Conformal Field Theory and Combinatorics

Transcription

1 Conformal Field Theory and Combinatorics Part I: Basic concepts of CFT 1,2 1 Université Pierre et Marie Curie, Paris 6, France 2 Institut de Physique Théorique, CEA/Saclay, France Wednesday 16 January, 2008

2 Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

3 GENERAL OVERVIEW Part I: Basic concepts of Conformal Field Theory Part II: Coulomb Gas construction and loop models Part III: Applications to Combinatorics SCOPE OF THE LECTURES Deals exclusively with two dimensions A short introduction to a huge field Key arguments given, but no detailed proofs No details on relation to Stochastic Loewner Evolution Mainly a physicist s point of view

4 GENERAL OVERVIEW Part I: Basic concepts of Conformal Field Theory Part II: Coulomb Gas construction and loop models Part III: Applications to Combinatorics SCOPE OF THE LECTURES Deals exclusively with two dimensions A short introduction to a huge field Key arguments given, but no detailed proofs No details on relation to Stochastic Loewner Evolution Mainly a physicist s point of view

5 HISTORICAL REMARKS Quantum Field Theory in statmech: since Onsager (1944) Wilson (1969): renormalisation group and ε expansion Integrability: 6 & 8 vertex models (Lieb 1967, Baxter 1971) Global conformal invariance in d dim. (Polyakov 1970) TWO DIMENSIONS: THE BREAK-THROUGH Local conformal invariance in d = 2 (Belavin-Polyakov-Zamolodchikov 1984) Applications to statmech (Cardy, Saleur, Duplantier,...) Extended CFT (Zamolodchikov), boundary CFT (Cardy), classification of rational CFT,... Coulomb Gas (den Nijs 1983, Nienhuis 1984, Di Francesco-Saleur-Zuber 1987)

6 HISTORICAL REMARKS Quantum Field Theory in statmech: since Onsager (1944) Wilson (1969): renormalisation group and ε expansion Integrability: 6 & 8 vertex models (Lieb 1967, Baxter 1971) Global conformal invariance in d dim. (Polyakov 1970) TWO DIMENSIONS: THE BREAK-THROUGH Local conformal invariance in d = 2 (Belavin-Polyakov-Zamolodchikov 1984) Applications to statmech (Cardy, Saleur, Duplantier,...) Extended CFT (Zamolodchikov), boundary CFT (Cardy), classification of rational CFT,... Coulomb Gas (den Nijs 1983, Nienhuis 1984, Di Francesco-Saleur-Zuber 1987)

7 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

8 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities CONFORMAL TRANSFORMATION IN d DIMENSIONS Invertible mapping x x with g µν (x ) = Λ(x)g µν (x) Angle preserving Generated by translations, dilatations, rotations, inversions Isomorphic to the pseudo-orthogonal group SO(d + 1, 1) with 1 2 (d + 1)(d + 2) real parameters CONNECTION BETWEEN QFT AND STATISTICAL MECHANICS Partition function and correlators are functional integrals Z = DΦ e S[Φ] φ 1 (x 1 )... φ k (x k ) = Z 1 DΦ φ 1 (x 1 )... φ k (x k )e S[Φ]

9 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities CONFORMAL TRANSFORMATION IN d DIMENSIONS Invertible mapping x x with g µν (x ) = Λ(x)g µν (x) Angle preserving Generated by translations, dilatations, rotations, inversions Isomorphic to the pseudo-orthogonal group SO(d + 1, 1) with 1 2 (d + 1)(d + 2) real parameters CONNECTION BETWEEN QFT AND STATISTICAL MECHANICS Partition function and correlators are functional integrals Z = DΦ e S[Φ] φ 1 (x 1 )... φ k (x k ) = Z 1 DΦ φ 1 (x 1 )... φ k (x k )e S[Φ]

10 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities QUASI-PRIMARY FIELD φ(x) Transforms covariantly: φ(x) φ (x ) = x x /d φ(x) = φ is the scaling dimension (critical exponent) CONFORMAL INVARIANCE FIXES 2 AND 3-POINT FUNCTIONS Setting x ij = x i x j φ 1 (x 1 )φ 2 (x 2 ) = δ 1, 2 x φ 1 (x 1 )φ 2 (x 2 )φ 3 (x 3 ) = C 123 x x x Structure constants C 123 to be fixed by other means

