Superconformal index of higher derivative $\mathcal N=1$ multiplets in four dimensions

Supersymmetric partition function of $\mathcal N=1$ superconformal theories on $S^1_{\beta} \times S^3$ is related to the superconformal index receiving contributions from short representations. The leading coefficients in the small $\beta$ (high"temperature") expansion of the index were previously related to the conformal anomaly coefficients of the theory. Assumptions underlying universality of these relations were tested only for simplest low-spin unitary multiplets. Here we consider examples of higher derivative non-unitary $\mathcal N=1$ multiplets that naturally appear in the context of extended conformal supergravities and compute their superconformal index. We compare the coefficients in the small $\beta$ expansion of the index with those proposed earlier for unitary multiplets and suggest some modifications that should apply universally to all types of theories. We also comment on the structure of subleading terms and the case of $\mathcal N=4$ conformal supergravity.


Introduction
Unitary representations of N = 1 superconformal algebra play an important role in many aspects of supersymmetric quantum field theory.Applications of non-unitary representations are less studied.They appear, in particular, in extended conformal supergravities [1][2][3][4] written in terms of N = 1 multiplets.Recently, such multiplets were considered in the computation of conformal anomalies (in 4 and 6 dimensions) in the context of AdS/CFT [5][6][7][8][9]; similar massless and massive multiplets were also discussed in [10].Non-unitary superconformal theories may have other interesting applications (see, e.g., [11]).
Given a CFT one may define the standard partition function on S1 β × S3 (with fermions treated as antiperiodic on S 1 of length β to have usual thermodynamic interpretation).When CFT is also a superconformal theory one may formally define also another -"supersymmetric" -partition function Z susy on S 1 β ×S 3 by (i) taking fermions to be periodic on S 1 β and (ii) introducing extra R-symmetry gauge-field couplings in the action to preserve global supersymmetry on S 3 [12][13][14].While having no thermodynamic interpretation 1 Z susy will instead be related to the superconformal index I(β) [15][16][17].It will thus be protected by supersymmetry, receiving contributions only from short multiplets (thus being computable exactly, e.g., using localization, see [18] for a comprehensive review).
In this paper we shall study the properties of the superconformal index for higherderivative (and higher spin) N = 1 non-unitary multiplets.We shall compute the coefficients in the small β expansion of the index I(β) and compare with their expected expressions in terms of conformal anomaly coefficients proposed earlier on the basis of studies of unitary low-spin examples [19,20].We shall find that some modifications of these expressions are required in the non-unitary cases. 2e shall start in Section 2 with a short review of the definition of the superconformal index for a 4d N = 1 theory and its relation to the supersymmetric partition function on S 1 β × S 3 .We shall then discuss what is known about the leading coefficients in their small β expansion, emphasizing that the assumptions used to derive the general expressions for the coefficients were checked only in models with simplest unitary (chiral and vector) multiplets.
In Section 3 we shall introduce four basic higher-derivative N = 1 superconformal multiplets for which we shall later compute the superconformal index.These non-unitary multiplets are the N = 1 building blocks of extended conformal supergravities.We shall discuss the superfield structure of the multiplets and check the relation between their chiral and conformal anomalies as predicted by the superconformal invariance.
In Section 4 we shall compute the superconformal index I(β) of these free non-unitary multiplets by an explicit "letter"-counting algorithm.The multiplets that contain higher spin gauge fields (conformal gravitino and graviton) require careful treatment of equation of motion constraints and Bianchi identities for the field strengths.In Section 4.2 where we present an efficient method to extract the small β expansion of the index, including all possible subleading terms.
In Section 5 we shall compare the expressions for the coefficients in the expansion of I(β) with those proposed earlier for unitary multiplets and propose some modifications that should apply universally to all types of theories.We also comment on subleading terms and the case of finite N = 4 conformal supergravity.
In Appendix A we shall present the free action of the [ 1  2 ] tensor multiplet.In Appendix B we shall discuss the results for the chiral gravitational and gauge anomalies of the conformal gravitino and non-gauge antisymmetric tensor.In Appendix C we shall review the expression for the conformal higher spin partition function on S 1 β × S3 b and work out its small β expansion.Appendix D will contain a discussion of how to one may compute the constant and log β term in the partition function using the direct expansion and ζ-function regularization in terms of spectrum of dimensionally reduced theory on S 3 .In Appendix E we shall repeat the computation of the superconformal index and its small β expansion for non-unitary multiplets in case of unequal fugacities which is related to supersymmetric partition function on S 1 β × S 3 b (with S 3 b being a squashed 3-sphere).Finally, in Appendix F we demonstrate how to perform similar analysis of the index of six-dimensional theories with (1, 0) supersymmetry discussing considering the examples of scalar, tensor and non-unitary higher derivative vector multiplets.
Let us now review what was claimed in the past about each coefficient in the expansion (2.6) or (2.7).
Leading term ∼ 1/β: It was argued in [19] that the coefficient of the leading Cardy-type [50] term in (2.7) can be expressed in terms of the conformal anomaly coefficients as 6 (2.8) The proof in [19] was based on the expected form of the effective action of the dimensionally reduced 3d theory corresponding to the limit of small radius of S 1 .It contains a Chern-Simons term k S 3 a ∧ F where a is the Kaluza-Klein graviphoton (mixed component of the metric tensor in reduction to 3d) and F is the R-symmetry gauge field strength.The 4 For a conformal theory on curved space, the coefficient of the logarithmic UV divergence in the standard partition function is Here we ignored possible ∇ 2 R term, C 2 is the square of the Weyl tensor and R R = C 2 − 2R 2 µν + 2 3 R 2 is 32π 2 times the Euler number density.Note that in contrast to the standard Casimir energy [44][45][46] the "supersymmetric" one may be viewed as "scheme-independent" [43,47] as supersymmetry should prohibit adding extra local counterterms that may modify the the expression for Esusy. 5The fact that the index encodes Esusy was first suggested in [41].The relation between the index and Z susy was later clarified in [29] who showed that the e −β Esusy factor in (2.4) is a normal-ordering effect like for the standard Casimir-like contribution.Further discussion of the universality of the relation (2.4) appeared in [34]. 6Related observations appeared previously in [49,51,52] with further developments in [53,54,20,55,56].
coefficient k ∼ C 1 is then proportional to the R-current gravitational anomaly.It was computed by considering the example of a 4d Weyl fermion leading to where Tr(R) is the sum of r-charges.In general, the N = 1 superconformal symmetry relates the gravitational R-current anomaly to the trace anomaly coefficients as [57] ∇ (2.12) Here we use the somewhat loose notation Tr(R) and Tr(R 3 ) for the gravitational and gauge anomaly coefficients: they are literally the sum of r-charges and their cubes only in the case of the standard chiral fermions but in general contain also field-dependent coefficients, i.e.
Constant term ∼ β 0 : Motivated by the study of explicit examples of standard chiral and vector multiplets, the constant C 2 in (2.6) was conjectured in [20] to be equal to the logarithm of the supersymmetric partition function of the dimensionally reduced theory on S 3 7 Logarithmic term ∼ log β: The coefficient C 3 of the logarithmic term in (2.6) was conjectured to be [20] again motivated by the examples of chiral and vector multiplets. 8  Linear term ∼ β: As already mentioned, the coefficient of the linear term in (2.7) is the "supersymmetric" analog of Casimir energy. 9The relation (2.4) may be studied in the low temperature β → ∞ 7 To be precise, the Ansatz in [20] is slightly different by a multiple of log(2π) because the logarithmic term in (2.6) or (2.7) is written as log β 2π .We claim that the identification (2.14) is correct if the logarithmic term is simply log β, see Appendix D. 8 Let us note that the combination 2 a − c plays a special role in the analysis of the N = 2 models in [58,59] where it is essentially the sum of dimensions of operators parametrizing the Coulomb branch. 9The effective Hamiltonian appearing in (2.3) is Hsusy = ∆ − 1 2 R. Normal ordering :Hsusy: = Hsusy − Hsusy = Hsusy − Esusy is implicitly understood, and is essentially the reason why Z susy is not equal to the index in (2.4) (see [29] for details).This is a general feature of the relation between the QFT partition function and the thermodynamic partition function Tr(e −βH ) [60].
or high temperature β → 0 limit.For β → ∞ the leading contribution comes only from the energy exponential, i.e.

