Relaxed hypermultiplet in four dimensional N=2 conformal supergravity

Superconformal matter multiplets play a crucial role in the construction of Poincare supergravity invariants. Off-shell multiplets allow for construction of general matter couplings in supergravity. In Nucl. Phys. B214 (1983) 519-531, relaxed hypermultiplet was constructed in rigid supersymmetry which on coupling with the real scalar multiplet allowed for an off-shell formulation of the rigid hypermultiplet. In this paper, we extend the relaxed hypermultiplet to conformal supergravity. For consistency with the superconformal algebra, we find that the fields have to be allowed to transform in a non-canonical way under SU(2) symmetry. We find suitable field redefinitions to obtain fields which are irreducible representations of SU(2) R-symmetry and present the full non-linear transformation rule.

Superconformal matter multiplets play a crucial role in the construction of Poincaré supergravity invariants. Off-shell multiplets allow for construction of general matter couplings in supergravity. In [1], relaxed hypermultiplet was constructed in rigid supersymmetry which on coupling with the real scalar multiplet allowed for an off-shell formulation of the rigid hypermultiplet. In this paper, we extend the relaxed hypermultiplet to conformal supergravity. For consistency with the superconformal algebra, we find that the fields have to be allowed to transform in a non-canonical way under SU (2) R-symmetry. We find suitable field redefinitions to obtain fields which are irreducible representations of SU (2) R-symmetry and present the full non-linear transformation rule.

I. INTRODUCTION
Conformal supergravity plays an important role in constructing physical supergravity theories with higher derivative corrections. The higher degree of symmetries in conformal supergravity allows for the superconformal symmetries to be realized off-shell for certain multiplets and this plays a crucial role in the construction of higher derivative corrections to the physical supergravity theories with lesser number of symmetries than conformal supergravity. The higher derivative corrections are important for several reasons. They can be studied as supersymmetric counter-terms for removing divergences as well as anomalies of some tree level global symmetries at the loop level. They can also be used to study the higher derivative as well as quantum corrections to the entropy of black holes using techniques such as localization [2] and entropy function [3,4] and use the results to take the comparison with the microstate counting in string theory beyond the leading order Bekenstein-Hawking entropy.
One of the crucial ingredients of conformal supergravity is the Weyl multiplet which is a multiplet of fields that contains the graviton and its superpartner gravitino. The superconformal algebra closes on the multiplet without using any equations of motion while the structure constants of the algebra become field dependent. Such an algebra is known as a soft algebra. Further, one finds various representations of the soft algebra, which are known as matter multiplets, and consider their coupling to conformal supergravity using a set of procedures which goes by the name of superconformal multiplet calculus. Upon using some of these matter multiplets as compensators to gauge fix the additional symmetries in conformal supergravity, one gets the physical supergravity theory with symmetries belonging to the super-Poincare group. In the case of extended supergravity, one has to add auxiliary fields to the Weyl multiplet so that the off-shell degrees of freedom for bosons and fermions match. There is a standard choice of the auxiliary fields which lead to the standard Weyl multiplet (see [5] for the N = 2 and N = 4 standard Weyl multiplet in four dimensions). There is a different choice of auxiliary fields that contains a scalar field of non vanishing mass dimension and is known as the dilaton Weyl multiplet (see [6] for the N = 2 dilaton Weyl multiplet in four dimensions).