11 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities QUASI-PRIMARY FIELD φ(x) Transforms covariantly: φ(x) φ (x ) = x x /d φ(x) = φ is the scaling dimension (critical exponent) CONFORMAL INVARIANCE FIXES 2 AND 3-POINT FUNCTIONS Setting x ij = x i x j φ 1 (x 1 )φ 2 (x 2 ) = δ 1, 2 x φ 1 (x 1 )φ 2 (x 2 )φ 3 (x 3 ) = C 123 x x x Structure constants C 123 to be fixed by other means

12 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

13 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities IN TWO DIMENSIONS: Conf. transformation satisfies Cauchy-Riemann eqs Use complex coordinates z x 1 + ix 2 and z x 1 ix 2 Conformal mapping z z = w(z) is analytic FROM GLOBAL TO LOCAL CONFORMAL INVARIANCE Global conformal transformations are projective w(z) = a 11z + a 12 a 21 z + a 22 Here a ij C and det a ij = 1 (6 real parameters) Analytic maps actually only conformal locally Relaxing to this gives us ly many parameters!

14 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities IN TWO DIMENSIONS: Conf. transformation satisfies Cauchy-Riemann eqs Use complex coordinates z x 1 + ix 2 and z x 1 ix 2 Conformal mapping z z = w(z) is analytic FROM GLOBAL TO LOCAL CONFORMAL INVARIANCE Global conformal transformations are projective w(z) = a 11z + a 12 a 21 z + a 22 Here a ij C and det a ij = 1 (6 real parameters) Analytic maps actually only conformal locally Relaxing to this gives us ly many parameters!

15 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities TRANSFORMATION LAW REVISITED φ (w, w) = ( dw dz ) h ( ) d w h φ(z, z) d z Quasi-primary φ: satisfies this for projective maps Primary φ: satisfies this for any analytic map SOME NOMENCLATURE The real parameters (h, h) are called the conformal weights = h + h is the scaling dimension s = h h is the spin

16 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities TRANSFORMATION LAW REVISITED φ (w, w) = ( dw dz ) h ( ) d w h φ(z, z) d z Quasi-primary φ: satisfies this for projective maps Primary φ: satisfies this for any analytic map SOME NOMENCLATURE The real parameters (h, h) are called the conformal weights = h + h is the scaling dimension s = h h is the spin

17 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

18 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities STRESS TENSOR = NOETHER CURRENT OF CONF. SYMM. Response to anisotropy (like in integrable systems) PROPERTIES OF T µν : T µν (x) = 1 2π δ log Z δg µν (x) T z z = T zz = 0 (use trans. and rot. inv.) z T zz = z T z z (use further scale inv.) So T(z) T zz (resp. T( z) T z z ) is (anti)analytic REMARK ON EXTENDED CFT T(z) coexists with further analytic currents Keywords: Super CFT, parafermions, W algebra, WZW

19 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities STRESS TENSOR = NOETHER CURRENT OF CONF. SYMM. Response to anisotropy (like in integrable systems) PROPERTIES OF T µν : T µν (x) = 1 2π δ log Z δg µν (x) T z z = T zz = 0 (use trans. and rot. inv.) z T zz = z T z z (use further scale inv.) So T(z) T zz (resp. T( z) T z z ) is (anti)analytic REMARK ON EXTENDED CFT T(z) coexists with further analytic currents Keywords: Super CFT, parafermions, W algebra, WZW

20 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities STRESS TENSOR = NOETHER CURRENT OF CONF. SYMM. Response to anisotropy (like in integrable systems) PROPERTIES OF T µν : T µν (x) = 1 2π δ log Z δg µν (x) T z z = T zz = 0 (use trans. and rot. inv.) z T zz = z T z z (use further scale inv.) So T(z) T zz (resp. T( z) T z z ) is (anti)analytic REMARK ON EXTENDED CFT T(z) coexists with further analytic currents Keywords: Super CFT, parafermions, W algebra, WZW

