Conformal Field Theory

Home , Central charge, Correlation function

typeset by walker h. stern

The following notes were taken from a course given at Universit¨atBonn during the summer semester 2016 by Dr. Hans Jockers 2

Contents

1 Conformal Field Theories in d dimensions 4 1.1 Conformal group in dimension d (d > 2) 4 1.2 Correlation function of Quantum Fields (d > 2) 8 2 Conformal Field Theories in 2 Dimensions 12 2.1 Conformal Algebra in 2 Dimensions 12 2.2 Correlation functions of (quasi-)primary ﬁelds 16 2.3 Radial Quantization 18 2.4 Operator product expansions 23 2.5 Energy-momentum tensor and conformal Ward identities 25 2.6 Examples of CFTs (Free CFTs) 29 2.7 The central charge c 33 2.8 The Hilbert Space of States 40 2.9 Conformal Families and Operator Algebra 42 2.10 Conformal Blacks & Crossing Symmetries 47 3 Minimal Models 50 3.1 Warm-up: Representations of su(2) 50 3.2 Reducible Verma Modules & Singular Vectors 52 3.3 Kac Determinants 55 3.4 Fusion Rules 58 3.5 The Critical Ising Model 60 3.6 Minimal Model Characters 64 4 Modular Invariance 65 4.1 Partition Function 66 4.2 Modular Invariance 68 4.3 Free Boson Parition Function 70 4.4 Interlude: Mode Expansions of Free Fermions 71 4.5 Partition Function of the Free Fermion 75 5 Applications 79 5.1 Aﬃne Kac-Moody algebras and WZW models 79 5.2 Coset Constructions 84 3

Introduction

Aim: to understand CFT’s in 2d.

Why? (A) Second order phase transitions in 2d systems (so-called critical phenomena). For example, the 2d Ising model. Take a 2d lattice of critical sites (in principal, this is assumed to be inﬁnite):

1 σ spin 2

These spins are equipped with a nearest neighbor interaction. The energy for the system is given by X E({σ}) = − σiσj adjacent lattice sites

1 where σi ∈ ± 2 . There are ground states for the system: all states having the same spin (ie either all | ↑i or all | ↓i). The partition function is X Z = exp (−E({σ})β) {σ} 1 where β = T is the inverse temperature. We can then compute the correlation functions: 1 X −|i − j| hσ , σ i − exp (−E({σ})) σ σ ∼ exp i j Z i j ζ(T ) {σ} where |i − j| is the distance from the site i to the site j.

If T → ∞, then ζ(T ) → 0, so that hσi, σji → 0. If T → Tc, ζ(T ) → ∞, and the system reaches criticality.

At criticality, T = Tc, we have that 1 hσi, σji ∼ 1 |i − j| 4 This theory is scale-invariant (for scales a), and can be solved by use of a 2-dimensional CFT1 1 Onsager won the ’68 Nobel prize in Chemistry for his solution of the 2d Ising model at criticality. 4

(B) String theory provides other examples where CFT’s can be given meaning. It concerns itself with the time evolution of 1d objects (”strings“):

2d world sheet

time t

closed string

This time evolution is described by the Polyakov action, which is reparametrization invariant and Weyl invariant. Together these give us conformal invariance, which leads to a 2d CFT description of 2d worldsheets.

1 Conformal Field Theories in d dimensions

1.1 Conformal group in dimension d (d > 2)

Definition (Local Conformal Transformations). Let M,N be smooth d-dimensional manifolds with metrics g and h respectively2. A local 2 Very generally, we can take (M, g) diffeomorphism of open sets and (N, h) to be semi-Riemannian. That is, the ‘metrics’ can be taken to give symmetric, nondegenerate, φ : U ⊂ M → φ(U) ⊂ N bilinear pairings on the tangent spaces (not necessarily positive definite). is called a local conformal transformation if

φ∗h = Λ · g for a smooth scale function

Λ: U → R>0

Remark. (local) conformal transformations preserve angles. Given two vectors X,Y ∈ TpM, for p ∈ U, we then have

h(φ∗X, φ∗Y ) cos ∠(φ∗X, φ∗Y ) = |φ∗X|h|φ∗Y |h φ∗h(X,Y ) = |X|φ∗H |Y |φ∗h Λg(X,Y ) = √ √ Λ|X|g Λ|Y |g = cos ∠(X,Y ) 5

Remark. • Conformal transformations deﬁne ’new‘ metrics on a space for M = N g˜ = φ∗g = Λ · g for φ : M → M. This implies, with a little computation, that the Weyl Tensor remains invariant under conformal transformations, but the Riemann tensor does not. We will, for now, work with conformally ﬂat space M = N = Rd, with the metric

η = diag(−1,..., −1, 1,..., 1) | {z } | {z } n m where n + m = d. So we can write a conformal transformation φ : Rd → Rd Xµ 7→ φα(Xµ), by requiring the relation

∂φα ∂φβ η = Λη αβ ∂Xµ ∂Xν µ,ν

• The 2d/3d Weyl tensor always vanishes on any 2d/3d manifold that is locally conformally ﬂat.

• In 2d, we can see from cartography that we can map a globe minus a point to a sheet of paper by preserving angles.

We will now consider an inﬁnitesmal transformationXµ 7→ Xµ + µ(X). Which gives us

2 2 µ ν ds → ds + (∂µν + ∂ν µ)dX dX as the transformation of line elements3 3 The line element is, as usual, deﬁned by This transformation is conformal if 2 µ ν ds = ηµν dX dX

∂µν + ∂ν µ = ηµν C

By taking the trace of this relation, we can ﬁnd that

2 ∂ + ∂ = div()η (∗) µ ν ν µ d µν Furthermore

(∗) 2 ∂ ∂ (div) = ∂ ∂ρ(∂ + ∂ − ∂ ) = ∂ ∂ (div) − ∂ µ ν µ ν ρ ρ ν ρ ν d µ ν µ ν Since this expression is symmetric under the interchange of µ and ν, we can simplify to 2 1 ∂ ∂ (div) − (∂ + ∂ ) d µ ν 2 µ ν ν µ and, applying (∗) again, we get

(ηµν + (d − 2)∂µ∂ν )(div) = 0 6

Taking the trace, we then get

(div) = 0

Ansatz: For the inﬁnitesmal transformations:

µ ∂µ∂ν (div) = 0 ⇒ div = A + BµX

Which implies ν ν ρ µ = aµ + bµν X + CµνρX X where Cµνρ = Cµρν From this, we can classify inﬁnitesmal conformal transformations:

(i) µ = aµ, called translations

ν (ii) µ = bµν X . In (∗), we get: 2 b + b = η (ητσb ) µν µν d µν τσ

ν a) µ = wµν X , where wµν = −wνµ, which we call rotations. ν b) µ = λ · ηµν X = λXν which we call dilations.

ν ρ 4 4 1 ρσ (iii) µ = CµνρX X . In (∗), we get : Where bµ = d Cρσµη

Cµνρ = ηµρbν + ηµν bρ − ηνρbµ

so that 2 µ = 2(b · X)Xµ − bµ|X| which we call special conformal transformations (SCTs).

By integrating these inﬁnitesmal versions, we can obtain expressions for ﬁnite conformal transformations:

(i) Translations: X˜ µ = Xµ + aµ

(ii) Rotations: Poincar´eSubgroup ˜ µ µ ν X = Mν X µ for Mν ∈ SO(n, m).

(iii) Dilations: X˜ µ = ΛXµ

(iv) SCTs: µ µ 2 ˜ µ X − b |X| X = µ 2 2 1 − 2bµX + |b| |X| 7

As it happens, the conformal group acting on Xµ is precisely SO(n + 1, m + 1) d+1 Interlude: Projective space RP = Rd+2 \{0}/R∗. Consider the map

d d+1 ι : R → RP 1 1 Xµ 7→ (1 + |X|2): X1 : ··· : Xd : (1 − |X|2) =: X 2 2 P Properties:

(i) ι(R) is on the projective light cone: d+1 {0 = ηP(XP,XP)} ⊂ RP where

ηP = diag(−1,..., −1, 1,..., 1) | {z } | {z } n+1 m+1

d+1 (ii) SO(n + 1, m + 1) acts on RP in the canonical way XA = M A XB P P B P where M A ∈ SO(n + 1, m + 1) P B (iii) Transformations in SO(n + 1, m + 1) induce the following conformal transformations on Rd a) Rotations:   1 0 0 M A =  µ  B 0 Mν 0 0 0 1 with M µ ∈ SO(n, m) ⊂ SO(n + 1, m + 1). ν Poincar´eSubgroup b) Translations (Xµ → Xµ + aµ)

 1 µ 1  1 + 2 |a| −a − 2 |a| A  1 µ  MB =  aµ −a  1 µ 1 − 2 |a| a 1 − 2 |a|

c) Dilations (Xµ → rXµ)

 1+r2 1−r2  2r 0 2r A  1  MB =  0 0  1−r2 1+r2 2r 0 2r

µ µ 2 µ X −b |X| d) SCT’s (X → µ 2 2 ) 1−2bµX +|b| |X|

 1 µ 1  1 + 2 |b| −b − 2 |b| A  1  MB =  −bµ bµ  1 µ 1 1 + 2 |b| −b 1 − 2 |b| 8

Conformal Invariants d For points xi ∈ R

• If we require translational and rotational invariance, we can take as our invariant

|x1 − x2|

• If we additionally require scale invariance, it suﬃces to add another point: |x1 − x2| |x1 − x3| • Applying an SCT, we get

1 1 2 2− 2 2 2− 2 |x1 −x2| → 1 − 2bx1 + |b| |x1| 1 − 2bx2 + |b| |x2| |x1 −x2|

Therefore, conformal invariants are anharmonic ratios5: 5 Also called cross ratios

|x1 − x2||x3 − x4| |x1 − x3||x2 − x4|

N Remark. There are a total of 2 (N − 3) distinct anharmonic invariants for a set of N points6. 6 Warning: There can be complicated algebraic relations among such invariants. 1.2 Correlation function of Quantum Fields (d > 2)

We take the Euclidean signature in d dimensions, so that our conformal group is SO(1, d + 1). We then get the commutation relations

[JAB,JCD] = i (ηABJBC + ηBC JAB − ηAC JBD − ηBDJAC )

And JAB = −JBA,

η = diag(−1, 1, 1,..., 1) | {z } d+1

and A, B, C, D range through −1, 0, 1, 2,...,D. For the Conformal Generators7: 7 µ, ν are assumed to be strictly positive in the deﬁnitions below. • Lµν = Jµν Rotations Poincar´eAlgebra • Pµ = J−1,µ Translations

• D = J−1,0 Dilations

• Kµ = −J−1,µ + J0,µ SCTs 9

The Conformal algebra is then the Poincar´ealgebra plus the commutation relations

[D,Pµ] = iPµ

[D,Kµ] = −iK + µ

[D,Lµ,ν ] = 0

[Kµ,Pν ] = 2i(ηµν D − Lµν )

From these algebraic deﬁnitions, we can then get conformal transformations, eg:

µ • exp (−ib Kµ) ∈ SO(1, d + 1) SCT

µ • exp (−ia Pµ) ∈ SO(1, d + 1) Translations

• exp (−iλD) ∈ SO(1, d + 1) Dilation

i µ,ν • exp − 2 ω Lµν ∈ SO(1, d + 1) Rotation

Quantum Fields as ‘∞-dimensional representations’ of conformal group

Example. We can think of functions {g} as representations of SO(1, d + 1) via the map

d d ρg : SO(1, d + 1) × Maps(R → R) → Maps(R → R)

(Λ, g) 7→ g ◦ fΛ−1 where

µ µ fΛ : x 7→ f ◦ Λ(x ) µ −1 µ fΛ−1 : x 7→ f ◦ Λ (x )

We then have the inﬁnitesmal generators of the conformal group in the representation ρg

µ µ µ fΛ : x 7→ x + (x) µ µ µ fΛ−1 : x 7→ x − (x)

Example. As an example, we can look at the inﬁnitesmal dilation lambda Λλ µ µ µ fΛλ : x 7→ x + λx So we then have ∂g ρ (Λ )(g)(xµ ) = g(xµ − µ (x)) = g(xµ ) − λxµ =! e−iλD g(xµ ) g λ ∂xµ µ which implies Dµ = −ix dµ 10

In a similar fashion, we can ﬁnd that

Pµ = −i∂µ

Lµν = i(xµ ∂µ − xν ∂µ ) ν 2 Kµ = −i(2xµ x ∂ν − |x| ∂µ )

A Quantum Field ψ is then given by φA such that, for a ﬁnite dimensional vector space V ,

d d ρψ : SO(1, d + 1) × Maps(R → V ) → Maps(R → V ) A A B (Λ, ψ ) 7→ ρV (Λ) B ψ ◦ fΛ−1

Quasi-Primary Fields We consider a ﬁeld ψ(x) at x = 0 8. We 8 x = 0 can be thought of as the then have invariant locus with respect to the action of the lorentz group, dilations, and SCTs. See, eg, [1]. Lµν ψ(0) = Sµ,ν ψ(0) Dψ(0) = −i∆ψ(0)

Kµψ(0) = −Kµψ(0)

We make the further assumption that we have an irreducible representation of the Lorentz group, to get

[∆,Sµν ] = 0

Which, by Schur’s lemma, implies ∆ ∼ 1, ie a number, which we will call the scaling dimension. We also get

[∆,Kµ] = −Kµ so that Kµ = 0 We can also compute the ‘scaling dimension at other locations’

µ Dψ(xµ) = De−ix Pµ ψ(0)

h µ i µ = D, e−ix Pµ + e−ix Pµ D ψ(0)

[D,Pµ]=iPµ µ µ −ix Pµ µ −ix Pµ = e (x Pµ) − i∆e ψ(0)

µ µ = (X Pµ − i∆)ψ(x )

If we assume our field is spinless, ie Sµν = 0, then the field is characterized entirely by the scaling dimension D. We can then derive the finite transformation behavior of a spinless field φ(x).