In N = 2 supergravity there are basically two matter multiplets, with spin less than 2, which has distinct propagating degrees of freedom. The multiplet which has one propagating spin-1 degree of freedom with helicity = ±1, two propagating spin-0 degrees of freedom and two propagating spin-1/2 degree of freedom with helicity = ±1/2 is known as the vector multiplet. Another multiplet which has four propagating spin-0 degrees of freedom and two propagating spin-1/2 degrees of freedom with helicity = ±1/2 is known as the scalar multiplet or the hypermultiplet. One can add auxiliary fields to the vector multiplet to obtain an off-shell N = 2 vector multiplet with 8+8 (bosonic+fermionic) off-shell degrees of freedom. This has been very well studied along with its coupling to N = 2 conformal supergravity and has also been used as a compensator to get N = 2 Poincare supergravity. However, to have an off-shell completion of hypermultiplet with a finite number of auxiliary fields is not very straightforward. One of the ways to get an off-shell completion of the N = 2 hypermultiplet is to introduce a non trivial central charge in the supersymmetry algebra [7]. Another way to get an off-shell completion of the N = 2 hypermultiplet without a central charge is to consider a different multiplet where one of the propagating spin-0 degree of freedom is represented by a dual 2-form gauge field. Such a multiplet is known as a linear multiplet or a tensor multiplet [8,9]. Its off-shell completion has 8 + 8 degrees of freedom. It is also very well studied along with its coupling to N = 2 conformal supergravity [10] and has also been used as a compensator to get N = 2 Poincare supergravity. However, the dualization of a 2-form gauge field to a scalar will generally not work if we have higher number of derivatives. At the level of two derivatives, the couplings of the tensor multiplet are very restrictive in the sense that they do not couple to a spin-1 gauge field and do not have self coupling. These drawbacks of the tensor multiplet necessitates the formulation of an off-shell hypermultiplet without dualizing any spin-0 degree of freedom. Such an attempt was made in [1] for supersymmetric field theory, where the authors introduced two new multiplets: the 32 + 32 components relaxed hypermultiplet and the 24 + 24 components real scalar multiplet. They further considered a supersymmetric invariant action for the relaxed hypermultiplet and the coupling between the relaxed hypermultiplet and the real scalar multiplet. Upon using the equations of motion they showed that the combined system of the relaxed hypermultiplet and the real scalar multiplet describes the physical degrees of freedom of a hypermultiplet.
In [11], the extension of the 24 + 24 components real scalar multiplet to conformal supergravity was studied. At the linearized level, the components of this multiplet along with the 24 + 24 components of an N = 2 standard Weyl multiplet coupled with the 48+48 components current multiplet of a rigid N = 2 tensor multiplet. It was also shown that one can impose a consistent set of constraints on the real scalar multiplet to reduce its components to an 8+8 components restricted real scalar multiplet which can be mapped to the 8 + 8 components of a N = 2 tensor multiplet coupled to conformal supergravity. Further, in [12] an invariant action for the real scalar multiplet was constructed using a new density formula based on a fermionic multiplet which was used to find new higher derivative couplings of the tensor multiplet in N = 2 conformal supergravity.
In this paper, we would like to extend the 32 + 32 components relaxed hypermultiplet to N = 2 conformal supergravity. This will pave the way for an off-shell treatment of the hypermultiplet in N = 2 conformal supergravity. In [13], extension of relaxed hypermultiplet to conformal supergravity was studied in harmonic superspace by relating it to a well known rigid supersymmetry multiplet albeit with additional global SU (2) indices. In this paper, we explictly couple the relaxed hypermultiplet to conformal supergravity. In appendix-B, we will comment on the relation between our results and the harmonic superspace discussion in [13].
The plan of the paper is as follows. In section-II, we will review the relaxed hypermultiplet and real scalar multiplet in supersymmetric field theory as done in [1]. We will also discuss how the two multiplets were used to describe the physical degrees of freedom of a hypermultiplet. In section-III, we will discuss the extension of the relaxed hypermultiplet to conformal supergravity. In section-IV we will end with some conclusions and future directions. We will also present the details of an N = 2 standard Weyl multiplet, that are relevant for the paper, in appendix-A.

II. REVIEW OF THE RIGID RELAXED HYPERMULTIPLET
As we discussed in the previous section, while considering specific actions the tensor/linear multiplet provides an off-shell description of the hypermultiplet degrees of freedom using dualization, this mapping between the 2form gauge field and the 0-form scalar field does not hold true for more general actions.
In [1], the authors tried to evade the use of 2-form gauge fields by generalizing the superspace constraints corresponding to the linear multiplet so that there is no conserved vector that can be idenitified as the field strength of a 2-form gauge field. The resulting 32 + 32 multiplet was dubbed the relaxed hypermultiplet. In this section, we will elaborate on this construction and explain how when coupled with another 24 + 24 multiplet, this gives the precise on-shell degrees of freedom as the hypermultiplet. As this construction was carried out in N = 2 flat space supersymmetry, components of various multiplets under consideration can be organized as irreducible representations of global SU (2) symmetry. In this section, this global SU (2) symmetry is simply referred to as SU (2).