21 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities CONFORMAL WARD IDENTITY Let X = j φ j(z j, z j ) with φ j primary Let C be any counterclockwise contour encircling {z j } Apply infinitesimal conformal transformation z = z + ε(z) C T(z)X ε(z) dz = j ( hj ε (z j ) + ε(z j ) zj ) X, Better known after application of Cauchy theorem: h j T(z)φ j (z j, z j ) = (z z j ) 2φ j(z j, z j ) + 1 zj φ(z j, z j ) + O(1). z z j OUR FIRST EXAMPLE OF AN operator product expansion (OPE) Exact identity, valid under expectation value Note singularities upon approaching two fields

22 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities CONFORMAL WARD IDENTITY Let X = j φ j(z j, z j ) with φ j primary Let C be any counterclockwise contour encircling {z j } Apply infinitesimal conformal transformation z = z + ε(z) C T(z)X ε(z) dz = j ( hj ε (z j ) + ε(z j ) zj ) X, Better known after application of Cauchy theorem: h j T(z)φ j (z j, z j ) = (z z j ) 2φ j(z j, z j ) + 1 zj φ(z j, z j ) + O(1). z z j OUR FIRST EXAMPLE OF AN operator product expansion (OPE) Exact identity, valid under expectation value Note singularities upon approaching two fields

23 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities CONFORMAL WARD IDENTITY Let X = j φ j(z j, z j ) with φ j primary Let C be any counterclockwise contour encircling {z j } Apply infinitesimal conformal transformation z = z + ε(z) C T(z)X ε(z) dz = j ( hj ε (z j ) + ε(z j ) zj ) X, Better known after application of Cauchy theorem: h j T(z)φ j (z j, z j ) = (z z j ) 2φ j(z j, z j ) + 1 zj φ(z j, z j ) + O(1). z z j OUR FIRST EXAMPLE OF AN operator product expansion (OPE) Exact identity, valid under expectation value Note singularities upon approaching two fields

24 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities TRANSFORMATION LAW OF T µν Quasi-primary with h = 2, but not primary: T(z 1 )T(z 2 ) = c/2 (z 1 z 2 ) 4 + 2T(z 2) (z 1 z 2 ) 2 + T(z 2) z 1 z 2 + O(1). Here c is the central charge or conformal anomaly Measures # quantum degrees of freedom in the CFT T (w) = ( ) dw 2 [ T(z) c ] dz 12 {w; z}. Use associativity, and {w; z} = 0 for projective transf. Result: {w; z} is the Schwarzian derivative {w; z} = d3 w/dz 3 dw/dz 3 ( d 2 w/dz 2 ) 2. 2 dw/dz

25 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities TRANSFORMATION LAW OF T µν Quasi-primary with h = 2, but not primary: T(z 1 )T(z 2 ) = c/2 (z 1 z 2 ) 4 + 2T(z 2) (z 1 z 2 ) 2 + T(z 2) z 1 z 2 + O(1). Here c is the central charge or conformal anomaly Measures # quantum degrees of freedom in the CFT T (w) = ( ) dw 2 [ T(z) c ] dz 12 {w; z}. Use associativity, and {w; z} = 0 for projective transf. Result: {w; z} is the Schwarzian derivative {w; z} = d3 w/dz 3 dw/dz 3 ( d 2 w/dz 2 ) 2. 2 dw/dz

26 Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

27 OPERATOR FORMALISM Quantisation scheme: chose space and time directions Transfer matrix e H propagates in time direction Radial quantisation: take H = dilatation operator MODE OPERATORS L n = 1 2πi C z n+1 T(z) dz, Ln = 1 2πi C z n+1 T( z) d z. Then H = (2π/L)(L 0 + L 0 c/12) The L n satisfy the Virasoro algebra (use OPE of TT ) [L n, L m ] = (n m)l n+m + c 12 n(n2 1)δ n+m,0 Decoupling [L n, L m ] = 0 we can focus on L n only Caveat: on the torus L n and L n couple (modular invariance)