µ ∆/d µ Λ µ ∂x˜ µ φ(x ) −→ φ˜(˜x ) = φ(x ) ∂xν µ µ µ x 7→ x˜ = fΛ(x )

we call such ﬁelds Quasi-Primary Scalar Fields. 11

Correlation Functions of Quasi-Primary Fields From a CFT, we get the path integral measure

[dΦ] eS[Φ]

Where Φ is the ﬁelds in our theory, and S is the action9 functional. 9 Which must be conformally invari- The correlation function is then ant. 1 Z hφ (x ) ··· φ (x )i = [dΦ]e−S[Φ]Φ (x ) ··· Φ (x ) 1 1 N N Z 1 1 N N

where Z Z = [dΦ]e−S[Φ]

Applying a conformal transformation, 1 Z hφ (˜x ) ··· φ (˜x )i = [dΦ]˜ e−S[Φ]˜ Φ˜ (˜x ) ··· Φ˜ (˜x ) 1 1 N N Z 1 1 N N which implies

µ −∆1/d µ −∆N /d ∂x˜1 ∂x˜N hφ1(˜x1) ··· φN (˜xN )i = ν ··· ν hφ1(x1) ··· φN (xN )i ∂x˜1 ∂x˜N

2-point Correlation Functions

Take two ﬁelds φ1 and φ2. From the requisite symmetries, we can deduce some of the form of the correlation function

• Rotation+Translation invariance implies

hφ1(x1)φ2(x2)i = f(|x1 − x2|)

• Dilationx ˜µ 7→ λxµ invariance implies

hφ1(˜x1)φ2(˜x2)i = λ−∆1−∆2 hφ1(x1)φ2(x2)i

Together, these computations show us that

C1 hφ1(x1)φ2(x2)i = ∆ +∆ |x1 − x2| 1 2 Now applying invariance under SCTs we notice that

∂x˜k 1 µ = µ 2 2 d =: Sk(b ) ∂xµ (1 − 2bµxk + |b| |xk| ) and then compute 1 hφ1(˜x1)φ2(˜x2)i = µ ∆ µ ∆ hφ1(x1)φ2(x2)i S1(b ) 1 S2(b ) 2 C = 12 ∆1 ∆2 ∆1+∆2 S1 S2 |x1 − x2| 12

but, on the other hand

C12 hφ1(˜x1)φ2(˜x2)i = ∆ +∆ |x˜1 − x˜2| 1 2 C = 12 (∆ +∆ )/2 (∆ +∆ )/2 1 2 1 2 ∆1−∆2 S1 S2 |x1 − x2|

Hence C12 6= 0 only if 1 (∆ + ∆ ) = ∆ = ∆ 2 1 2 1 2 or, more precisely  C12  2∆ ∆1 = ∆2 = ∆ |x1−x2| hφ1(x1)φ2(x2)i = 0 else

One can perform similar simpliﬁcations for 3 and 4 point correlation functions. It is left as an exercise to see

hφ1(x1)φ2(x2)φ3(x3)i =

C123 ∆ +∆ −∆ ∆ +∆ −∆ ∆ +∆ −∆ |x1 − x2| 1 2 3 |x1 − x3| 1 3 2 |x2 − x3| 2 3 1 and

hφ1(x1)φ2(x2)φ3(x3)φ4(x4)i =

Y ∆/3−∆k−∆` f(anh. ratios) |xk − x`| 1≤k<`≤4 P where ∆ = i ∆i.

2 Conformal Field Theories in 2 Dimensions

2.1 Conformal Algebra in 2 Dimensions

2 Local conformal transformations in R with metric δαβ = ηαβ are given by diﬀerentiable maps

2 2 ϕ : U ⊂ R → V ⊂ R where ∗ ϕ η = Λη, Λ: UR>0 This implies ∂ϕα ∂ϕβ η = Λη αβ ∂xµ ∂xν µν where we assume the transformation is orientation preserving, ie

1 2 ∂ϕ(x , x ) > 0 ∂(x1, x2) 13

We then obtain from η11 and η22

1 2 2 2 1 2 2 2 Λ = (∂1ϕ ) + (∂1ϕ ) = (∂2ϕ ) + (∂2ϕ )

and from η12 and η21

1 1 2 2 0 = (∂1ϕ )(∂2ϕ ) + (∂1ϕ )(∂2ϕ )

We can combine these two equations into a single complex equation to get 1 1 2 2 2 2 (∂1ϕ ) − i(∂2ϕ ) = (∂2ϕ ) + i(∂1ϕ ) or, equivalently

1 1 1 2 ∂1ϕ − i∂2ϕ = ±(∂2ϕ + i∂1ϕ )

We then get solutions Orientation Preserving Orientation Reversing (+) (-) 1 2 1 2 ∂1ϕ = ∂2ϕ ∂1ϕ = −∂2ϕ 1 2 2 1 ∂2ϕ = −∂1ϕ ∂1ϕ = ∂2ϕ 1 2 1 2 ∂ϕ(x ,x ) > 0 ∂ϕ(x ,x ) < 0 ∂(x1,x2) ∂(x1,x2)

Metric in homolorphic coordinates The (+) solution gives precisely the Cauchy-Riemann Diﬀerential Equation, so that local conformal transformations in complex coordinates z = x1 + ix2 are local biholomorphic functions ϕ(z). In terms of the holomorphic and antiholomorphic coordinates z, z, we can rewrite the metric

ds2 = (dx1)2 + (dx2)2 = dzdz where dz = dx + idy and dz = dx − idy. Under biholomorphic mappings, we get ∂ϕ ∂ϕ dz 7→ dz, dz 7→ dz ∂z ∂z We then have that the metric transforms as 2 2 φ ∂ϕ ds = dzdz 7→ dzdz ∂z so that 2 ∂ϕ ∆(z, z) = ∂z We thereby see that local conformal transformations are local holomorphic coordinate changes.

Remark. • Often we regard z and z as independent coordinates, and enhance (x1, x2) ∈ R2 to complex coordinates (x1, x2) ∈ C2. • ‘Physical Condition’. We impose z = z∗, the complex conjugate. 14

Conformal Algebra If we take an inﬁnitesmal conformal transformation z 7→ z˜ = z + (z), and take a Laurent expansion ∞ X n+1 (z) = cnz n=−∞

where the cn are assumed to be inﬁnitesmal, we can compute the action on functions ϕ(z, z) ∈ Map(C → R). ecn`n+cn`n ϕ(z, z) = ϕ(z − (z), z − (z))

which is equivalent to

(1 + cn`n + cn`n)ϕ(z, z) = (1 − (z)∂z − (z)∂z)ϕ(z, z) So the generators of the conformal algebra (in ‘function representation’) are n+1 n+1 `n = −z ∂z, `n = −z ∂z The commutation relations these generators satisfy (Witt Algebra relations)

[`n, `m] = (n − m)`n+m

[`n, `m] = (n − m)`n+m Remark. • Treating z and z as independent variables yields two copies of the Witt Algebra A ⊕ A.

• Imposing the ‘physical condition’ leaves us with the subalgebra of

A ⊕ A generated by `n + `n and i(`n − `n for all n • For a Quantum Theory, we need a central extension of the Witt Algebra, which is the so-called Virasoro Algebra.

Global Conformal Transformations On S2 = C∪{∞}, global holomorphic transformations are generated by global vector ﬁelds. We take as an Ansatz ∞ X v(z) = − an`n n=−1 ∞ X n+1 = anz ∂z n=−1 ∞ z→w= 1 z X −n+1 7→ − anw ∂w ∂ →w2∂ z w n=−1 With a global vector ﬁeld

v(z) = −(a−1`−1 + a0`0 + a1`1) We then have generators (applying the physical condition) 15

• `−1 + `−1 and i(`−1 − `−1) translations

• `0 + `0 dilation

• i`0 − i`0 rotation

• `1 − `1 and i(`1 − i`1) SCT

Finite Global Conformal Transformations For S2 = C ∪ {∞}, a, b, c, d ∈ C, ad − bc = 1 we have " # a b ∈ SL(2, C)/Z2 = SO+(1, 3) c d which is the conformal group in d = 2 dimensions. From this, we have the group of M¨obiustransformations of the sphere az + b z 7→ cz + d Here, our generators are

• Translations " # 1a ˜ 1

• Dilations " # λ1/2 λ−1/2

• Rotations " i θ # e 2 −i θ e 2

• SCT " # 1 0 ˜b 1

Remark. • For four points z1, . . . , z4, we can deﬁne invariant 4-point cross ratios (z − z )(z − z ) η = 1 2 3 4 (z1 − z3)(z2 − z4) There are relations among these cross ratios. In fact, for four points, there is only one independent cross ratio. This can be seen by noting that there is always a transformation in SL(2, C)/Z2 which maps     z1 ∞ z   1   2     7→   z3  η  z4 0 16

Physical States are characterized by the eigenvalues of the dilation

operator `0 + `0 and the rotation operator i`0 − i`0. These can be written in terms of the quantities h, h10 10 Sometimes known as the conformal weights of the state |ψi.

`0|ψi = h|ψi

The scaling dimension is the ∆ = h + h (the dilation operator eigenvalue) and the spin11 is s = h − h (the ‘rotation operator eigenvalue’). 11 As one might expect, for bosonic states, we have s ∈ Z. For Fermionic 1 states, s ∈ Z + 2 . We could also take, 2.2 Correlation functions of (quasi-)primary fields more generally, a parafermionic state s ∈ Q. Definition. A field φ(z, z) of conformal weight (h, h) is a primary field if it transforms under local conformal transformations

z 7→ φ(z) =z ˜ locally as

−h ∂φ−h ∂φ φ(z, z) 7→ φ˜(˜z, z˜) = φ(z, z)(∗) ∂z ∂z

Remark. • A Field is called a quasi-primary field if (∗) holds for 12 12 global conformal transformations Sometimes also called SL(2, C)- primaries • A field wtih a different transformation behavior is called a secondary field.

If we take an inﬁnitesmal variation of a primary ﬁeld

z˜ = z + (z) z˜ = z + (z)

We can compute

˜ δ,φ(z, z) = φ(z, z) − φ(z, z) = φ˜(˜z − (z), z˜ − (z)) − φ(z, z)

Taylor expanding to ﬁrst order we get

˜ ˜ ˜ = φ(˜z, z˜) − (z)∂zφ − (z)∂zφ − φ(z, z)

h −h −h i = (1 + ∂z((z)) (1 + ∂z(z)) − (z)∂z − (z)∂z − 1 φ(z, z)

which then becomes δ,φ(z, z) = −(h(∂z(z)) + (z)∂z) − (h(∂z − (z)) + (z)∂z) × φ(z, z) (∗∗) 17

2-point correlation function of quasi-primary fields We consider a quasi-primary field as above13. We can then compute 13 The equation (∗∗) holds for quasi- primary fields where is infinitesmal and a generator of a global conformal 0 = δ,hφ1(z1, z1)φ2(z2, z2)i transformation. = hδ,φ1(z1, z1)φ2(z2, z2)i + hφ1(z1, z1)δ,φ2(z2, z2)i

We can write our inﬁnitesmal conformal transformations as

2 = c−1 + c0z + c1z 2 = c−1 + c0z + c1z

for these and ,(∗∗) holds for any quasi-primary ﬁeld. This implies that

0 = [h1(∂z1 (z1)) + (z1)∂z1 + h2(∂z2 (z2)) + (z2)∂z2 + c] × hφ1φ2i

We can then use the coeﬃcients of

2 (z) = c−1 + c0z + c1z

c−1: We have (∂z1 + ∂z2 )hφ1φ2i = 0 So

hφ1φ2i = c(z1 − z2)

c0: We have

(h1 + h2 + z1∂z1 + z2∂z2 )hφ1φ2i = 0 so, substituting our ﬁrst result, we get

0 (h1 + h2)c(z1 − z2) + (z1 − z2)c (z1 − z2) = 0

and solving this diﬀerential equation:

C12 c(z1 − z2) = h +h (z1 − z2) 1 2 c1: We have

2 2 (2h1z1 + 2h2z2 + z1 ∂z1 + z2 ∂z2 )hφ1φ2i = 0

substituting in

C12 2 2 C12 (2h1z1+2h2z2) h +h −(h1+h2)(z1 −z2 ) h +h +1 = 0 (z1 − z2) 1 2 (z1 − z2) 1 2 so C12(h1 − h2) h +h +1 = 0 (z1 − z2) 1 2

which means that either C12 = 0 or (h1 = h2). 18

If we add in complex conjugation, we see that  C12  h1 = h2, h1 = h2 (z −z )h1+h2 (z −z )h1+h2 hφ1(z1, z1)φ2(z2z2)i = 1 2 1 2 0 else

Remark. • We have a single-values correlator for s = h − h ∈ Z or in 1 Z + 2 . This is owing to the face that

−2h −2h −2h 2s (z1 − z2) (z1 − z2) = |z1 − z2| (z1 − z2)

If we let s ∈ Q, we get a multiple-valued correlation function for parafermions.

• If we let spin equal zero, ie h = h, we recover our previous d- dimensional result.

More generally, we can apply the same techniques to compute 3- and 4- point correlation functions, ﬁnding, when we do, that

C123 hφ1(z1, z1)φ2(z2, z2)φ3(z3, z3)i = h +h −h h +h −h h +h −h (z1 − z2) 1 2 3 (z1 − z3) 1 3 2 (z1 − z2) 2 3 1 1 × h +h −h h +h −h h +h −h (z1 − z2) 1 2 3 (z1 − z3) 1 3 2 (z2 − z3) 2 3 1 and

hφ1(z1, z1)φ2(z2, z2)φ3(z3, z3)φ4(z4, z4)i

Y h/3−hk−h` h/3−hk−h` = f(η, η) (zk − z`) (zk − z`) 1≤≤`≤4

where (z − z )(z − z ) η = 1 2 3 4 (z1 − z3)(z2 − z4) and 4 X h = hi i=1 4 X h = hi i=1

2.3 Radial Quantization

First Example We now return to the context of string theory, with 2-dimensional Minkowski space. Suppose we have a conformally ﬂat world-sheet 19

2d world sheet

time x t

closed string

where we have coordinates t ∈ R (‘time’) and x ∈ S1 = R/(L·Z) where L is the length of the string. In 2d Minkowski space we can take the light cone coordinates t ± x, and apply a Wick Rotation τ := it So that we can deﬁne complex coordinates

ζ = τ + ix

ζ = τ − ix with the identiﬁcation ζ ∼ ζ + iL We then have a conformal map to the punctured conformal plane C∗ −∞ ∞

τ1 τ2

2πζ ζ 7→ exp( L )

τ1 τ2

τ = −∞ τ = ∞ z = 0

In Quantization we have

• A Hilbert state space at each ﬁxed spatial slice (ie, ﬁxed time τ)

• Dynamics given by a propagator amounst such slices in time. 20

On the conformal plane, we have • Hilbert state space deﬁned on circles about the origin

• Propagation of states in the radial direction

i) Dilation operator is the Hamiltonian for the string ii) Rotation operator is a spatial translation along the string.

For such strings, a radial quantization scheme is very natural.

Second Example We again consider the Euclidean space of statistical mechanics. The standard quantization is comprised of: • Hilbert Spaces of states along 1d slices (for example in a 2d lattice).

• Transfer matrices describing a correlation orthogonal to the quan- tized slices. In the limit where the lattice spacing goes to zero, and at criticality, we have a conformal symmetry. So taking radial slices of the plane and a radial propagator gives the same result14. Radial quantization is 14 We could also use something a convenient choice because of the use of the radial operator product stranger, like, for example expansion.

In and Out States We make the assumptions: i) There is a vacuum state |0i from which we can construct the Hilbert space of states in terms of creation operators (positive frequency modes) but in practice, the radial and euclidean schemes prove easiest to work ii) As τ → ±∞, that is, as (z, z) → ∞, 0, the Hilbert space of states with. looks like the Hilbert space of states for a theory of free ﬁelds. We can deﬁne an in-state

|φini = lim φ(z, z)|0i z,z→0

This equality is known as the state-ﬁeld correspondence 21

Hermitian Product In string theory, Hermition conjugation has no eﬀect on the 2d Minkowski space

• Hermitian conjugation on Euclidean time:

τ = it : τ 7→ −τ

• Hermitian conjugation on radial coordinates 1 z 7→ z∗

On quasi primary ﬁelds, the Hermitian conjugation is given by:

1 1 φ(z, z)† = z−2hz−2hφ( , ) z z

We can then deﬁne an out-state by

hφout| := |φini that is † hφout| = lim h0|φ(z, z) z,z→0 This defines a good Hermitian product for quasi-primary fields. As a first example, we can compute:

hφout|φini = C More generally, if we consider multiple ﬁelds, we ﬁnd

 C12 h1 = h2, h1 = h2 hφ1,out|φ2,ini = 0 else

In our computation above, one step went unexplained: The radial ordering (?). We now delve into its meaning 22

Recall that in QFT, the N-point correlator of quantum ﬁelds is given by a time-ordering prescription.

hφ1(x1, t1) ··· φN (xN , tN )i = h0|T (φ1(x1, t1) ··· φN (xN , tN )) |0i

where

T (φ1(x1, t1) ··· φN (xN , tN )) = ±φσ(1)(xσ(1), tσ(1)) ··· φσ(N)(xσ(N), tσ(N))

The sign is due to Bose/Fermi statistics, and the permutation σ is such that

tσ(1) > tσ(2) > ··· > tσ(N) We can perform a similar ordering for radial quantization: the radial ordering.  φ1(z, z)φ2(w, w) |z| > |w| R (φ1(z, z)φ2(w, w)) = ±φ2(w, w)φ1(z, z) |z| < |w|

The positive sign comes in the case where at least one of the ﬁelds in question is bosonic, and the minus sign in the case where both ﬁelds are fermionic. With the radial ordering, we have the operator-state correspondence

hφ1(z1, z1) ··· φN (zN , zN )i = h0|R (φ1(z1, z1) ··· φN (zN , zN )) |0i

Similarly to a Fourier expansion, we have the Mode expansion for quasi-primary ﬁelds of dimension (h, h).