The superspace constraint for the linear multiplet can be written as: where L ij is a triplet of real superfields. Here the superspace covariant derivative D αi is given as: where (x µ , θ αi ,θα i ) are the superspace coordinates written in two component Weyl spinors notation. The above constraint in component form means that there is no field that transforms in the 4 representation of SU (2) in the transformation of L ij which is the lowest mass dimension field of the linear multiplet. That L ij is real triplet of scalar fields means that the field L ij satisfies the pseuo-reality constraint, By an investigation of the possible SU (2) representations that appear in the transformation of L ij and higher mass dimension fields that appear in its transformation rules, consistent with the above constraint and the supersymmetry algebra, one finds that the linear multiplet fields are L ij , G, ϕ i and H a . Here, G is a complex scalar field and ϕ i is an SU (2) doublet of Majorana fermions. Supersymmetry algebra implies that H a is a real conserved vector, which can also be seen by trying to match the bosonic and fermionic off-shell degrees of freedom. The resulting multiplet has 8 + 8 off-shell degrees of freedom.
In [1], a 24+24 multiplet was presented with the superfield L ijkl satisfying the constraint that the lowest mass dimension field L ijkl does not contain in its transformation any fermion that transforms in the 6 representation of SU (2). This can be analogously written as the constraint, Analogous to L ij , the field L ijkl satisfies the pseudoreality condition, We will not review this multiplet here, but relevant to us is the fact that this multiplet contains a fermion ψ ijk that transforms in the 4 representation of SU (2). To extend the linear multiplet to a multiplet with no conserved vectors, the authors realxed the constraint (1) such that a fermion in 4 representation appears in the transforma-tion of L ij and it is identified with the fermion ψ ijk from the multiplet that corresponds to L ijkl , while keping the constraint (4) intact. Relaxation of the constraint (1) allows for a rearrangement of the off-shell degrees of freedom and H a need not be conserved. The two multiplets coupled this way form a multiplet which contains 32 + 32 off-shell degrees of freedom, with no conserved vectors. This multiplet is referred to as the relaxed hypermultiplet. Transformation rule for the components of the multiplet is given as: where we have given the results of [1] in four component formalism. The SU (2) representation of the various fields of the relaxed hypermultiplet is obvious from the index structure in the above transformation rule. M ij and N are complex triplet and singlets under SU (2). V ij µ and G µ are real triplet and singlet vector fields respectively. We have chosen the normalization of the fields in the relaxed hypermultiplet such that where X is any field in the relaxed hypermultiplet.
The above 32 + 32 multiplet when coupled to a 24 + 24 multiplet corresponding to a superfield contragradient to the L ijkl , was shown to contain the appropriate on-shell degrees of freedom as the hypermultiplet. This 24 + 24 multiplet is based on a real scalar superfield V satisfying the constraint: The detailed transformation rule for the components of the multiplet in flat space was presented in [1] and was generalized to conformal supergravity recently in [11] following which we will refer to this multiplet as the real scalar multiplet. The supersymmetric invariant action with coupling between the real scalar multiplet and relaxed hyper multiplet in flat space was presented in [1] and is given as: where I 1 is the kinetic action for relaxed hypermultiplet and I 2 is the coupling of the relaxed hypermultplet with the real scalar multiplet. V, K ij , A µij , C ijkl , ψ i and ψ ijk are the field components of the real scalar multiplet. V is a real scalar field, K ij is a complex triplet of scalars.
A µij and C ijkl have the same pseudo reality properties as V µij and L ijkl . Equation of motion for L ij leads to Thus L ij correspond to three scalar on-shell degrees of freedom. Equations of motion for G µ and V µij lead to Thus V corresponds to one scalar on-shell degree of freedom. The field ψ i is identified with λ i from using ξ i equations of motion. Equation of motion for λ i gives: Thus λ i corespond to four on-shell spin-1/2 degrees of freedom. The remaining equations of motion set L ijkl , C ijkl , ψ ijk , ξ ijk and ξ i to zero. Thus the bosonic on-shell degrees of freedom are contained in V, L ij as four scalar degrees of freedom and fermionic on-shell degrees of freedom are contained in λ i (or ψ i ) as four spin-1/2 degrees of freedom. Thus the 56 + 56 multiplet with the above action has the appropriate on-shell degrees of freedom for the 4 + 4 hypermultiplet. As mentioned earlier, the 24 + 24 real scalar multiplet was generalized to N = 2 conformal supergravity in [11]. In the next section, we will generalize the 32 + 32 relaxed hypermultiplet to N = 2 conformal supergravity.