28 OPERATOR FORMALISM Quantisation scheme: chose space and time directions Transfer matrix e H propagates in time direction Radial quantisation: take H = dilatation operator MODE OPERATORS L n = 1 2πi C z n+1 T(z) dz, Ln = 1 2πi C z n+1 T( z) d z. Then H = (2π/L)(L 0 + L 0 c/12) The L n satisfy the Virasoro algebra (use OPE of TT ) [L n, L m ] = (n m)l n+m + c 12 n(n2 1)δ n+m,0 Decoupling [L n, L m ] = 0 we can focus on L n only Caveat: on the torus L n and L n couple (modular invariance)

29 OPERATOR FORMALISM Quantisation scheme: chose space and time directions Transfer matrix e H propagates in time direction Radial quantisation: take H = dilatation operator MODE OPERATORS L n = 1 2πi C z n+1 T(z) dz, Ln = 1 2πi C z n+1 T( z) d z. Then H = (2π/L)(L 0 + L 0 c/12) The L n satisfy the Virasoro algebra (use OPE of TT ) [L n, L m ] = (n m)l n+m + c 12 n(n2 1)δ n+m,0 Decoupling [L n, L m ] = 0 we can focus on L n only Caveat: on the torus L n and L n couple (modular invariance)

30 STRUCTURE OF HILBERT SPACE Vacuum 0 invariant under projective transf.: L ±1 0 = 0 Also fix ground state energy by L 0 0 = 0 Eigenstates of H from primaries: h, h = φ(0, 0) 0. Highest-weight condition: L n h, h = L n h, h = 0 for n > 0 Descendents of φ at level {N, N}, where N = k i=1 n i: φ {n, n} L n1 L n2 L nk L n1 L n2 L n k h, h Verma module (or highest-weight representation) = one primary and its descendents Not in general irreducible (see later)

31 STRUCTURE OF HILBERT SPACE Vacuum 0 invariant under projective transf.: L ±1 0 = 0 Also fix ground state energy by L 0 0 = 0 Eigenstates of H from primaries: h, h = φ(0, 0) 0. Highest-weight condition: L n h, h = L n h, h = 0 for n > 0 Descendents of φ at level {N, N}, where N = k i=1 n i: φ {n, n} L n1 L n2 L nk L n1 L n2 L n k h, h Verma module (or highest-weight representation) = one primary and its descendents Not in general irreducible (see later)

32 STRUCTURE OF HILBERT SPACE Vacuum 0 invariant under projective transf.: L ±1 0 = 0 Also fix ground state energy by L 0 0 = 0 Eigenstates of H from primaries: h, h = φ(0, 0) 0. Highest-weight condition: L n h, h = L n h, h = 0 for n > 0 Descendents of φ at level {N, N}, where N = k i=1 n i: φ {n, n} L n1 L n2 L nk L n1 L n2 L n k h, h Verma module (or highest-weight representation) = one primary and its descendents Not in general irreducible (see later)

33 CORRELATION FUNCTIONS OF DESCENDENTS Related to corresponding primary corr. func. (use Ward id.) ( L n φ ) (w)x = L n φ(w)x L n { } (n 1)hj (w j w) n wj (w j j w) n 1 GENERAL FORM OF OPE OF TWO PRIMARIES φ 1 (z, z)φ 2 (0, 0) = p C {n, n} 12p {n, n} {, } C 12p zhp h 1 h 2 +N z h p h 1 h 2 + Nφ {n, n} p (0, 0) C {n, n} 12p (with C {, } 12p 1) obtained by acting with L n for n > 0 C 12p coincide with those of 3-point function Found by conformal bootstrap (associativity of 4-point fct.)

34 CORRELATION FUNCTIONS OF DESCENDENTS Related to corresponding primary corr. func. (use Ward id.) ( L n φ ) (w)x = L n φ(w)x L n { } (n 1)hj (w j w) n wj (w j j w) n 1 GENERAL FORM OF OPE OF TWO PRIMARIES φ 1 (z, z)φ 2 (0, 0) = p C {n, n} 12p {n, n} {, } C 12p zhp h 1 h 2 +N z h p h 1 h 2 + Nφ {n, n} p (0, 0) C {n, n} 12p (with C {, } 12p 1) obtained by acting with L n for n > 0 C 12p coincide with those of 3-point function Found by conformal bootstrap (associativity of 4-point fct.)

