# On functional representations of the conformal algebra

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## Abstract

Starting with conformally covariant correlation functions, a sequence of functional representations of the conformal algebra is constructed. A key step is the introduction of representations which involve an auxiliary functional. It is observed that these functionals are not arbitrary but rather must satisfy a pair of consistency equations corresponding to dilatation and special conformal invariance. In a particular representation, the former corresponds to the canonical form of the exact renormalization group equation specialized to a fixed point whereas the latter is new. This provides a concrete understanding of how conformal invariance is realized as a property of the Wilsonian effective action and the relationship to action-free formulations of conformal field theory. Subsequently, it is argued that the conformal Ward Identities serve to define a particular representation of the energy-momentum tensor. Consistency of this construction implies Polchinski’s conditions for improving the energy-momentum tensor of a conformal field theory such that it is traceless. In the Wilsonian approach, the exactly marginal, redundant field which generates lines of physically equivalent fixed points is identified as the trace of the energy-momentum tensor.

## 1 Introduction

### 1.1 Conformal field theories

The essential information content of any quantum field theory (QFT) is encoded in its correlation functions. As such, the various different approaches to the former ultimately amount to different strategies for computing the latter. In an ideal situation, it could be imagined that the correlation functions are determined entirely by some symmetry, allowing one to concentrate solely on the representation theory of the appropriate algebra, having dispensed with standard notions such as an action and corresponding path integral.

In general, such a strategy is not available. However, a partial realization occurs for QFTs exhibiting conformal symmetry – the conformal field theories (CFTs). If we suppose that the correlation functions involve a set of local objects, \(\{\mathcal {O}(x)\}\), then a special set of ‘conformal primary fields’, \(\mathscr {O}_i(x)\), can be identified for which the correlation functions exhibit covariance under global conformal transformations. Focussing on the conformal primaries, the various two- and three-point correlation functions are determined by the conformal symmetry in terms of the a priori unknown CFT data: the scaling dimensions, \(\Delta _i\), spins and three-point coefficients, \(C_{ijk}\). At the four-point level and beyond, the direct constraints of conformal symmetry are weaker, still.

*n*-point correlation functions can be determined in terms of \(n-1\) point correlation functions. In this manner, the content of conformal field theories can, in principle, be boiled down to the CFT data introduced above.

However, to determine the various combinations of CFT data which correspond to actual CFTs (possibly subject to constraints, such as unitarity) requires further input. One approach is to exploit associativity of the OPE to attempt to constrain the CFT data (this technique is known as the conformal bootstrap). In general, the task is extremely challenging since one can expect an infinite number of conformal primaries, the scaling dimensions of which must be self-consistently determined. Nevertheless, substantially inspired by work of Dolan and Osborn [1, 2, 3], remarkable recent progress has been made in this area [4, 5, 6, 7, 8, 9, 10].

In two dimensions, additional structure is present. Whilst the global conformal group is always finite dimensional, in \(d=2\) there exists an infinite dimensional local conformal algebra, the Virasoro algebra. Fields can now be classified according to their transformations under local (rather than just global) conformal transformations and, as such, arrange themselves into multiplets comprising a Virasoro primary and its descendants. In the seminal paper [11] it was shown that there are a set of special theories, the ‘minimal models’ for which there are only a finite number of Virasoro primaries possessing known scaling dimensions. This simplification is sufficient for the bootstrap procedure to determine the CFT data and for all correlation functions to be expressible as solutions to linear partial differential equations.

However, even in this situation, there are some natural questions to pose, at least coming from the perspective of the path integral approach to QFT: is it possible to encode the dynamics of these theories in an action and, if so, is there a concrete recipe for doing so? Is this procedure possible for all such theories or only some of them? Are the resulting actions guaranteed to be local and, if so, why? In this paper, it will be attempted to provide answers to some of these questions and hopefully to offer a fresh perspective on others.

These questions are equally valid (and perhaps less academic) in situations where the conformal bootstrap is insufficient to provide a complete understanding of the theories to which it is applied. We will take the point of view that, in this situation, one approach is to try to introduce an action formulation of the theory in question. Notice that this has been deliberately phrased so as to reverse the logical order compared to the path integral approach. Typically, within the path integral paradigm, the first thing that one does is write down a (bare) action. The conformality or otherwise of the resulting theory must then be determined. In our approach, however, we envisage starting with correlation functions which are conformally covariant by assumption and then introducing the action as an auxiliary construction.

