In three dimensions, pure gravity with a negative cosmological constant can be described by Chern–Simons gauge theory [21,22]. As a natural extension, we can construct a higher-spin gauge theory using Chern–Simons theory based on a higher-rank gauge algebra [32]. We are interested in Chern–Simons theory, whose action is given by (26)

Here is the level of Chern–Simons theory and are one-forms taking values in . The generators of sl can be decomposed in terms of the adjoint action of embedded sl as (27)

Here denotes the spin- representation of sl, and we have adopted the principal embedding of sl. The generators in sl (adjoint representation) and are denoted as and , respectively.

For the application to higher-spin AdS gravity, we need to assign an asymptotic AdS condition to the gauge fields. We use the metric of Euclidean AdS as , where the boundary is at . In a gauge choice, we can set (28)

We have a similar expression for but suppress it here and in the following. The configuration corresponding to AdS background is given by . The asymptotic AdS condition restricts the form of as [33–36] (29)

There are residual gauge symmetries preserving the condition (29), and some of them generate W symmetry near the AdS boundary. We can define classical Poisson brackets for the reduced phase space. Moreover, we can see that in Eq. (29) generate the W symmetry in terms of the Poisson brackets. At the classical level, the relation between the Chern–Simons level and the central charge of the dual conformal field theory is given by the Brown–Henneaux one as [37] (30)

See Refs. [33–36] for more details.

At the leading order in , the rules for computing conformal blocks from the Chern–Simons theory with open Wilson lines were given in Ref. [16]; see also Ref. [38] for . For the two- and three-point functions in Eq. (3), we use (31)

Here hw and lw denote the highest and lowest weight states in finite-dimensional representations of sl, respectively, and represents the path ordering. Moreover, we remove the dependence in the gauge field as using a gauge transformation. We include corrections by extending the analysis in Refs. [23,24] for . At the leading order in , we treat the coefficient in Eq. (29) as a function of . At higher orders in , we regard as an operator, and the expectation values of open Wilson lines are evaluated by using the correlators of , which are uniquely fixed by the W symmetry.

In this subsection we explain our prescription to compute the matrix elements of sl algebra for evaluating the expectation values of open Wilson lines as in Eq. (31). We start with the simplest case with and then extend the argument for generic . For , there are several previous works in Refs. [23–25], and we start by clarifying the representation with -derivatives in Ref. [23].

For two-point functions we evaluate (32) where belongs to the spin- representation of sl(2) with . We set the norm of these states as (33)

With these states, the sl(2) generators in the Wilson line are described by matrices.

As in Refs. [23,25], it would be convenient to map the expression as (34) then the sl generators can be written as (35)

In Ref. [23], they proposed that the wave functions are given by (36)

We would like to give a derivation such that it can be extended for generic . It is easy to obtain as a solution to the equation . The others follow as (37)

The dual states should satisfy (38) which leads to (39)

In particular, we have as in Eq. (36). The normalization is set to be a convenient value.

We then apply the analysis to the case with generic . A way to represent the generators of is using matrices, and sl(2) generators can be embedded as described, e.g., in Appendix A of Ref. [13]. Then the other generators may be obtained as (40) where of are inserted. The fundamental representation of sl can be described by an -dimensional vector, which behaves as a spin- representation under the action of the embedded sl. Therefore, the description with matrices can be given by Eq. (32) with and open Wilson lines based on sl algebra. In this specific case, we can map the matrix representation to the one with -derivatives using Eqs. (35) and (40). In the representation with -derivatives, the generators of sl should be given by [39] (41) where (42) with . The wave functions are precisely those in Eq. (36). The generators (41) with Eq. (42) are those of higher-spin algebra hs for , and sl can be realized by hs with as an ideal, which removes generators with N$]]>.

With the realization of generators, in Eq. (29) are computed as (43) where the first few expressions are (44)

In particular, we have for .

In order to compute the correlators in Eq. (3), we need to consider the expectation values of open Wilson lines with corresponding to the highest weight in the fundamental representation of sl. As explained above, they can be expressed for as (45) with . Here the generators are written in terms of -derivatives as in Eq. (42). We would like to treat them perturbatively in (or ). Following the analysis in Ref. [24], we compute (46)

Integrating over , we find (47) where (48) see Eq. (3.3) of Ref. [23] for .