Z susy β→∞
∼ e −β Esusy , where the asymptotics of the index corresponds to the vacuum contribution.Equivalently, E susy = − lim β→∞ d dβ log Z susy .On the other hand, in the β → 0 limit the partition function (and the index) is governed by the high-energy part of the spectrum.This leads to the singular Cardy-type term in (2.6). 10 The relation (2.5) of E susy to the conformal anomaly coefficients was demonstrated explicitly in the case of the chiral and vector multiplets [29,34].In addition, a general derivation was proposed in [61] based on a representation of E susy in terms of the anomaly polynomial and assuming the standard relations (2.11) between the anomaly coefficients.
Higher order terms ∼ β n : It was claimed in [20] that the O(β 2 ) corrections are exponentially suppressed, i.e. schematically go as e −1/β for β → 0. Let us recall, as an analogy, that absence of β n (n ≥ 2) corrections for in the case of the logarithm of the thermodynamic partition function Z(β) was observed in the past for the standard conformal fields [62] where this follows from a simple modular transformation property of Z(β).It is unclear if similar modular properties play a role in the supersymmetric context.
The above discussion leaves several open questions.One is whether the prediction (2.9) and thus (2.8) for the leading-order coefficient C 1 is completely universal, i.e. holds for general N = 1 superconformal theories, including also non-unitary and higher spin multiplets.The argument in [19] relying on Chern-Simons term in the reduced 3d effective theory appears to be specific to case of standard Weyl fermions.As we shall see below, the relation (2.9) indeed requires a generalization in the non-unitary case.
The second question is about the expression (2.15) for coefficient C 3 of the log β term which was checked only for standard multiplets.We shall find that (2.15) needs a modification in the case of higher spin superconformal multiplets.We shall propose an alternative universal expression for the log β term in terms of the integer numbers of conformal Killing tensors associated with each conformal gauge field in the multiplet.
Finally, it is not clear a priori if the non-unitary multiplets will also have exponentially decaying corrections in their index expansion, i.e. if all power β 2 , β 3 , ... corrections in (2.7) will be absent.Indeed, we shall find that such power corrections will survive for multiplets containing higher spin (s > 1) conformal spins.
To address these questions, below we shall consider several non-unitary (higher derivative and higher spin) superconformal multiplets that appear in the context of extended conformal supergravities.For each of these free multiplets we will explicitly compute the index (2.3),obtain the coefficients in its small β expansion and compare them with their expected values based on relations found earlier in the studies of unitary multiplets.
3 Higher derivative N = 1 superconformal multiplets Here we will describe the content of the four basic higher-derivative N = 1 superconformal multiplets for which we will later compute the index (2.3).They naturally appear in the decomposition of N ≤ 4 conformal supergravities in terms of N = 1 multiplets.
Here we indicated how the SU (4) representations of each field in N = 4 theory is decomposed, resulting N = 1 R-charges and the N = 1 multiplets each field belongs to.For example, the 4 gravitinos of the N = 4 supergravity split into the one belonging to [2] and having r = 1, and three from the three [ 3  2 ] multiplets with r = − 1 3 .The resulting R-charge values are in agreement with the representation theory of N = 1 superconformal algebra discussed below.
For future reference in Table 1 we list the conformal anomaly coefficients of the individual fields (see [63,64,4]).In Table 2 we present the resulting values of a and c for the basic unitary and non-unitary multiplets introduced above.We also give particular combinations of a and c corresponding to the expected values of (i) the coefficients Tr(R) and Tr(R 3 ) of the gravitational and R-symmetry chiral anomalies computed according to (2.12) (ii) E susy as given by (2.5), and (iii) the coefficient C 3 as defined in (2.15).It is interesting to observe [4] that the values for the [1] and [0 ] multiplets are exactly opposite to each other.Note also that the combinations 3 Tr(R) and C 3 are always integer.

Structure of the N = 1 multiplets
Let us now discuss the N = 1 superfield description of the above multiplets that allows one to independently fix the R-charges of the individual fields which will be in agreement the R-charge assignment in (3.3) following from the N = 4 conformal supergravity.
As a consistency check, we will then be able to show that the resulting chiral anomaly coefficients Tr(R) and Tr(R 3 ) computed directly from (2.13) will be in agreement with the values in Table 2 found assuming the supersymmetry-implied relations (2.12).