III. EXTENSION OF THE N = 2 RELAXED HYPERMULTIPLET TO CONFORMAL SUPERGRAVITY
In N = 2 conformal supergravity, multiplets form representations of the following soft algebra [14]: where δ Q , δ S , δ M , δ K , δ D , δ A and δ V are respectively the infinitesimal transformations corresponding to Qsupersymmetry, S-supersymmetry, local Lorentz transformation, special conformal transformation, dilatation, U (1) and SU (2) R-symmetry. The infinitesimal transformations δ gauge on the right hand side of [δ Q , δ Q ] correspond to the additional gauge symmetries that a multiplet may possess. The infinitesimal transformation δ (cov) (ξ), is the covariant general coordinate transformation defined as: where δ gct is the general coordinate transformation. In the summation, T runs over all the superconformal transformations (including the additional gauge symmetries of the multiplet) except local translation, and h µ (T ) is the corresponding gauge field with appropriate factors as listed in table-III (see appendix-A). The parameters for the transformations listed on the RHS of first of the equations (12) are given as: We will obtain the superconformal transformation rule for the relaxed hypermultiplet, by demanding the consistent closure of the superconformal algebra (12) on the relaxed hypermultiplet. The fields L ij , L ijkl , V ij a and G a are real. Therefore, the chiral weights of these fields are 0. Let us assign an arbitrary Weyl weight w to the field L ij . From the supersymmetry transformation rule (6) and from the knowledge of the weights of the supersymmetry parameters (see equation (A7) in appendix-A), the chiral weights of the remaining fields get completely fixed, and the Weyl weights get determined in terms of w. The fields L ij and L ijkl are the lowest Weyl weight components in the multiplet and therefore must be invariant under Stransformations. For the S-transformation of ψ ijk and λ i we will consider terms allowed by the Weyl weights, chiral weights, and the global SU (2) index structure as follows: where x, y and z are arbitrary coefficients which will be determined using the Q − S commutation relation. To do this we need to know how L ij and L ijkl transform under V -transformation. At the level of flat space supersymmetry, we had discussed in the previous section that the fields in the relaxed hypermultiplet are irreducible rep-resentations of the global SU (2) symmetry. Assuming this to be true even for SU (2) R-symmetry in conformal supergravity, we evaluate the S − Q commutator on L ij : We also calculate the expected result of the commutator: For the superconformal algebra to close consistently equations (16) and (17) should match. But we clearly see that there is mismatch between the equations due to the presence of L ijkl terms in equation (16). Setting the coefficient y = 0 might seem to resolve this mismatch; however, it leads to similar inconsistency in the [δ S , δ Q ]L ijkl equation. These mismatches that arise are rooted in the way in which relaxed hypermutliplet was constructed. The superspace constraint (1) was relaxed which led to the intertwining of the 24 + 24 multiplet and the 8 + 8 linear multiplet through the supersymmetry transformation. This suggests that for consistent coupling of the relaxed hypermultiplet to conformal supergravity, we should allow the two multiplets to get intertwined not just through Q-SUSY, but also through other superconformal transformations. Thus we come up with the following prescription to accommodate the required changes in the superconformal transformation: (I) We allow any field in the relaxed hypermultiplet to mix with any other field in the multiplet (or even the derivatives of the fields) under Vtransformation, given that such a mixing is consistent with Weyl weights, chiral weights and Lorentz structure.
(II) We also allow for covariant derivative terms consistent with the weights to appear in the Stransformation of the fields.
(III) In addition to the soft algebra relations (12), we make use of the following commutation relations involving V-transformations to fix the coefficients in the tranformations.
