"Induced"N=4 conformal supergravity

We consider an abelian N=4 super Yang-Mills theory coupled to background N=4 conformal supergravity fields. At the classical level, this coupling is invariant under global SU(1,1) transformation of the complex ("dilaton-axion") supergravity scalar combined with an on-shell vector-vector duality. We compute the divergent part of the corresponding quantum effective action found by integrating over the super Yang-Mills fields and demonstrate its SU(1,1) invariance. This divergent part related to the conformal anomaly is one-loop exact and should be given by the N=4 conformal supergravity action containing the Weyl tensor squared term. This allows us to determine the full non-linear form of the bosonic part of the N=4 conformal supergravity action which has manifest SU(1,1) invariance.


Introduction
The N = 4 conformal supergravity (CSG) as formulated in [1] should have global SU (1,1) or SL(2, R) symmetry acting on the singlet complex scalar (described by a 4-derivative analog of the SU(1, 1)/U(1) coset sigma model). 1 While the complete N = 4 superconformal transformation laws were written down in [1], the full non-linear action of such N = 4 conformal supergravity was not explicitly constructed so far. The aim of this paper is to find the full bosonic part of such action.
This manifest SU(1, 1) symmetry is in general broken if one couples the N = 4 CSG to N = 4 super Yang-Mills (SYM) theory [2,3]. It is, however, preserved in an weaker "on-shell" form in the case when the N = 4 SYM theory is abelian: the resulting equations of motion are invariant under the SU(1, 1) acting not only on the complex scalar but also on the Abelian SYM vector via vector-vector duality transformation. 2 This symmetry is then inherited by the equations of motion of the N = 4 Poincare supergravity [5] as it can be obtained [2] from a system of 6 abelian vector multiplets coupled to the N = 4 conformal supergravity multiplet. 3 As was found in [7,8], the SU(1, 1) invariant N = 4 CSG of [1] has non-zero beta-function or conformal anomaly and is thus inconsistent at the quantum level unless it is coupled to four N = 4 vector multiplets (see [9] for a review). This conclusion was confirmed in [10] on the basis of analysis of the local SU(4) chiral anomaly (which is in the same multiplet with trace anomaly).
At the same time, it was suggested in [7,8] that there might exist an alternative version of N = 4 CSG without the SU(1, 1) invariance in which a non-minimal coupling of the singlet scalar to the square of the Weyl tensor may be present. For a particular value of such coupling the resulting "non-minimal" N = 4 CSG can be made UV finite by itself, i.e. without adding extra N = 4 vector multiplets [7]. 4 Curiously, a similar type of "non-minimal" N = 4 conformal supergravity seems to emerge [11] in the twistor-string [12] context.
The coupling between N = 4 SYM and N = 4 CSG multiplets appears also in the context of the AdS/CFT correspondence [13,14,15]: the N = 4 SYM path integral with the CSG fields as external "sources" may be interpreted as a generating functional for correlators of particular 1/2 BPS operators (dimension 2 chiral primary operator and its supersymmetry descendants, i.e. the fields of the stress tensor multiplet dual to N = 8, d = 5 supergravity fields). After integrating over the quantum SYM fields, the conformal supergravity action should then be the coefficient of the logarithmic divergence in the resulting effective action. In that limited sense 1 To make this symmetry linearly realized one may introduce also a spurious local U (1) symmetry. 2 This on-shell symmetry can be promoted to a manifest symmetry of the action (at the expense of manifest Lorentz symmetry) if one uses a phase-space type formulation where one doubles the number of vectors, see, e.g., [4]. 3 This can be done by partial gauge fixing and solving for some of the CSG fields that in the absence of the pure CSG action play a role of auxiliary fields [2,3]. Potential importance of superconformal formulation of N = 4 Poincare supergravity was recently emphasised in [6]. 4 It is not clear, however, how this conjecture can be reconciled with the SU (4) anomaly cancellation study [10] which does not seem to be sensitive to such non-minimal terms. That suggests a potential problem with realization of supersymmetry which should be requiring that all superconformal anomalies should belong to one supermultiplet.
the N = 4 CSG may be interpreted as an "induced" theory. 5 Since the superconformal anomaly should be 1-loop exact, the result for the logarithmic divergence should be given just by the 1-loop contribution. 6 This also means that the divergent term is not sensitive to the non-Abelian structure of the SYM theory, i.e. it is sufficient to consider just one abelian N = 4 vector multiplet coupled to the external N = 4 CSG multiplet and do the gaussian integral over the N = 4 vector multiplet fields.
As the full non-linear form of the coupling between the N = 4 SYM and CSG multiplets is known [2,3], and since the one-loop logarithmic divergence of the N = 4 vector multiplet fields is determined by a relevant Seeley coefficient of the corresponding 2nd order matrix differential operator (with coefficients depending on the external CSG fields) it should thus be straightforward to reconstruct the full non-linear form of the resulting N = 4 CSG action using the standard algorithm [16], i.e. one should get [14] where N is the number of N = 4 vector multiplets, Λ is a UV cutoff. Here I N =4 CSG should be the CSG action as it starts with the Weyl tensor squared C 2 term (up to total derivative Euler density term): since I N =4 CSG should inherit all the symmetries of N = 4 conformal supergravity by construction 7 and contains the C 2 term it must represent the complete non-linear action of N = 4 conformal supergravity.
In particular, since the coupling between an Abelian N = 4 SYM and N = 4 CSG multiplets preserves the scalar SU(1, 1) symmetry combined with a duality rotation of the N = 4 SYM vector [2] and since the latter is integrated over in the path integral, the resulting "induced" CSG action should have manifest (off-shell) SU(1, 1) symmetry. 8 This was already demonstrated in [19] in the subsector of the standard SL(2, R) invariant scalar-vector coupling (e −σ F mn F mn − iCF mn F ⋆ mn ). Here we will demonstrate this for the full N = 4 vector -CSG coupling case, thus determining the full SU(1, 1) invariant form of the bosonic part of the N = 4 CSG action.
This computation is of interest as the complete non-linear form of the N = 4 CSG action was not explicitly given before. The terms in the CSG action which are quadratic in the non-metric fields (but non-linear in the metric) can be reconstructed [7,9] by requiring the Weyl symmetry and reparametrization invariance, but higher order terms are hard to determine directly. 9 The non-linear terms of N = 4 CSG action should of course reduce to the corresponding terms in the full N = 2 CSG action which was found in [1]; this provides a non-trivial check.
As the "induced" CSG action we find below is manifestly SU(1, 1) invariant, an apparent absence of an alternative to the SU(1, 1) invariant coupling [2] between the Abelian N = 4 SYM and N = 4 CSG multiplets appears to rule out the possibility of some SU(1, 1) non-invariant "non-minimal" conformal supergravity model.
We shall start in section 2 with a review of the Lagrangian of an Abelian N = 4 vector multiplet coupled to (bosonic part of) N = 4 conformal supergravity background. In section 3 we shall compute the UV divergent part of the effective action found by integrating over the vector multiplet fields and show that the resulting SU(1, 1) invariant expression has the expected structure of the N = 4 CSG action. A short summary will be given in section 4.