Unitary scalar [0] and vector [1] multiplets
The [0] chiral multiplet containing one complex scalar and one Weyl fermion corresponds to a chiral superfield Φ = φ + θψ + θ 2 ϕ.Omitting the auxiliary field ϕ and using that θ α has R-charge 1 we then find the following dimensions ∆ and R-charges The N = 1 superconformal algebra requires that for the superconformal primary or the lowest chiral superfield component (here the scalar field) one should have r = 2 3 ∆ (see, e.g., [70]).This fixes r = 2 3 , consistently with the SU (4) decomposition in (3.3).The resulting chiral anomaly coefficients in (2.13),(3.4) are then in agreement with the values in Table 2. Similar agreement will be found for all other multiplets discussed below.The vector multiplet [1] is related to the chiral spinor field strength superfield W α = ψ α + θ β F + αβ + ... so that we find The symmetric tensors F + αβ and F − α β are the (anti) self-dual parts of the Maxwell field strength in spinor notation.Here the lowest component is ψ α so that r = 2 3 ∆ ψ = 1.This gives, in agreement with Table 2,

Higher derivative scalar multiplet [0 ]
A general discussion of N = 1 non-unitary multiplets can be found in [71] (at the level of states), and in [72] (at the level of fields).We need to embed the [0 ] multiplet in (3.2) into a chiral superfield with the superconformal primary being the 4-derivative scalar φ (4) of dimension 0. Other fields are then obtained by applications of the supersymmetry generator This multiplet may be viewed as resulting from the application of extra to the standard chiral multiplet, i.e. (φ, ψ, ϕ) → (φ (4) , ψ (3) , φ), where, in particular, the auxiliary field ϕ becomes a (complex) dynamical scalar.As this multiplet is non-unitary, it is not a priori obvious how to fix the value of r.Nevertheless, from the analysis of [72] (see also [73]) the vanishing scaling dimension ∆ = 0 should imply r = 0, i.e. vanishing chiral weight.In practice, this is still consistent with the rule r = 2 3 ∆ φ (4) = 0. From the point of view of N = 4 supergravity (cf.(3.3)) this is also consistent with the higher derivative scalar being an SU (4) singlet.The direct evaluation of Tr(R) and Tr(R 3 ) then gives in agreement with the values in Table 2.

Tensor multiplet [ 1 2 ]
The superfield embedding of this multiplet can be found, e.g., in [74]. 11The field content in (3.2) may be organized into a chiral superfield, see Appendix A. Its lowest component is ψ α and other components are built by acting with Q α decreasing r by one (or, in conjugate case, with Q α increasing r by one).As a result, we find The R-charge is again determined by r = 2 3 ∆ ψ (3) = 1 3 .The direct computation of the chiral anomaly coefficients Tr(R) and Tr(R 3 ) based on (2.13),(3.4)involves summing up contributions from all the fields but the scalars (3.12)

Conformal graviton multiplet [2]
The superfield description of the linearized N = 1 conformal supergravity multiplet [2] was discussed in [2,3].The corresponding real superfield starting with graviton contains (in a Wess-Zumino gauge) the components corresponding to the fields in (3.2) The expected value of the graviton R-charge is r = 0. To confirm this one may consider the corresponding chiral field strength superfield [77,78] (cf.(3.19)) where Ψ αβγ is the gravitino field strength and Φ αβγ is the "second" gravitino field strength (self-dual part of the strength of so that r Ψ = 1 and thus in (3.21) we should have r = 0 (equivalently, this follows from the fact that Weyl tensor The chiral anomaly coefficients Tr(R) and Tr(R 3 ) here receive contributions only from the Weyl gravitino (see (2.13),(3.4)) These values, like those in (3.12) and (3.20), are once again in agreement with the corresponding values in Table 2 demonstrating consistency with the supersymmetry which underlies the relations (2.12).

The superconformal index of N = 1 multiplets
The explicit evaluation of the index (2.1) in a free superconformal theory can be done in terms of the plethystic exponential where the single-particle index i(p, q) can be computed by letter counting [15][16][17].We shall consider the special case of p = q = e −β ≡ t corresponding to (2.3) and use the notation, cf.
Below we will first compute the single-particle index i(β) for the multiplets introduced in the previous section and then discuss the β → 0 expansion.
12 Eq.(4.1) is a standard way to build symmetric multi-particle states in terms of the single-particle states.This is made explicit by the illustrative relation Table 3. Contributions to the index of the [0 ] multiplet.j 1 and j 2 (which can take positive and negative values) stand for third components of the two SU (2) spins which label the states in the superconformal index.Each undotted (dotted) ± index contributes ± 1 2 to j 1 (j 2 ).

Computing the single-particle index
For the familiar unitary multiplets [0] and [1] one finds [70] i For the [0 ] multiplet in (3.9) the analysis goes as follows.One has to consider the fields X ("letters") with Applying derivatives ∂ α α one builds new letters.In the following it will be convenient to denote the spinor indices α = 1, 2 by ±: 1 → + and 2 → − (and similar for dotted indices).The +/− notation is convenient because each type of index (dotted or undotted) with such a value increases/decreases the third component of the associated Lorentz spin by 1 2 .The derivatives ∂ ± + do not change δ.They can be applied repeatedly leading to a universal factor 1/(1 − t) 2 in the single-particle index i.Instead, ∂ ± − increase δ by two units.Any derivative ∂ α α does not change r so, on the δ = 0 states, it increases 2j 2 + r δ=0 = ∆ − r 2 by one unit.This gives the set of contributions in Table 3.The descendants (obtained by the application of ∂ ± + leaving δ invariant) that have the form of equations of motion are This gives the index (cf.(4.3)) A similar analysis can be carried out for the tensor multiplet [ 1  2 ] in (3.11).In this case, the list of non-zero contributions is collected in Table 4.The contribution of the last line of this table should not be included: due to the spinor equations of motion, this derivative Table 4. Contributions to the index of the [ 1  2 ] multiplet.The components of the (anti) self-dual tensor are symmetric in the (dotted) undotted indices.may be replaced by another one with δ = 0.This gives the contributions −2 t from the "left" and "right" chiral fields.Next, we have to take into account the equations of motion for the (j 1 , j 2 ) = (1, 0) and (j 1 , j 2 ) = (0, 1) chiral components.One can check that there are no contributions with δ = 0 and thus To find the index for the gravitino multiplet [ 3  2 ] in (3.18) we should take into account that relevant letters should be gauge invariant, i.e. use the gravitino field strength in (3.19) and also the "second" gravitino field strength Φ αβγ (cf.(3.22)).The latter obeys the Bianchi identity and thus its dimension is 5 2 while the R-charge is opposite to that of Ψ αβγ .The non-zero contributions are collected in Table 5.The Bianchi identity for the (complex) Maxwell field strength lead to additional vector contribution −2 t .Finally, we should account for the equations of motion of T + αβ , i.e. ∂ α α∂ β β T +αβ = 0 getting another −t 2 3 +2 contribution.As a result, the final expression for the index is For the graviton multiplet [2] in (3.21) the computation of the index should be again done in terms of the gauge invariant field strengths appearing in (3.22).The resulting non-zero contributions are collected in Table 6.Before taking into account Bianchi identities, the index is simply −4t (1−t) 2 .The Bianchi identities may only contribute a term proportional to (± 1 2 , 1) Table 5. Contributions to the index of the 12) The summary of the computed indices is presented in Table 7. 14