X −m−h −n−h φ(z, z) = z z φm,n m,n∈Z where the φm,n are operators.

Using the calculus of residues, we can compute the φm,n I dz I dz φ = zm+h−1 zn+h−1φ(z, z) m,n 2πi 2πi which then gives us the operator product expansion for the Hermitian conjugate

1 1 X φ(z, z)† = z−2hz−2hφ( , ) = z−m−hz−n−hφ z z −m,−n n,m

Looking at † X −m−h −n−h † φ(z, z) = z z φm,n m,n we see † φm,n = φ−m,−n 23

A well-deﬁned in-state implies that

lim φ(z, z)|0i z,z→0 must be ﬁnite. This, in turn, requires that we have

φm,n|0i = 0 for m > −h, n > −h

Remark. The mode expansion gives rise to string Fourier modes on the cylinder 2πi 2πi z = exp (t + x) z = exp (t − x) L L giving the expansion

X 2πi 2πi φ(z, z) = φ exp − (∆ + m + n)t exp − (s + m + n)x m,n L L m,n∈Z where s = h − h is the spin, ∆ + m + n is the energy eigenstate of the mode, and s + m − n is the wave propagation along the spatial direction of the string.

2.4 Operator product expansions

Correlators of 2 or more ﬁelds typically exhibit singularities as their insertion points coincide. For example, for quasi-primaries

hφk(z1, z2)φ`(z2, z2)i = h0|R(φk(z1, z1)φ`(z2, z2)|0i δ = `k 2h 2h (z1 − z2) k (z1 − z2) k the last expression is called the canonical norm for quasi-primaries. The operator product expansion (OPE): describes the behavior of radially ordered quantum ﬁelds as the coincide. For example, for A(z, z) and B(z, z)

N M X X {A, B}n,m(w, w) R(A(z, z)B(w, w)) = (z − w)n(z − w)m n=−∞ m=−∞ with N,M ≥ 0. In this case, we get a ﬁnite number of singular terms. These singular terms play a special role:

N M X X {A, B}n,m(w, w) R(A(z, z)B(w, w)) ∼ = A(z, z)B(w, w) (z − w)n(z − w)m n=1 m=1 The degree zero term, known as the normal order product, is written I dz I dz R(A(z, z)B(w, w)) : A(w, w)B(w, w): = {A, B}0,0(w, w) = w 2πi w 2πi (w − z)(w − z) 24

Thus singular terms normal order product R(A(z, z)B(w, W )) = A(z, z)B(w, w) +: A(w, w)B(w, w): +O(z−w, z−w)

Remark. • Fields φ(z) that depend only on z are called chiral ﬁelds. Fields φ(z) that depend only on z are called anti-chiral ﬁelds

• OPEs of chiral ﬁelds will play an important role.

• Notation: In OPEs the radial order symbol R is often dropped

R(A(z, z)B(w, w)) = A(z, z)B(w, w)

OPEs and commutators Let A(z) and B(z) be bosonic chiral ﬁelds. We can deform the contour of a small circle about w as seen below Im(z)

Re(z)

Im(z)

w Re(z)

So we can rearrange the contour integral of the radially ordered product: I dz I dz I dz R(A(z),B(w)) = A(z)B(w) − B(w)A(z) w 2πi |w|+ 2πi |w|− 2πi

= QAB(w) − B(w)QA

= [QA,B(w)] 25

H dz where QA = 2πi A(z). The result is that contour integrals encode (anti-)commutators at equal time15 15 Whether we get a commutator or I dz an anti-commutator is dependent on R(A(z)B(w) = [QA,B(w)]± the Bose/Fermi statistics. If at least χ 2πi one of the operators is bosonic, we More generally, we can take have [−, −]+ = [−, −] I dw I dz R(A(z),B(w)) = [Q ,Q ] If both are Fermionic, we have 2πi 2πi A B ± [−, −]− = {−, −} with QA and QB as above.

2.5 Energy-momentum tensor and conformal Ward identities

Interlude: Conserved charges and inﬁnitesmal transformations of ﬁelds. If we have a (d + 1) dimensional QFT with a conserved Noether µ µ current j , (ie, ∂µj = 0) then we get a conserved charge (the Noether Charge) Z d Q = dx [j0(x)]

For inﬁnitesmal symmetry transformations acting on a quantum ﬁeld φ(x), ˜ δφ(x) = φ(x) − φ(x) = −[Q, φ(x)] where the last term is the equal time commutator. In the context of a 2d CFT with radial quantization, the conserved charge Q of the radial component of a conserved chiral current yields I dz δφ(w, w) = −[Q, φ(w, w) = − R(A, φ) w 2πi

Energy-Momentum tensor

The energy-momentum tensor Tµ,ν generates local coordinate trans- formations16. For Lorentz-invariant theories it is conserved 16 In Lagrangian theories, it is the response of the Lagrangian to the µ ∂ Tµν = 0 variation of space-time.

so we get a conserved charge Z d P µ = dx Tµ,nu

which are the momentum operators, and are translationally symmetric.

By rotational invariance, Tµν can also be chosen to be symmetric

Tµν = Tνµ

In conformal theories, we call the conserved current

D ν jµ = x Tνµ 26

the dilation current because the associated charge D is the dilation operator. Scale invariance (generated by the dilation operator) leads to

µ D µ 0 = ∂ jµ = Tµ so that the energy-momentum tensor is traceless in CFTs. Back in the 2d setting, we can take holomorphic coordinates

z = x1 + ix2 z = x1 − ix2 1 1 ∂ = (∂ − i∂ ) ∂ = (∂ + i∂ ) z 2 1 2 z 2 1 2 We can then compute the energy-momentum tensor

∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x2 T = T + T + T z,z ∂z ∂z 1,1 ∂z ∂z 2,2 ∂z ∂z 2,2 1 = (T − 2iT − T ) 4 1,1 1,2 2,2 1 T = (T + 2iT − T ) z,z 4 1,1 1,2 2,2 ∂x1 ∂x1 ∂x1 ∂x2 ∂x2 ∂x1 ∂x2 ∂x2 T = T = T + + T + T z,z z,z ∂z ∂z 1,1 ∂z ∂z ∂z ∂z 1,2 ∂z ∂z 2,2 1 = (T + T ) 4 1,1 2,2 From the metric

gz,z = gz,z = η(∂z, ∂z) = η(∂z, ∂z) = 0 1 g = g = η(∂ , ∂ ) = z,z z,z z z 2 we can compute conservation laws:

i) Translation invariance gives

∂zTz,z + ∂zTz,z = 0

ii) Because of scale invariance

Tz,z = Tz,z = 0

so that

∂zTz,z = 0

17 17 Implying that Tz,z and Tz,z are chiral and anti-chiral currents , Notation: the chiral energy- respectively. momentum current is

T (z) := −2πTz,z(z) and the anti-chiral is

T (z) := −2πTz,z(z) 27

Conformal Transformations of primary fields Let φ(w, w) be a chiral ﬁeld of conformal weight (h, h). Taking a local inﬁnitesmal coordinate transformation

z 7→ z˜ = z + (z)

we get

δφ(w, w) = −[Q, φ(w, w)] I dz = − (z)R(T (z), φ(w, w)) w 2πi

= − (h(∂w(w))φ(w, w) + (w)∂wφ(w, w)) (∗)

We notice that I dz (z)φ(w, w) 2 = ∂w((w))φ(w, w) w 2πi (z − w) I dz (z)φ(w, w) = (w)∂wφ(w, w) w 2πi (z − w) so that, with (∗) we ﬁnd that a primary ﬁeld φ(w, w) is characterized by its OPE with T (z) and T (z).

h 1 R(T (z)φ(w, w)) ∼ φ(w, w) + ∂ φ(w, w) (z − w)2 z − w w h 1 R(T (z)φ(w, w)) ∼ φ(w, w) + ∂ φ(w, w) (z − w)2 z − w w

Remark. Theres OPEs of φ(z, z) with T (z) and T (z) are often used as the deﬁnition of a primary ﬁeld.

Conformal Ward identities

Let φk(wk, wk) be any primary ﬁelds located at |wk| < R, k = 1,...,N. For z 7→ z + (z), we can use the contour

Im(z)

R `R

w2 Re(z)

w3 28

to compute

I dz (z)T (z)φ1(w1, w1) ··· φN (wN , wN ) `R 2πi N X I dz = φ (w , w ) ··· (z)T (z)φ (w , w ) ··· φ (w , w ) 1 1 1 2πi k k k N N N k=1 wk which implies

hT (z)φ1(w1, w1) ··· φN (wN , wN )i N X hk 1 ∂ ∼ 2 + hφ1(w1, w1) ··· φN (wN , wN )i (z − wk) z − wk ∂wk k=1 which is known as the conformal Ward identity for primary ﬁelds. The conformal Ward identities imply I dz δhφ1 ··· φN i = − (z)hT (z)φ1(w1, w1) ··· φN (wN , wN )i ` 2πi N I X dz hk 1 ∂ (z) 2 + hφ1 ··· φN i 2πi (z − wk) z − wk ∂wk k=1 ` in particular

δhφ1 ··· φN i = 0 for an inﬁnitesmal global conformal transformation

2 (z) = c−1 + c0z + c1z allows us to recover diﬀerential equations for correlators as discussed before. There are some technical assumptions that were needed to derive the Ward identities:

• T (z) must be regular in the conformal plane18. 18 It is worth recalling that T (z) should not be thought of as a func- • Poles must only arise when the ﬁelds coincide. tion, but rather as a holomorphic section of a bundle speciﬁed by the • T (z) is an energy density, that is scaling dimension. As a result, we can require more strongly that T (z) 1 −2 be regular on P without necessarily Tµν ∼ (length) imposing the condition that T (z) be constant. From which we can calculate the scaling dimension ∆ = 2 2 which implies T (z) has conformal weight h = 2, and T (z) has conformal weight h = 2. 29

If we take the coordinate transformation 1 z 7→ w = − z we see that

dw −2 T (z) 7→ T˜(w) = T (z) = z4T (z) dz which tells us that T (z) must decay as z−4 if regularity is to hold at z = ∞.

2.6 Examples of CFTs (Free CFTs)

Recall from QFT:

(A) Noether’s Theorem (See, eg, di Francesco et al): For a QFT with an action Z S[φA] = ddxL[φA]

and an inﬁnitesmal symmetry transformations δxµ x˜µ = xµ + δwa

A A ˜A µ δF [φ ] φ (˜x ) + wa δwa there is a conserved current

µ a ∂ jµ = 0

given by

ν A a ∂L A δx ∂L δF jµ = µ A ∂ν φ − ηµν L + µ A ∂(∂ φ ) δwa ∂(∂ φ ) δwa

(B) Wick’s Theorem: A time-ordered product19 of ﬁelds equals 19 For CFT, we replace the time- the sum of normal ordered products with all possible contrac- ordered product with the radially ordered product, ie, τ(··· ) with tions, eg R(··· ).

τ(φ1, φ2, φ3, φ4) =: φ1φ2φ3φ4 : +: φ1φ2 φ3φ4 : + ··· +: φ1φ2 φ3φ4 :

+: φ1φ2 φ3φ4 : + ··· +: φ1 φ2φ3 φ4 :

where the contractions are given by20 20 Here I use the notational convention that φˆi means omit the entry ˆ ˆ φi : φ1 ··· φk1 ··· φk2 ··· φn : = (±): φ1 ··· φk1 ··· φk2 ··· φN : × hφk1 φk2 i 30

Example I: The Free Boson Take the free, real ﬁeld φ : R2 → R with g Z S[φ] = d2x∂ φ(x)∂µφ(y) 2 µ 1 Z Z = d2x d2yφ(x)A(x, y)φ(y) 2

as the action21 where 21 Notice that this is a scalar ﬁeld whose action is obviously invariant µ µ µ A(x, y) = −gδ(x − y )∂µ∂ with respect to translations, rotations, and scale trasformations. A The two point correlation function is based upon little work shows that it is also SCT invariant. R(φ(x)φ(y)) = GF (x, y) the Feynman propagator, which is deﬁned to be

−1 Gf (x, y) = A (x, y) in the sense that22 22 That is, in a very loose sense, Z inverse under the inner product on 2 µ µ 2 d yA(x, y)GF (y, z) = δ(x − z ) distributions of R . This implies that

µ µ ∂ ∂ δ(x − z ) = −g µ Gf (x, z) (∗) ∂x ∂xµ As we saw previously, rotational and translational invariance means

GF (x, y) = GF (|x − y|) = GF (r) and, expressing the equation (∗) in polar coordinates and integrating, we get Z 1 1 = 2π dr r −g ∂ r∂ G (r) r r r F which, when solved, yields 1 G0 (r) = − F 2πgr so that 1 G (x, y) = − log |x − y| + const F 2πg

In complex coordinates23, we can rewrite the action as 23 Where Z i 2 d2z = dzdz = dx2 S[φ] = 2g d z∂zφ(z, z)∂zφ(z, z) 2

so that 1 R(φ(z, z)φ(w, w)) = − (log(z − w) + log(z − w) + const) 4πg

24 24 For a chiral (holomorphic) ﬁeld, we have ∂zφ(z, z) = 0, so Note that, for free ﬁelds there is only a single singular term in the 1 1 R(∂ φ(z, z)∂ φ(w, w)) ∼ − OPE. z w 4πg (z − w)2 31

Energy-Momentum tensor We take an inﬁnitesmal symmetry transformation

x˜µ = xµ + µ

acting on S[φ]. φ˜(˜xµ) = φ(xµ) Then we get ∂L T = ∂ φ − η L µν ∂(∂µφ) ν µν g = g∂ φ∂ φ − η ∂ φ∂ρφ µ ν 2 µν ρ so the tensor is symmetric and traceless (as expected from a scale invariant theory). In complex coordinates, we can compute (cf section 2.5) 1 T (z) = −2π (T − 2iT − T ) 4 1,1 1,2 2,2

= −2πg∂zφ(z, z)∂zφ(z, z)

From quantum theory, we have

h0 | T (z) | 0i = 0 so that

T (z) = −2πg : ∂zφ(z, z)∂zφ(z, z :

OPEs

(A) We compute the radial ordering

R(T (z)∂wφ(w, w)) = −2πg : ∂zφ∂zφ: ∂wφ ∼ −2πg : ∂z φ∂zφ: ∂w φ+: ∂zφ∂z φ: ∂w φ ∂ φ(z, z) ∼ z (z − w)2 ∂ φ(w, w) ∂ φ(w, w) ∼ z + z (z − w)2 (z − w)

Which tells us that ∂zφ(z, z) is a chiral primary ﬁeld with con-

formal weight h = 1. Similarly, we have that ∂zφ(z, z) is an anti-chiral primary with conformal weight h = 1. 32

(B) We now work on the energy-momentum tensor

2 2 R(T (w)T (w)) ∼ 4π g (: ∂zφ∂zφ:: ∂wφ∂wφ:) ! ∼ 4πg 2(: ∂z φ∂z φ∂wφ∂w φ: ) + 4: ∂z φ∂zφ∂w φ∂wφ:

1/2 : ∂ φ∂ φ: ∼ − 4πg z z (z − w)4 (z − w)2 1/2 2(−2πg : ∂ φ∂ φ:) : ∂2 φ∂φ : ∼ + w w − 4πg w w (z − w)4 (z − w)2 (z − w) Where the last step comes by Fourier expanding. In conclusion, we then have

c/2 2T (w) ∂ T (w) R(T (z)T (z)) ∼ + + w (z − w)4 (z − w)2 (z − w)

where c = 1.