At heart, the underlying philosophy of this paper is to take an intrinsically quantum field-theoretic starting point i.e. the symmetry properties of the correlation functions of the theory at hand. The idea then is to show that, perhaps given certain restrictions, this *implies* that (should one so desire) a local action can be constructed, from which the correlation functions can, in principle, be computed. If one is to take QFT as fundamental, this seems more philosophically satisfactory than taking a classical action as the starting point. Moreover, it clarifies the relationship between two largely disassociated textbook approaches to QFT.

### 1.2 The exact renormalization group

As will be exposed in this paper, the formalism which binds together the classic CFT approach to field theory with its path integral counterpart is Wilson’s exact renormalization group (ERG) [12]. Starting from conformally covariant correlation functions, the strategy is to encode the information thus contained in various functional representations.^{1} Each representation will yield different expressions both for the conformal generators and for the conformal primaries and their descendants. The most direct representation follows from introducing sources and embedding the correlation functions in the Schwinger functional, \(\mathcal {W}\) (the subtleties of doing this in the presence of infrared (IR) and ultraviolet (UV) divergences are discussed later). Associated with the Schwinger functional is a representation of the conformal algebra; for CFTs, each of the generators annihilates \(\mathcal {W}\).

The form of the shift (1.2) may seem a little odd. Ultimately, it can be traced back to Morris’ observation that the Wilsonian effective action naturally generates the correlation functions with an extra factor of momentum squared on each leg [13].

Next, we introduce a deformation of the Schwinger functional obtained by adding an apparently arbitrary functional of \(\varphi \) and \(J\), which, amongst other things, means that the deformed functional may depend separately on the field and the source. The motivation for this follows from previous studies of the ERG: this deformed functional is recognized as something which can be generated from a Wilsonian effective action, \(\mathcal {S}\). To put it another way, we know the answer that we are looking for!

Recall that, in the Schwinger functional representation, a CFT is such that the generators annihilate \(\mathcal {W}\). This is translated into the statement that the generators in the new representation annihilate \(\exp {-\mathcal {S}}\). However, whereas the generators in the Schwinger functional representation are linear in functional derivatives, in the new representation the generators associated with dilatations and special conformal transformations are quadratic. The upshot of this is that two of the linear conditions implied by the annihilation of \(\exp {-\mathcal {S}}\) can be replaced two by non-linear conditions on \(\mathcal {S}\). The first of these is identified with the fixed-point version of an ERG equation and the second – constituting one of the central results of this paper – is recognized as a new analogue of the special conformal consistency condition discovered long ago by Schäfer [14]. Associated with these conditions is a representation of the conformal algebra in which the generators depend explicitly on the Wilsonian effective action.

The non-linearity of the ERG equation seems to be crucial (as emphasised by Wegner [15]). In the linear Schwinger functional representation, the scaling dimensions of the fields must be determined in a self-consistent fashion using the bootstrap equations. In the Wilsonian approach, a different strategy is used. One field,^{2} with a priori unknown scaling dimension, \(\delta \), is separated from the rest and used to formulate an ERG equation (as anticipated above, we identify this field as the fundamental field). As such, it appears that an ERG equation contains two unknowns: the Wilsonian effective action and \(\delta \). The correct interpretation is that an ERG equation is a non-linear eigenvalue equation [16]; however, this hinges on one further ingredient: we demand that the solutions to the ERG equation are quasi-local.^{3}

It is the combination of non-linearity and quasi-locality which allows, in principle, for the spectrum of \(\delta \) to be extracted from the ERG equation. Indeed, by demanding quasi-locality, the spectrum of possible values of \(\delta \) can be shown to be discrete [17]. Let us emphasise that the spectrum of \(\delta \) does not correspond to the spectrum of fields within a given CFT; rather, each value of \(\delta \) obtained by solving the ERG equation corresponds to a different CFT.

Presuming that some solution to the ERG equation has been obtained, the second step would be to compute the spectrum of fields. With both \(\mathcal {S}\) and \(\delta \) now known, the dilatation generator has a concrete form. It now provides a linear eigenvalue equation for the fields and their scaling dimensions. In general, the spectrum is rendered discrete by the condition of quasi-locality: this is illustrated for the Gaussian fixed point in [15, 17]. Within the derivative expansion approximation scheme, see [16] for an excellent description of how discreteness of the spectrum arises for non-trivial fixed points.