According to the current prescription, the two-point function of in Eq. (3) should be computed as (49) where is evaluated by the correlators of in the W theory. The leading-order expansion in leads to (50) as expected.

We are also interested in the three-point functions in Eq. (3), which should be obtained as (51)

The first nontrivial contributions come from the terms of order . At this order, we need to compute (52)

The normalization of higher-spin currents in Eq. (29) corresponds to (see, e.g., Ref. [40]) (53)

Using (54) we find (55)

The result is consistent with Eq. (22) in the convention of Eq. (53). In fact, it is the same as Eq. (1.3) of Ref. [40] up to a factor if we set (or ), and this is related to the triality relation discussed in Ref. [14].

The corrections of the two- and three-point functions in Eq. (3) can be evaluated from higher-order contributions in Eq. (47) using the Wilson line method. However, integrals over diverge when two (or more) currents collide. Therefore, we need to decide how to deal with these divergences, and we explain our prescription in this subsection.

Let us start with the correlators of higher-spin currents, which are uniquely fixed by the W symmetry in terms of central charge . In particular, we use the two-point functions (56) which reduce to Eq. (53) if we use the relation in Eq. (30). At finite , the relation of Eq. (30) should be modified, and corrections to higher-spin propagators are automatically included by expanding in instead of ; see Ref. [24] for some arguments. Divergence would arise at the coincident point , and we need to decide how to regularize it. We introduce a regulator as (57) by shifting the conformal weight of the higher-spin current as . This choice is reasonable since it does not break the scaling symmetry. Analogously, we introduce the regulator to other correlators of higher-spin currents by shifting the conformal wights of the current.

Introducing the regulator , integrals over become finite but have terms diverging at . In the usual quantum field theory with a renormalizable Lagrangian, we can remove divergences by renormalizing the overall normalization of quantum fields and the parameters of interactions. In the current case, we offer to remove divergences in a similar manner. We first use the fact that the normalization of a two-point function can be chosen arbitrarily by the redefinition of the operator. We remove a kind of divergence by changing the overall factor of the open Wilson line such that the corresponding two-point function becomes the normalized one as in Eq. (6). We then notice that the three-point interactions between two scalars and a higher-spin field are governed by the coefficients in front of in Eq. (45). We introduce parameters such that Eq. (45) becomes (58)

In terms of expansion, Eq. (47) is changed as (59) where are given by Eq. (48). At the leading order in , as in Eq. (30) and . From the next order in , we shift the values of to remove divergences. Namely, we expand in as (60) and absorb divergences in order by order. We conjecture that all divergences can be removed by these two methods of renormalization.

As explained above, we decide to remove divergences by properly choosing the “bare” values of parameters . However, we have still freedom to choose the terms independent of . Here we fix them such that the three-point functions in Eq. (3) are reproduced from the Wilson line method as in Eq. (51). Since the three-point functions can be fixed by the W symmetry as shown in the previous section, we would say that the regularization scheme is determined by making use of the boundary symmetry. This is expected to fix all the ambiguities left, and the other physical quantities should be predictable. In the following two sections, we examine concrete examples with and show that the corrections in the conformal dimensions of scalar operators can be reproduced from the bulk viewpoint up to the two-loop level applying the prescription described above.

For the two-point function of , we need to evaluate the expectation value of the open Wilson line as in Eq. (49). With , the expansion of the open Wilson line in Eq. (59) becomes (63) with (64) and so on. Here are defined in Eq. (48). Since the one-point function vanishes as , the nontrivial contribution starts from . The contribution corresponds to the one-loop correction in the two-point function of as in Fig. 1.

The integrals in over diverge, and we introduce a regulator as in Eq. (57), i.e., (65) for spin-two current. With the regulator, we obtain a finite result after the integration over as (66)

Using Eq. (63) and , the above expression leads to (67) up to the terms of order and .