Remark. Note that T (z) is not primary because of the circled term. It is ‘almost’ a chiral primary ﬁeld of conformal weight h = 0

c is called the central charge. For the free boson, as we have seen, c = 1.

Example II: The Free Fermion CFT We take the Euclidean action of a real (Majorana) fermion25 25 Where the γµ are Pauli matrices. In principle, we could take any, by Z g 2 † 2 µ here we take S[ψ] = d xΨ γ γ ∂µΨ 2 0 −i γ1 =   i 0 Z ∂z = g d2xΨ†  ddots  Ψ and   0 1 γ2 = ∂z 1 0 Z so that 2 = g d z ψ∂zψ + ψ∂zψ {γµ, γν } = 2ηµν where Ψ = (ψ, ψ). The equations of motion are then

∂zψ = 0 ⇒ ψ is a chiral (holomorphic) ﬁeld

∂zψ = 0 ⇒ ψ is an anti-chiral ﬁeld

We can then calculate the Feynman Propagator26 26 As before, the idea is to write the action in terms of a kernel which we 1 Z Z then invert S[Ψ] = d2x d2yΨ†(x)A(x, y)Ψ(y) 2 µ µ 2 µ A(x, y) = gδ(x − y )γ γ ∂µ 33

We then obtain the Feynman propagators27 27 Notice that the fermi statistics is encoded within the propagator. 1 1 R(ψ(z, z)ψ(w, w)) ∼ 2πg z − w 1 1 R(ψ(z, z)ψ(w, w)) ∼ 2πg z − w R(ψ(z, z)ψ(w, w)) = 0

We can also compute the energy-momentum tensor using Noether’s theorem

T (z) = −πg : ψ∂zψ :

OPEs

(A) We have two possible contractions to worry about in computing the radial ordering

R(T (z)ψ(w, w) ∼ πg : ψ∂zψ : ψ(w, w) 1 ∂ ψ(z, z) 1 1 ∼ z − ψ(z, z)∂ 2 (z − w) 2 z z − w | {z } | {z } red orange 1 ∂ ψ(w, w) ∂ ψ(w, w) ∼ w + w 2 (z − w)2 (z − w)

which implies that ψ(z, z) is a chiral primary with conformal 1 weight h = 2 . Similarly, ψ(z, z) can be shown to be an anti-chiral 1 primary with conformal weight h = 2 (B) We also have, for the energy-momentum tensor

1/2 2T (w) ∂ T (w) R(T (z)T (w)) ∼ 2 + + w (z − w)4 (z − w)2 (z − w)

which tells us that the free Majorana fermion has central charge 1 c = 2 .

2.7 The central charge c

Observation: c 2T (w) ∂ T (w) R(T (z)T (w)) ∼ 2 + + w (z − w)4 (z − w)2 (z − w)

We call c the central charge. As we have already calculated, the free 1 boson has c = 1, and the free Majorana fermion has c = 2 . 34

Properties

• For dilations z 7→ λz, we have

T (z) 7→ λ−2T (z)

which, in turn, implies

R(T (z)T (w)) 7→ λ−4R(T (z)T (w))

so the central charge is consistent with the expected scaling relation.

• For a sum of decoupled CFTs CFT1,..., CFTN , the energy- momentum tensor is

N X Ttotal(z) = T1(z) i=1 And, since the theories are decoupled, this means

N X R(Ttotal(z),Ttotal(w)) = R(Tk(z)Tk(w)) k=1 In turn, this means that we get

ctotal = c1 + ··· + cN

ie, the central charge is an extensive quantity.

• Transformation behavior under inﬁnitesmal conformal transformations xµ 7→ x˜µ = xµ + µ(x) yields I dz δ T (w) = −[Q ,T (w)] = − (z)R(T (z)T (w)) 2πi I dz (z) c z(z)T (w) (z)∂ T (w) = 2 + + w 2πi (z − w)4 (z − w)2 z − w c = − ∂3 (w) − 2(∂ (w))T (w) − (w)∂ T (w) 12 w w w

A ﬁnite local conformal transformation takes the form

z 7→ z˜ = w(z) so that dw −2 c T˜(˜z) = T (z) − {w(z); z} dz 12

with the Schwarzian derivative28 28 Interestingly, while the invariance properties proved below related the Schwarzian derivative to the complex plane, it also appears to bear some relation to solutions of the hypergeometric equation. 35

w000(z) 3 w00(z)2 {w(z); z} = − w0(z) 2 w0(z)

For z 7→ u(z) and u 7→ w(u) we have the following ‘chain rule’ for the Schwarzian derivative du {w(u(z)); z} = {u(z); z} + {w(u); u} (∗) dz

Now, moving on to global conformal transformations in PSL(2, C), we have az + b z 7→ z˜ = cz + d Such transformations are generated by

(i) z 7→ λz, which has Schwarzian derivative

{λz; z} = 0

(ii) z 7→ z + c, which has Schwarzian derivative

{z + c; z} = 0

1 (iii) z 7→ − z , which has Schwarzian derivative

1 −3!z−4 3 −2z−3 2 { ; z} = − z −z−2 2 −z−2 = 6z−2 − 6z−2 = 0

As a result, we see that {w(z); z} = 0 for any PSL(2, C) transformation. This implies that for such transformations

dw −2 T˜(w) = T (z) dz and thus, T (z) is a chiral quasi-primary ﬁeld with conformal weight h = 2. 36

Conformal anomaly and central extensions

Symmetry group G on some space lifts to action lifts to a on phase unitary space? rep?

Quantum Mechanics Classical Mechanics Want to impose G Poisson Algebra act- action is unitary ing on phase space

Anomaly Obstruction to lifts

In some cases of anomalies, we can ﬁnd a group extension Gˆ of G by some group H29 such that a representation can be constructed 29 That is, a Gˆ ﬁtting into a short with respect tog ˆ exact sequence 0 → H → Gˆ → G → 0 Example. In quantum mechanics in three dimensions, consider the 1 symmetry group SO(3), and the state space of spin 2 states 1 1 , ± 2 2 The action of 2πiJ 1 1 2πi(± 1 ) 1 1 e 3 , ± = e 2 , ± 2 2 2 2 1 1 = − , ± 2 2 does not quite agree with identites. However, there is an extension of SO(3) by Z/2, ie

SU(2)/(Z/2) =∼ SO(3) Which implies that, taking Gˆ = SU(2), and considereding " # −1 ∈ SU(2) −1 we get the desired result. Moreover, this extension is central. That is, Z/2 is in the center of SU(2). 37

Example (Classical Mechanics). We can ﬁnd an explicit central extension of the Galileo group, see [2] for details.

Central Extensions of the deWitt Algebra We consider the OPE of T (z), the inﬁnitesmal generator for local conformal transformations. c/2 zT (w) ∂ T (w) R(T (z),T (w)) ∼ + + w (z − w)4 (z − w)2 z − w Recall that T (w) has the expansion

X −n−2 T (z) = z Ln n∈Z I dz L = zn+1T (z) n 2πi which gives us the commutation relations I I dw dz m+1 n+1 [Ln,Lm] = w z R(T (z)T (w)) 0 2πi w 2πi

From these relations, we retrieve the Virasoro Algebra Vir ⊕ Vir30 30 This algebra is completely characterized by the commutation relations Claim. Vir is a central extension of the deWitt algebra31 A by . C c 2 [Ln,Lm] = (n − m)Ln+m + n(n − 1)δn+m,0 That is, there is a short exact sequence of Lie algebra homomorphisms 12 c 2 f g [Ln, Lm] = (n − m)Ln+m + n(n − 1)δn+m,0 0 → C → Vir → A → 0 12 [Ln, Lm] = 0 where C is generated by c, and there includes into the center of Vir, 31 Whose commutation relations are of and f, g respect the Lie bracket of Lie algebras. the form 32 Remark. Vir is the unique non-trivial central extension of A. [`n, `m] = (n − m)`n+m 32 To see why this is the case, we ﬁrst comment that non-trivial cen- Non-trivial meaning the sequence does not split, that is, is not equiva- tral extensions are classiﬁed by the cohomology group lent to the sequence Z2(g, a) 0 → a → a ⊕ g → g → 0 H2(g, a) = B2(g, a) Where (a)Θ ∈ Z2(g, a) means that

Θ: g ⊗ g → a Θ(X,Y ) = −Θ(Y,X) Θ(X, [Y,X]) + Θ(Z, [X,Y ]) + Θ(Y, [Z,X]) = 0

we call such Θ cocycles

(b)Θ ∈ B2(G, a), the exact cocycles, are characterized by the property that there exists µ : g → a such that Θ(X,Y ) = µ([X,Y ]) 38

Given a representative of an equivalence class [Θ] ∈ H2(g, a), we see that Θ deﬁnes a Lie bracket for h (The direct sum of g and a as vector spaces) by the formula33 33 More generally, the second Lie Al- gebra Cohomology classiﬁes extensions 0 0 0 g [L ⊕ Θ,L ⊕ Θ]h = [L, L ] + Θ([L, L ]) of a Lie algebra by a module over g. See, for example [3] pp. 234 or [4] Since the Θ is exact, we have pp. 153 for a purely mathematical treatment.

L˜ = L ⊕µ L and hence 0 0 [L,˜ L˜ ]h = [L,˜ L˜ ]g Now, returning to the case of the de Witt algebra A, one can compute that 2 H (A, C) =∼ C =∼ hΘi where c Θ(L ,L ) = n(n2 − 1)δ n m 12 n+m,0 This tells us that Vir is unique as a non-trivial central extension34. 34 For a more thorough treatment, see [5] ch. 4 & 5. The physical interpretation of c

(1) Casimir Energy35: Previously we mapped the cylindrical world- 35 The Casimir eﬀect is a phenomenon sheet of a string to the (punctured) conformal plan. If we now do whereby a force is observed between two plates positioned extremely close the reverse, we get a map together. L z 7→ ζ = log(z) 2π where L is the circumference of the cylinder. In these new coordinates,

 Schwarz. Deriv. 2π 2 c z }| { 2   Tcyl(ζ) = z T (z) − {log(z), z}  L 12 | {z } 1 −2 | {z } 2 z dζ 2 ( dz ) So that 2π 2 c T (z) = z2T (z) − cyl L 24

Assuming that the expectation values satisfy

hT (z)i = T (z) = 0

then 1 hT i = − hT (ζ)i + T (ζ) 0,0 2π cyl cyl π(c + c) = 12L2 39

This allows us to compute an energy for the cylinder

Z L π(c + c) Ecyl = − hT0,0i = − 0 12L which is the Casimir energy.

Ecyl

Notice that, as expected, if L → ∞, we have Ecyl → 0 Casimir Energy of the Free Scalar Field (Heuristic deriva- tion): We can compute the ground state energy of a sum of harmonic oscillators X 1 E = ω |0i 2 p p∈{Harm. Osc.} which yields36 36 We can think of our harmonic ∞ X 1 2π oscillators as merely being waves on a E = 2 · k τ-slice. The factor of 2 appearing in |0i 2 L k=1 the equation can then be said to arise as a result of left- and right-moving To attempt to remove this divergence, we introduce the UV regu- waves (sin and cos respectively). lator L = 1 ΛUV and add an exponential damping term

∞ X 1 2π E = 2 · k e−k |0i 2 L k=1 ∞ ! 2π d X = e−k L d k=0 2π 1 1 = − + O() L 2 12

We then deﬁne the regularized energy to be

2π π Ereg = lim E − = − |0i →0 |0i L2 6L

And, since for the scalar ﬁeld c = c = 1, π Ereg = − (c + c) |0i 12L 40

(2) Trace anomaly on a closed oriented Riemann Surface Cg of genus g: µ On the classical level, Tµ = 0. On the quantum level, we get

µ hTµ iCg ∼ c · R

where R is the Ricci scalar37. 37 The term on the left hand side is called the trace anomaly. Notice The proportionality constant can be computed from the free that R is a local quantity which fermionic CFT. See [6] for more details. This yields depends locally on the metric and has scaling dimension 2. This dependence on the metric is acceptable since a µ c T µ = R conformal transformation is also a 24π metric transformation.

2.8 The Hilbert Space of States

As before we consider the OPE c/2 zT (w) ∂ T (w) T (z)T (w) ∼ + + w (z − w)4 (z − w)2 z − w

with X −n−2 T (z) = z Ln n and the commutation relations c [L ,L ] = (n − m)L + n(n2 − 1) n m n+m 12 In this system the vacuum, T (z)|0i for z → 0 must be well deﬁned, which implies that38 38 This follows from the fact that T (z)|0i must not have singular terms. Ln|0i = Ln|0i = 0 for n > −2. In particular, the vacuum |0i invariant under global conformal transformations, which are generated by L±1, L0, L±1, and L0.

Raising and Lowering operators of primaries We consider a primary of conformal weights (h, h)

X −m−h −k−h φ(w, w) = w w φm,k k,m

And the commutation relations39 39 Following from the OPE

I hT (w) ∂wT (w) dw n+1 T (z)φ(w, w) ∼ + [Ln, φ(w, w)] = R(T (z)φ(w, w))z (z − w)2 (z − w) w 2πi n n+1 − h(n + 1)w φ(w, w) + w ∂wφ(w, w)

40 40 Which, in turn, implies Because φ−m,−k|0i vanishes for m < h, k < h. 41

[Ln, φm,k] = (n(h − 1) − m)φn+m,k

We deﬁne an asymptotic state

|h, hi := φ(0, 0)|0i = φ−h,−h|0i

And get the action of L0

L0|h, hi = [L0, φ(0, 0)]|0i = h|h, hi

analogously, we see that

L0|h, hi = h|h, hi

We then see that |h, hi is the energy eigenstate of the Hamiltonian

H = L0 + L0

The commutation relations

[L0, φm,k] = −mφm,k

tell us that φm,k are raising operators for m < 0 and lowering operators for m > 0.

Verma Module

The Virasoro raising operators L−m (m > 0) acting on |h, hi yield

[L0,L−m] = mL−m

⇒ L0L−m|h, hi = (m + h)|h, hi

By similar arguments, we see that Lm for m > 0 are lowering operators

Lm|h, hi = Lmφ−h,−h|0i

= (m(h + 1) + h)φ−h+m,−h|0i = 0

where the last equality holds because φ−m,−h|0i = 0 for m < h.