From the perspective of different representations of the conformal algebra, it is locality, together with non-linearity, which singles out the ERG representation as special. Remember that the reason for considering elaborate representations of the conformal algebra is more than just academic: part of the motivation is to provide tools for understanding conformal field theories for which the conformal bootstrap seems intractable. In the ERG approach, scale-invariant theories can be picked out from an equation by applying a constraint which is easy to implement on the solutions: that of quasi-locality. The price one pays for this is the introduction of a considerable amount of unphysical scaffolding, notably the UV regularization. One can imagine other representations of the conformal algebra which entail similar complication but without the redeeming feature of a simple condition which can be imposed on solutions of the scale/special conformal consistency conditions.

An interesting question to ask is whether the set of conformal field theories (perhaps subject to constraints of physicality) is in one-to-one correspondence with the set of equivalence classes of quasi-local actions. It is tempting and perhaps not too radical to speculate that, at the very least for the sorts of theories which form the focus of this paper – non-gauge theories, on a flat, static background, for which the energy-momentum tensor exists and is non-zero – the answer is yes. While there are some suggestive numerical results [18], it is desirable to have a proof, one way or the other. The thrust of this paper gives some clues as to how this can be achieved (as discussed further in the conclusion) but at a rigorous level the question remains unanswered, for now.

### 1.3 The energy-momentum tensor

It is worth emphasising that the approach advocated above is precisely the opposite of the standard path integral approach: our starting point is the correlation functions; then we introduce sources; next we introduce a field and arrive at an action! Everything within this picture, with the exception of the correlation functions, themselves, is an auxiliary construction. As such, this sits rather uncomfortably with standard expositions of the role of the energy-momentum tensor in QFT: for these tend to start with the classical action.

*defining*the energy-momentum tensor in a particular representation of the conformal algebra.

^{4}To see how this comes about, consider the Ward identity associated with translation invariance. Taking \(T_{\alpha \beta }\) to denote a quasi-local representation of the energy-momentum tensor, we have [19]

*n*and observe that the result can be cast in the form

This analysis of the improvement procedure will be seen to have close parallels to that of Polchinski’s classic paper [21]. Indeed, as in the latter, a sufficient condition for this improvement is essentially that primary vector fields of scaling dimension \(d-1\) are absent from the spectrum. In the same paper, Polchinski completed an argument due to Zamolodchikov [22] showing that, in two dimensions, the energy-momentum tensor of a scale-invariant theory can be rendered traceless if the theory is unitary and the spectrum of fields is discrete. The veracity of this for \(d>2\) has been much debated [23].

As additional confirmation of the consistency of our approach it will be shown in Sect. 4.2.2 that \(\mathcal {T}_{\alpha \beta }\) has the same properties under conformal transformations as a tensor, conformal primary field of scaling dimension, \(d\). (We use the term field with care since the associated representation is non-local, but henceforth will be less assiduous.) Let us emphasise that we are not claiming \(\bigl \langle T_{\alpha \beta } \bigr \rangle \) is a representation of the energy-momentum tensor in the sense of having the correct transformation properties under the appropriate representation of the conformal group; only for the full \(\mathcal {T}_{\alpha \beta }\) does this hold.

While this section began with the standard representation of the energy-momentum tensor – i.e. a quasi-local object – from the perspective of this paper we view as more primitive the non-local representation furnished by the Schwinger functional, \(\mathcal {T}_{\alpha \beta }\). For theories supporting a quasi-local representation, \(T_{\alpha \beta }\) can be recovered via the ERG, as will become apparent in Sect. 4.4. With this achieved, another of the central results of this paper will become apparent: in the ERG representation, the trace of the energy-momentum tensor is nothing but the exactly marginal, redundant field possessed by every critical fixed point. (Redundant fields correspond to quasi-local field redefinitions.) It is the existence of this field which causes quasi-local fixed-point theories to divide up into equivalence classes: every fixed-point theory exists as a one-parameter family of physically equivalent theories [15, 16, 17, 24, 25]. This is the origin of the quantization of the spectrum of \(\delta \).

The construction of the energy-momentum tensor will be illustrated in Sect. 4.5 using the example of the Gaussian fixed point; Sect. 4.6 demonstrates how the construction breaks down for a simple, non-unitary theory.

## 2 Conformal symmetry in QFT

### 2.1 Elementary properties of the conformal group

^{5}

Let us emphasise that, at this stage, the representation of the \(\{\mathcal {P}_\mu , \mathcal {M}_{\mu \nu }, \mathcal {D}, \mathcal {K}_\mu \}\) and the \(\mathscr {O}(x)\) are yet to be fixed; a key theme of this paper will be the exploration of certain representations thereof, some of which are non-standard.