35 Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

36 HILBERT SPACE OF COMPLETE CFT n h, hv(c, h) V(c, h) h, h V(c, h) = Verma module of highest-weight state h Multiplicities n h, h to be fixed by modular invariance For minimal models the sum h, h is finite Character χ (c,h) OF MODULE V(c, h) χ (c,h) (τ) = Tr q L0 c/24, Trace needs inner product on V(c, h): defined by L n = L n τ C is the modular parameter, and q = e 2πiτ

37 HILBERT SPACE OF COMPLETE CFT n h, hv(c, h) V(c, h) h, h V(c, h) = Verma module of highest-weight state h Multiplicities n h, h to be fixed by modular invariance For minimal models the sum h, h is finite Character χ (c,h) OF MODULE V(c, h) χ (c,h) (τ) = Tr q L0 c/24, Trace needs inner product on V(c, h): defined by L n = L n τ C is the modular parameter, and q = e 2πiτ

38 IF V(c, h) WERE IRREDUCIBLE, THINGS WOULD BE SIMPLE... # descendents at level N is # integer partitions p(n) χ (c,h) (τ) = qh c/24 P(q) NULL FIELD χ 1 with P(q) 1 1 q n = q1/24 η(τ) = p(n)q n n=1 Linear combination of level-n descendents of h Is both descendent and primary: L n χ = 0 for n > 0 Happens only for particular values of h Inclusion of Verma modules: V χ (c, h) V(c, h) V χ (c, h) V(c, h), hence χ χ = 0 n=0 OBTAINING AN IRREDUCIBLE MODULE M(c, h) FROM V(c, h) Quotient out null fields: set χ = 0

39 IF V(c, h) WERE IRREDUCIBLE, THINGS WOULD BE SIMPLE... # descendents at level N is # integer partitions p(n) χ (c,h) (τ) = qh c/24 P(q) NULL FIELD χ 1 with P(q) 1 1 q n = q1/24 η(τ) = p(n)q n n=1 Linear combination of level-n descendents of h Is both descendent and primary: L n χ = 0 for n > 0 Happens only for particular values of h Inclusion of Verma modules: V χ (c, h) V(c, h) V χ (c, h) V(c, h), hence χ χ = 0 n=0 OBTAINING AN IRREDUCIBLE MODULE M(c, h) FROM V(c, h) Quotient out null fields: set χ = 0

40 IF V(c, h) WERE IRREDUCIBLE, THINGS WOULD BE SIMPLE... # descendents at level N is # integer partitions p(n) χ (c,h) (τ) = qh c/24 P(q) NULL FIELD χ 1 with P(q) 1 1 q n = q1/24 η(τ) = p(n)q n n=1 Linear combination of level-n descendents of h Is both descendent and primary: L n χ = 0 for n > 0 Happens only for particular values of h Inclusion of Verma modules: V χ (c, h) V(c, h) V χ (c, h) V(c, h), hence χ χ = 0 n=0 OBTAINING AN IRREDUCIBLE MODULE M(c, h) FROM V(c, h) Quotient out null fields: set χ = 0

41 IF V(c, h) WERE IRREDUCIBLE, THINGS WOULD BE SIMPLE... # descendents at level N is # integer partitions p(n) χ (c,h) (τ) = qh c/24 P(q) NULL FIELD χ 1 with P(q) 1 1 q n = q1/24 η(τ) = p(n)q n n=1 Linear combination of level-n descendents of h Is both descendent and primary: L n χ = 0 for n > 0 Happens only for particular values of h Inclusion of Verma modules: V χ (c, h) V(c, h) V χ (c, h) V(c, h), hence χ χ = 0 n=0 OBTAINING AN IRREDUCIBLE MODULE M(c, h) FROM V(c, h) Quotient out null fields: set χ = 0