Deﬁnition. A descendant state is given by acting with a chain of

raising operators L−m1 ,...,L−kn on the asymptotic state |h, hi. Any descendant state can be written in the form

L−k1 ··· L−kn |h, hi

with the ordering convention 1 ≤ k1 ≤ k2 · · · ≤ kn

The Hilbert space of states arising from |h, hi is called a Verma modules. It forms a representation of the Virasoro algebra41 41 ie a module over the Virasoro algebra. 42

2.9 Conformal Families and Operator Algebra

Recall that a descendant state42 42 Recall: X −n−2 T (z) = z Ln L−n|h, hi = [L−n, φ(0, 0)]|i n I dz R(T (z)φ(0, 0) = n−1 |0i 0 2πi z for n ≥ 1 yields a definition of a descendant field (secondary field): I {u} dz R(T (z)φ(w, w)) φ (w, w) = n−1 w 2πi (z − w) for n ≥ 1. Recursively, we have descendant states:

I dz R(T (z)φ{ks−1,...,k1}(w, w) φ{ks,ks−1,...,k1}(w, w) = ks−1 w 2πi (z − w) where ki ≥ 1 Remark. • We use the short-hand notation ~ φ{k}(w, w) = φ{ks,ks−1,...,k1}(w, w)

43 43 • If we include anti-holomorphic descendants , we get states Using the operator L−n instead of L−n. ~ ~ φ{k},{k}(w, w)

We can also compute correlation functions of descendants, for instance: I {k} dz 1 hφ0 (w0)φ1(w1) ··· φN (wN )i = k−1 hT (z)φ0(w0) ··· φN (wN )i w0 2πi (z − w0) I dz 1 = k−1 w0 2πi (z − w0) N ! X 1 hn × ∂ + hφ (w ) ··· φ (w )i (z − w ) wn (z − w )2 0 0 N N n=0 n n where the second equality follows from the Ward identity (cf section 2.5). The correllators

hT (z)φ0(w0) ··· φN (wN )i can have poles only at z → wk. This implies that N I {k} X dz 1 hφ0 (w0)φ1(w1) ··· φN (wN )i = − k−1 2πi (z − w0) `=1 w` N ! X 1 hn × ∂ + hφ ··· φ i z − w wn (z − w )2 0 N n=0 n n N X ∂w` hell = − k−1 + (−(n − 1)) k (w` − w0) (w` − w0) `=1

× hφ0 ··· φN i 43

So that

{k} hφ (w0) ··· φN (wN )i = L−khφ0(w0) ··· φN (wN )i where N X (k − 1)h` 1 L−k = k − k−1 ∂` (w` − w0) (w` − w0) `=1

In general, any correlator

{k~0} {k~N } hφ0 ··· φN i can be rewritten in terms of diﬀerential operators acting on

hφ0 ··· φN i

with the help of the Ward identity. Therefore,

hφ0 ··· φN i

determines all descendant correlators.

Definition. The set of a primary field φ together with all its descendant fields is called the conformal family of φ, which we denote by [φ].

Conformal families have the following properties:

• Members of the conformal family [φ] transform under (local) conformal transformations among themselves. That is, OPEs of T (z) with any ﬁeld of [φ] will involve only ﬁelds of [φ].

• We have the operator state correspondence

{Conformal families} 1:1↔ {Verma Modules}

The Operator Algebra For a given CFT, the OPEs among all its primaries (including regular terms) form the so-called operator algebra. The knowledge of the operator algebra determines all correlators of the CFT (it ‘solves’ the CFT). Given a CFT (with countably many primaries), we normalize primaries such that

δk` hφk(w, w)φ`(z, z)i = (z − w)2hk (z − w)2h`

Then44, 44 See, for instance, the deﬁnitions of in and out states in section 2.3. 2hk 2hk lim w w hφk(w, w)φ`(z, z)i = hhk, hk|h`, h`i = δk` (w,w)→∞ (z,z)→0 44

The orthogonality of primaries therefore implies the orthogonality of entire Verma modules. To see this, we can use the Virasora algebra for descendant states. For example:

L−n|h`, h` and L−n2 L−n3 |hk, hki (ni > 0)

gives us

hh , h |L† L† |h , h i = hh , h |L L L |h , h i k k −n3 −n2 ` ` k k n3 n2 −n1 ` `

= hhk, hk|Ln3 [Ln2 ,L−n1 ]|h`, h`i

= hhk, hk|Ln3 ((n2 + n1)Ln2−n1 ) c + n2(n2 − 1)δ |h , h i 12 2 2 n2−n1,0 ` ` = (n + n )hh , h |L L |h , h i non-vanishing 2 1 k k n3 n2−n1 ` ` for n1>n2

n1>n2 = (n2 + n1)hhk, hk|[Ln3 ,Ln2−n1 ]|h`, h`i

= (n2 + n1)hhk, hk|(n3 + n1 − n2)Ln2−n3−n1 c + n (n2 − 1)δ |h , h i 12 3 3 n2+n3−n1,0 ` `

= (n2 + n1)δn2+n3,n1 ((n3 + n1 − n2)h` c + n (n2 − 1))hh , h |h , h i 12 3 3 k k ` ` = 0

Scale invariance determines the structure of the operator algebra:

~ ~ ~ X X X s{k}{k} hs−hk−h`+|k| φk(z, z)φ`(0, 0) = Ck` z ~ [φs] {k} {~k}

~ ~ ~ hs−hk−h`+|k {k}{k} ×z φs (0, 0)

where ~ X |k| = kn n and ~ k = (k1, k2,...) k1 ≤ k2 ≤ k3 ···

s{~k}{~k} The constants Ck` are the structure constants of the operator algebra.

Three-Point Correlator 45

We can compute the correlator

2hs 2hs hφs|φk(z, z)|φ`i = lim w w hφs(w, w)φk(z, z)φ`(0, 0)i w,w→∞ w2hs = lim Csk` w,w→∞ zhk+h`−hs whs+h`−hk (w − z)hs+hk−h` anti-holomorphic × part C = sk` h +h −h zhk+h`−hs z k ` s or, alternately X X X hφs|φk(z, z)|φ`i = hφs| [...] |0i ~ [φ`] {k} {~k}

X r{}{} hs−hk−h` hs−hk−h` = Ck` hφs|φriz z r δ Cr{}{} = sr k` h +h −h zhk+h`−hs z k ` s so that s{}{} s Ck` = Ck` = Csk` Since all correlators of descendants arise from correlators of primaries, one can show that:

~ s{~k}{~k} s s{~k} s{k} Ck` = Ck`βk` βk`

where

s{~k} s{~k} βk` = βk` (hs, hk, h`, c) s{} βk` = 1

Example. Consider two chiral primaries φk(z) and φ`(z) with h =

hk = h` (for simplicity), then

X s hs−2h φk(z)φ`(0) = Ck`z Xs(z) s where ∞ X X N s{~k} Xs(z) = z βk` L−{~k}φs(0) N=0 {~k} |~k|=N Hence ∞ X N Xs(z)|0i = z |N; hsi N=0 where |H; hsi is the level N state descending from

|φsi = |hsi 46

We can then compute:

Lnφk(z)φ`(0) = [L − n, φk(z)]|h`i n+1 n = z ∂z + (n + 1)hz φk(z)|h`i

so that ∞ X N Ln (Xs(z)|0i) = z Ln|N; hsi N=0 ∞ ! X n+n N+n = (N + hs − 2h)z + (n + 1)hz |N; hsi N=0 Which gives us that

Ln|N + n; hsi = (hs + (n − 1)h, +N)|N; hsi (∗)

For low N, we can then begin to examine what descendant states are produced:

Level 1 There is one descendant state

s{1} |1; hsi = βk` L−1|hsi

(∗) (i) L1|1; hsi = hs|0; hsi = hs|hsi s{1} s{1} (ii) L1|1; hsi = βk` [L1,L−1] |hsi = 2βk` h2|h2i | {z } 2L0 Hence, 1 βs{1} = kl 2 Level 2 There are 2 descendant states

s{2} s{1,1} |2, hsi = βk` L−2|hsi + βk` L−1L−1|hsi

(i) We have

(∗) 1 L |2; h i = (h + 1)|1; h i = (h + 1)L |h i 1 s s s 2 s −1 s (∗) Ls|2; hsi = (hs + h)|hsi

From (i) and (ii), we ﬁnd a pair of linearly independent equations s{1,1} s{2} for βk` and βk` : 2 s{1,1} c − 12h − 4hs + chs + 2hs βk` = 2 4(c − 10hs + 2chs + 16hs) 2 s,{2} 2h − hs + 4hhs + hs βk` = 2 c − 10hs + 2chs + 16hs 47

Remark.

s{~k} • All of the coeﬃcients βk` (hk, h`, hs, c) can, in principle, be recursively determined. As a result, the operator algebra of a CFT is determined completely by

a) conformal families [φs]

b) three-point correlators of the primaries, ie Csk`

• Csk` must be obtained separately (eg, by dynamic input or crossing symmetries).

2.10 Conformal Blacks & Crossing Symmetries

Having now treated three-point correlation functions, we an in a posi- tion to revisit four-point and higher correlation functions, ie

hφk(z1, z1)φ`(z2, z2)φm(z3, z3)φn(z4, z4)i

We apply a conformal transformationwhich sends

z1 → ∞, z2 → 1, z3 → η, z4 → 0 where, η is, as usual, the cross ratio

(z − z )(z − z ) η = 1 2 3 4 (z1 − z3)(z2 − z4)

We can then write down a correlation function matrix element45

`k 2hk 2hk Gmn(η, η) = lim z z hφk(z, z)φ`(1, 1)φm(η, η)φn(0, 0)i 45 z,z→∞ Where we use the indexing convention that the indices are read = hhk, hk|φ`(1, 1)φm(η, η)|hn, hni counterclockwise around the symbol, as in `k If we then insert the operator product expansion of φm(η, η) and Gmn φn(0, 0), we get * `k X X p{~k}{~k} hp−hm−hn−|~k| Gmn(η, η) = hk, h)k | φ`(1, 1) Cmn η [φp] ~k,~k hp−hm−hn−|~k| ×eta × L ~ L | hp, hp −{k} −{~k}

We can then compute

`k X p `k `k Gmn(η, η) = CmnCpk`Fmn(p|η)F mn(p|η)

[φp] 48

where

hhi|φj(1)L ~ |hpi `k hp−hm−hn X |~k| p{~k} −{k} Fmn(p|η) = η η βmn hhi|φj(1)|hpi {~k}

× L ~ L |hp, hpi −{k} −{~k} The `k Fmn(p|η)

are called conformal blocks of the conformal family [φp].

`k hn+hm−hp Remark. •F mn(p|η)η are regular holomorphic functions at η = 0.

`k • The coeﬃcients of the expansion of Fmn(p|η) depend only on

c, hp, h`, hk, hm, hn that is, there is no dependence on the 3-pt structure constants.

`k • For explicit expressions for Fmn(p|η), see, eg, [7]

Crossing Symmetries ij The Gk` depend on a particular order of primaries, which results in one particular OPE. We could, however, think of reordering the primaries to get a diﬀerent expression. We can diagrammatically represent the conformal blocks in the following way, permuting the indices each time46: 46 Only the index circled in orange must be ﬁxed under this permutation. 1) This is because it determines both the out state and the ηhi··· regularization (1) j i (∞) factor in the limit.

j i p Fk` (p|x) =

(η) k ` (0) which, in turn, implies that we can write the s-channel expres-

sion47 47 From quantum ﬁeld theory one     would also expect a T-channel and a j i j i U-channel, which, as we will shortly     see, also exist.     ji X p  p   p  Gk`(x) = CijCpk`     p      (η) k `   (η) k ` 

Here, we have (∞ − 1)(η − 0) x = = η (∞ − η)(1 − 0) 49

2) (1) j i (∞) p ji Fk`(p|x) = (η) k ` (0) giving the t-channel     p p `i X p Gkj(1 − η, 1 − η) = Ci`Cpkj     p η η

Here we have (∞ − 0)(η − 1) x = = 1 − η (∞ − η)(0 − 1) 3) (1) j i (∞)

ki F`j (p|x) = p (η) k ` (0) giving the u-channel     `i 1 1 X p G , = C Cp`j  p   p  kj η − 1 η − 1 ik p η η

here, (∞ − η)(0 − 1) 1 x = = (∞ − 0)(η − 1) η − 1 We would expect these diﬀerent interpretations to agree, that is, we impose the crossing symmetries

1 1 Gji (η, η) = G`i (1 − η, 1 − η) = Gki , k` kj `j η − 1 η − 1 | {z } | {z } s-channel t-channel | {z } u-channel

Bootstrap approach: Specify the dynamics of a CFT by consistency using crossing symmetries. Given a CFT with central charge c and N conformal families we have N 3 + N |{z} |{z} Ck`m h` parameters, for k, `, m ∈ {1,...,N}. The crossing symmetries provide N 4 constraints, so naive counting suggests that crossing symmetries ﬁx all unknown parameters. This is true in some cases, eg for minimal models. 50

Remark. Higher point correlators can be calculated analogously to the 4 point case.

3 Minimal Models

Definition. A minimal model conformal field theory is a conformal field theory with a finite numbe rof conformal families.

Example. The 2d Ising model at criticality.

Remark. For the purposes of this lecture, we will focus on unitary minimal models.

3.1 Warm-up: Representations of su(2)

48 49 48 The Lie algebra su(2) has generators J3 and J± , satisfying More commonly studied are representations of the Lie group SU(2), but for our purposes, we need the [J3,J±] = ±J± corresponding Lie algebra. [J+,J−] = 2J3 49 It also admits a Casimir Operator

2 1 J := J3 + {J+,J−} To ﬁnd representations of su(2), we start with a highest weight 2 state |ji with

J3|ji = j|ji

J+|1i = 0

k We can then deﬁne a decendant state J−, where

This implies that, for j ∈ 2N0, the value k = 2j + 1 yields a highest weight state, ie 2j+1 J+ J− |ji = 0 The state 2 |χ2j+1i := J−j + 1|ji is called a singular vector. |χ2j+1i generates a subrepresentation of the highest weight representation generated by |ji.

Hilbert Space of States & Unitary Representations 51

Let |ψi be a state with conjugate hψ|,

† J3 = J3 † J± = J∓

and take the normalization hj|ji

(i) For 2j ∈ N0, if we have

k1 k2 hj|J+ J− |ji= 6 0

then k1 = k2. Additionally

hχ2j+1|χ2j+1i = 0

and k k hχ2j+1|J+J−|χ2j+1i = 0

so that the subrepresentation generated by |χ2j+1i yields only null states. We can therefore take our Hilber space of states to be the quotient of our representation by the relations

k |ψi ∼ |ψi + αJ−|χ2j+1i

That is k H = J−|ji k=0,...,2j = h|jii / h|χ2j+1i]i

So we get a ﬁnite spin representation of su(2)50 50 Which is, in addition, unitary.

(ii) Let j > 0, 2j∈ / N0: We then get a non-ﬁnite representation of su(2). Pick k = d2j + 1e, then

k k k−1 k−1 hj|J+J−|ji = hj|J+ J− |ji ×k (2j + 1 − k) | {z } | {z } >0 <0

We easily see that we get a negative norm state at level d2j + 1e, giving us a non-unitary representation of su(2).

Observation. • Singular vectors give rise to unitary (ﬁnite dimensional) representations of su(2).

• Highest weight representations without singular vectors yield (inﬁ- nite dimensional) non-unitary representations of su(2). 52

3.2 Reducible Verma Modules & Singular Vectors

We explore highest weight representations of the Virasora algebra with central charge c. Let |hi be a highest weight state:

Ln|hi = 0 for n > 0

L0|hi = h|hi

The space of states will then be

L−{~k}|hi

with inner product.

Deﬁnition. A singular vector is a descendant state |χi with Ln|χi = 0 for all n > 0, that this, |χi is a highest weight state.

Observation. A singular vector and its descendants are orthogonal to any other state.

