Logarithmic conformal field theories have a vast range of applications, from critical percolation to systems with quenched disorder. In this paper we thoroughly examine the structure of these theories based on their symmetry properties. Our analysis is model-independent and holds for any spacetime dimension. Our results include a determination of the general form of correlation functions and conformal block decompositions, clearing the path for future bootstrap applications. Several examples are discussed in detail, including logarithmic generalized free fields, holographic models, self-avoiding random walks and critical percolation.
The ABC (in any D) of Logarithmic CFT
Matthijs Hogervorst, Miguel Paulos, Alessandro Vichi
[2cm] C.N. Yang Institute for Theoretical Physics, Stony Brook University, USA
Theoretical Physics Department, CERN, Geneva, Switzerland
- 1 Introduction
- 2 Consequences of conformal invariance
- 3 Conformal block decompositions
- 4 Holographic logCFT
- 5 Examples of logCFTs
- 6 Discussion
- A Simplifications for the two-point function
- B Logarithmic OPE
- C Partial wave decomposition
- D Conformal block identities
- E Free field limit of logarithmic GFF
Conformal field theories are scale invariant, which seems to require that two-point functions behave as power laws. But this is not quite true: conformal correlation functions can in fact have logarithms [Saleur:1991hk, Rozansky:1992td, Gurarie:1993xq], which contain a scale. Many interesting models turn out to have this property, such as percolation [Cardy:1999zp], self-avoiding walks [Duplantier:1987sh, Saleur:1991hk], spanning forests [Ivashkevich:1998na], as well as systems with quenched disorder [Caux:1995nm, Maassarani:1996jn, Caux:1998sm, Cardy:1999zp, Cardy:2013rqg]. This surprising fact and its consequences will be examined at length in this paper, where we will study logarithmic Conformal Field Theories (logCFTs) starting from first principles and especially without fixing a particular spacetime dimension.
We should begin by remarking that there is already a vast literature on the subject in the two-dimensional case. In the seminal work [Gurarie:1993xq], Gurarie was the first to point out that logarithmic terms in CFT correlation functions are caused by reducible but indecomposable representations of the two-dimensional conformal group. Subsequent work focused on the constraints coming from conformal symmetry on (chiral) three- and four-point functions, and Operator Product Expansions (OPEs) [RahimiTabar:1996ub, Flohr:2000mc, Flohr:2001tj, Flohr:2005qq, Rasmussen:2005ta, Nagi:2005sb]. Infinitely many two-dimensional logCFTs were later constructed by extending the Kac table of the ordinary Virasoro minimal models [Pearce:2006sz]. Both the representation content and the fusion rules of these “logarithmic minimal models” have been studied in detail [Feigin:2005xs, Feigin:2006iv, Feigin:2006xa, Rasmussen:2007su, Rasmussen:2008xi]. A partially overlapping direction of research has focused on realizing 2 logCFTs as continuum limits of lattice models, see e.g. [Read:2007qq, Gainutdinov:2012qy, Gainutdinov:2012mr, Gainutdinov:2013tja]. In spite of these developments, it is fair to say that 2 logCFTs are significantly less understood than their non-logarithmic counterparts. In particular, the computation of non-chiral (also known as bulk or local) correlation functions remains a difficult problem [Do:2007dn, Vasseur:2011ud, Ridout:2012ew, Fuchs:2013lda, Santachiara:2013gna, Gaberdiel:2007jv, Runkel:2012rp, Fuchs:2016wjr]. The references given above can serve as a starting point for the reader. More comprehensive discussions can be found in the review articles [Flohr:2001zs, Gaberdiel:2001tr, Creutzig:2013hma, Gurarie:2013tma]. A special class of 2 logCFTs, in the form of WZW and sigma models on superspaces, is reviewed in [Quella:2013oda].
Higher-dimensional logCFTs have received much less attention, apart from the determination of constraints on some scalar two- and three-point functions [Ghezelbash:1997cu]. This state of affairs is unfortunate, since interesting logarithmic theories are certainly not confined to two dimensions. As already mentioned, CFTs coupled to quenched disorder generically flow to logCFTs at long distances. Likewise, the -state Potts critical point in and the model in dimensions become logarithmic in certain limits of their parameters. These logCFTs describe theories with non-local actions, like percolation and polymer statistics. Logarithmic theories are also known to arise as limits of quantum field theories with instantons, like 4 super-Yang-Mills theory [Frenkel:2006fy, Frenkel:2007ux, Frenkel:2008vz]. An additional reason to be interested in logCFTs comes from holography, see e.g. [Grumiller2013]. Most of the work in this direction has so far focused on the AdS/CFT correspondence or does not go beyond the level of three point functions.
