closed contour surrounding the origin... derivatives symmetry decay truncation [γ] z[p] m z [q] [dz] δ = πi δ p+q, residue calculus [γ] z{p} m z [q] [dz] δ = πi δ p+q, where z m {p} = 4 x {± δ δ x)[p],±i }(zm

27 Discrete monomial functions (example ) values of z []

28 Discrete monomial functions (example ) + i 3 + i + i + i i + i + i 3 + i + i + 3i 3 + 3i + 3i + 3i 3i + 3i + 3i 3 + 3i + 3i + i 3 + i + i + i i + i + i 3 + i + i + i 3 + i + i + i i + i + i 3 + i + i 3 3 i 3 i i i i i i 3 i i i 3 i i i i i i 3 i i 3i 3 3i 3i 3i 3i 3i 3i 3 3i 3i i 3 i i i i i i 3 i i values of z []

29 4 4 Discrete monomial functions (example ) 8i 6i 3 4i 4 i i 3 + 4i + 6i 8i 6i 9i 3i 3i + 3i + 3i 9i + 6i 3 4i 3i i i + i i + 3i 3 + 4i 4 i 3i i i i + i + 3i 4 + i 4 + i + 3i + i i i i 3i 4 i 3 + 4i + 3i i + i i i 3i 3 4i + 6i 9i + 3i + 3i 3i 3i 9i 6i 8i + 6i 3 + 4i 4 + i 4 4 i 3 4i 6i 8i values of z []

30 Discrete monomial functions (example 3) 8 + 8i 5 + 6i 6 6i 6i 6 6i 5 + 6i 8 + 8i 6 + 5i 6 + 6i 6 + 3i 3 3i 3i 3 3i 6 + 3i 6 + 6i 6 + 5i i 3 + 6i 3 + 3i i 3 + 6i i 6 + 6i 3 + 3i 3i 3i 3 + 3i 6 + 6i i 3 3i 3i 3i 3 3i 6 6i i 3 6i 3 3i i 3 6i i 6 5i 6 6i 6 3i 3 + 3i 3i 3 + 3i 6 3i 6 6i 6 5i 8 8i 5 6i 6 + 6i 6i 6 + 6i 5 6i 8 8i values of z [3]

31 Discrete monomial functions (example -) π 3 5iπ 6i 4 3 π + 4i 3 iπ 8i 3 iπ i + iπ 3 π 4 π + 3iπ 5i 3 i 3 iπ 4i 8 3 π π 3 iπ 4i π + iπ i i i + iπ 3 π 4 5π 6 3π 5 + i iπ iπ i iπ 8i 3iπ i i 4iπ 4 π 6 5π 5 3π + i iπ π + iπ i iπ 4i i 3 π + iπ 3 iπ 4i 4 π π + 3iπ 5i 3 i 3 π 8 3 i 3 π + iπ 3 5iπ 6i π i 3 iπ 8i 3 iπ 4π 8 3π π π π π 3π 8 4π iπ 8i π + iπ 4i 3 6i 5iπ π 3 + i 3 iπ 8 3 π 3 + i 3 π π 4 π 3 + i 3 iπ 3iπ + 5i π + i iπ 4i iπ 5π 6 4i iπ π 4 3π 5 + iπ i + i 4iπ i 3iπ 8i iπ i 6 5π iπ 4 π 5 3π + iπ i + i 4i iπ 3 π + i 3 iπ 4i iπ π + i iπ π i 3 π 3iπ + 5i 4 π iπ 8i 3 π iπ 4i 3 6i 5iπ i 3 π + 3 iπ values of z [ ]

32 Discrete monomial functions (example -) 3π iπ 8i 3 π + 56i 3 6iπ 6π π + 5iπ iπ i 3 8i 56i 3 π + 6iπ 3 3 π i 4iπ 3π i 3 iπ π 3π 4 3 6π π π 4 3 3π + 4i 3 iπ 4iπ i 4 3 3π 5iπ + 8i i 3 iπ 3π iπ 4i 3 π + 3iπ 4i 8 π 3iπ + 4i π i 3 6iπ π 3iπ + 4i i 4iπ π 3 4i iπ 3π π 3 5iπ + 8i π 4 3 3π + iπ 4i 3 π π 8 π + 3iπ 4i iπ 4i π 4iπ i 56i + 6iπ 3 π 3 4iπ i π iπ 4i π + 3iπ 4i 56i + 6iπ 3 π 8 π 4 3 3π + iπ 4i π 4 3 3π 3π 4i iπ π 3 π i 3 6iπ i 4iπ π 3iπ + 4i π 8 π π + 3iπ 4i 3π iπ 4i 3 i 3 iπ 3π 4 3 5iπ + 8i 4 3 3π + 4i 3 iπ 3iπ + 4i 4iπ i 3π 6π iπ + 8i i 4iπ π iπ i 3 3π i 3 iπ 3 π 56i 3 π + 6iπ π + 5iπ 8i 6π π + 56i 3 6iπ 3π iπ 8i values of z [ ]

