A CFT is a physical theory invariant under the group of infinitesimal conformal transformations. In particular, as anticipated already, there is no intrinsic notion of length scale or of massive ex- citations.
Even though the definition of a CFT is much more general, let us first take the lagrangian perspective, whose logic is familiar from the definition of general QFTs:
• Our starting point is a classical theory defined by an action S[φi(x)] with the specific
requirement that this action is invariant under infinitesimal conformal transformations. • The basic objects of this theory are the fields Oi(x). By this we mean, by slight abuse of
notation, any local expression built out of the φi(x) appearing in the action and derivatives
thereof. Local fields are e.g. products or power series such as exponentials of φi(x). These
Oi(x) are also called local operators.
• The quantum theory is determined by the correlation functions hO1(x1) . . . On(xn)i =
1 Z
Z
Dφie−S[φi]O1(x1) . . . On(xn). (4.29)
Importantly, the expression inside h. . .i is always time-ordered as is familiar from the treat- ment of correlation functions in usual QFTs.
• In writing equations involving the local Oi we will always think of operator equations in
the full quantum theory, i.e. think of the operators as inserted into a time-ordered path integral as above. For instance an equation of the form
O1(x1) O2(x2) = f (O1(x1), O2(x2)) (4.30)
is shorthand for
hO1(x1) O2(x2) . . .i = hf (O1(x1), O2(x2)) . . .i (4.31)
with . . . representing arbitrary operator insertions at a distance bigger that |x1− x2|.
Now comes an important, though probably unfamiliar point: Conformal symmetry allows for a rather different definition of the theory than the one above starting from a classical action. A generic CFT in d dimensions need not have a description via an action. Rather it is defined by a ’complete set’ of local fields Oi and their correlation functions. If we do have a lagrangian
description, these correlation functions are given as above. More generally, however, we can think of the correlators as maps from the space of operators to C consistent with conformal invariance. We will see that this is very constraining. If really all correlators are known in terms of a finite amount of input data the theory is solved completely and defined through these data.1
There are essentially two reasons why this can work: First, because there is a special notion of a ’complete set of operators’ available in a CFT which does not exist in a general QFT - the set of quasi-primary fields - and second, because the operator product expansion (OPE) of two such quasi-primaries has remarkable properties. Let us introduce both concepts in turn.
1) Primary and quasi-primary fields
Since in this course we are mainly interested in the applications two-dimensional CFTs to string theory we restrict the following discussion, unless states otherwise, to a d = 2 CFT on S2= C ∪ ∞, with general fields of the form O(z, ¯z).
1) If a field Φ(z, ¯z) transforms under z 7→ z0= λz, λ ∈ C as
Φ(z, ¯z) 7→ Φ0(z0, ¯z0) = λ−h¯λ−¯hΦ(z, ¯z) (4.32) then it has conformal dimension (h, ¯h). Note that in general ¯h 6= h∗.
2) A primary field Φ(z, ¯z) transforms as a tensor under conformal tranformations z 7→ z0 =
f (z): Φ(z, ¯z) 7→ Φ0(z0, ¯z0) = ∂f ∂z −h ∂ ¯f ∂ ¯z −¯h Φ(z, ¯z). (4.33) Note: In particular, and in the spirit of the discussion around (4.31), we require as part of the defining property of a primary field that any correlation function involving primary fields transforms as hY i Φ(zi, ¯zi)i 7→ Y i ∂f ∂z|zi −hi ∂ ¯f ∂ ¯z|¯zi −¯hi hY i Φ(zi, ¯zi)i. (4.34)
1A CFT in d = 2 dimensions is indeed exactly solvable in this sense. This is because the two-dimensional
Virasoro algebra is infinite-dimensional. In higher dimensional CFTs the computation of all correlators in terms of finite data is possible in principle, but much harder in practice due to the lack of this extra symmetry.
Expanding f (z) = z + (z) + . . . gives the infinitesimal scaling behaviour for primary fields δ,¯Φ(z, ¯z) = −(h∂z + ∂z+ ¯h∂¯z¯ + ¯∂z¯)Φ(z, ¯z). (4.35)
3) A quasi-primary field satisfies (4.33) for f ∈ P SL(2, C). In particular every primary is a quasi-primary, but not the other way round.
4) A chiral field is a field Φ(z), an anti-chiral field is a field Φ(¯z). Remarks:
• Quasi-primaries are tensors under the group of globally defined conformal transformations. In a d-dimensional CFT with d > 2 (on Rm,n with m + n = d), these are the fields with specific transformation behaviour under SO(m + 1, n + 2).2 All the statements we make about the quasi-primaries based on their transformations under P SL(2, C) transformations in a two-dimensional CFT carry over analogously to quasi-primaries in higher-dimensional CFTs.
• What has no analogue in higher dimensions is the concept of a primary field, which exploits the infinitesimal structure of the two-dimensional Virasoro algebra.
Mode Expansions
Before we proceed let us define the mode expansion of two-dimensional primaries. Consider the transformation from the worldsheet on the cylinder to S2= C ∪ ∞.
Suppose on the cylinder a field Φ is purely left-moving and has the mode expansion ΦL(ξ−) = X n φne−i nξ − (2π`)h. (4.36) Then it is easy to show that if Φ is primary of weight h, the corresponding expansion on S2 is
Φplane(z) =
X
n
z−n−hφn. (4.37)
See Ass. 8 for a proof. The generalisation to fields with chiral and anti-chiral pieces is obvious. The modes φn can be obtained with the help of the residue theorem as
φn =
1 2πi
I
dz Φ(z) zn+h−1. (4.38) 2) The Operator Product Expansion
The second important concept in a CFT is that of the operator product expansion (OPE). In a general (Lorentz-invariant) QFT, the OPE is defined as an approximative expansion of two operators Oi(xi) and Oj(xj) valid in the limit xi− xj→ 0,
Oi(xi)Oj(xj) = X k Cijk (|xi− xj|) | {z } functions in C Ok(xk). (4.39)
OPEs of the above type are defined generally in QFT as convergent series in a certain open neighbourhood of the operators.
In a d-dimensional CFT the structure of the OPE is much more powerful. This is because the OPE In a d-dimensional CFT satisfies three key properties that rely on conformal invariance, which we state here without proof:
• In a d-dimensional CFT the OPE of two quasi-primaries involves only other quasi-primaries and their derivatives, the so-called descendent fields.
• The functional dependence of the Ck
ij(|xi− xj|) is completely fixed by conformal invariance.
• The OPE is an exact expression, i.e. an asymptotic series with radius of convergence the distance to the next field insertion when viewed as an operator equation.
To illustrate this let us specify to a general d = 2 CFT. It can be shown that the OPE of two quasi-primaries φi(z) and φj(w) (taken to be chiral for brevity) can be written as
φi(z)φj(w) = X k,n≥0 Cijk an ijk n! 1 (z − w)hi+hj−hk−n∂ nφ k(w), (4.40) anijk = 2hk+ n − 1 n −1h k+ hi− hj+ n − 1 n , (4.41)
where the sum over k on the right involves only quasi-primaries. For a proof see e.g. [BP], Chapter 2.6.3. What is important for us is the appearance of the so-called structure con- stants Ck
ij. The structure constants directly determine the 3-point functions of the CFT. Then,
by successive application of the OPE we have a chance of reducing all higher n-point functions to lower correlators. This is the idea behind defining the CFT in terms of a finite amount of data. To summarise, of special importance in a (two-dimensional) CFT is the set of quasi-primaries (primaries) and their OPE. We will now learn how to deduce the OPE for certain fields.