We compare the above expression in Eq. (67) with the expansion of the two-point function in Eq. (10). We can see that the term correctly explains in Eq. (61), as shown in Ref. [24]. The expression in Eq. (67) has a term proportional to , which diverges for . We can remove the divergence by changing the overall factor of the open Wilson line as (68)

With the normalization, we have (69) for . In other words, we choose the -independent part such that the corresponding two-point function has unit normalization as in Eq. (6).

We have proposed that three-point functions can be computed with open Wilson lines as in Eq. (51) and reproduced the tree-level results as in Eq. (55). In this subsection, we examine the next leading order in . There are two types of contribution at the order as in Fig. 2 and we would like to examine them in turn.

The first one is from (70) which is represented as diagram (a) in Fig. 2. Here we need to introduce the regulator to the three-point function of spin-two current. Our prescription is to shift the conformal weight from to , so we use (71)

The integral becomes simpler by taking as (72) up to the term of order .

The second one is from (73)

At the leading order in , the four-point function is given by a sum over the products of the two-point function as (74)

Denoting (75) we find (76) up to the terms of . The integrals , , correspond to the diagrams (b), (c), (d) in Fig. 2, respectively.

Combining the results so far, we find (77)

The expression diverges for , and we remove the divergence by properly choosing in Eq. (60) as (78)

Here is an arbitrary constant, which shall be fixed shortly. With this choice of the parameter , there arises a contribution of order from the following term as (79) up to the terms of order . Here is given in Eq. (64). With this prescription, we have (80) for . Therefore, setting , we reproduce the expected result as Eq. (62) with in Eq. (61). In summary, we choose the parameter in Eq. (58) as (81) in order to absorb a divergence from the one-loop diagram and also reproduce the result from the conformal Ward identity.

In the previous subsections we have regularized divergences arising up to the one-loop order. Our claim is that other quantities are predictable after the renormalization. Here we would like to examine the two-point function at two-loop order. Generically, two-loop diagrams have one-loop sub-diagrams, and there would appear non-local divergences from the sub-diagrams. After all one-loop divergences are removed by the renormalization procedure, we should have no non-local divergences at the two-loop order. There would be local divergences remaining, which can be renormalized as for the one-loop computations. As discussed in Ref. [24], a two-point function without proper renormalization does not reproduce the correct dependence on and at the two-loop order because of non-local divergences as .

Since now it is not expected to have such divergences after the renormalization, it should be possible to reproduce the correct shift of conformal weight even at the order. We shall show that this is indeed the case in this subsection.

We first evaluate the expectation value of the open Wilson line at the order without renormalization, then we consider its effects. A contribution comes from in Eq. (64) as (82) which is expressed as diagram (a) in Fig. 3. The integral is computed as

(83)

Here we neglect the terms of and write down only the terms depending on or . In the rest of this subsection, we include only such terms. Another type of contribution arises from in Eq. (64). Defining (84) we find (85)

These integrals correspond to diagrams (b), (c), (d) in Fig. 3, respectively. Summing over all contributions we find (86)

Therefore, a non-locally divergent term as remains, and the expression cannot be compared with Eq. (10).

Now we include the effects of renormalization, namely, the change of overall factor as in Eq. (68) and the shift of parameter as in Eq. (81). These effects lead to an extra contribution as (87) where (88)

The extra contribution can be evaluated as (89)

Thus in total we arrive at (90) which does not have any non-local divergence. Compared with the expansion of the two-point function in Eq. (10), the coefficients in front of and at the order are correctly reproduced with in Eq. (61).

As for , we start by examining the two-point function at the order. Since spin-three current is involved along with spin-two current , there are two types of corrections as (92) at this order. They are represented in Figs. 1 and 4, respectively. Here is defined in Eq. (64) and

(93)

Since we have already computed as in Eq. (66), we just need to evaluate . The prescription in Eq. (57) leads us to adopt (94) with the shift of conformal dimension of from to . Using this expression, we find (95) up to the term of order . Inserting , we obtain (96) up to the terms of orders and . In particular, the -order correction of conformal weight is read off as , which is consistent with Eq. (91). In order to remove the divergence at up to the order, we renormalize the Wilson line operator as (97) which leads to the corresponding two-point function of canonical form as in Eq. (6).