33 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

34 Discrete Gaussian Free Field (definition again) 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, Ω δ C δ, Ωm δ Cm δ, Ωc δ Cc δ centered Gaussian field on vertices of discrete domain Ω δ probability density p(φ) exp ( E(φ)) E(φ) = ( ) z w φ(z) φ(w) Dirichlet energy covariance E [ Φ(z) Φ(w) ] = G Ωδ (z, w) G Ωδ (z, w) = expected time at w for random walk from z before exiting Ω δ δ G(, w) = δ w ( ) Green s function

35 Discrete holomorphic current (definition) Discrete Gaussian Free Field (dgff): Φ = ( Φ(z) ) z C δ originally defined on Ω δ C δ extend as zero to C δ \ Ω δ and C δ centered Gaussian field on C δ covariance E [ Φ(z) Φ(w) ] = G Ωδ (z, w) Discrete holomorphic current J = ( J(z) ) z C m δ J(z) := δ Φ(z) = ( Φ ( z + δ ) ( δ ) Φ z } {{ } vanishes if z on vertical edge ) ( i Φ ( z + iδ ) ( iδ ) ) Φ z } {{ } vanishes if z on horizontal edge centered complex Gaussian field (... and a local field of dgff!) purely real on horizontal edges, imaginary on vertical edges covariance E [ J(z) J(w) ] = (z) δ (w) δ G Ωδ (z, w)

36 Discrete holomorphic current (correlations) Φ = ( Φ(z) ) z C δ Wick s formula for centered Gaussians: [ n ] E Φ(z j ) = j= P pairing {k,l} P of {,...,n} E [ Φ(z k )Φ(z l ) ] }{{} G Ωδ (z k,z l ) Discrete holomorphic current J = ( J(z) ), J(z) = z C m δ Φ(z) δ Proposition (harmonicity of Φ, holomorphicity of J) E [( δ Φ ) (z) n j= Φ(w j) ] = when z w j > δ for all j E [( δ J ) (z) n j= Φ(w j) ] = when z w j > δ for all j δ J = δ Φ is a null field

37 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

38 Discrete Laurent modes of the current F(w) = F [ (Φ(w +xδ)) x V ] local field of dgff γ sufficiently large counterclockwise closed path on C c δ surrounding origin and V δ For j Z define a new local field of dgff ( J j F ) (w) by ( Jj F ) () := J ( ) [ j ] z m z F() [dz] δ π [γ] Lemma (discrete current modes) J j : F/N F/N is well-defined independent of choice of... add null field to F() add null field to ( J j F ) () change γ add null fields δ J(z) ( ) to ( J j F ) () γ V z... representative... contour

39 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

40 Commutation of modes of the discrete current Proposition (commutation of discrete current modes) [J i, J j ] = i δ i+j, id F/N γ γ V z ζ γ γ V = ζ z γ V γ z ζ [( ) ] E J i J j F() J j J i F() [ ] = i δ i+j, E F() (residue calculus)

41 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

42 Sugawara construction with the dgff current Verify assumptions: V vector space space F/N of local fields modulo null fields a j : V V linear for each j Z discrete current Laurent mode J j : F/N F/N ( Jj F ) () := π [γ] J(z m) z [ j ] F() [dz] v V N Z : j N = a j v = monomial truncation: z C δ D : j D = z[j] = [a i, a j ] = i δ i+j, id V Laurent mode commutation [J i, J j ] = i δ i+j, id F/N Theorem (Virasoro action for dgff) L n := J j J n j + J n j J j j< defines Virasoro representation with c = on the space F/N of correlation equivalence classes of local fields of the dgff. j