Small β expansion of the index I(β)
Let us now use the above results for i(β) to compute the small β expansion of the superconformal index I(β) in order to compare with the expected expansion (2.7).A generalization 13 This extra +4 t 3 comes from the field in the 4th line of the Table 6 (which enters a Bianchi identity contributing +2t 3 ) and from which is another Bianchi identity similar to (4.9) conserving δ = 0 when α = β = − and γ = ± is arbitrary). 14The supersymmetric index is the n = 1 case of similar indices for theories on the lens space S 1 β × S 3 /Zn that have been computed in [32] for unitary theories.In the n > 1 cases the index receives contributions from twisted sectors.It would be interesting to extend the analysis to the non-unitary multiplets considered here and explore the n → ∞ limit when S 3 /Zn → S 2 and the index reduces to that of a 3d theory.
to a 2-parameter family of unequal p and q in (2.1) will be discussed in Appendix E.
The usual approach to derivation of the small β expansion of the index for models involving unitary multiplets starts from the summation in (4.1) in terms of elliptic Γ function.Modular properties of the resulting expressions [79] are then exploited to discuss the small β limit.Here we propose a simpler approach based on the techniques developed for studying similar limit of standard partition functions using that I(β) has a formal structure of a partition function.Let m be a label a particular multiplet and let us define the Mellin transform of the single-particle index i(β) as Then for I(β) in (4.2) one finds 15   log which is valid when v is sufficiently large.When the vertical contour in (4.14) is moved to the left, we pick up residues of poles at integer u and this has the form of a small β expansion with the coefficients involving the residues of z m (u). 16In general, given a term t q (1−t) 2 in the index, we may use that and taking residues in (4.14) we immediately obtain the expansion of the index.
Let us apply this method to the known cases of the chiral and vector multiplets.For the [0] chiral multiplet we have from (4.3) 15 Eq.(4.14) follows from the Mellin inversion formula and may be checked for a typical single-particle contribution to the single-particle index starting from the relation e −β = 1 2πi v+i∞ v−i∞ du β −u Γ(u) valid for v > 0 (see, e.g., [80]).Here ζ(u + 1) is the Riemann zeta function.Note that the relations below apply to both fermions and bosons (as fermions are treated as periodic on the circle). 16In some cases symmetry properties of the integrand allow one to relate the contour associated with −v to that at +v.Then the remainder of the small β (high temperature) pole expansion can be found explicitly and gives rise to a "temperature inversion" relation [80].
Multiplying this by Γ(u)ζ(u+1)β −u (to get the integrand in (4.14)) and taking the residues of the poles, we get contributions to (4.14) coming from u = −1, 0, 1 only.As a result, where Taking residues in (4.14), we obtain Applying the same method also to the four non-unitary multiplets with single-particle indices given in Table 7 the final results can be summarized as follows 18   [0] = (2 φ, ψ) : log 17 According to [81] log Z susy ).The relation of the index to 3d partition function in the β → 0 limit after the removal of singular terms is a non-trivial fact depending on regularization, see [82][83][84][85][86]49].For a discussion of this relation in the case of non-supersymmetric conformal partition functions see Appendix D. 18 Let us note that for the [0 ] multiplet one has to be careful with the contribution of the higher derivative scalar φ (4) .This field has canonical dimension 0 (like a scalar in 2d) and one finds terms of the form ∞ n=0 n −u .The n = 0 term is ambiguous and we used the natural analytical continuation 0 −u ≡ 0 for all (complex) u.
[ 1  2 ] = (ψ, 2ϕ, T µν , ψ (3) ) : log where These constant terms will be further discussed in Appendix D.2. 19   5 Structure of small β expansion of the index of non-unitary multiplets Let us now compare the explicit values of the coefficients appearing in the small β expansion (2.6) of the indices in (4.21) with their expected values discussed in Section 2, i.e. with the previously suggested relations (2.9),(2.14),(2.5). 20We shall denote the true values of the coefficients as C i with C i being the expected values:  2, we find agreement for the [0], [1], [0 ] multiplets but discrepancies for the non-unitary multiplets [ 1  2 ], [ 3  2 ], [2] containing the antisymmetric tensor or conformal gravitino: ) 19 Let us note that part of the expansions in (4.21) can be found by a naive procedure of first expanding the single-particle index and then applying (4.1) term by term.In general, i(β) = is indeed the coefficient of the leading term in (4.21) in all cases.The same agreement is found for the linear in β term.The term proportional to A0 is ill-defined but a heuristic replacement rule ζ(1) → − log β reproduces indeed the log β term in (4.21).However, all other subleading corrections are not captured correctly by this procedure. 20Here we will not attempt to compare the expressions in (4.22) with their expected (2.14) values log Z susy since to compute the latter requires first the construction of the explicit supersymmetric Lagrangians for the non-unitary multiplets on S 1 × S 3 that should contain extra couplings to the R-symmetry gauge field background.Nevertheless, we remark that k[ ), where (R) was defined in footnote 17. (5.3) Remarkably, the correction terms ν m are all integer multiples of 8.For the collection of multiplets appearing (3.1) in the N -extended conformal supergravities we then get ν N =1 = 24, ν N =2 = 8, ν N =3 = ν N =4 = 0 .The N = 3 and N = 4 conformal supergravities also have a = c [4] or tr R = 0 and thus C 1 = C 1 = 0.The same result is found also for N = 4 vector multiplet or [1] + 3[0], i.e.
The reason why the relation (2.9) between the 1/β term and Tr(R) suggested in [19] fails to be universal may be due to the fact that the argument in [19] may not directly apply to theories containing more complicated chiral fields (self-dual tensors, conformal gravitions, etc.) rather than just the standard Weyl fermions. 21One possibility is that in reconstructing 3d effective action by matching anomalies there is an integer-shift ambiguity in the coefficient of the 3d Chern-Simons term used in [19] leading in general to the presence of the correction term ν in (5.2).Note that a shift of C 1 from its value C 1 in (2.8) was also discussed for non-abelian gauge theories in [21].
Let us note also that the presence of the correction ν is similar to what happens in non-unitary 2d CFT.In a generic CFT the partition function Z(β) is related to the density of states ρ(E) dE.Writing the energies in terms of the conformal dimensions and the 2d central charge, E = 2π (∆ + ∆ − 1 12 c), the modular invariance Z(β) = Z(β −1 ) implies that the limit β → 0 is related to the β → ∞ one in which Z is dominated by the lowest-energy states Z(β → ∞) ∼ A β −λ e −β E min .Here λ is non-zero for gapless systems so that [89,90] where ∆ min ≡ 1 2 min(∆ + ∆).In unitary theories one has ∆ min = 0 and c eff = c.Instead, in non-unitary theories, ∆ min is typically negative and c eff > c.This inequality may be violated in the case of supersymmetric partition function where fermions are taken to be periodic and contribute with a negative sign.In non-unitary case one has generically [90] c eff (B) − c eff (F) = c(B) − c(F).The correction ∆ min in c eff = c − 24 ∆ min is analogous to the parameter ν in (5.2).There are, however, important differences: modular invariance is not available in general and the role of the 2d central charge is played by c − a (cf.(2.8)).Nevertheless, it is tempting to relate the presence of the non-zero ν in (5.2) with the existence of negative norm states in the case of non-unitary multiplets.