42 CONSTRUCT unitary representations OF VIRASORO ALGEBRA No state of negative norm occurs 1st step: Compute Kac determinant det M (N) of inner products between level-n descendents Roots det M (N) = 0 parametrised by c(m) = 1 6 m(m + 1) h(m) = h r,s (m) [(m + 1)r ms]2 1 4m(m + 1) Here r, s Z with r, s 1 and rs N Unitarity condition for c < 1 (Friedan-Qiu-Sherker 1984): m, r, s Z with m 2 and 1 r < m and 1 s m

43 CONSTRUCT unitary representations OF VIRASORO ALGEBRA No state of negative norm occurs 1st step: Compute Kac determinant det M (N) of inner products between level-n descendents Roots det M (N) = 0 parametrised by c(m) = 1 6 m(m + 1) h(m) = h r,s (m) [(m + 1)r ms]2 1 4m(m + 1) Here r, s Z with r, s 1 and rs N Unitarity condition for c < 1 (Friedan-Qiu-Sherker 1984): m, r, s Z with m 2 and 1 r < m and 1 s m

44 CONSTRUCT unitary representations OF VIRASORO ALGEBRA No state of negative norm occurs 1st step: Compute Kac determinant det M (N) of inner products between level-n descendents Roots det M (N) = 0 parametrised by c(m) = 1 6 m(m + 1) h(m) = h r,s (m) [(m + 1)r ms]2 1 4m(m + 1) Here r, s Z with r, s 1 and rs N Unitarity condition for c < 1 (Friedan-Qiu-Sherker 1984): m, r, s Z with m 2 and 1 r < m and 1 s m

45 CORR. FUNCTIONS SATISFY LINEAR DIFFERENTIAL EQS. Generic descendent state χ(w) = α Y L Y φ(w) Y, Y =N Y = {r 1, r 2,..., r k } Y = r 1 + r r k L Y = L r1 L r2 L rk Correlators of χ must vanish: χ(w)x = α Y L Y (w) φ(w)x = 0 Y, Y =N Most useful for 4-point functions (linear ODE of η)

46 CORR. FUNCTIONS SATISFY LINEAR DIFFERENTIAL EQS. Generic descendent state χ(w) = α Y L Y φ(w) Y, Y =N Y = {r 1, r 2,..., r k } Y = r 1 + r r k L Y = L r1 L r2 L rk Correlators of χ must vanish: χ(w)x = α Y L Y (w) φ(w)x = 0 Y, Y =N Most useful for 4-point functions (linear ODE of η)

47 CORR. FUNCTIONS SATISFY LINEAR DIFFERENTIAL EQS. Generic descendent state χ(w) = α Y L Y φ(w) Y, Y =N Y = {r 1, r 2,..., r k } Y = r 1 + r r k L Y = L r1 L r2 L rk Correlators of χ must vanish: χ(w)x = α Y L Y (w) φ(w)x = 0 Y, Y =N Most useful for 4-point functions (linear ODE of η)

48 MINIMAL MODELS (Kac table h r,s OF CONFORMAL WEIGHTS) Require that OPE closes over a finite # primaries Main idea: consistency between OPE and differential eqs. c = 1 6(m m ) 2 mm h r,s = (mr m s) 2 (m m ) 2 4mm Here m, m, r, s Z with 1 r < m and 1 s < m FUSION ALGEBRA φ (r1,s 1 )φ (r2,s 2 ) = r,s φ (r,s) r from 1 + r 1 r 2 to min(r 1 + r 2 1, 2m 1 r 1 r 2 ) s from 1 + s 1 s 2 to min(s 1 + s 2 1, 2m 1 s 1 s 2 )

49 MINIMAL MODELS (Kac table h r,s OF CONFORMAL WEIGHTS) Require that OPE closes over a finite # primaries Main idea: consistency between OPE and differential eqs. c = 1 6(m m ) 2 mm h r,s = (mr m s) 2 (m m ) 2 4mm Here m, m, r, s Z with 1 r < m and 1 s < m FUSION ALGEBRA φ (r1,s 1 )φ (r2,s 2 ) = r,s φ (r,s) r from 1 + r 1 r 2 to min(r 1 + r 2 1, 2m 1 r 1 r 2 ) s from 1 + s 1 s 2 to min(s 1 + s 2 1, 2m 1 s 1 s 2 )