In this paper we will perform a careful and systematic study of the formal structure of logCFTs in any spacetime dimension. One motivation is the expectation that a broader look at these theories can help us better understand the two-dimensional case. More importantly, we hope that these structural results improve our knowledge about higher-dimensional logarithmic fixed points. This is especially urgent in the light of the conformal bootstrap [Rattazzi:2008pe], which in recent years has proved to be a powerful tool in analyzing CFTs in any dimension. It has been applied in many contexts, for example in computing critical exponents of the 3 Ising and models to high precision [ElShowk:2012ht, El-Showk:2014dwa, Kos:2014bka, Kos:2016ysd] but also for understanding structural properties of CFTs analytically [Komargodski:2012ek, Fitzpatrick:2012yx, Hartman:2015lfa, Hofman:2016awc]. Our work clears the path for any future bootstrap applications to logCFTs.
The outline of this paper is as follows. In section 2 we discuss general consequences of (logarithmic) conformal invariance for -point correlation functions. We start with a discussion of radial quantization in these theories. The Hilbert space contains reducible but indecomposable representations, which means that the dilatation operator cannot be made hermitian. This inevitably leads to the appearance of logarithms in correlation functions. We work out the Ward identities and their solutions, spelling out in detail the general form of two, three and four point functions. Three- and four-point functions must satisfy further constraints from Bose or crossing symmetry.
Section 3 is concerned with the derivation of the conformal partial wave and conformal block decompositions of four-point functions. Our main result is to show that conformal blocks of logarithmic primaries in the four point function of logarithmic operators can be determined by computing derivatives of ordinary, non-logarithmic conformal blocks. We show this by solving the Casimir equation à la Dolan and Osborn [DO2] for a few cases, and then in full generality via radial quantization methods. In order to illustrate the formalism, we work out a few explicit examples at the end of the section.
In section 4 we reconsider and extend previous holographic approaches to logarithmic theories. LogCFTs can be modeled holographically by actions containing higher derivatives, which we motivate by coupling bulk theories to bulk disorder. We then provide a thorough discussion of scalar theories, computing all two point functions without recourse to holographic renormalization. We discuss interactions, and show how the resulting structure is consistent with the results of sections 2 and 3. Next we introduce and discuss the holographic version of logarithmic spin-1 multiplets, described by models with higher derivatives of the Maxwell tensor. We finish with some comments on spin-2 models. These holographic toy models should prove useful in future AdS/CMT applications to strongly coupled disordered systems.
In section 5 we analyze a number of concrete logCFTs. We begin with a 2 example, the triplet model, which is the bosonic sector of the theory of symplectic fermions. Many results are known for this theory, and we show they are in full agreement with our formalism. Next we consider what we call the logarithmic generalized free field, the logCFT analog of mean field theory. We discuss in detail the four point functions in this model and their conformal block decompositions. Two further examples are considered in a more limited way: the self-avoiding random walks, described by the model with , and critical percolation, given by the limit of the Potts model. Both theories have a Lagrangian description in the UV, which allows for computations using the epsilon expansion. We reconsider some existing results in our framework.
We finish this paper with a discussion of several issues and an outlook on future work. Several appendices complement and complete calculations done in the main bulk of the paper.
2 Consequences of conformal invariance
A CFT is characterized by its symmetry under the action of the conformal group, which in Euclidean signature is . Logarithmic CFTs are also invariant under the action of the same group, but what sets these theories apart is that they contain reducible but indecomposable representations, which we call logarithmic multiplets. In this section, we shall examine the constraints imposed by conformal invariance on correlation functions with insertions of logarithmic operators. For normal CFTs, such constraints and their solutions are well-known, see for instance [Osborn:1993cr]. In logCFTs the Ward identities satisfied by the correlation functions of the associated operators take an unusual form. Nevertheless, we shall solve them in full generality, with particular attention paid to the two, three and four-point correlation functions.
2.1 Logarithmic multiplets
We begin by recalling the form of the conformal algebra, for the sake of completeness but also to set our conventions. The algebra contains as generators for dilatations, for translations, for special conformal transformations and for -dimensional rotations, satisfying non-trivial commutation relations:
Next we consider representations of this algebra. In logarithmic CFTs, states are organized in logarithmic multiplets of rank . Such a multiplet is built on top of primary states , , obeying the highest-weight condition
The states can have arbitrary spin, although we are suppressing indices for simplicity. A full representation of the conformal algebra consists of the -primary states and their infinite descendants. The latter are obtained by acting an arbitrary number of times with on the primary states, exactly like for a standard conformal multiplet with .
The generator of dilatations acts on primary states in the following way:
The Jordan block form of the matrix in (2.3) means that for such representations are indecomposable but reducible. It is from this simple fact that the entire peculiar structure of logarithmic theories will emerge [Gurarie:1993xq].