43 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

44 The critical Ising model Ω C open, -connected lattice approximation Ω δ C δ, Ω δ C δ, Ωm δ Cm δ, Ωc δ Cc δ Ising model: random spin configuration σ = ( σ z )z C δ {+, } C δ σ Cδ \Ω δ + (plus-boundary conditions) P [ {σ} ] ( ) exp β E(σ) (Boltzmann-Gibbs) E(σ) = z w σ z σ w (energy) β = β c = log ( + ) (critical point)

45 Local fields of the Ising model Local fields F(z) of Ising V Z finite subset P : {+, } V C a function { P( σ) = P(σ) even parity P( σ) = P(σ) odd F(z) = P ( ) (σ z+δx ) x V σ = ( σ z )z Ω δ Ising F space of local fields F = F + F by parity N F space of null fields ( zero in correlations ) { F + F even odd F/N equivalence classes of local fields (same correlations)

46 Disorder operators in Ising model p Disorder operator pair: ( (µ p µ q ) λ := exp β z,w λ σ z σ w ) q λ p, q C δ dual vertices λ path between p and q on C δ disorder line Remark: a single disorder operator is NOT a local field a disorder operator pair is a local field (with fixed disorder line λ)

47 Corner fermions in Ising model p c x c, d C c δ corners x, y C δ adjacent to c, d, respectively p, q C δ adjacent to c, d, respectively d y q λ c ν(c) := x p phase factor x p λ c path between c and d on C δ W(λ c : c d) cumulative angle of turning of λ c Corner fermion pair: (Ψ c cψ c d ) λ c := ν(c) exp Remark: one corner fermion is NOT a local field ( ) i W(λ c : c d) (µ p µ q ) λ σ x σ y a corner fermion pair is a local field (with fixed disorder line)

48 Discrete holomorphic fermions in Ising model z w λ m Remark: (as before) z, w C m δ midpoints of edges λ m path between z and w on C δ c, d C c δ adjacent to z, w, respectively λ c,d c path between c and d on C δ obtained by local modification of λ m Holomorphic fermion pair: (Ψ(z)Ψ(w)) λm := 8 c,d one holomorphic fermion is NOT a local field a holomorphic fermion pair is a local field (Ψ c cψ c d ) λ c,d c (with fixed disorder line)

49 Properties of the fermion pairs z w λ m Lemma (disorder line independence mod ±) If λ m, λ m are disorder lines between z, ζ C m δ n then ] n E [(Ψ(ζ)Ψ(z)) λm σ wj = ( ) N ] E [(Ψ(ζ)Ψ(z)) λ σ m wj j= where N is the number of points w j in the area enclosed by λ m and λ m. j= Lemma (antisymmetry of fermions) (Ψ(ζ)Ψ(z)) λm = (Ψ(z)Ψ(ζ)) λm Lemma (holomorphicity and singularity of fermion) [ n E ( ] δ Ψ(ζ )Ψ(z m )) σ wj = δ ζ,x E [ n ] σ wj 4 x z m j= j=

50 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

51 Laurent modes of fermions in even sector F(w) = P [ (σ w+xδ ) x V ] even local field of Ising γ, γ large nested counterclockwise closed paths on C c δ For k, l Z + define a new local field ( (Ψ k Ψ l ) F ) (w) by ( (Ψk Ψ l ) F ) () := ( ζ [ k ] z [ l ] Ψ ( ) ( ) ) ζ m Ψ zm F() [dz] δ [dζ] δ π [ γ] Lemma (discrete fermion mode pairs) (Ψ k Ψ l ): F + /N + F + /N + is well-defined [γ] Remark: (as before) one fermion Laurent mode is NOT defined a fermion Laurent mode pair is defined, and acts on (even) local fields γ γ V z ζ

52 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable on even sector 4.) Anti commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

53 Anticommutation of fermion modes in even sector Proposition (anticommutation of fermion modes) (Ψ k Ψ l ) + (Ψ l Ψ k ) = δ k+l, id F + /N + γ γ V z ζ γ γ V = ζ z γ V γ z ζ [( ) ] E (Ψ k Ψ l ) F() + (Ψ l Ψ k ) F() [ ] = δ k+l, E F() (residue calculus)

54 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Anti commutation relations of Laurent modes 5.) Virasoro action through Sugawara construction