50 MINIMAL MODELS (Kac table h r,s OF CONFORMAL WEIGHTS) Require that OPE closes over a finite # primaries Main idea: consistency between OPE and differential eqs. c = 1 6(m m ) 2 mm h r,s = (mr m s) 2 (m m ) 2 4mm Here m, m, r, s Z with 1 r < m and 1 s < m FUSION ALGEBRA φ (r1,s 1 )φ (r2,s 2 ) = r,s φ (r,s) r from 1 + r 1 r 2 to min(r 1 + r 2 1, 2m 1 r 1 r 2 ) s from 1 + s 1 s 2 to min(s 1 + s 2 1, 2m 1 s 1 s 2 )

51 CONSTRUCT IRREDUCIBLE MODULES M r,s V r,s contains V m +r,m s and V r,2m s at levels rs and (m r)(m s) The 2 submodules intersect Hence need series of inclusions-exclusions END RESULT FOR CHARACTERS OF MINIMAL MODELS χ (r,s) (τ) = K (m,m ) r,s (q) K (m,m ) r, s (q) K (m,m ) r,s (q) = q 1/24 P(q) n Z q (2mm n+mr m s) 2 /4mm

52 CONSTRUCT IRREDUCIBLE MODULES M r,s V r,s contains V m +r,m s and V r,2m s at levels rs and (m r)(m s) The 2 submodules intersect Hence need series of inclusions-exclusions END RESULT FOR CHARACTERS OF MINIMAL MODELS χ (r,s) (τ) = K (m,m ) r,s (q) K (m,m ) r, s (q) K (m,m ) r,s (q) = q 1/24 P(q) n Z q (2mm n+mr m s) 2 /4mm

53 Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

54 WHAT HAS BEEN ACHIEVED Classification of series of minimal models No lattice model / continuum action was ever defined! All critical exponents from Kac table All corrections to scaling from characters HOW TO IDENTIFY WITH MICROSCOPIC MODEL Get some h r,s by other means (exact solution...) Get c from numerics (FSS of free energy on cylinder) Deduce fusion algebra from symmetry of lattice model E.g. Ising model (m = 2): c = 1 2, h σ = 1 16, h ε = 1 2, εε = 1, σσ = 1 + ε, σε = σ

55 WHAT HAS BEEN ACHIEVED Classification of series of minimal models No lattice model / continuum action was ever defined! All critical exponents from Kac table All corrections to scaling from characters HOW TO IDENTIFY WITH MICROSCOPIC MODEL Get some h r,s by other means (exact solution...) Get c from numerics (FSS of free energy on cylinder) Deduce fusion algebra from symmetry of lattice model E.g. Ising model (m = 2): c = 1 2, h σ = 1 16, h ε = 1 2, εε = 1, σσ = 1 + ε, σε = σ

56 WHAT HAS BEEN ACHIEVED Classification of series of minimal models No lattice model / continuum action was ever defined! All critical exponents from Kac table All corrections to scaling from characters HOW TO IDENTIFY WITH MICROSCOPIC MODEL Get some h r,s by other means (exact solution...) Get c from numerics (FSS of free energy on cylinder) Deduce fusion algebra from symmetry of lattice model E.g. Ising model (m = 2): c = 1 2, h σ = 1 16, h ε = 1 2, εε = 1, σσ = 1 + ε, σε = σ

57 CFT defined on a torus Compactified boson Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

58 CFT defined on a torus Compactified boson HOW DO L n AND L n COUPLE ON A TORUS? Identify points wrt the lattice ω 1 Z + ω 2 Z Take periods such that ω 1 R and Iω 2 > 0 Z depends on modular parameter τ ω 2 /ω 1...but not on ω 1,ω 2 if same lattice is spanned! USE CARTHESIAN QUANTISATION SCHEME Z(τ) = Tr exp [ (Iω 2 )H (Rω 2 )P] H P = (2π/ω 1 )(L 0 + L 0 c/12) = (2π/iω 1 )(L 0 L 0 c/12) Z(τ) = Tr (q ) L 0 c/24 q L 0 c/24 = h, h n h, h χ (c,h) (τ) χ (c, h) (τ) with q = e 2πiτ