### 2.2 Correlation functions

*n*,

^{6}now for all \(a_1, \ldots , a_n\)

At certain stages, we will simply assume that the Schwinger functional involving solely \(J\), \(\mathcal {W}[J]\), is well defined. To be precise, when we talk of existence of the Schwinger functional, it is meant that the correlation functions of the field conjugate to \(J\) can be directly subsumed into \(\mathcal {W}[J]\) and so the naïve identities (2.8a)–(2.8d) hold. Note that existence of \(\mathcal {W}[J]\) is considered a separate property from \(\mathcal {W}[J]\) being non-zero.

By definition, we take \(J_i\cdot \mathscr {O}_i \) to have zero scaling dimension; this implies that \(J_i(x)\) has scaling dimension \(d-\Delta _i\). This leads us to the first of several functional representations of the conformal generators that will be presented in this paper.

### Representation 1

*V*being the volume of the space on which the field theory lives. For this paper, however, we will generally ignore volume terms; as such, we henceforth understand equality in functional equations to hold only up to volume terms. This issue will be addressed more fully in [26].

### 2.3 From sources to the fundamental field

Up until this point, our functional representation has utilized sources. The transition to fields proceeds in several steps, along the way giving new representations of the conformal algebra. The first such step is provided by the shift (1.2). Clearly, at this stage, the dependence on \(J\) and \(\varphi \) will not be independent. However, the link will be severed in a subsequent representation. To prepare for this severing, our aim in this section is, given the shift (1.2), to construct a representation in which the generators involve functional derivatives with respect to \(\varphi \) (rather than \(\partial ^2 \varphi \)).

As mentioned earlier, we anticipate that \(\varphi \) will play the role of the fundamental field. Before proceeding, it is worth pointing out that there is a subtlety over precisely what is meant by the latter. Strictly speaking, both the Wilsonian effective action and the field to which \(J\) couples are *built* out of \(\varphi \). Within the ERG representation (and assuming sufficiently good IR behaviour), \(\varphi \) coincides with a conformal primary field only up to non-universal terms, which vanish in the limit that the regularization is removed. While this subtlety will be largely glossed over since it seems to have no great significance, the issue of theories for which bad IR behaviour prevents \(\varphi \) from corresponding to a conformal primary in any sense will be returned to later.

### Representation 2

Before introducing the next representation, it is worth mentioning that the functional \(\mathcal {W}[J+ \partial ^2 \varphi ]\) may have different (quasi)-locality properties with respect to \(J\) and \(\varphi \). For non-trivial fixed points this will not be the case, as can be seen at the two-point level. In momentum space, the two-point correlation function goes like \(1/p^{2(1-\eta /2)}\). For non-trivial fixed points, \(\eta /2\) is some non-integer number. While multiplying by a factor of \(p^2\) removes the divergence as \(p^2 \rightarrow 0\), it does not remove the non-locality. For trivial fixed points, however, non-locality may be ameliorated. This can be convenient and is exploited in Sect. 4.5.

### 2.4 From the fundamental field to the ERG

Before moving on let us not that, in general, the Wilsonian approach deals not just with scale (or conformally) invariant theories but with theories exhibiting scale dependence. Scale-independent actions are typically denoted by \(\mathcal {S}_\star \) and solve the fixed-point version of an ERG equation. However, since this paper will only ever deal with fixed-point quantities, the \(\star \) will henceforth be dropped.

*g*, and a representation of this generator, denoted by \(\mathscr {G}\), define

### Representation 3

^{7}

*b*. However, in this case \(\mathcal {S}[\varphi ]\) is related to \(\mathcal {W}_\mathcal {U}\) by rescaling each leg of every vertex of the latter by a factor of \(e^{-b}\), which gives us nothing new. Instead, we solve the constraints by introducing a kernel, \(\mathcal {G}\bigl ((x-y)^2\bigr )\), and taking

^{8}Strictly speaking, this suggests that in \(d=2\) we should work in finite volume, at least at intermediate stages.

### Representation 4

*bona-fide*ERG equation, \(\mathscr {D}_U\) must on the one hand be quasi-local and, on the other, must be such that (up to vacuum terms), (2.33a) can be cast in the form [27]

## 3 Polchinski’s equation from the conformal algebra

Our treatment so far has been very general; in this section we will provide a concrete realization of our ideas by showing how to derive what is essentially Polchinski’s equation [28]. Mimicking the previous section, we will take the Auxiliary Functional Representation as our starting point, deriving the generators in this representation. The constraint equation (2.33a) will then be seen to produce the desired ERG equation, with (2.33b) producing its special conformal partner. Finally, we will give the expressions for the associated conformal generators corresponding to the ERG representation.