Logarithmic term ∼ log β:
The comparison between the coefficients of log β in (4.21) and the suggested values C 3 = −4 (2a−c) in (2.15) [20] given in Table 2 again implies the presence of an integer correction: ) (5.7) (5.8) Considering the collections of multiplets appearing in N -extended conformal supergravities (3.1) one finds (5.9) Instead of trying to understand why C 3 is, in general, different from C 3 = −4 (2a − c) proposed in [20] let us suggest an alternative general expression for it.Let us start by noting that the logarithmic term in the expansion of the index is the same as in the expansion of the supersymmetric partition function (2.6) and thus may have a universal origin.One may attempt to interpret the singular log β term appearing in the β → 0 limit as associated with the KK modes that become "massless" in the limit of shrinking S 1 .In practice, this relation is not straightforward and depends on regularization, see also Appendix D. 22  One may then expect that the coefficient of the log β term should be the same as in the standard partition function for the conformal gauge fields on S 1 β × S 3 (with both bosons and fermions taken to be periodic on the circle).Then the log β term should receive contributions only from the conformal gauge fields in each multiplet.These can be found from the conformal higher spin partition functions derived in [5] and reviewed in Appendix C below.
The analysis in Appendix C shows that the log β contribution comes from a specific SO(4, 2) conformal character and is determined by a particular integer equal up to sign to the number n CKT of conformal Killing tensors for the bosons and the number n CKS of conformal Killing spinor-tensors for the fermions. 23 We propose that the coefficients C 3 of the log β term in the expansion of the supersymmetric partition function on S 1 β × S 3 for a generic superconformal multiplet should be given by the sum of the contributions from the conformal higher spin gauge fields in this multiplet, i.e. (5.10) In the case of the multiplets discussed in this paper the relevant conformal gauge fields are the standard vector V µ (s = 1), the conformal graviton h µν (s = 2) and the conformal gravitino Ψ µ (s = 1) for which we find from (C.4),(C.6): (5.11) 22 Let us note also that the log β term in the case of unitary non-abelian gauge theories was discussed in [21] where its coefficient was related to the dimension of the space of flat directions (with no curvature coupling) in the 3d theory. 23Note that the UV finite partition function on S 1 β × S 3 where S 3 has radius R (set to 1 in the above discussion) contains the log β term as part of the dimensionless log(β/R) term.This suggests (see also Appendix D) that may be the dependence on R coming from some zero modes is determined by n CKT .The total power of R depends on a regularization of the contribution of all other modes (for a related discussion on S 3 see [91]).

Linear term ∼ β:
For this term there is full agreement between the expected values of the supersymmetric energy (2.5) in Table 2 and the coefficients in (4.21).

Higher order corrections:
Higher order corrections in the small β expansion of log I(β) can be found by taking residues in (4.A similar pattern is found for the non-supersymmetric partition function of the conformal higher spin fields.For instance, in the case of a spin s bosonic conformal field the correction to (C.3) can be written as where R s (β) is a infinite series that can be found in a closed form .16)This vanishes for the Maxwell field (s = 1), while for s > 1 one finds To summarize, the above analysis of non-unitary N = 1 multiplets shows that in general (cf.(2.7)) Here ν is a integer multiple of 8 which is non-zero for non-unitary multiplets with higher spin fields and n = C 3 given by (5.10) is another integer which is non-zero only for multiplets with gauge fields.k ≡ C 2 is a constant that should be related (2.14) to the partition function on S 3 and R(β) contains power-like corrections for multiplets with conformal higher spin gauge fields but is O(β k e −1/β ) otherwise.
It is of interest to consider special combinations of the basic N = 1 multiplets [0], [1] and [2], [ 3  2 ], [ 1  2 ], [0 ] that have vanishing leading β −1 and β coefficients in the expansion of the index (5.18),i.e. have vanishing total a, c and ν coefficients Here k and k are integers and we assumed that there is just one graviton multiplet.As k > −1 (for the number of [0] not to be negative) the simplest solutions are k = 0 and k = 11, 12, ... for which there are 4 conformal gravitini as in N = 4 conformal supergravity.Using (5.13) we then find that the coefficient n of the log β term in (5.18 , corresponds to the familiar case of the N = 4 conformal supergravity coupled to four copies of which is a superconformal theory not only at the quadratic but also the interacting level [92,4].The small β expansion of the superconformal index of this theory (which does not depend on the conformal supergravity and the SYM coupling constants) is given by where the infinite series of power corrections come only from the N = 4 conformal supergravity contribution (cf.(5.14)).
The minimal solution for the superconformal theory (5.19) having k = 0, k = 11 has a smaller field content (it corresponds to removing the pair [0 ] + [1] that has zero anomalies, see Table 2): It has similar expansion of the index: log I(β)= − log β + k + ....This combination cannot be written as a collection of N = 3 or N = 2 multiplets; it was not included in classification of finite theories in section 6.3 of [4] as having separate [ 3  2 ] multiplets that are not part of an extended conformal supergravity theory is not expected to lead to a classically consistent theory at a non-linear level.
A Free action for the [ 1  2 ] multiplet Let us first recall the action for the standard chiral multiplet described by a chiral superfield Φ = φ(y) + θψ(y) + θ 2 ϕ(y) (with If instead one starts with a chiral spinor superfield where χ α is a spinor with dimension 1 2 , ψ α is a standard dimension 3 2 spinor and the boson Q αβ is a combination of symmetric tensor T αβ (corresponding to self-dual part of the antisymmetric tensor T µν ) and a complex scalar φ.The corresponding conformally invariant action is then In components this gives where the bosonic term may be written explicitly as Eqs. (A.4),(A.5)give the action for the superconformal [ 1  2 ] multiplet in (3.2) (after the renaming χ → ψ (3) ).The action for the antisymmetric tensor in (A.5) in spinor notation is equivalent to ∂ µ T + µν ∂ λ T −λν in vector notation with T + → T αβ ∼ (1, 0), T − → T α β ∼ (0, 1).