59 CFT defined on a torus Compactified boson HOW DO L n AND L n COUPLE ON A TORUS? Identify points wrt the lattice ω 1 Z + ω 2 Z Take periods such that ω 1 R and Iω 2 > 0 Z depends on modular parameter τ ω 2 /ω 1...but not on ω 1,ω 2 if same lattice is spanned! USE CARTHESIAN QUANTISATION SCHEME Z(τ) = Tr exp [ (Iω 2 )H (Rω 2 )P] H P = (2π/ω 1 )(L 0 + L 0 c/12) = (2π/iω 1 )(L 0 L 0 c/12) Z(τ) = Tr (q ) L 0 c/24 q L 0 c/24 = h, h n h, h χ (c,h) (τ) χ (c, h) (τ) with q = e 2πiτ

60 CFT defined on a torus Compactified boson HOW DO L n AND L n COUPLE ON A TORUS? Identify points wrt the lattice ω 1 Z + ω 2 Z Take periods such that ω 1 R and Iω 2 > 0 Z depends on modular parameter τ ω 2 /ω 1...but not on ω 1,ω 2 if same lattice is spanned! USE CARTHESIAN QUANTISATION SCHEME Z(τ) = Tr exp [ (Iω 2 )H (Rω 2 )P] H P = (2π/ω 1 )(L 0 + L 0 c/12) = (2π/iω 1 )(L 0 L 0 c/12) Z(τ) = Tr (q ) L 0 c/24 q L 0 c/24 = h, h n h, h χ (c,h) (τ) χ (c, h) (τ) with q = e 2πiτ

61 CFT defined on a torus Compactified boson Summary 1 Introduction 2 Global conformal invariance Two dimensions and local conformal invariance Stress tensor and Ward identities 3 4 CFT defined on a torus Compactified boson

62 CFT defined on a torus Compactified boson MODULAR INVARIANCE APPLIED TO FREE compactified BOSON End result will be crucial for the Coulomb Gas approach Helical boundary conditions (with m, m az) S[φ] = g 2 d 2 x ( φ) 2 φ(z + kω 1 + k ω 2 ) = φ(z) + 2πR(km + k m ) Functional integral can be done explicitly Use ζ-function regularisation to deal with zero-mode Z only modular invariant when summed over m, m : Z(τ) R 2 Z 0 (τ) m,m az exp ( 2π 2 gr 2 mτ m 2 ) Iτ

63 CFT defined on a torus Compactified boson MODULAR INVARIANCE APPLIED TO FREE compactified BOSON End result will be crucial for the Coulomb Gas approach Helical boundary conditions (with m, m az) S[φ] = g 2 d 2 x ( φ) 2 φ(z + kω 1 + k ω 2 ) = φ(z) + 2πR(km + k m ) Functional integral can be done explicitly Use ζ-function regularisation to deal with zero-mode Z only modular invariant when summed over m, m : Z(τ) R 2 Z 0 (τ) m,m az exp ( 2π 2 gr 2 mτ m 2 ) Iτ

64 CFT defined on a torus Compactified boson COMPACTIFIED BOSON (continued) End result (after Poisson resummation in m ): Z(τ) = 1 η(τ) 2 e Z/a, m az q he,m q h e,m h e,m = ( ) 2 1 e 2 R 4πg + mr 4πg 2 h e,m = ( ) 2 1 e 2 R 4πg mr 4πg 2 Modular inv. has completely specified the operator content! Electric/magnetic charges e, m on mutually dual lattices Note e m under Ra 2πg (Ra 2πg) 1

65 CFT defined on a torus Compactified boson COMPACTIFIED BOSON (continued) End result (after Poisson resummation in m ): Z(τ) = 1 η(τ) 2 e Z/a, m az q he,m q h e,m h e,m = ( ) 2 1 e 2 R 4πg + mr 4πg 2 h e,m = ( ) 2 1 e 2 R 4πg mr 4πg 2 Modular inv. has completely specified the operator content! Electric/magnetic charges e, m on mutually dual lattices Note e m under Ra 2πg (Ra 2πg) 1