Proof. Let |χi be a singular vector at level N, so that it has an expansion as X |χi = c~kL−{~k}|hi ~k |~k|=N Then the state

hψ|L−{~`}|χi can only be non-vanishing if ψ is a state at level N + |~`|. Let

|ψi = L−{ ~m}|hi where |~m| = N + |~`|. Then

hψ|L{vec`}|χi = hh|L{ ~m}L−{~`}|χi h i = hh| L{ ~m},L−{~`} |χi = 0 because h i X L{ ~m},L−{~`} = D~sL~s ~s where |~s| = |~m| − |~`| = N > 0.

Reducible Verma Module

Deﬁnition. A Verma module V (c, h) is reducibleif there is a singular vector at some level N, ie X |χi = c~kL−{~k}|hi ~k |~k|=N 53

such that, for all n > 0, Ln|χi = 0. In this case, |χi is a highest weight state generating a Verma submodule Vχ of V (c, h). Remark. •| χi is a primary state of conformal weight h + N, and it is also a descendant state of |hi

• Vχ is orthogonal to V (c, h) with respect to the deﬁned inner product.

• Let V (c, h) be a Verma Module with singular vectors |χii, then we can construct irreducible51 Verma module M(c, h) by quotienting 51 See previous note. out all Verma submodules

M(c, h) = V (c, h)/ ∼

Singular Vectors & Negative Norm States at Low Levels

Level 0 hh|hi = 1: normalization condition.

Level 1 L−1|hi has norm

hh|L1L−1|hi = 2hhh|hi = 2h

so that the Gram matrix at level 1 is

(1) M = (hh|L1L−1|hi) = (2h)

52 52 That means we get a null state for h = 0 , and a necessary condi- That is, L−1|0i is a singular vector. tion for unitary representations is that This null state corresponds to the vacuum. h ≥ 0

Level 2 We now have two possible states

2 |ψ1,1i = L−1|hi

|ψ2i − L−2|hi

We can compute the 2 × 2 Gram matrix at level 2, " # hψ |ψ i hψ |ψ i M (2)(c, h) = 1,1 1,1 1,1 2 hψ2|ψ1,1i hψ2|ψ2i " # 4h(2h + 1) 6h = 1 6h 4h + 2 c 54

We compute that the trace of this matrix is 1 trM (2)(c, h) = 8h(h + 1) + c 2

and the determinant53 53 Known as the Kac determinant.

(2) det M (c, h) = 32(h − h1,1)(h − h1,2)(h − h2,1)

Each zero of this determinant will correspond to a singular vector. the ﬁrst,

h1,1, = 0 corresponds to the singular vector we have already found. The other two 1 h = 5 − c − p(1 − c)(25 − c) 1,2 16 1 h = 5 − c + p(1 − c)(25 − c) 2,1 16 are new

Exercise. There are two singular vectors at level 2, |χ1,2i and |χ2,1i associated to h = h1,2 and h = h2,1 respectively.

A First Glance at Unitary Representations

We have singular vectors h = h1,1 = 0, and 4 2 4 c 3 h = h , h ⇔ 1 = h + − h − + 1,2 2,1 3 3 3 6 2

h c = 1

5 h = 8 Unitary repr. still possible from anal- generic ysis up to level 2, points non- BUT: we expect 1 unitary further constraints. h = 4

manifest non-unitary representation (level 1) 55

3.3 Kac Determinants

The Kac determinants are the determinants of the Gram Matrices, and display a number of useful properties:

(i) A singular vector |χi at level K yields null states at level N > K:

L−{~k}|χi with |~k| = N − K. Each singular vector at level K therefore yields p(N − K)54 null states at level N. 54 Where p(m) is the number of partitions of m. More precisely, it is (ii) The order of the entries of the Gram determinants M (N)(c, h) at the number of distinct non-decreasing level N in h is given by sequences of natural numbers which add up to m. † • Since each element in L = L ‘commutes to L0’, −{~k} {~k}

† ~ ordh hh|L L |hi = length(k) −{~k} −{~k}

• For ~k0 6= ~k

† ~ ordh hh|L L |hi < length(k) −{~k0} −{~k}

Since now not all generators in L† = L do not commute −{~k} {~k} to L0.

This tells us that the diagonal terms in the Gram matrix at level N give rise to the leading contribution in h55: 55 The second step of this calculation is a purely number theoretic identity, (N) X ~ not a physical principle. ordh det M (c, h) = length(k) ~k |~k=N X = p(N − r · s) r,s∈Z 1≤r,s r·s≤N

(iii) At level N we have

# ((r, s) | 1 ≤ r, s and r · s = N)

new singular vectors, labelled by hr,s

We can compute a formula for the Kac determinants

(N) Y p(N−r·s) det M (c, h) = αN (h − hr,s(c)) r,s≥1 r·s≤N where αN is some numerical (non-vanishing) constant independent of h and c. 56

56 56 Conjecture (Kac). The values of hr,s and c for minimal models Later proved by Feign and Fuchs. are given by ((m + 1)r − ms)2 − 1 h = r,s 4m(m + 1) 6 c(m) = 1 − m(m + 1) Note. There are two possible solutions for the second equation

1 r25 − c m = − ± 1/2 2 1 − c However,

hr,s(m1(c)) = hs,r(m2(c))

We take the Convention that, for m real, we choose m ∈ R≥0

Unitary representations of Vir We begin with a few observations

(i) For level 1, if det M (1) (c, h) ≥ 0 then h ≥ 0

† c 2 (ii) hh|L−n L−n |hi = 2nh + 12 n(n − 1) > 0 This must hold for all n, which implies that c ≥ 0

(iii) If we consider the Kac determinants 1 < c < 25, we get that

m1/2 ∈/ R, which implies that hr,s ∈/ R for r 6= s. If r = s, then hr,r < 0 for r >. Similarly, for c ≥ 25, hr,s < 0, so there are no positive solutions for h. Taken all together, this implies that det M (k) is non-vanishing and positive deﬁnite for c > 1 and h ≥ 0 (see exercises).

(iv) In the region 0 ≤ c ≤ 1, h ≥ 0 we have the Kac determinant √ √ 2 96hr,s + 4(1 − c) = 1 − c(r + s) ± 25 − c(r − s) ≥ 0

a) near c = 1, take c = 1 − 6. For r 6= s 1 1 √ h () = (r − s)2 ± (r2 − s2 ) + O() r,s 4 4 For r = s 1 − c h = (r2 − 1) r,r 24 Diagrammatically, we can represent what we know so far about unitary representations and minimal models in a somewhat ﬁner version of the graph from last section: 57

h4,2 h3,1

35 24 h4,3 = 4 h1,4 m min. model 1 1

h3,2

non-unitary unitary representations representations

5 h2,1 8 = 3 h1,3 m

1 min. model 3 h3,3 1 4 h2,4 1 h2,2 8 h3,4 h1,2 h2,3 0 h 1 7 1 c 1,1 2 10 = 2

m non-unitary min. model representations

Returning to the region 0 ≤ c < 1, h ≥ 0,

• A generic point in this region gives rise to non-unitary representations only.

• Frienan, Qiu, Shenkar, in [8] found unitary representations on ‘first intersections’. The idea is that the resulting singular vectors at ‘first intersections’ are sufficient to make the associated Verma module Unitary. These ‘first intersections’ are given by precisely the formulae from Kac’s conjecture above, where 1 ≤ r < m and 1 ≤ s < r

Note. • hr,s (m) = hm−r,m+1−s (m) for m ∈ Z≥2 • We denote the associated conformal families

[φr,s ] ≡ [φm−r,m+1−s ] 58

We can codify the distinct conformal families with the use of conformal diagrams. For example, when m = 2, we have c = 0 57 57 h1,2 = h1,2 = 0 or, diagrammatically That is, this minimal model gives the vacuum CFT. r

φ1,1

φ1,2 s

1 In a more complicated case, we can take m = 3, so that c = 2 . Looking at the diagram r

φ2,1 φ2,2 φ2,3

φ1,1 φ1,2 φ1,3

we see that

[φ1,1] = [φ2,3], h1,1 = 0 1 [φ ] = [φ ], h = 2,1 1,3 2,1 2 1 [φ ] = [φ ], h = 2,2 1,2 2,2 16 The conformal diagrams list unitary represenations for a given central charge c = c(m).

Goal: Construct, for a given c(m), unitary CFTs constructed frm the conformal families associated to unitary representations of the Virasora algebra for the given c(m). The resulting CFT’s are unitary minimal models.

3.4 Fusion Rules

Singular vectors give rise to selection rules for non-vanishing three point correlators. 59

Example. We work at level 2 (cf Exercise 8.3). We have a singular vector 3 2 L−2 − L−1 |h2,1i = |χ2,1i 2(2h2,1 + 1) with the null ﬁeld

{2} 3 {1,1} χ2,1(z) = φ2,1 (z) − φ2,1 2(2h2,1 + 1) where58 58 This holds because 1,1 2 I φ2,1(z) = ∂z φ2,1(z) {1} dz φ2,1 (w) = R(T (z)φ2,1(w)) w 2πi Since |χ i is singular, we have 2,1 = ∂wφ2,1(w)

0 = hχ2,1(z)φ1(z1)φ2(z2)i C(z,z ,z ) 1 2 3 2 z }| { = L−2 − ∂z hφ2,1(z)φ1(z1)φ2(z2)i 2(2h2,1 + 1) where

h2−h1−h2,1 h2,1−h1−h2 h1−h2,1−h2 C(z, z1, z2) = C(z − z1) (z1 − z2) (z2 − z)

(cf section 2.9). We then have the diﬀerential equation for C(z, z1, z2), given by h1 ∂z1 h2 ∂z2 3 2 0 = 2 − + 2 − − ∂z (z1 − z) z1 − z (z2 − z) (z2 − z) 2(2h2,1 + 1)

× C(z, z1, z2)

This gives us a constraint for C 6= 0:

2(h2,1 + 1)(h2,1 + 2h2 − h1) = 3(h2,1 − h1 + h2)(h2,1 − h1 + h2 + 1)

Inserting h2,1 = h2,1(m), h1 = hr,s(m) and h2 = hr0,s0 (m) yields the solution r0 = r ± 1 s0 = s So the selection rule is

0 0 hφ2,1φr,sφr0,s0 i= 6 0 ⇒ r = r ± 1, s = s 0 0 hφ1,2φr,sφr0,s0 i= 6 0 ⇒ r = r, s = s ± 1

Fusion rules, on the other hand, summarize the conformal families

appearing in the OPE. For a CFT with states [φp], we can deﬁne the fusion product: X ` [φp] × [φj] = Nkj[φ`] ` where Nkj ∈ Z≥0. For minimal models in particular, we can go further, and say ` Nij ∈ {0, 1} 60

The three point selection rules at level 2 then tell us

(r0,s0) 0 0 N(2,1),(r,s) = 0 for (r , s ) 6= (r ± 1, s)

and (r0,s0) 0 0 N(1,2),(r,s) = 0 for (r , s ) 6= (r, s ± 1)

A dynamical analysis of minimal models yields59 for a unitary 59 See, for example, [6] or [9]. minimal model c = c(m)

k=r+r0−1 `=s+s0−1 X X [φ(r,s)] × [φ(r0,s0)] = φ(k,`) k=1+|r−r0| `=1+|s−s0| k+r+r0≡1 mod 2 `+s+s0≡1 mod 2

3.5 The Critical Ising Model

Recall the Ising model from the ﬁrst lecture:

We have N × N lattice of critical sites:

1 σ spin 2

These spins are equipped with a nearest neighbor interaction. The energy for the system is given by X E({σ}) = − σiσj adjacent lattice sites

1 where σi ∈ ± 2 . There are ground states for the system: all states having the same spin (ie either all | ↑i or all | ↓i). The partition function is X Z = exp (−E({σ})β) {σ}

1 where β = T is the inverse temperature. The energy at a site is X k = σiσk q n o i∈ nearest neigh. of k 61

We can take a continuum limit, a → 0, where Na = const. In this case, we get

σ(z, z) spin ﬁeld (z, z) energy density ﬁeld

The correlation functions for these ﬁelds will be given by

−1 X −βE({σ}) hσ(z, z))σ(0, 0)i = lim Z σkσ0e a→0 Na=const {σ} −1 X −βE({σ}) h(z, z))(0, 0)i = lim Z k0e a→0 Na=const {σ}

The Ising model has a critical temperature Tcrit

h0i = ±1 hσi ' 0 T

0 Tcrit and the correlator goes as

hσ(z, z)σ(0, 0)i ∼ e−|z|/ζ(T ) for T > Tcrit. Where ζ goes as:

Correlation Length

Tcrit

At Tcrit, Onsager showed local conformal invariance, and computed the correlators 1 hσ(z, z)σ(0, 0)i = |z|1/4 1 h(z, z)(0, 0)i = |z|2

1 with c = c = 2 . He also computed the ﬁeld content of the Ising model60 60 It is a spinless theory, ie h = h. 62

1 1 σ(z, z):(h, h) = , spin operator 16 16 1 1 (z, z):(h, h) = , energy operator 2 2 1 :(h, h) = (0, 0) (in any CFT)

1 The m = 3 minimal model has c = 2 , and 1 [φ ] = [φ ]: h = 2,2 1,2 2,2 16

[φ1,1] = [φ2,3]: h1,1 = 0 1 [φ ] = [φ ]: h = 2,1 1,2 2,1 2 The physical ‘Ising minimal model’ includes the anti-holomorphic sector to match with the operators

σ(z, z) = φ2,2(z)φ2,2(z)

(z, z) = φ2,1(z)φ2,1(z)

The fusion rules for the Ising model can be determined using the order-disorder symmetry and the symmetry

σ 7→ −σ

This yields

[σ] × [σ] = [1] + [] [σ] × [] = [σ] [] × [][1]

We would like to determine the 3 point structure constants. From the above, we have

Cσσ1 = C1 = 1

We need to determine Cσσ by dynamics. To do this, we ﬁrst examine the leading terms in the Operator Product

1 1 σ(z)σ(0) = 1 ( + ··· + |z|Cσσ(0, 0) + ··· ) (∗) |z| 4

The m = 3 minimal model has a singular vector at level 2 4, 3 2 |χ1,2i = L−2 − |h2,2i L −1

(|χ2,2i = |χ1,2i)

Which implies that 4 L − L2 hσ(z, z)φ (z , z ) ··· φ (z , z )i = 0 (∗∗) −2 3 −1 1 1 1 N N N 63

We would like to determine the four-point correlation function

1 σσ 4 Gσσ = lim |z| hσ(z, z)σ(1, 1)σ(η, η)σ(0, 0)i z,z→∞

the idea is to take (∗∗) with φ1 = φ2 = φ3 = σ and derive a diﬀerential equation in η. We obtain the equation

d2 1 d 1 η(1 − η) + ( − η) + f (η) = 0 dη2 2 dη 16 k

for k = 1, 261. The matrix element can be expressed in terms of these 61 That is, since this is a second order 62 diﬀerential equation (with singular fk as: 2 points at 0,1, and ∞) we have two 1 X Gσσ(η, η) = c f (η)f (η) linearly independent solutions. σσ 1 k` k ` 62 4 And the f , which turn out to |η(1 − η)| k,`=1 k satisfy the same diﬀerential equations. The solutions fk are given by

1 p 2 f1/2 = 1 ± 1 − η However, there are multiple branch cuts involved in this deﬁnition. σσ Since we want Gσσ to be single valued, there is only one choice we can make for the constants ck`, which gives us

σσ C p p Gσσ(η, η) = 1 |1 + 1 − η| + |1 − 1 − η| |η(1 − η)| 4 Expanding around η = 0, we get (to leading order) σσ C 1 Gσσ = 1 2 + |η| + ··· (∗ ∗ ∗) |η| 4 2

σσ 63 63 We can also obtain an expression for Gσσ using the OPE , If we Corresponding to the diagram apply (∗), we ﬁnd that (1) σ σ (z) 1 σσ 4 Gσσ = lim |z| hσ(z, z)σ(1, 1)σ(η, η)σ(0, 0)i z,z→∞ P = 1 · 1 hσ(z,z)σ(1,1)i |z| 4 = lim hσ(z, zσ(1, 1)1i+ 1 (η) σ σ (0) z,z→∞ |η| 4 +|η|Cσσhσ(z, z)σ(1, 1)(0, 0)i + ···

1 2 |z| 4 1 C = lim + |η| σσ 1 1 1 + 1 −1 1 +1− 1 z,z→∞ |η| 4 |z − 1| 4 |z − 1| 8 8 |z| 8 8

1 2 = 1 1 + |η|Cσσ |η| 4 Comparing this result with (∗ ∗ ∗), we get that 1 1 c = C = 2 σσ 2 As a result, we have solved the Ising model by consistency, and the CFT we have found completely determines the dynamics of the system. 64

Remark. Remarkably enough, we did not at any point need a La- grangian of our theory!