On the cylinder , we normally think of states as energy eigenstates, with playing the role of the Hamiltonian. In logarithmic CFTs, the states are actually generalized eigenstates, meaning that they satisfy
Passing to flat space, the states correspond to insertions of local operators at the origin. To insert them elsewhere we simply act with the generator of translations. Under rotations and translations, local operators transform as they would in a CFT. However, (2.3) implies that the action of the and generators is now:
where is a matrix representation of the -dimensional rotation group, acting on the indices of . We see that both dilatations and special conformal transformations lead to a mixing between different operators in the multiplet. This mixing is an inevitable consequence of the reducible but indecomposable property of these logarithmic multiplets. The action of the generators above, together with translations and rotations, determine the Ward identities for correlation functions in the usual way:
with an arbitrary generator of the conformal algebra and the arbitrary local operators. We see that in general these identities relate correlators of different components of the same multiplet.
There is a formal way of understanding the origin of equations (2.5), see e.g. [RahimiTabar:1996ub]. Let us start with the action of the dilatation generator on a rank-1 primary state . Formally we have
Similarly one deduces
It follows that if we make the identification
we recover the transformation laws (2.3) and (2.5) for a rank- multiplet. This relation is a formal trick that can be useful in solving the Ward identities. Nevertheless, many known logCFTs are limits of one-parameter families of CFTs [Cardy:1999zp, Cardy:2013rqg] and in those cases the identification (2.9) is more than a bookkeeping tool. Indeed, when tuning a parameter to a special value , a logarithmic multiplet can arise when operators collide to the same scaling dimension. In order to cancel divergences in , one is forced to consider linear combination of operators which become derivatives with respect to the scaling dimension in the limit. Some examples of this phenomenon will be presented in section 5.
In the next subsections, we will use the Ward identities to constrain the form of -point functions of logarithmic operators. Before we do so, we may ask what happens when we consider a finite conformal transformation with scale factor
By exponentiating the action of the generators it is easy to show that, say, a rank-two scalar multiplet of dimension transforms as:
Generalizations are straightforward, but here already we see the feature that gives logarithmic CFTs their name, namely the appearance of logarithms. Such logarithms will abound in correlation functions. As an immediate consequence we notice that, in radial quantization, conjugate states have to be defined in an unusual way. In a CFT, such states can be obtained by performing an inversion which maps , for some scale :
Usually the scale is set to one implicitly. However, the scale is important in a logarithmic theory since now the conjugate states become:
We see that, while we may get rid of the overall factor, the scale survives inside the logarithm.
This might seem paradoxical: how can a scale invariant theory contain a scale? To understand how this can be, consider performing a change of basis of states of the form
for some fixed coefficients .
Since , this leaves both the action of the conformal generators and the Ward identities unchanged.
2.2 Two-point functions
Here we will derive the form of the two-point functions of logarithmic operators. Let’s consider two scalar multiplets: of dimension and rank , and of dimension and rank . Without loss of generality, we can assume that . By Poincaré invariance, their two-point function can be written as
where and is a matrix of size that we wish to determine. The Ward identities imply that
for all . Here and in what follows we use the convention that if the labels are unphysical, i.e. if or . Combining both equations, we obtain the useful relation:
From this it is easy to determine that . Indeed, if this wasn’t the case we would get immediately , and using the same relation successively determines that all other elements would also be zero. Proceeding then with , Eq. (2.17) now implies that the matrix element only depends on and thus we set
In the new variables, Eqs. (2.16) implies
Summarizing, the Jordan block form of the representation (2.3) forces various logarithmic terms to be present in the correlation function.
There are several important simplifications possible at this stage. First, we remark that after a suitable change of basis, all two-point functions of operators in different multiplets can be made to vanish. Since the proof of this statement is slightly technical, we refer to Appendix A for details. Next, we remark that for the correlator of two identical multiplets, we can always assume that . If this is not the case, the bottom component of the multiplet completely decouples from the theory. But if , there exists a field redefinition which allows us to set . Indeed, the undetermined constants precisely match the number of free parameters in the field redefinition matrix in (2.14), and we may use this freedom to set such parameters to zero.
In conclusion, the two-point functions of a logarithmic multiplet of dimension can always be brought to the canonical form
for some constant . In particular, if is of rank , we have
which is a standard result in dimensions [Gurarie:1993xq].
One particular consequence of these results is that unitarity is broken. Reflection positivity would require the two point functions to be positive, for all and (hermitian) fields . However it is evident from (2.21) that this is not possible unless all multiplets have rank and . The same conclusion can be drawn by inspecting the matrix of inner products . As shown in appendix C, this matrix always has negative eigenvalues.
The generalization to traceless symmetric tensors of spin is straightforward.
To simplify correlators of spinning operators , we use a coordinate-free notation [Costa:2011mg]:
where is an auxiliary vector satisfying . With this notation, the two-point functions of a logarithmic spin- multiplet can be brought into the following form:
again for some undetermined constant .