We move to three-point functions with one conserved current. For , there are two choices of currents, i.e., spin-two current and spin-three current . We start by computing up to the order by following the previous analysis for . With the convention of in Eq. (56), the corresponding three-point function is given by Eq. (62) with Eq. (91). At the leading order in , we have already computed as in Eq. (55) with and . In the following we shall examine the next nontrivial order in .

At the order in , there are several types of contribution as in Fig. 2 and Fig. 5.

One comes from (98) where and were introduced in Eqs. (64) and (93), respectively. The first contribution corresponds to the diagram (a) in Fig. 2 and it has been computed as in Eq. (72) with . For the second one corresponding to the diagram (a) in Fig. 5, we find (99) where we have used (100) with the shifts of conformal weight both for and .

Other types of contribution include four conserved currents. One of them involves four spin-two currents as (101)

They are represented in Fig. 2 and have already been evaluated in Sect. 4.2. Others involve two spin-two and two spin-three currents, and the correlator of them is factorized at the leading order in as (102)

Therefore, we need to evaluate (103) which correspond to diagrams (b), (c), (d) in Fig. 5, respectively. Explicitly performing the integrals, we find (104)

Combining all contributions so far, we have (105)

The above expression reduces to (106)

As before, we choose the parameter in Eq. (58) as (107)

This leads to an extra contribution up to the order from (108) with in Eq. (64). We can see that this contribution cancels the divergence in Eq. (106) as (109) for . The constant term in Eq. (107) is chosen in order to reproduce Eq. (62) with as in Eq. (91).

We would like to compute another correlator as with spin-three current at the order. Using the leading-order result in Eq. (55) with , , and the correction as obtained in Sect. 2, the corresponding three-point function is given by (110)

Following the prescription discussed in Sect. 3.4, we choose the parameter in Eq. (58) such that the Wilson line computation reproduces this expression.

With two current insertions from an open Wilson line, we have the following type of contribution as (111) which come from diagrams (a), (b) in Fig. 6. For the correlator of the three currents, we have used Eq. (100). There are contributions with two spin-two and two spin-three currents. With the correlator in Eq. (102), they are given by

(112) which correspond to diagrams (c), (d), (e) in Fig. 6. Integrating over the variables , we find (113)

Furthermore, we need to consider a contribution of the form as (114) with (115)

Denoting (116) we find (117)

Here , , correspond to diagrams (f), (g), (h) in Fig. 6.

Combining the results so far as (118) we find (119)

We remove the divergent term by properly choosing the parameter in Eq. (58) as before. We propose to use (120) which leads to an extra contribution at the order as (121)

Here is defined as (122)

Including the effect, we obtain (123) for as in Eq. (110).

As for , we examine the two-point function up to the order and see whether we can reproduce the correction of conformal weight as in Eq. (91) after adopting the regularization. As before, we first evaluate the correction without renormalization and then include its effects.

There are contributions involving only spin-two currents, which were already evaluated in Eq. (86). We find (124) by setting . In this subsection, we only keep the terms involving or and not vanishing at .

Furthermore, we include the effects of spin-three current . In order to make our notation simpler, we adopt the following rule. If comes from the open Wilson line, then we use index . If enters instead of , then we replace the index by . We first compute those with three currents as (125) and , , which are represented by diagrams (a), (b), (c) in Fig. 7, respectively. Integrations over yield

(126)

There are also contributions with two spin-two and two spin-three currents such as (127) and so on. They are computed as (128) which correspond to diagrams (d)–(i) in Fig. 7. Finally, those with four spin-three currents are (129) and others with different products of the two-point function. They are obtained as (130) which are represented in diagrams (j), (k), (l) in Fig. 7, respectively. Summing up all contributions we have (131) which includes a non-locally divergent term.

Let us then examine the effects of renormalization. There are two types of -order corrections as in Eq. (92) before the renormalization. Multiplying the terms due to renormalization, some contributions at the order arise. With Eqs. (107) and (97), the contribution with spin-two current becomes (132) see Eq. (88) for the previous case with . The contribution with spin-three current is (133) where we have used Eqs. (120) and (97). Thus the - and -dependent terms in the total contribution are (134)

The coefficients in front of and are precisely those in Eq. (10) with Eq. (91). We would like to emphasize again that there is cancellation among non-local divergences.