B Chiral anomalies of conformal gravitino and non-gauge tensor field
To compute anomalies of a higher spin field one needs to couple it to a gravitational and gauge fields (assuming certain chiral transformation properties), take into account contribution of ghosts, etc.One may use, e.g., a perturbative approach, computing triangle diagrams corresponding to the matrix element of the chiral current between the vacuum and a two-graviton state, or two chiral symmetry gauge fields.An alternative topological approach is based on relating the anomaly of the chiral current to the Atiyah-Singer index of a certain elliptic operator mapping fields to fields of opposite chirality.This approach is somewhat heuristic and is practically useful only if the starting (higher-spin) field theory is consistent.For a detailed comparison of the perturbative and topological methods for standard gravitino see [69].
Here we shall discuss the chiral gravitational and gauge anomalies of the conformal gravitino and the antisymmetric non-gauge tensor field T µν justifying the values of their coefficients κ 1 and κ 2 given in (3.4). 24Their embedding into conformal supergravity means that it is possible to consistently couple them to gravity and the chiral gauge field.Using topological approach we shall assume the existence of a suitable elliptic operator whose index computes the chiral anomaly.Earlier results for the chiral anomaly coefficients in (3.4) can be found in [69,67,66,94].
Following [95], let us consider a a compact 4-manifold M 4 and a field belonging to the general spinor bundle X mn ≡ X m,n , i.e. a tensor X (α 1 ...αm)( β1 ... βn) with m symmetric spinor indices and n symmetric dotted spinor indices.In particular, the gravitino Ψ µ corresponds to X 2,1 +X 1,2 (modulo gauge symmetry) while the tensor T µν to X 2,0 +X 0,2 (i.e. to the sum of the self-dual and antiselfdual parts in spinor notation).The index theorem computes the analytical index of the universal chirality-swapping elliptic operator D mn : X mn → X nm in terms of topological quantities.In 4 dimensions, it reads where T M is the tangent bundle, and td, e, ch are the Todd class, the Euler class and the Chern character.It is understood that one has to extract the part of degree 4 and evaluate it on M 4 .If the fields transform in a non-trivial U (1) gauge bundle V , we have to multiply the index by the Chern character ch(V ) = 1 − c 2 + . . . .The degree 4 terms give the gravitational and pure gauge contributions to the divergence of the corresponding chiral current (see, e.g., [96,95,97,98]) where c 2 = − 1 8π 2 Tr(F ∧F ) is the second Chern class and p 1 = 1 8π 2 tr(R∧R) is the Pontryagin class.For a Weyl fermion we get (cf.(2.10),(2.13),(3.4)) For the conformal gravitino (taking into account the ghost subtraction [67]) and the selfdual 2-tensor or symmetric bispinor, we get Thus in units of the Weyl fermion anomaly the chiral gravitational anomaly of the conformal gravitino is 5 6 : (− 1 24 ) = −20 while its gauge anomaly is 4.25 For the self-dual antisymmetric tensor T αβ the chiral gravitation anomaly is (−1) 1  3 : (− 1 24 ) = 8 while its gauge anomaly is −4 (where we have included a −1 factor due to the different statistics of the tensor with respect to the Weyl fermion).These values are in agreement with those given in (3.4).
The reason why the real T µν tensor contributes [66] to the chiral anomaly can be understood from the analogy of its kinetic operator in spinor basis in (A.5) with the (square of) Dirac operator for a Weyl spinor.Its chiral gravitational anomaly can be also obtained by adapting the analysis of [99], i.e. by observing that the antisymmetric tensor anomaly related to a chiral rotation between the self-dual and anti-selfdual parts is the same as the electromagnetic duality anomaly of a Maxwell field (cf.also [100,68]).