*h*will be specified momentarily. Recalling (2.20c), (2.23) and (2.24c), it is apparent that

*h*such that

*G*according to

*G*and \(\mathcal {G}^{-1}\) – related via (3.7) – are quasi-local. Typically, \(\mathcal {G}\) is chosen according to (2.29), with the cutoff function conventionally normalized so that \(K(0) = 1\) (further details can be found in [17, 20]). Volume terms, discarded in this paper, are carefully treated in [20].

### Proposition 1

### Proof

The result follows from the form of \(D^{(\Delta )}\) and \({K^{(\Delta )}}_{\mu }\) given in (2.4b) and (2.4c).

*H*,

*x*and

*y*. Consequently, we arrive at the following analogue of (3.5):

The generators in the ERG representation are constructed from (3.9) and (3.22) using the recipe in (2.32c) and (2.32d). The resulting expressions can be simplified by utilising the constraint Eqs. (3.10) and (3.28).

### Representation 5

*G*and \(G_\mu \) are defined in terms of \(\mathcal {G}\) via (3.6) and (3.11), the volume terms have been neglected and \(\mathcal {S}\) satisfies (3.10) and (3.28).

Though the analysis up to this point has been phrased in terms of conformal primary fields, we are at liberty to consider non-conformal theories: this can be done simply by taking the fields to which \(J_i\) couple as not being conformal primaries.

## 4 The energy-momentum tensor

### 4.1 Proposal

With this in mind, there are two assumptions at play in the statement that \(\mathscr {O}^{(\delta )}\) and \(\mathscr {O}^{(d-\delta )}\) are conformal primaries. First, it is assumed that \(\mathcal {W}[J]\) exists and is non-zero; we will encounter theories for which one or other of these conditions is violated in subsequent sections. More subtly, it is assumed that \(\mathscr {O}^{(d-\delta )}\) is amongst the spectrum of fields. As will be seen in Sect. 4.4, if the ERG representation is quasi-local then \(\mathscr {O}^{(d-\delta )}\) is present in the spectrum as a redundant field.

Note that there are interesting theories for which the assumption that \(\mathscr {O}^{(d-\delta )}\) is in the spectrum of fields does not hold, in particular the mean field theories. This class of theories (recently featuring in e.g. [7, 8, 30, 31]) are such that the *n*-point functions are sums of products of two-point functions and cannot be represented in terms of a quasi-local action. The latter restriction amounts to defining mean field theories so as to exclude the Gaussian theory, plus its quasi-local but non-unitary cousins [17] (see also Sect. 4.6); this is done for terminological convenience. Thus, for mean field theories, (4.1b) amounts to minor notational abuse since, strictly, \(\mathscr {O}\) should be reserved for conformal primaries. Accepting this, we henceforth interpret \(\mathscr {O}_{J}^{(d-\delta )}\) as an object we are at liberty to construct, that in most – though not all – cases of interest is indeed a conformal primary. A surprising feature of mean field theories is that the energy-momentum tensor is not amongst the spectrum of conformal primary fields.^{9}

*x*only via the field (the ellipsis represent higher derivative terms). With this in mind, let us consider (4.8) in a quasi-lcoal representation. Quasi-locality implies that any terms which vanish when integrated must take the form of total derivatives establishing that, for theories which permit a quasi-local formulation, (4.9) is correct as it stands. Note that by focussing on theories supporting a quasi-local representation excludes mean field theories, in particular, from the remainder of the discussion.

*the Ward Identities can be interpreted as defining a non-local representation of the energy-momentum tensor*. Denoting the energy-momentum tensor in an arbitrary representation – which may or may not be quasi-local – by \(\mathcal {T}_{\alpha \beta }\), we tentatively define this object via

^{10}:

### 4.2 Justification

In this section, we justify, for \(d>1\), the proposal encapsulated in (4.10a), (4.10b) and (4.10c), which comprises three steps. First it is shown that an object which satisfies these equations is implied by translation, rotation and dilatation invariance so long as Polchinski’s conditions [21] for the improvement of the energy-momentum tensor are satisfied. Secondly, it is shown how both the traceful and longitudinal components of \(\mathcal {T}_{\alpha \beta }\) transform in a manner consistent with \(\mathcal {T}_{\alpha \beta }\) being a conformal primary field of dimension \(d\). Finally, the extent to which (4.10a), (4.10b) and (4.10c) serve to uniquely define \(\mathcal {T}_{\alpha \beta }\) is discussed.