55 Sugawara construction for Ising even local fields V vector space space F + /N + of even local fields modulo null fields b k : V V linear for each k Z + fermion Laurent mode pairs (Ψ k Ψ l ): F + /N + F + /N + π [ γ] [γ] ζ[ k ] z [ l ] v V N Z : l N = b l v = ( Ψ ( ζ m ) Ψ ( zm ) ) ( ) [dz] δ [dζ] δ monomial truncation: z C δ D : l D z[l ] = [b k, b l ] + = δ k+l, id V anticommutation (Ψ k Ψ l ) + (Ψ l Ψ k ) = δ k+l, id F + /N + and (Ψ p Ψ k )(Ψ l Ψ q ) + (Ψ p Ψ l )(Ψ k Ψ q ) = δ k+l, (Ψ p Ψ q ) Theorem (Virasoro action for Ising even sector) L n := ( ) + k (Ψ n k Ψ k ) ( ) + k (Ψ k Ψ n k ) k> k< defines Virasoro repr. with c = on the space F + /N + of correlation equivalence classes of Ising even local fields.

56 Outline / steps For the Ising model and discrete GFF:.) Define model and local fields.) Suitable discrete contour integrals and residue calculus.) Introduce discrete holomorphic observable 3.) Define Laurent modes of the observable 4.) Anti commutation relations of Laurent modes 5.) Apply Sugawara construction to define Virasoro action on odd local fields

57 Odd sector: Discrete half-integer monomials Proposition (discrete half-integer monomial functions) functions z z [p], p Z +, defined on the double cover [C δ ; ] [Cm δ ; ] ramified at the origin, such that δ z [p] = whenever... discrete holomorphicity p > and z [C δ ; ] [Cm δ ; ] p < and z [C δ ; ] [Cm δ ; ], z > R p δ δ z [p] = p z [p ] z [p] has the same 9 rotation symmetry as z p for p < we have z [p] as z for any z there exists D z such that z [p] = for p D z For γ large enough counterclockwise closed contour surrounding the origin... derivatives symmetry decay truncation [γ] z[p] m z [q] [dz] δ = πi δ p+q, residue calculus [γ] z{p} m z [q] [dz] δ = πi δ p+q, where z m {p} = 4 x {± δ δ x)[p],±i }(zm

58 Odd sector: Laurent modes of fermions F(w) = P [ (σ w+xδ ) x V ] odd local field of Ising γ, γ large nested counterclockwise closed paths on C c δ γ γ V z ζ For i, j Z define a new local field ( (Ψ i Ψ j ) F ) (w) by ( (Ψi Ψ j ) F ) () := ( ζ [ i ] z [ j ] Ψ ( ) ( ) ) ζ m Ψ zm F() [dz] δ [dζ] δ π [ γ] Lemma (discrete fermion mode pairs) (Ψ i Ψ j ): F /N F /N is well-defined [γ]

59 Odd sector: Anticommutation of fermion modes Proposition (anticommutation of fermion modes) (Ψ i Ψ j ) + (Ψ j Ψ i ) = δ i+j, id F /N γ γ V z ζ γ γ V = ζ z γ V γ z ζ

60 Odd sector: Fermionic Sugawara construction Proposition (fermionic Sugawara, Ramond sector) Suppose: V vector space, b j : V V linear for each j Z v V N Z : j N = b j v = Def.: Then: [b i, b j ] + = δ i+j, id V L n := ( ) + j b n j b j ( ) + j b j b n j (n Z \ {}) j j< L := j b j b j + 6 id V j> L n : V V is well defined [L n, L m] = (n m) L n+m + n3 n 4 δ n+m, id V Theorem (Virasoro action on Ising odd local fields) The space of odd Ising local fields modulo null fields becomes Virasoro representation with central charge c =.

61 Conclusions and outlook Lattice model fields of finite patterns form Virasoro repr. TODO TODO discrete Gaussian free field: L n on F/N by bosonic Sugawara Ising model: L n on F + /N + F /N }{{} Neveu-Schwarz Ramond by fermionic Sugawara Many CFT ideas rely on variants of Sugawara construction Wess-Zumino-Witten models symplectic fermions coset conformal field theories CFT minimal models Coulomb gas formalism CFT fields lattice model fields of finite patterns - correspondence via the Virasoro action on lattice model fields? correlations of lattice model fields with appropriate renormalization converge in scaling limit to CFT correlations? conceptual derivation of PDEs for limit correlations via singular vectors?

62 THANK YOU! γ V γ z ζ The end