We will now illustrate our prescription. Consider the fields L ij and L ijkl . They have the same Weyl weight, chiral weight and are the lowest Weyl weight components of the multiplet with no Lorentz indices. Hence we propose the following ansatz for their V-transformation: where α, β, µ, θ are arbitrary coefficients which need to be fixed by using the soft algebra (12) and the commutation relations (18). On evaluating the V-V commutator on L ij and L ijkl , we find the following two possibilities for the coefficients: (i) α = 5 2 , µβ = 3 4 and θ = 3. (ii) α = − 1 2 , µβ = 3 4 and θ = 1. On operating Q-V commutator on L ij , we find the second possibility to be ruled out. Further by making use of the S-Q commutator on L ij and L ijkl , the coefficients β, µ and x, y, z get fixed. In addition the lowest Weyl weight value w also gets determined.
Proceeding analogously, using the aforementioned prescription and algebraic relations we determine the S-and V-transformation of all the fields in the multiplet. Having determined the S-transformation of all the fields in the multiplet, we make use of the S −S commutation relation from (12) to determine the K-transformation for fields in the multiplet. The linearized superconformal transformation rule for the relaxed hypermultiplet is found to be: where the derivative D a that appears in the above transformations is the superconformal covariant derivative (defined by the equation (A6) in appendix-A). The transformation parameters Λ D , Λ M , Λ Ka , Λ A and Λ i V j correspond to dilatation, local Lorentz transformation, special conformal transformation, U (1) and SU (2) R-symmetry respectively. Notice that the S-transformation of ξ i and the V -transformations of V ij a , G a , and ξ i involves covari-ant derivative terms. As was shown above, the fields in the relaxed hypermultiplet had to be allowed to transform in a noncanonical way under SU (2), when we extend the multiplet to conformal supergravity. Moreover, we note in the above transformation rules that there are fields in the relaxed hypermultiplet which are not K-invariant as contrary to other known matter multiplets in the litera-ture. But, we will show that by appropriate redefinition of the relaxed hypermultiplet fields we can switch to a new SU (2) basis wherein all the fields transform in a way that SU (2) irreps do. In addition, the fields will also be K-invariant in the new basis. Consider the Vtransformation of ξ i . We note that it transforms into a linear combination of γ a D a λ i and γ a D a ψ ijk , and the diagonal elements (i.e terms containing ξ i ) are absent in the transformation. Thus by taking appropriate linear combinations (which we list out in appendix-B) of the components of the fields ξ i , γ a D a λ i and γ a D a ψ ijk we define two new fields Ω and ξ which are K-invariant and transform as SU (2) scalars. Along the same lines we carry out the redefinition of the other fields as well.
Table-I summarizes the field content and properties of the relaxed hypermultiplet in the new basis. The linearized transformation of the relaxed hypermultiplet in terms of the redefined fields is given as follows 1 : where A ijk = ε il ε jm ε kn A lmn . The subscript L in ξ L and Ω L denotes the left chirality, and the fields ξ and Ω have been defined with right chirality.
Finally we obtain the complete non-linear superconformal transformation by introducing non-linear covariant terms into the Q-and S-transformation rules. The non-linear terms are constructed by taking simple products of standard Weyl multiplet fields, curvatures and relaxed hypermultiplet fields. On careful analysis, we find that there is no such non-linear term consistent with Weyl weight, chiral weight and the representations, which can be added to the S-transformation. Hence, the Stransformation rules remain unchanged. However, for the Q-transformation there are non-linear terms consistent with weights and representations. We add all such possible terms and fix the coefficients using the algebra 1 See appendix-B for the details of the field redefiniton. (12). We give the final result below: Field SU (2) Irreps Properties Weyl Chiral weight (w) weight (c) complex, Lorentz vector 4 0 to black hole entropy.
In this paper we have extended the 32 + 32 relaxed hypermultiplet constructed in [1] for flat space to N = 2 conformal supergravity. In doing this, we find the peculiar feature that the fields which are irreducible representations of the global SU (2) symmetry must be allowed to transform in a non-canonical way under the local SU (2) R-symmetry in order to be consistent with the superconformal algebra. We perform suitable field redefinitions so that the redefined fields are irreducible representations of the SU (2) R-symmetry as well as invariant under Ksymmetry. In appendix-B, we present the precise field redefinitions as well as the relation of our results to the constructions in harmonic superspace [13] and projective superspace [15].