#### 4.2.1 Existence

The most basic requirement for the existence of a non-null \(\mathcal {T}_{\alpha \beta }\), as defined via (4.10a), (4.10b) and (4.10c), is that the fields \(\hat{\mathscr {O}}^{(d-\delta )}\) and \(\mathscr {O}^{(\delta )}\) exist and are non-zero. There is some degree of subtlety here since it is conceivable that \(\mathscr {O}^{(\delta )}\) does not exist in the Schwinger functional representation but does exist in a quasi-local representation. An example would be the Gaussian theory in \(d=2\). The IR behaviour of this theory is sufficiently bad that the lowest dimension conformal primary is not \(\varphi \) but rather \(\partial _\mu \varphi \). One method for dealing with this theory would be to perform the analysis of this section in terms of vector fields. An alternative, however, is to implicitly work within a quasi-local representation; note that though \(\mathscr {O}^{(\delta )}\) exists, we are accepting a degree of notational abuse since it is not a conformal primary in the standard sense. (Later, where more care must be taken, the symbol \(\phi \) used, instead). With this in mind, for the duration of this section we assume that at least one representation of \(\hat{\mathscr {O}}^{(d-\delta )}\) and \(\mathscr {O}^{(\delta )}\) exists, and at least one of these representations is quasi-local.

*first*two indices, (4.9) is left invariant by the shift

With these points in mind, consider the sufficient conditions for translation, rotation and dilatation invariance to imply (4.10a), (4.10b) and (4.10c). In \(d>2\), absence of a primary vector field of scaling dimension \(d-1\) is sufficient. In \(d=2\) this condition must be supplemented by at the absence of vector fields that can be written as \(\partial _\sigma f_{\sigma \lambda }\), where \(f_{\sigma \lambda }\) does not reduce to the Kronecker-\(\delta \). In some sense, this is academic since it cannot be realised for unitary theories. Moreover, as we shall see in Sect. 4.3, this additional condition is relevant only for theories with sufficiently bad IR behaviour.

To conclude this section, let us mention an alternative way to recover Polchinksi’s conclusion as to the conditions under which improvement of the energy-momentum tensor is possible. A version of the analysis below forms a key part of [32], in which an argument is given as to why scale invariance is automatically enhanced to conformal invariance for the Ising model in three dimensions.

#### 4.2.2 Conformal covariance

In this section we will show that (4.10a), (4.10b) and (4.10c) are consistent with \(\mathcal {T}_{\alpha \beta }\) being a candidate for a conformal primary field of dimension \(d\). By ‘candidate field’ we mean an object that has the desired properties under conformal transformations but may or may not turn out to be amongst the spectrum of conformal primaries for a given theory. Let us start by noting that, by construction, \(\mathcal {T}_{\alpha \alpha }\) is a candidate for a conformal primary field (cf. (4.2)). The rest of this section will be devoted to showing that the longitudinal parts of \(\mathcal {T}_{\alpha \beta }\) also transform correctly.

#### 4.2.3 Uniqueness

Though we will not rely on the following restriction in this paper, it is expected that the energy-momentum tensor is unique for unitary theories.^{11} In \(d=4\), \(C_{\alpha \rho \beta \sigma }\) transforms under the \((2,0) \oplus (0,2)\) representation. However, Mack rigorously established that, for a representation of type (*j*, 0), unitarity demands that the scaling dimension \(\Delta > 1 + j\) [34]. This implies that the scaling dimension of \(\partial _\rho \partial _\sigma C_{\alpha \rho \beta \sigma }\) is greater than five and so this field cannot contribute to the energy-momentum tensor which, in the considered dimensionality, is of scaling dimension four. A similar result is expected to hold in higher dimensions.

### 4.3 Conformal invariance

### 4.4 Quasi-local representation

The defining equations for the energy-momentum tensors, (4.10a), (4.10b) and (4.10c) are independent of any particular representation. As already apparent, a prominent role is played by the Schwinger functional representation; the other representation of particular interest is furnished by the ERG, which provides a quasi-local framework. For the sake of definiteness, in this section we will explore the energy-momentum tensor using the canonical ERG equation, discussed in Sect. 3. Recall that \(\delta \) can be written in terms of the anomalous dimension of the fundamental via (2.19).