3.6 Minimal Model Characters

The character of a (potentially reducible) Verma module V (c, h) is deﬁned to be64 64 Where #(N) is the number of ∞ ∞ states at level N. h− c X N h− c Y N −1 χV (c,h) = q 24 #(N)q = q 24 (1 − q ) N=0 N=1 This last equality means that we can represent the character in terms of the Euler function ∞ Y φ(q) = (1 − qN ) n=1 as h− c q 24 χ = V (c,h) φ(q) Turning our attention to the characters of a unitary minimal model representation M(c, h), we ﬁrst note that

hr,−s(m) − hr,s(m) = h−r,s(m) − hr,s(m) = r · s

Additionaly, we have a ‘symmetry of indices’ so that

hr,s(m) = h−r,−s(m) = hr+m,s+(m+1)(m)

We can use these to ﬁnd out more about our singular vectors from ﬁrst intersections Singular vectors Level Conf. Weight

|χr,si r · s hr,s + r · s = h−r,s = hm+r,m+1−s

hr,s + (m − r)(m + 1 − s) = hm−r,s−m−1 |χm−r,m+1−si (m − r)(m + 1 − s) = hr,2(m+1)−s The irreducible Verma module is therefore given by M(c(m), hr,s(m)) = Vr,s/ Vr+m,−s+m+1 ∪ Vr,2(m+1)−s

To compute the character, we want to subtract contributions from each submodule, but this leads to the possibility of double counting sub-submodules, and so on. We therefore need to look at the structure of submodules:

Vr,s = V−r,−s = Vr+m,s+m+1

Vr+m,m+1−s = V−r−m,s−m−1 = Vm−r,s+(m+1)

Vr,2(m+1)−s = V−r,s−2(m+1) = V2m−r,s

so we have 65

Vr+m,m+1−s ⊃ Vr+2m,s ∪ V2m−r,−s ≡ ≡

Vr,2(m+1)−s ⊃ Vr+m,s−(m+1) ∪ V3m−r,(m+1)−s

These are the only common submodules, but they might have non- trivial intersections. Considering higher submodule chain structure gives us65 65 Where each arrow in the diagram goes from a module to a submodule.

k Vr+m,m+1−s Vr+2m,s ··· V k 1−(−1) r+km,(−1) s+ 2 (m+1)

Vr,s

k Vr,2(m+1)−s Vr,2(m+1)+s ··· V k 1−(−1) r,k(m+1)+(−1) s+ 2 (m+1)

The character of M(c(m), hr,s(m)) thus becomes

−c/2π ∞ h 1−(−1)k (m) q h X k r+km,(−1)ks+ χ (q) = q r,s + (−1) · q 2 r,s φ(q) k=1

k !! r,k(m+1)+(−1)ks+ 1−(−1) (m+1) + q 2

Or, if we deﬁne the functions −1/24 q X 2 K(m) = q(2m(m+1)k+r(m+1)−ms) /(4m(m+1)) r,s φ(q) k∈Z then (m) (m) (m) χr,s = Kr,s − Kr,−s (m) Remark. χr,s are the generating functions for the (physical) states at level N 66. 66 Shifted by qhr,s−c/24 4 Modular Invariance

Up to this point, we have studied CFTs on the conformal plane. We now aim to study CFT’s on the 2-torus. Motivation: • Consistent 2d CFTs describing critical phenomena should be locally independent of the 2d geometry. In particular, it should be independent of the quantization scheme (on the torus T 2).

• In String Theory, CFTs on Riemann surfaces form part of perturba- tive string theory. T 267 gives rise to a 1 loop correction term. 67 A genus 1 Riemann surface. 66

4.1 Partition Function

We can consider our map on the punctured plane from section 2.3, and extend it to the torus:

L z 7→ ζ = 2π log(z)

identifying boundaries

2 On T , our local symmetries are still the Virasora generators Ln. 68 68 Our global symmetries, however, are generated by L0 and L0 . So that the group of symmetries is U(1) × U(1), the isometries of T 2 with the ﬂat metric. Space-time Structure of the Torus We can obtain the torus as a quotient of the complex plane by a lattice generated by a pair of vectors ω1 and ω2. Diagrammatically, we take the quotient of: 67

Imz

ω1

ω2

ω2 ω1 Rez P to get:

Translations along the cycle are given by a exp − (H · Imω2 − iP Reω2) |ω2| Recall (cf. section 2.7) that we calculated the energy momentum tensor for the cylinder:

2 ! 2π X c T (ζ) = L e−2πn/Lζ − cyl L n 24 n∈Z This allows us to compute the operators H and P : 1 Z 2π 2 c 2π c H = L0 + L0 − = L0 + L0 − 2π ω1 L 12 L 12 Z 2 1 2π 2π P = L0 + L0 = L0 + L0 2π ω1 L L The partition function on T 2 is then given by

Z(ω1, ω2) = Tr (exp (− (HImω2 − iP Reω2)))

Using the modular parameter τ = τ1 + iτ2 where

Reω2 Reω2 τ1 = = ω1 L Imω2 Imω2 τ2 = = ω1 L 68

the partition function takes the form c c Z(τ) = Tr exp 2πi τ L − − τ L − 0 24 0 24 Or L − c L − c Z(τ) = Tr q 0 24 q 0 24 where q = exp(2πiτ).

Observation. The partition function only depends on the modular parameter τ (that is, on the shape of the torus/the twisted gluing condition). No dependence on the size of the torus exists, which is consistent with the conformal property of CFTs.

4.2 Modular Invariance

So far, we have singled out a particular choice of lattice (1-cycles) on T 2, however, we can think somewhat more generally. Consider the picture:

Imz

ω1 + ω2

H ω2

ω1 ω1

ω2

ω1 + ω2 ω2 ω1 Rez P

It is not hard to see that the lattice Λ generated by ω1 and ω2 will also be generated by ω1 + ω2 and ω1, and will be generated by ω1 + ω2 and ω2. Consequently, many diﬀerent choices of initial cycles will give us the same torus. In general, if (ω2, ω1) deﬁnes the lattice Λ ⊂ C, then

" 0 # " #" # ω2 a b ω2 0 = ω1 c d ω1 for " # a b ∈ SL(2, Z) c d describes the same lattice Λ. 69

Under such a transformation, the modular parameter transforms as aτ + b τ 0 = cτ + d where69 69 " # The group a b ∈ SL(2, Z)/(Z/2) PSL(2, Z) = SL(2, Z)/(Z/2) c d is the modular group of the torus. Partition functions of consistent CFTs must be modular invariant (ie, independent of the quantization scheme) due to conformal invariance. Therefore we require that

T 2 0 T 2 ZCFT (τ ) = ZCFT (τ) " # aτ+b a b for τ 0 = where ∈ PSL(2, Z). cτ+d c d

Remarks. (i) PSL(2, Z) is generated by the transformations

1 T : τ 7→ τ + 1 S : τ 7→ − τ

So Z(τ) is modular invariant if and only if

Z(τ + 1) = Z(τ) 1 Z − = Z(τ) τ

(ii) The shape of a torus is speciﬁed by a point in the upper half- plane τ ∈ H = {C | Imτ > 0}

Imz

Rez 1

tori with the ‘same shape’ are therefore identiﬁed by the action of PSL(2, Z) on τ. As a result the moduli space of ‘shapes of tori’70 is given by 70 More explicitly, this is the moduli space of complex structures on the Mcs ' /P SL(2, ) H Z surface of genus 1. 70

To describe distinct points on Mcs, we can look at the fundamental domain F0 ⊂ H, which is given explicitly by  1 Imτ > 0, 2 ≤ Reτ ≤ 0, |τ| ≥ 1 F0 = 1 Imτ > 0, 0 < Reτ < 2 , |τ| > 1 or, diagrammatically:

TF0

SF0

4.3 Free Boson Parition Function

Recall (Exercise 5.1). The Free Boson CFT has a continuum of primaries iαφ(z,z) Vα =: e :

71 71 1 for α ∈ R. These satisfy the relations Where g = 4π is the coupling constant. α2 L |V i = |V i 0 α 2 α α2 L |V i = |V i 0 α 2 α The free Boson CFT also has central charge

cBoson = 1

We now compute the partition function of the free Boson72 72 Where, as before,

c c 2πiτ L0− L0− q = e ZBoson(τ) = Tr q 24 q 24  

√ Z ∞ 2  X α +|~k|− 1  = 2 dα  q 2 24  |{z} −∞   conv. {~k} normal- | {z } ization int. k1

 2   X α +|~k|− 1  ×  q 2 24     {~k}  k1

√ Z ∞ −2piIm(τ)α2 − 1 − 1 = 2 dαe q 24 q 24 −∞

| {z 1 } (Im(τ)) 2 ∞ ∞ Y −1 Y −1 × 1 − qN × 1 − qN N=1 N=1

So that, with some simpliﬁcation73,we get 73 Notably, using Dedekind’s eta function 1 ∞ 1 Y N ZBoson(τ) = 24 − pIm(τ)|η(τ)|2 η(τ) = q (1 q ) N=1 to ease writing. To see modular invariance, we consider the modular properties of η(τ).

2πi T : η(τ + 1) = e 24 η(τ). √ 1 S: η − τ = −iτη(τ) (This can be computed by Poisson resumma- tion).

Returning to ZBoson, we see that

T : |η(τ)| 7→ |η(τ)|

and since Im(τ + 1) = Im(τ), we see

ZBoson(τ + 1) = ZBoson(τ)

Similarly, we note

2 1 2 S : η − = |τ||η(τ)| τ and 1 τ Im(τ) Im − = Im − = τ |τ|2 |τ|2 so that 1 Z − = Z (τ) Boson τ Boson Taken together, this shows that the free Boson CFT is modular invariant.

4.4 Interlude: Mode Expansions of Free Fermions

The mode expansion of the Boson and the Fermion CFT is given by

X −n−1 i∂zφ(z, z) = αnz n∈Z 72

The radial ordering is given by 1 (R (∂ φ(z, z)∂ φ(w, w))) ∼ − z w (z − w)2

And X −n− 1 ψ(z) = ψnz 2 n 1 where h = 2 and  1 n ∈ Z + 2 periodic boundary cond. n ∈ Z anti-periodic boundary cond.

We call the periodic boundary conditions the NS (Nevea-Schwartz)- sector and the anti-periodic boundary conditions the R (Raymond) sector. We can think of these two diﬀerent sectors in terms of mon- odromy about the origin:

In the NS sector, we get ψ(e2πiz) = ψ(z), whereas in the R sector, we get ψ(e2πiz) = −ψ(z). In more explicit geometric terms, we can think of ψ as sections of a spin bundle74 on the (punctured) complex plane. 74 As it turns out, there are precisely two spin bundles on the annulus The commutation relations for the modes are given by ∗ 1 C ' S up to isomorphism, corre- I I sponding to the NS and R sectors. 2 sz n dw m [αn, αm] = i z ∂z, w ∂wφ = nδn+m,0 The structure group for spin bundles 2πi 2πi is Spin(n), deﬁned by the short exact sequence and, taking the anticommutator because of Fermi statistics. 0 → Z2 → Spin(n) → SO(n) → 0 1 ∼ {ψn, ψm} = δn+m,0 For S , we consider SO(1) = 0, so ∼ that Spin(1) = Z2. The two principal 1 1 ∗ ∼ Z2-bundles over S (H (C , Z2) = We can also consider twist ﬁelds, which change the boundary condi- 1 1 ∼ H (S , ZZ ) = Z2) are then given by: tions of fermions from periodic to anti-periodic. 1 The map As an application, let us look at the spin operator of the Ising 1 1 S × Z2 → S model: (eiφ, σ) 7→ eiφ

7→ Ising Model c = 1 1: 1 1 2 Free Fermion CFT c = 2

2 The map [σ] × [σ] = [1] + [] S1 → S1 (z, z) = φ(2,1)(z)φ(2,1)(z) i : ψψ : 1 eiφ 7→ e2iφ h = h = 2

7→ 73

From [] × [σ] = [σ] we infer

σ(w, z) (z, z)σ(w, w) ∼ 1 1 2(z − w) 2 (z − w) 2

1 75 75 because Cσσ = 2 . This implies that Where we denote by µ the disorder operator of the Ising model, ie the µ(w, w) dual ﬁeld to σ. ψ(z)σ(w, w) ∼ √ 1 2(z − w) 2 and, dually, σ(w, w) ψ(z)µ(w, w) ∼ √ 1 2(z − w) 2

Important for us: The spin operator σ(w, w) changes the sign of ψ(z). In the NS and R sectors, we can then compute 2-point correlators. In the NS sector: ∞ ∞ X −n− 1 X −m− 1 hψ(z)ψ(w)iNS = h0| ψnz 2 ψmw 2 |0i 1 1 n= 2 m=− 2 ∞ X −n− 1 n− 1 {ψn,ψm}=δn+m,0 = z 2 w 2 1 n= 2 ∞ 1 X 1 = z−kwk = z (z − w) k=0 In the R sector (from the twist ﬁeld):

hψ(z)ψ(w)iR = h0|σ(∞)ψ(z)ψ(w)σ(0)|0i = hσ|ψ(z)ψ(w)|σi ∞ −∞ X X −n− 1 −m− 1 = hσ|ψnψm|σiz 2 w 2 n=0 m=0 ∞ 1 1 X −n− 1 −n− 1 = √ hσ| {ψ , ψ } |σi + z 2 w 2 zw 2 0 0 | {z2 } n=1 ψ0 1 p z + p w = 2 w z z − w Remarks. • The short distance behavior is unchanged between the sectors76 76 Does this correspond to the local triviality of (spin) bundles? • The boundary conditions modify the behavior at 0 and ∞

• Similarly, we can introduce µ(z, z) as a twist ﬁeld!

If we now examine the Fermionic zero mode ψ0 in the R-sector, we have (−1)F : fermionic mode counting operator 74

which satisﬁes the anti-commutation relation

F (−1) , ψn = 0

for all n. This implies that (−1)F has eigenvalues ±1 for even/odd numbers of fermionic creation and annihilation operators. In particular:

F (−1) , ψ0 = 0 n√ √ o 2ψ0, 2ψ0 = 2 (−1)F , (−1)F = 2 furnish a 2 dimensional Cliﬀord algebra:

{Γµ, Γν } = 2gµν where " # 1 gµν = 1 Furthermore F [L0, ψ0] = [L0, (−1) ] = 0 so the R-ground state must be a representation of the 2 dimensional

Cliﬀord algebra. The smallest representation is |±iR with

± (−1) |±i = ±|±iR

and √ ( 2ψ0) = |±iR Remarks. • We identify:

|+iR = σ(0, 0)|0i

|−iR = µ(0, 0)|0i

• Considering ψ(z) and ψ(z) together, there are two operators (−1)FL and (−1)FR .