C Conformal higher spin partition function on S 1
β × S 3 Here we shall review the expression for the conformal higher spin partition function Z(β) on S 1 β × S 3 and consider the small β expansion of log Z(β) focussing on the interpretation of the coefficient of the log β term.
Let us start with the bosonic fields.The Maxwell vector and the conformal graviton are simplest cases of the 4d conformal higher spin fields with s kinetic term for a spin s field [4].The standard partition function (or character of the corresponding representation of the conformal group) for the conformal higher spin s field on S 1 β × S 3 is given by [5] log Here Z s is the single-particle partition function playing the role as the single-particle index in (4.1).
Using the method sketched in (4.14)-(4.17)one finds that the small β expansion of log Z s has the form 26log Z s = π 4 s (s + 1) 45 Here the coefficient of the log β is simply minus the number n CKT of conformal conformal Killing tensors in 4 dimensions. 27E cas is the standard Casimir energy.The constant k s in (C.3) has a non-polynomial dependence on s and may be expressed as a linear combination of transcendental constants. 28 dimension.The reason why the first term in the corresponding expansion of the supersymmetric partition function in (2.6) has "softer" β −1 behaviour is due to supersymmetric cancellations. 27For a spin s field it is the dimension of the SO(4, 2) representation (s − 1, s − 1, 0) labelled by the Young tableau that has two rows of length s − 1. 28 It can be written as Similarly, for the fermionic conformal fields with spin s = s + 1 2 (with s = 0 corresponding to Weyl spinor, s = 1 to conformal gravitino, etc.) one finds [101] Z s (β) = 4 (s + 1) t The β → 0 expansion of the (periodic) fermionic partition function is given by Here n CKS is the number of the conformal Killing spinor-tensors.As in the bosonic case, the constant k s has a non-polynomial dependence on s.
To further investigate why the coefficient of log β is related to the integers n CKT and n CKS let us write a general single particle partition functions associated with a conformal field in 4d as where q runs over some finite set of integers or half-integers.This covers the cases of the bosonic and fermionic conformal higher spin fields in (C.2),(C.5)as well as other "matter" conformal fields in section 3.1 above [5] For a general P (t) one finds using (C.1) where Bendersky's generalised gamma function logΓ k (n) = n−1 j=1 j k log j comes from simplification of derivatives of Hurwitz zeta function by logΓ k 29 This expression is formally valid for for all fields as we assume that the fermions are also taken to be periodic on the "thermal" cycle S 1 β .
The transcendental constant k P is determined by the form of P (t), but cannot be expressed as a linear combination of its derivatives at t = 1.For a theory on S 1 β × S 3 , the β → 0 or t → 1 singularity of Z(β) is ∼ 1/(1 − t) 3 , i.e.P (t) should have an explicit 1 − t factor. 30 This implies the constraint p 0 = P (1) = 0. Besides, for all conformal fields we get one extra constraint on P (t) 31  P (1) − 3 P (1) = 0 , (C.10) that implies the vanishing of the coefficient of the 1/β 2 term in (C.9).Assuming these constraints, the expansion (C.9) simplifies to This has similar structure as the expansion of the superconformal index in (2.7) (apart from the leading 1/β 3 term that cancels in a supersymmetric combinations of fields).
The coefficient of the log β term is thus related to a particular combination p 4 −2p 3 +6p 1 of derivatives of P (t) at t = 1.To understand the meaning of this combination we may use the relation between the 4d conformal partition function and its AdS 5 counterpart. 32As discussed in [5] there exists a close relation between the single-particle partition function Z of a conformal field on S 1 β × S 3 and the partition function Z HS of the associated higher spin field in AdS 5 with quantum numbers determined by the Lorentz spins and conformal dimension of the 4d conformal field: 33 (C.12) Here the function σ(t) is a finite polynomial in t + t −1 and is generically present in the case of 4d conformal higher spin fields related to massless higher spin fields in AdS 5 .It is given by the character of the finite dimensional irreducible representation of SO(4, 2) corresponding, in bosonic case, to the conformal Killing tensors in 4 dimensions.Its value at t = 1 gives the dimension of this representation, i.e. the total number of conformal Killing tensors σ(1) = n CKT . 34Using the conformal group representation theory and (C.12) one can show that 30 This follows both from the conformal group representation theory and from the simple remark that 1/(1 − t) 3 factor takes into account the contributions of states on S 3 corresponding to all spatial derivatives of the field. 31This relation was discussed in [102] where it was related to the absence of suitable counterterms in the heat kernel calculation of the "energy" E(β) = −∂ β log Z(β). 32Let us note that use of the AdS connection is useful but is not really necessary for the final conclusion as the form of Z(β) used below can be justified purely on the basis of the conformal group representation theory discussed in Appendix F in [5]. 33Changing notation slightly here instead of β we use t = e −β as the argument of the partition functions. 34The general expression for σ(t) for a bosonic conformal spin s field is where Q(t) is a smooth function.The conditions P (1) = 0 and (C.10) can be checked to hold automatically and the coefficient of the log β term in (C.11) then reads as Thus the coefficient of the logarithmic term is simply minus the number of the conformal Killing tensors of a rank related to the spin of the conformal gauge field.Similar result is found for the fermionic conformal higher spin fields.
For non-gauge conformal fields (like the scalars φ, φ (4) , spinors ψ, ψ (3) , and the tensor T µν ) one finds that σ(t) = 0 and thus there is no log β term in the small β expansion of the corresponding log Z(β).

D Expansion of partition function in terms of regularized theory on S 3
It is possible to compute the constant and logarithmic contributions to the small β expansion of the partition function directly in terms of the spectrum of the dimensionally reduced 3d theory.Below we shall explain this starting with the example of the standard (nonsupersymmetric) partition function on S 1 β × S 3 .

D.1 Standard bosonic partition function
In general, for a free conformal field on S 1 β × S 3 , we can write the single particle partition function and the full partition function in terms of the (square roots of) eigenvalues λ n and their multiplicities d n of the corresponding Laplacian on S 3 (equal to energies of states or dimensions of CFT operators) Thus the coefficient of log β should be directly related to the (regularized) sum of the multiplicities, while the constant term k should be the partition function of the reduced 3d theory on S 3 .Note that as in a conformal theory Z depends only on dimensionless ratio β/R where R is the radius of S 3 (the square roots of eigenvalues λ n scale as R −1 ) the dependence on R is also controlled by the regularized total number of the eigenvalues or n. 36  The natural regularization is the spectral ζ-function one: if ζ ∆ (z) = n d n λ −z n then n = ζ ∆ (0) and k = ζ ∆ (0).This analytic regularization can be implemented simply by adding the factors e −ελn , doing the sums and then dropping all terms which are singular in the limit ε → 0.
As an example, let us consider the partition function of a conformally coupled scalar on S 1 β × S 3 where (t ≡ e −β ) Using the method of section 4.2 one can show that the exact small β expansion of log Z 0 is The same results of the coefficient n = 0 of log β term and the constant term are indeed found by using directly the corresponding regularized expressions in (D.3) (here Thus the constant in (D.5) may be identified with the partition function of the dimensionally reduced scalar 3d theory on S 3 computed using natural analytic regularization. 37imilar computation can be done for the Maxwell vector field where n+1) ).(D.8) Here λ n = n + 1 (n = 1, 2, ....) is the square root of the eigenvalue of the transverse 3-vector Laplacian on S 3 and 2n(n + 2) is its degeneracy [5].The exact small β expansion of Z 1 computed as in section 4.2 reads Using instead the direct expansion and (D.3) we find gives in agreement with the coefficients of the log β and the constant term in (D.9).Another non-trivial example is that of the conformal graviton for which [5] 38 in agreement with (5.11).Similar computations can be done also for other conformal fields appearing in the N = 1 multiplets discussed in the text, confirming that n = 0 for non-gauge fields and is always an integer (see (5.10),(C.4),(C.6),(C.14))for the gauge fields.This fact suggests that it should have some "zero-mode" interpretation which remains to be clarified (cf.[91]).