*h*appearing in (3.1) is (given appropriate boundary conditions) related to \(\varrho \) according to [17]

*b*, then

### 4.5 The Gaussian fixed point

An instructive illustration of many of the concepts discussed above is provided by the Gaussian fixed point, which describes a free theory for which the fundamental field has scaling dimension \(\delta = \delta _0 = (d-2)/2\). Before providing the canonical ERG representation of this theory, we will derive the expression for the energy-momentum tensor in the para-Schwinger functional representation. We do this since, for the special case of the Gaussian fixed point, this representation is, in fact, strictly local and, as such, equivalent to an unregularized action approach. Consequently, this should provide a familiar setting prior to our exposition of the less conventional ERG approach. Note that the difference between \(\mathscr {O}^{(d-\delta _0)}\) and \({\hat{\mathscr {O}}}^{(d-\delta _0)}\) amounts only to a vacuum term, which we ignore (the same is true in Sect. 4.6).

#### 4.5.1 Para-Schwinger functional representation

#### 4.5.2 ERG representation

### 4.6 A non-unitary example

Having observed the successful construction of the energy-momentum tensor in the simplest local, unitary theory, in this section we will provide a simple non-unitary example where the construction breaks down – at least in \(d=2\). (For a pedagogical exposition of various aspects of unitarity, see [38].) Specifically, we will consider a free theory for which \(\eta = -2\).

*b*parametrizes the line of equivalent fixed points and \(\varrho \) is given by (4.41). Note that the two-point function for the full action, cf. (2.37), starts at \(\mathrm{O}\bigl ( p^4 \bigr )\). It follows that

^{12}:

## 5 Conclusions

The essence of the philosophy advocated in this paper is a conservative one. At its heart is the desire to view, as far as possible, QFT as fundamental. (Whether or not this ultimately turns out to be the case is beside the point; the goal is partly to see how far one can go by pursuing this agenda.) Taking this seriously, we are driven to look for theories which make sense down to arbitrarily short distances (i.e. theories exhibiting non-perturbative renormalizability). Wilsonian renormalization teaches us that a sufficient condition for this is scale-invariant behaviour in the deep UV, suggesting that we investigate either fully scale-invariant theories or relevant/marginally relevant deformations, thereof.

Our exclusive focus has been on theories exhibiting invariance under the full conformal group. In this context, there are two largely disjoint approaches: one based on exploring the constraints implied by conformal invariance on the correlation functions and the other a path integral approach built upon a quasi-local action. True to our philosophy, the former is viewed as more primitive due to its inherently quantum field-theoretic nature whereas, through the action, the latter manifests its classical heritage. Ideally, then, what we would like is to be able to start with an approach based on the correlation functions and to show how an action-based description emerges.

This paper largely shows how to achieve this. Starting from the statements of conformal covariance of the correlation functions, the first step is to wrap these up into functionals of sources (accepting a degree of formality in this step). Associated with this is a functional representation of the conformal algebra. This forms the basis for constructing more elaborate representations, involving auxiliary functionals. These auxiliary functionals satisfy consistency conditions, and our development culminated with a representation in which the condition corresponding to dilatation invariance is nothing but the fixed-point version of the canonical ERG equation of [29]. The explicit form of the partner encoding special conformal invariance is a new result of this paper.

Nevertheless, it must be acknowledged that we were guided towards this representation because we knew what we were looking for. This, in of itself, is not an issue. More pressing is that, coming from the path integral perspective, it is expected that all physically acceptable solutions to ERG equations correspond to quasi-local actions [12, 17]. The step that is missing in this paper is to show that ERG representations of CFTs necessarily have a Wilsonian effective action that is, indeed, quasi-local. Or, to put it another way, of all the possible representations of the conformal algebra, what makes the ERG representation so special? Suggestively, as alluded to in Sect. 4.4, if the ERG representation of a CFT is quasi-local, then the energy-momentum tensor is amongst the spectrum of conformal primaries. Indeed, the ERG and the energy-momentum tensor share an intimate relationship revealed in this paper: lines of physically equivalent fixed points are generated by the trace of the energy-momentum tensor.