We consider:( −1)F := (−1)FL+FR and a two state subset of vac-

uum states (the non-chiral subsector) |±iR with

FL+FR (−1) |±iR = ±|±iR

1 with h = h = 16 If instead of the punctured plane we work on the cylinder, we must recall that primaries transform as

−h ∂φ−h ∂φ φ(z, z) 7→ φ(z, z) ∂z ∂z 75

The mapping to the cylinder is given by l z 7→ ζ(z) = log(z) 2π

∂ζ 2π L 1 L ζ so ∂z = 2π z and e So, for Bosons

2π X 2πζ n (i∂φ) (ζ) = α e L cyl L n n∈Z and for Fermions r 2π X − 2πζ n ψ (ζ) = ψ e L cyl L n ψn where  1 Z + NS sector n ∈ 2 Z R sector

77 77 c We can summarize the boundary conditions for fermions (L0)cyl = L0 − 24 . Conformal Plane Cylinder Periodic bound- NS sector: Anti-periodic boundary conditions. half-integral ary conditions. L |0i = 0 modes 0 (L ) |0i = − 1 |0i (−1)±|0i = 0 0 cyl 48 Anti-periodic boundary Periodic bound- R sector: ψ 0 conditions. ary conditions. zero mode 1 1 L0|±iR = 16 |±iR (L0)cyl|±iR = 24 |±iR ± (−1) |±iR = ±|±iR

4.5 Partition Function of the Free Fermion

For a torus of shape τ

Imz

Rez 1

we have 4 choices of boundary conditions: 76

P A R sector along spa- P P tial slices

P A NS sector along spa- A A tial slices

Observation. Except for P ,P , boundary conditions are not modular invariant

P T A

A A

Our expectation, therefore, is that a modular invariant partition function with anti-periodic boundary conditions must involve all sectors involving anti-periodic boundary conditions.

In the operator formalism, we look at hψ(z)Xi where X is some product of ﬁelds inserted at various positions. In this case hψ(z)Xi 6= 0 only if X is fermionic. We can then transport ψ(z) past all operators X. In anti-periodic boundary operators spatial conditions slice ψ(z + τ) = −ψ(z) ψ(z) In periodic boundary conditions, the insertion of (−1)F operator yields an additional minus sign

hψ(z)(−1)F Xi= 6 0 for X bosonic. ψ(z + τ) = ψ(z)

Thus, in the holomorphic sector, we get, in the spatial R sector

1 1 − F L0 P = √ q 48 trR((−1) q ) 2 P

1 1 − L0 A = √ q 48 trRq 2 P 77

And, in the spatial NS sector

1 − F L0 P = q 48 trNS((−1) q ) A

1 − L0 A = q 48 trNSq A

Where the √1 in the R sector is a convenient normalization factor 2 related to picking only two out of 4 ground state values. 1 In the NS sector: [0] + [h = 2 ] ' [φ1,1] + [φ2,1] conformal families.

4 ψ 5 ψ 3 |0i desc. of [0] level 4 (two states) − 2 − 2 so we have

3 5 7 L0 2 2 2 2 4 trNSq = 1 + q + q + q + q + q + 2q + ··· ∞ Y n+ 1 = 1 + q 2 n=0 and

1 3 5 7 F L0 2 3 4 trNS (−1) q = 1 − q 2 − q 2 + q − q 2 + q − q 2 + 2q + ··· ∞ Y n+ 1 = 1 − q 2 n=0 Remarks. • The structure of states is consistent with the singular vector structure:

[φ1,1]: state at level 1 singular vector

[φ2,1]: singular vector at level 2, so there is only one (rather than two) state at level 2.

• Projections to m = 3 minimal model characters − 1 1 f L χ = q 48 tr (1 + (−1) )q 0 [φ1,1] NS 2 − 1 1 F L χ = q 48 tr (1 − (−1) )q 0 [φ2,1] NS 2 78

In the RR sector, we have [σ] and [µ] dual conformal families

[φ1,2] ' [φ2,2]

L0 eigenvalue State 1 16 + 0 |±iR 1 16 + 1 ψ−1|±iR 1 16 + 2 ψ−2|±iR so we have

∞ L0 Y trRq = n=0 = 2 + 2q + ··· ∞ L0 Y n trR(−1)q = (1 − q ) = 0 n=0 because of the fermionic zero mode. 78 78 So, in summary, we ﬁnd : Where θi are the Jacobi theta functions: ∞ s X (n+ 1 )2/2 1 − 1 Y n θ1(τ) θ (τ) = q 2 P = √ q 24 (1 − q ) = = 0 2 η(τ) n∈Z 2 n=0 X n2/2 P θ3(τ) = q ∞ s n∈Z 1 − 1 Y n θ2(τ) X n n2/2 A = √ q 24 (1 + q ) = θ4(τ) = (−1) q η(τ) 2 n=0 n∈Z P Which display the modular proper- ∞ s ties − 1 Y n θ3(τ) P = q 48 (1 − q ) = iπ/4 η(τ) θ2(τ + 1) = e θ2(τ) n= 1 A 2 1 √ θ2(− ) = −iτθ4(τ) τ ∞ s − 1 Y n θ4(τ) θ3(τ + 1) = θ4(τ) A = q 48 (1 + q ) = η(τ) 1 √ n= 1 θ3(− ) = −iτθ3(τ) A 2 τ θ4(τ + 1) = θ3(τ) We can then write down the modular invariant partition function of 1 √ θ4(− ) = −iτθ2(τ) the fermion: τ 2 2 2 2

Zferm(τ) = P + A + P + A

P P A A

Or exchange wrt S

θ2(τ) θ3(τ) θ4(τ) Zferm(τ) = + + η(τ) η(τ) η(τ) wrt T is modular invariant. 79

Remarks. • The normalization projection onto ‘physical’ zero mode

ψ0 + ψ0 in RR rector means we choose 2 out of 4 values, so as to arrive at the √1 normalization factor. 2 • Ising model partition function has only [σ] family and no [µ] family, which implies

Zferm(τ) = 2ZIsing(τ) so the fermion description induces the ‘dual’ Ising model description.

5 Applications

5.1 Aﬃne Kac-Moody algebras and WZW models

Let g be a simple Lie algebra

a)A Lie algebra g is a vector space equipped with an anti-symmetric bilinear pairing [·, ·]: g × g → g called the Lie bracket satisfying the Jacobi identity. That is, such that for any x, y, z ∈ g

[[x, y], z] + [[z, x], y] + [[y, z], x] = 0 b)a simple Lie algebra g is one such that there are no proper subsets S ⊂ g such that [S, g] ⊂ S

These have be classiﬁed, and are precisely the Lie algebras An,Bn, 79 79 Cn, Dn,E6, E7, E8,F4, and G2. The dual Coexter numbers associated with these algebras are, c) Let J a be generators of g, then respectively:

a b X ab c g [J ,J ] = if cJ An n + 1 c Bn 2n − 1 Cn n + 1 ab where the f c are the real structure constants of g. The dimension Dn 2n − 2 of g is then given by E6 12 E7 18 E 30 dim g = #(generators) 8 F4 9 G2 4 d) The (non-degenerate) Killing Form for g is given by 1 K(x, y) = Tr (ad(x) · ad(y)) 2g Where g is the dual Coexter number and

ad(x): y 7→ [x, y] 80

From g, we can consider Aﬃne Kac-Moody algebras:

i) The loop algebra of g is the vector space of maps

Map(S1 → g)

Since these functions are periodic, we can consider the Fourier decomposition of J(θ) ∈ Map(S1 → g)

X inθ J(θ) = e Xn ,Xn ∈ g n

Identifying eiθ with t, we get an isomorphism given by the Fourier decomposition

1 −1 Map(S → g) ↔ g ⊗ C[[t, t ]]

The algebra on the right has generators

a n a J ⊗ t =: Jn

for n ∈ Z. Multiplying two periodic functions according to the Lie bracket yields another periodic function

a n b m ab c n+m [J ⊗ t ,J ⊗ t ] = if cJ t a b ab c [Jn,Jm] = if cJn+m

ii) There is a unique central extension of the loop algebra

a b X ab c ˆ a b [Jn,Jm] = if cJn+m + Kn K(J ,J ) δn+m,0 c | {z } δa,b which satisﬁes the commutation relations

a b X ab c ˆ a,b [Jn,Jm] = if cJn+m + Knδ δn+m,0 c a ˆ [Jn, K] = 0

This is the aﬃne Kac-Moody algebra of g.

Associated to an aﬃne Kac-Moody algebra, we have a current algebra kδab X J c(w) R(J a(z)J b(w)) ∼ + if ab (z − w)2 c z − w c where a X −n−1 a J (z) = z Jn n∈Z In modes, we recover

a b X ab c ˆ [Jn,Jm] := if cJn+m + Knδa,bδn+m,0 c 81

Suyawara Energy-Momentum Tensor: J a(z) are primary ﬁelds with respect to the energy-momentum tensor X a T (z) = γ : J Ja :(z) a With coupling constant γ, which is ﬁxed by the OPE c/2 2T (w) ∂T (w) R(T (z)T (w)) ∼ + + (z − w)4 (z − w)2 (z − w) So one obtains

1 k · dim g γ = ; c(g ) = 2(k + g) k k + g

Moreover:   b a I du X J (u)Jb(z)J (w) R(T (z)J a(w)) = γ   2πi  u − z  z b J a(w) ∂ J a(w) = ··· = 2γ(k + g) + w (z − w)2 z − w

Thus, J a(w) are chiral primary ﬁelds of conformal weight h = 1. The mode expansion yields

X −n−2 T (z) = Lnz n

where80 80 We define 1 X X a ( a Ln = : JmJa,n−m : J Ja` n < ` 2(k + g) : JaJ : = n a n∈ n a,` a Z J` Ja,n m ≥ ` So we see that a a [Ln,Jm] = −mJn+m In particular a a [L0,Jn] = −nJn We can then consider primary states. The representation theory of our affine Kac-Moody algebra yields affine heighest weight states |λi:

• For a simple Lie algebra g, we have generators E±α81 and Hi82 for 81 The Raising and lowering genera- i = 1, . . . , r where r = rank(g). They satisfy commutation relations tors. 82 The Cartan generators. [Hi,Hj] = 0 [Hi,E±α] = ±αE±α

where α ∈ ∆+ is a positive root. As an example, we can consider SU(2), which has Cartan generator J 3 and raising/lowering generators J ± satisfying

[J3,J±] = ±J± , [J+,J−] = +2J3 82

Recall that unitary representations are labeled by j, mi, 2j ∈ Z≥0, and m ∈ {−j, −j + 1, . . . , j − 1, j}. The highest weight state is then |j, ji.

±α i • Aﬃne Kac-Moody algebras have generators En and Hn. For a Wess–Zumino–Witten (WZW) highest weight state83 83 Also known as a WZW primary.

±α i α En |λi = Hn|λi = E0 |λi = 0

Note that |λi will also be a Virasora primary state, since, for n ≥ 0

X a Ln|λi = : JmJa,n−m : |λi = 0 m That said, a Virasora primary need not be a WZW primary.

± 3 Example (SU(2)k). We have generators Jn and Jn with

3 ± ± [Jn,Jm] = ±Jn+m 3 3 [Jn,Jm] = knδn+m,0 + − 3 [Jn ,Jm] = 2Jn+m + knδn+m,0

˜+ + ˜− − ˜3 3 k Observe that J := J+1, J := J−1 and J := J0 + 2 form a SU(2)-subalgebra. The highest weight states is

|j, ji

which has WZW descendants

|j, j − 1i,..., |j, −ji

The norm of the ﬁnal descendant is

+ − hj, −j| J+1J−1 |j, −ji = (−2j + k)hj, −j|j, −ji | {z } 2J˜3 So we derive a necessary condition for our representation to be unitary: k k ≤ , k ∈ 2 Z≥0 which must hold lest we have negative norm states.

To compute the conformal weight of the highest weight state, we note ﬁrst that g(SU(2)) = 2, so that 1 L = (J~ )2 + ... 0 k + 2 0 and hence j(j + 1) h = |j,ji h + 2 83

and 3 c = k + 2 According to our constraint derived in the example, the unitary WZW 1 k representations of SU(2)k can be labeled by [j] for j = 1, 2 ,..., 2

WZW Example (SU(2)1). We have two unitary representations: [0] 1 WZW 84 ˜− 84 and [ 2 ] ], both with c = 1 . All states are generated by J and For those familiar with string J +. theory, this can be associated with the 0 Boson on the self-dual circle. It arises from T-duality on the bosonic string.

WZW m = −2 m = −1 m = 0 m = 1 SU(2) [0] 3 1 1 3 m˜ = − 2 m˜ = − 2 m˜ = 2 m˜ = 2 rep.

0 |0i spin0 1 2 ^ spinJ˜− spin^1 + spin^3 ^1 2 2 spin 2 1 |−i | + −i | + +−i spin1 + J0

0 | − + + −i spin1 + 2 | − +−i | + + − +−i sing. vect | + − + −i spin0

| − + + − + −i |−−++−i | + − − + + −i 2spin1 + 3 | + − − + + −i | + − + − + −i |−+−+−i | + − + − + −i spin0

Virasora Conformal Families

Note. The commutation relation

3 [Ln,J0 ] = 0

ensures that the Virasora generators do not change the m-quantum numbers.85 85 I have not yet included the 1 WZW [ 2 ] chart. Remarks. • WZW models are unitary, which can be proven by:

a) Pure representation theoretic arguments on gk. 84

b) WZW models admit a non-linear σ-model description with manifest unitary action86 86 For g : S2 → G, the compact Lie group associated to G. Z k 2 −1 µ WZ SWZW = − d x K(g ∂ g, g∂µg) + 2πkS (g) 8π S2

where, for an extension of g g˜ : B3 → G87, the correction term is 87 Such an extension always exists given by since π2(G) vanishes for G a compact, simply-connected Lie group. Z WZ 1 −1 −1 µ ν ρ S (g) = − 2 K g˜ ∂µg,˜ g˜ ∂ν g,˜ g∂˜ ρg˜ dx ∧dx ∧dx 48π B3

• A WZW model has a ﬁnite number of conformal WZW families, but an inﬁnite number of Virasora conformal families.

• Roughly speaking, a CFT is called a Rational Conformal Field The- ory (RCFT) if it has a ﬁnite number of conformal families with respect to some current algebra (eg, minimal models, WZW models).

5.2 Coset Constructions

If we consider the aﬃne Lie algebra

gk ⊕ g`

we ﬁnd that gk+` is a diagonal aﬃne Lie subalgebra, that is

D a (1) a (2) a Jn = Jn + Jn D a D b ab D c ab Jn, Jm = if c Jn+m + (k + `)nδ δn+m,0

This allows us to construct the diagonal coset CFT.

gˆk ⊕ gˆ`/gˆk+`

is generated by the Virasora generators

coset gk g` gk+` Ln := Ln + Ln − Ln which implies that the central charge is

k ` k + ` ccoset − dim g + − k + g k + g k + ` + g

by construction, such coset CFTs are unitary because the gk and g` WZW models are unitary. Minimal model CFTs arise naturally in this framework, via

SU(2)k ⊕ SU(2)` SU(2)k+` 85

which has k 1 k + 1 c = 3 + − k + ` 3 k + 3 6 = 1 − (k + 1)(k + 3)

We can then identify these with m = 2, 3,... minimal models via m = k + 2. 86

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