In [1], in flat space, the relaxed hypermultiplet was coupled to a 24 + 24 real scalar multiplet to obtain an off-shell formulation for the hypermultiplet. The real scalar multiplet was extended to conformal supergravity in [11]. Further, it was shown to be reducible to an 8 + 8 restricted real scalar multiplet by imposing 16 + 16 constraints. This restricted real scalar multiplet admitted a tensor multiplet embedding. In [12], a new density formula in N = 2 conformal supergravity was presented with application to real scalar multiplet. This allowed for a new higher derivative invariant for the tensor multiplet in supergravity.
With the relaxed hypermultiplet in N = 2 conformal supergravity presented in this paper, it would be interesting to construct the analogue of the action (8) in conformal supergravity. This would allow for an off-shell formulation of hypermultiplet in four dimensional conformal supergravity. In [16], an extension of the analysis from [1] was generalised to six dimensional N = (1, 0) curved superspace to construct the off-shell representation for the hypermultiplet in six dimensional supergravity. This extension is different from that of this paper in the following sense. In [16], an additional real superfield T appeared in the superspace constraints along with the L ij and L ijkl superfields to define the relaxed hypermultiplet. The construction of [16] in six dimensions may have a straightforward generalization in four dimensions and one can have an alternate formulation of relaxed hypermultiplet with more number of components.
It will also be interesting to investigate if there exist constraints to reduce the relaxed hypermultiplet to a restricted multiplet. Whether the multiplet can be embedded into a density formula or can be mapped to other multiplets are the questions central to the techniques of superconformal tensor calculus, which need to be addressed for the case of relaxed hypermultiplet in conformal supergravity.

ACKNOWLEDGMENTS
This work is supported by SERB grant CRG/2018/002373, Govt of India. We thank Daniel Butter for useful discussion.  These fields are determined in terms of the independent fields in the Weyl multiplet by the following set of conventional constraints: where R(P ), R(Q), R(M ) and R(A) are respectively the super-covariant curvatures associated with local translation, Q-supersymmetry, local Lorentz transformation and U (1) R-symmetry, and˜ R(A) is the dual of R(A). The curvature R(V ) that appears in the superconformal transformation of the relaxed hypermutliplet (24) is the super-covariant curvature associated with SU (2) Rsymmetry. On imposing the above constraints, the soft algebra relations (12) are satisfied on the fields. Finally we end this section by defining the supeconformal derivative and listing out the Weyl weights and chiral weights of supersymmetry parameters: where the summation runs over all the superconformal generators T except local translation and h µ (T ) are the corresponding gauge fields as listed in table-III. The chiral weight and Weyl weight of the Q-and S-SUSY parameters are: In the new basis, the reality constraints of L ij , L ijkl , V ij a and G a translate into: Further to tidy up things, we make the following normalization choice for the fields: Also we define A ijk = ε il ε jm ε kn A lmn , which implies We will now discuss the relation between our results and the discussion in [13] 2 . In Table-IV, we see that for each mass dimension we have two sets of fields which transform under the same representation of SU (2). At the the lowest mass dimension level, we have A ijk and B ijk fields that transform under 4 representation of SU (2) R-symmetry. In (24), we see that in the transformation rule for A ijk there is no fermionic field that transforms under 5 representation of SU (2). We can write this constraint in superspace, analogous to the discussion in section-II, as follows: where a runs over 1, 2 and is an additional global SU (2) index carried by the superfield L ijka whose lowest mass dimension components are L ijka = (A ijk , B ijk ). At each mass dimension, the two fields with the same representation under SU (2) R-symmetry are now placed inside a 2 representation of the additional global SU (2) using the 2 We thank Daniel Butter for useful correspondence regarding this.
index a, upto field redefinitions. Clearly, the field redefinitions carried out in the present section are valid even in the case of rigid supersymmetry. Therefore at the level of rigid supersymmetry, relaxed hypermultiplet can be identified as the superfield L ijka . Such an identification was made at the rigid supersymmetry level in [13]. In this language, the linear multiplet defined by the constraint (1) is referred to as the O(2) multiplet. Coupling of O(n) multiplets to conformal supergravity was discussed in [13] in harmonic superspace and in [15] in projective superspace. Thus in [13], it was argued that extension of relaxed hypermultiplet to conformal supergravity would be the same as the O(3) superfield. In this paper, we explicitly coupled the relaxed hypermultiplet to conformal supergravity and indeed, we find that the only way to extend it to conformal supergravity is through an O(3) superfield with an additional flavor SU (2) index.