Therefore, referring to the questions posed in the introduction, the state of affairs is as follows. A concrete recipe has been provided for encoding conformal dynamics in an object recognizable as the Wilsonian effective action. This assumes (2.22) which, given the choice (2.28), amounts to an assumption of the existence of a path integral. As noted, in \(d=2\) it may be necessary to work in finite volume. Plausibly, the rather formal process presented will work for theories possessing an energy-momentum tensor, in which case we expect the Wilsonian representation to furnish a quasi-local formulation of the theory in question. It is clearly desirable, however, to place all of this on a more rigorous footing.

Beyond this matter, it is worth posing the question as to whether there may exist theories supporting representations of the conformal algebra for which the constraint which picks out physically acceptable theories is entirely different from quasi-locality, but equally powerful. Besides exploring this theme further, several avenues of future research suggest themselves. Transcribing our approach to supersymmetric theories should be largely a matter of working with the appropriate potential superfields, as in [39], and being mindful that the conformal algebra is enhanced to be superconformal. A controlled environment in which to further explore the CFT/ERG link is in \(d=2\), where it may be profitable to investigate functional representations of the entire Virasoro algebra. As indicated earlier, a rigorous treatment in \(d=2\) may entail a careful finite-volume treatment. Gauge theories present special problems [17, 40] and it is my belief that appropriately extending the ideas of this paper will require new ideas.

Furthermore, it is desirable to extend the scope of the analysis to include scale-dependent theories. It is anticipated that the renormalizability (or otherwise) of such theories is tied up with the renormalization of composite operators. Since, for irrelevant/marginally irrelevant perturbations, renormalizability is lost this suggests that, in order to properly define the fixed-point Schwinger functional involving the corresponding sources, some form of point splitting should be performed on the composite operators, to improve their UV behaviour. It may be that this engenders a natural way to uncover the operator product expansion within the ERG formalism, raising the hope of making concrete links between the ideas of this paper and recent developments in the application of the conformal bootstrap.

Finally, it is worth considering the question as to whether theories exist in which there are multiple, distinct quasi-local representations. This leads naturally to the subject of dualities and it is hoped that the ideas of this paper and the fresh perspective it gives on the nature and origins of Wilsonian renormalization will offer new insights in this area.

## Footnotes

- 1.
While the correlation functions themselves are conformally

*co*variant, we refer to the functionals as representing a conformally*in*variant theory. - 2.
In much of the ERG literature, ‘operator’ is used in place of field; however, following CFT conventions, we shall generally use the latter.

- 3.
By quasi-local it is meant that contributions to the action exhibit an expansion in positive powers of derivatives. Equivalently, in momentum space, vertices have an all-orders expansion in powers of momenta. Loosely speaking, at long distances a quasi-local action has the property that it reduces to a strictly local form. Quasi-locality is discussed, at length, in [17].

- 4.
Actually, there are CFTs, such as the Mean Field Theories, for which the energy-momentum tensor does not exist; this will be discussed further in Sect. 4.1.

- 5.
Were we to work in an operator formalism, the expressions on the left-hand sides would appear as commutators.

- 6.
Indices near the beginning of the alphabet are understood to label all fields, rather than just the conformal primaries.

- 7.
It is tempting to speculate that relaxing this requirement may be illuminating in the context of lattice theories.

- 8.
Indeed the true propagator at the Gaussian fixed point in \(d=2\) has logarithmic behaviour, emphasising that the interpretation of \(\mathcal {G}\) as a regularized propagator must be taken with a pinch of salt.

- 9.
- 10.
Note that since the correlation functions involved in our proposed definition of the energy-momentum tensor involve only scalar fields, we expect symmetry under interchange of indices.

- 11.
I would like to thank Osborn for informing me of this and for providing the argument as to why.

- 12.
I am very grateful to Hidenori Sonoda for supplying me with this solution.

## Notes

### Acknowledgements

This paper is dedicated to the memory of my friend, Francis Dolan, who died, tragically, in 2011. It is gratifying that I have been able to honour him with work which substantially overlaps with his research interests and also that some of the inspiration came from a long dialogue with his mentor and collaborator, Hugh Osborn. In addition, I am indebted to Hugh for numerous perceptive comments on various drafts of the manuscript and for bringing to my attention gaps in my knowledge and holes in my logic. I would like to thank Yu Nakayama and Hidenori Sonoda for insightful correspondence following the appearance of the first and third versions on the arXiv, respectively. I am firmly of the conviction that the psychological brutality of the post-doctoral system played a strong underlying role in Francis’ death. I would like to take this opportunity, should anyone be listening, to urge those within academia in roles of leadership to do far more to protect members of the community suffering from mental health problems, particularly during the most vulnerable stages of their careers.

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