2.3 Three-point functions
We will now study constraints on three-point functions in a similar fashion to the previous section, restricting our analysis to scalar-scalar-spin correlators for simplicity. Let us first consider a normal CFT with two scalar primaries with scaling dimensions , and a spin- primary of dimension . Conformal invariance forces their three-point function to take the following form:
where is an OPE coefficient,
We want to generalize this to the case where all operators are part of logarithmic multiplets, where have rank and has rank . This logarithmic three-point function takes the form:
We can obtain constraints on the functions using the and Ward identities. To simplify the resulting expressions, let’s introduce the variables
The Ward identities look now extremely simple:
Conformal invariance does not constrain the different OPE coefficients . However, when two or more of the fields are identical, additional constraints will come from Bose symmetry. Below, we will spell out these constraints for the case where the two scalars belong to rank-two multiplets.
As the simplest example of the formulae above, let us consider the case of two rank-1 scalars , of dimension , and one rank- scalar field of dimension . In this case, the three-point function reads
where are the relevant OPE coefficients. In more detail, for we find
For a more complicated example, consider the three-point function of a rank-two scalar primary and a rank- primary of spin :
where and . As a starting point we consider the general solution (2.33). However, since there are two insertions of the same multiplet, we have to take Bose symmetry into account, which requires
In particular, and will be even (resp. odd) under the exchange if is even (resp. odd). Consequently, we will treat the cases where is even and odd separately.
First, we consider the case of odd . Concretely, Eq. (2.37) implies that the coefficients obey
Consequently, there are only undetermined OPE coefficients. After defining the functions
it is possible to write the functions in the following compact form:
Notice that the correlator is generally nonzero, despite the fact that is odd in this case. Although this might seem paradoxical, an explanation comes by inspecting the OPE. As shown in appendix B, this expansion does not contain the primary operator , but it does include contributions from its descendants, which have a different spin and consequently a different parity under Bose symmetry.
Second, we consider the case of even . Here Bose symmetry only requires that
so there are undetermined OPE coefficients. Introducing the quantities
the functions can be written in the compact form:
Finally, as a special case of the above, consider the three-point function of itself:
In this case, Bose symmetry is even more constraining, and the most general solution to the Ward identities will only depend on four coefficients , :
All other (e.g. and ) are related to the above solutions by cyclic permutations of the .
So far, we have considered constraints coming from conformal symmetry alone on three-point functions. Some additional constraints apply to conserved currents, which are spin- operators whose correlators are conserved at non-coincident points:
for arbitrary insertions of . Current conservation puts a constraint on the dimension of :
This is a consequence of conformal invariance and holds both for ordinary and logarithmic CFTs. We will see that in logarithmic CFTs current conservation forces various three-point functions to vanish.
For definiteness, we consider the case where itself is a rank-one tensor operator — i.e. has no logarithmic partners. Furthermore, we will specialize to the three-point function where is a rank-two scalar of dimension . The strategy to derive these constraints is the following. The correlator can be written as
for some matrix determined in Sec. 2.3.1. Then it is shown in [Osborn:1993cr] that
at . We must then have also
This is the equation that we will use to get concrete constraints on OPE coefficients. In what follows, we will consider odd and even separately.
First, for odd , there is only one OPE coefficient, namely . We have
The only constraint comes from applying Eq. (2.50) to , which requires that , i.e. does not couple to at all. A different way to arrive at this conclusion comes from the OPE . Consider for definiteness the case , where for arbitrary we have
In the limit the second term in the OPE blows up, hence requiring that the OPE remains finite forces .
For even there are three OPE coefficients , and , and the three-point functions are
Applying Eq. (2.50) to does not give any constraints. However, applying it and shows that must vanish. We find no additional constraints on the coefficients and . As above, this argument is buttressed by analyzing the OPE, taking for definiteness. For arbitrary we have
If the term proportional to blows up in the limit . A similar argument applies to the OPE.
It may be interesting in future work to generalize this argument and to find constraints on OPE coefficients for conserved currents of rank in more general three-point functions.
2.4 Four-point functions
Let us now turn to the constraints of conformal symmetry on scalar four-point functions. We first discuss the non-logarithmic case, considering four different rank-1 scalar primaries of dimension . We recall that their four-point function can always be written as
where is a function depending on two independent cross ratios
and is a scale factor:
We want to generalize Eq. (2.55) to the case of four logarithmic scalars of rank . In this case, we write
and we wish to determine the constraints on the tensor imposed by conformal invariance. It will be useful to introduce four new variables :
In terms of the variables, the Ward identities take a particularly simple form:
This is proved in Sec. 2.5. Notice that the Ward identities do not restrict the dependence of on and , since the latter are conformally invariant. An immediate consequence of Eq. (2.61) is that the functions are polynomials in the , the degree of which will depend on the different ranks