N = 4 Abelian vector multiplet coupled to external N = 4 conformal supergravity
Let us start with a review of the action [2] for an Abelian N = 4 vector multiplet in a background of N = 4 conformal supergravity. We shall denote the vector multiplet fields as In what follows m, n, r, s = 1, 2, 3, 4 are space-time indices and i, j, k, l = 1, 2, 3, 4 are SU(4) indices. The scalar fields satisfy the conditions For the fermions ψ i = P + ψ i transforms as 4 of SU(4), and In what follows we shall consider only the bosonic CSG background. Here e a m is the vierbein, V i j m is SU(4) gauge field potential, T −ij mn are complex antisymmetric antiselfdual tensors of dimension 1 transforming in 6 of SU(4) (T −ij are Lorentz scalars of dimensions 0, 1 and 2 respectively (i.e. they have 4, 2 and 0 derivatives in their kinetic term in CSG action [1,9]). The complex scalars by adding a local U(1) gauge symmetry. Then φ α transforms under global SU(1, 1) as well as has the U(1) chiral weight −1. 10 Then only φ α transforms 9 In principle, they can be reconstructed using the Noether procedure given that the full non-linear supersymmetry transformation rules are known (and close off shell on CSG fields) [1]. 10 Other CSG fields having non-zero chiral weights are: . The Q-susy parameter ǫ i has weight 1/2. under SU(1, 1) but other fields with non-zero chiral weights transform under local U(1), i.e. all fields with derivative couplings and non-zero chiral weights couple to the scalar U(1) connection through the covariant derivative (w is the chiral weight) The scalar connection a m is invariant under the SU(1, 1) and transforms by a gradient under the U(1). The general form [2] of the N = 4 vector multiplet Lagrangian (before U(1) gauge fixing) may be written as [2] 5) and the fermionic part In general, the derivative D m contains the gravitational ∇ m part as well as the SU(4) gauge potential (V m ), in addition to the U(1) term (a m ) in (2.3) (note that the bosonic vector multiplet fields have zero chiral weights while ψ i has weight -1/2). While the F mn (A) dependent part of the action (2.4) is not invariant under SU(1, 1) acting on φ α , it was shown in [2] that the corresponding equations of motion (written in first order form) are invariant provided one also "duality-rotates" the vector field strength as in the closely related case of the Poincare supergravity [5].
Our aim will be to integrate over the vector multiplet fields {A m , ϕ ij , ψ i } in (2.4),(2.6) and compute the divergent part of the resulting effective action. For this we do not need to fix the local U(1) symmetry and may treat the scalar functions τ (φ), Φ(φ) and a m as arbitrary background fields. Equivalently, we may choose to fix the spurious local U(1) by a "physical" gauge, e.g., 7) 11 We use Euclidean signature with imaginary time (fourth) component, with ε 1234 = 1. For simplicity we shall often ignore trivial metric factors not distinguishing between coordinate and target-space indices (which are always contracted with Euclidean signature metric so we will often not raise them in the contractions). Self-dual parts of 2nd rank tensors are defined as where the complex scalar ζ (taking values in the disc |ζ| ≤ 1) is an independent degree of freedom. Then a m is no longer a invariant of a redefined SU(1, 1) acting on ζ (that preserves the gauge condition) but it changes only by a gradient. Explicitly, 12 Instead of ζ it is useful to use the complex scalar which is directly equal to the scalar-vector coupling τ (φ) in (2.4) The transformation from ζ to τ in (2.9) maps a unit into half-plane, so that τ transforms as τ → aτ +b cτ +d under the corresponding SL(2, R) equivalent to original SU(1, 1) (see, e.g., [20]). One has in (2.5) (2.12)