D.2 Supersymmetric case
The supersymmetric partition function is the same as the superconformal index up to the normal ordering supersymmetric Casimir energy factor in (2.4).This means that we may use the expansion of the index to extract the analogs of d n and λ n in (D.1).These will have again the meaning of multiplicities and eigenvalues of the single particle (free) supersymmetric spectrum.
Application of (D.3) is expected to give the constant and logarithmic terms in the expansion of the index.Let us check this claim for few examples of N = 1 multiplets using the expressions in Table 7.For the chiral multiplet, we have i Hence, from (D.3), 3 ) (n + 1) ) 38 Here to determine the effective λn = n + 2, dn = 2 (3 n 2 + 12n + 5) we re-expanded the final expression for the single-particle partition function of the conformal graviton on S 1 × S 3 in Eq. (3.22) of [5] that was obtained by combining the contributions of the transverse graviton and vector Laplacians on S 3 .In the notation of [5], these are respectively Z2,0 For the vector multiplet i and applying again (D.3) we find in agreement with (4.21).
In the case of the graviton multiplet [2] we have i and then the values of n [2] , k [2] are, again, in agreement with (4.21) Similar agreement with values in (4.21) is found also for other multiplets.
E Superconformal index corresponding to N = 1 multiplets on squashed S 3 In this Appendix we shall consider the generalized 2-parameter superconformal index (2.1),(4.1)which happens to be related [48,14,49] to the supersymmetric partition on S 1 β × S 3 b where b is the squashing parameter of the 3-sphere.The corresponding choice of the fugacities generalizing The small β expansion of the corresponding index for the scalar chiral multiplet [0] was found in [20] log The supersymmetric Casimir energy, entering the general relation (2.4), is expected to have the general form [29,34] that reduces to (2.5) for b → 1.We can easily obtain the expansion (E.2) by the ζ-function methods described in Section 4.2.Let us first recall that for general p, q the single-particle superconformal index in (4.1) is [70] (reducing to (4.3) for b = 1 or p = q = t)) The Mellin transform (4.13) of the index gets the following contribution from a term of the form where we adopted the standard notation for the Barnes double zeta function [103].Expanding around u = 1, 0, −1 we have (cf.(4.14)) Using [103,104] Res we obtain from (E.4) the following expression for z m (u) defined in (4.13) Combining the results in (E.7) with (E.6) we reproduce (E.2).
A similar computation can be done for the vector multiplet index where i In this case and using again (E.7) with (E.6) we find which matches the expression in Eq. (A. 19) in [20].Notice that the -1 coefficient of log β is independent of b, i.e. is the same as in (4.20), supporting its "topological" interpretation.
The same analysis can be repeated for the non-unitary multiplets.For the higher derivative multiplet [0 ] using the data in Table 3 to sum (−1) F p j 1 +j 2 +r/2 q −j 1 +j 2 +r/2 and dividing by , (E.12) leading to the small β expansion (E.15)For the gravitino multiplet [ 3  2 ] in Table 5 we get (E.17)A similar result is found for the graviton multiplet [2] in Table 6 i The pattern is thus the same as in the previous cases.In particular, the coefficient of log β does not depend on b and has the same (integer) value as we found in the undeformed case (see (4.21)).

F Small β expansion of superconformal index for (1, 0) multiplets in six dimensions
The above discussion of the index in four dimensions can be readily extended to six dimensional (1, 0) superconformal theories.Here we shall briefly outline what one finds for the unitary scalar and tensor multiplets, as well as for a non-unitary higher derivative vector multiplet.
The superconformal index for a (1, 0) 6d theory is defined similarly to (2.1) [28] I(t, u, v) = Tr (−1) F t ∆− r 2 u j 1 v j 2 ∆=2 r+ 1 2 (j 1 +2j 2 +3j 3 ) (F.1) Here (∆, r, j 1 , j 2 , j The constant k and log β term were not, in fact, discussed in [19,107] (the values of n and k given below will thus be new) but they should be expected from the general analysis we gave above in four dimension.
For the 6d scalar and tensor multiplets the single-particle superconformal index is given by [111] i S (β) = 2 t 3 2 − t A check of the consistency of (F.10) is that the coefficients of the leading ∼ β −3 , ∼ β −1 and β terms are again in agreement with (F.4),(F.5)and values in Table 8.In addition, we find that (F.12) The constant term k has a natural interpretation of the logarithm of the partition function of the 5-dimensional reduced theory found in the limit β → 0.
As for the log β term, it is present for the tensor and non-unitary vector multiplets.To understand its origin, let us briefly explain how to generalize the 4d analysis of Appendix C to the present 6d case.As in the 4d case, we expect that the log β correction should come only from gauge fields in the multiplets.These are the gauge tensor B − in the tensor multiplet and the 4-derivative gauge vector V (4) (with kinetic term (∂ µ F µν ) 2 ) in the vector multiplet.We also expect it to be related to the 6d analog of the σ(t) term in (C.12),(C.14),i.e. n = σ(1).
To check this claim, let us begin with the V (4) field.The associated dual higher spin field in AdS 7 (see discussion before (C.12)) has the SO(2, 6) content [7] (5; 1, 0, 0) − (6; 0, 0, 0), (F.The analog of the representation (C.12) then implies that the 6d partition function of V (4)  is given by The 15 t 2 term is associated with the lowest dimension field which is the field strength F µν (which has indeed dimension 2 and 6 2 = 15 components).Hence, we need σ(t) = +1 in order to cancel the spurious −1 and get the correct 6d partition function for the conformal field V (4) . 40Thus, the expansion of log I V should contain the term −σ(1) log β = − log β (same as for the 4d vector multiplet in (4.21)).