Divergent part of N = 4 SYM effective action in conformal supergravity background
The UV divergent part of the SYM effective action in the CSG background is related to conformal anomaly and thus should be given to all orders by the 1-loop logarithmically divergent term. To determine the latter one may just consider a single Abelian vector multiplet action (2.4),(2.6) quadratic in A = {A m , ϕ ij , ψ i } but keeping full dependence on the (bosonic) background fields G = {e a m , V i jm , T −ij mn , ζ, E ij , D ij kl }. As already mentioned, while it is not necessary to fix the U(1) gauge for concreteness we will be expressing all the scalar functions in terms of the complex scalar τ in (2.9)-(2.12). where H are second-order matrix differential operators, depending on the background fields G. Then where the diagonal DeWitt-Seeley coefficient a 2 of the generic operator has the following form [16] a 2 = tr 1 180 Here∇ m is given by the gravitational covariant derivative ∇ m plus possible extra gauge (SU(4) and U(1)) field potentials for unmixed fields, while h m AB accounts for the mixing between different types of fields.
The vector-scalar operator originating from from (2.4) may be written as where g = e σ/2 is a coupling function (see (2.11)), D m = ∇ m + ia m − V m and The fermionic operator can be found by squaring the first-order operator in (2.6) Here D m = ∂ m + 1 2 σ ab ω ab m + i 2 a m − V m and P ± are chiral projectors.

Vector-scalar sector
Let us start with the contribution of the vector-scalar sector (in which we will include also the ghost contribution). Ignoring first the vector-scalar mixing due to the T −ij mn background in (2.4) one is to account for the presence of a non-trivial scalar background-dependent factor in the vector kinetic operator H 1 . This issue was dealt with already in [19] in the case of a simple vector coupling in the first line of (2.4) and we will follow the same approach here.
Choosing the gauge fixing term as g 2 [∇ m ( 1 g 2 A m )] 2 where g = e σ/2 and redefining A m → gA m the vector operator H 1 may be written as (here C is the real part of τ in (2.9)) H 1mn = g mn (−∇ 2 + Π) + Π mn , (3.8) The corresponding ghost operator is there is also a non-trivial scalar background contribution [19] (∇ m τ = ∂ m τ ) The quadratic part of this 4-derivative action is the same as found for the singlet scalar kinetic term in the CSG action [9]. The full non-linear expression (3.13) is invariant under the SL(2, R) acting on the local scalar coupling τ = C + ig −2 [19] (note, e.g., that 1 Im τ D 2 τ → cτ +d cτ +d 1 Im τ D 2 τ ). To compute the scalar contribution we need to account for the reality constraints (2.1): we may solve them explicitly 15 or formally do the summation over i, j in (3.5), adding extra 1/2 factor in the final result. 14 We include the ghost contribution and ignore the scheme-dependent total derivative term ∇ 2 R. 15 A solution to these constraints may be chosen as The operator (3.5) has the form (3.3) where Applying the algorithm in (3.4) to this operator we find the total vector-scalar sector (1 vector, 6 real scalars) contribution to the logarithmic divergence coefficient (a 2 ) 1,0 = ( 1 10 + 6 120 )C 2 − ( 31 180 Here M and S were defined in (3.6),(3.13).

Fermionic sector
Let us now determine the fermionic contribution to (3.2). Squaring the operator in (3.7) and putting it into the form (3.3) gives The corresponding matricesP andF in (3.4) arê This gives (for the number n F = δ i i of Weyl fermions) 16 Then finally we get for the corresponding a 2 coefficient in (3.4) (here n F = 4 and we include the minus sign in front of the fermionic contribution in (3.1)) (a 2 ) 1/2 = 1 10 This expression is obviously SU(1, 1) invariant.
The resulting CSG Lagrangian is invariant under the global SU(1, 1), supporting the proposal [1] about the existence of the full non-linear N = 4 CSG action with such symmetry.

Summary
The above computation of divergent term in the N = 4 SYM effective action in conformal supergravity background allowed us to find the complete SU(1, 1) symmetric action of N = 4 conformal supergravity in the bosonic sector. We used that the divergent part of the effective action is local, preserves all the symmetries of the underlying classically superconformal theory and starts with the Weyl tensor squared term. The fermionic part of the N = 4 conformal supergravity action can be found by the same method. Indeed, the N = 4 SYM -CSG coupling given in [2] contains all the required fermionic terms. This is still straightforward but technically more involved.