Kahler Potential and Ambiguities in 4d N=2 SCFTs

The partition function of four-dimensional N=2 superconformal field theories on S^4 computes the exact Kahler potential on the space of exactly marginal couplings [1]. We present a new elementary proof of this result using supersymmetry Ward identities. The partition function is a section rather than a function, and is subject to ambiguities coming from Kahler transformations acting on the Kahler potential. This ambiguity is realized by a local supergravity counterterm in the underlying SCFT. We provide an explicit construction of the Kahler ambiguity counterterm in the four dimensional N=2 off-shell supergravity theory that admits S^4 as a supersymmetric background.


Introduction
Recent years have witnessed remarkable progress in obtaining the exact partition function of supersymmetric field theories in various background geometries. When the geometry is S 1 × M d−1 the partition function admits a standard Hilbert space interpretation as a supertrace over the states of the theory on M d−1 . In other geometries, such as on a sphere S d , the physical interpretation of the partition function must be sought.
In [1] it has been shown that the partition function of 4d N = 2 superconformal field theories (SCFTs) on S 4 computes the exact Kähler potential K on the space of exactly marginal couplings, also referred to as the conformal manifold. This result was proven both by using supersymmetric localization [2] and by conformal dimension regularization on S 4 , and extends the proof in [3] that the S 2 partition function of 2d N = (2, 2) SCFTs computes the exact Kähler potential on the conformal manifold, as conjectured by [4] based on the exact formulae in [5,6] (see also [3,7]). In detail, [1] demonstrated that Z S 4 = e K/12 . (1.1) These identifications provide a physical and geometrical interpretation of the sphere partition function of 4d N = 2 and 2d N = (2, 2) SCFTs. These results also provide a computational pathway for obtaining the exact metric in the conformal manifold, which are interesting new observables in these theories, acted on by dualities (see e.g. recent work [8] [9]).
Here we present an elementary proof of the formula (1.1) using supersymmetry Ward identities. This new proof does not require localization nor that the 4d N = 2 SCFT admits a Lagrangian description. By virtue of the relation (1.1) identifying the S 4 partition function with the Kähler potential K on the conformal manifold, it follows that the partition function is subject to the Kähler ambiguity transformations K(τ,τ ) → K(τ,τ ) + F (τ ) +F (τ ) , (1.2) where F is an arbitrary holomorphic function and τ are holomorphic coordinates on the conformal manifold. This ambiguity implies that the partition function is a section over the space of exactly marginal couplings. We also give the microscopic realization of the Kähler ambiguity (1.2) by constructing the local supergravity counterterm in 4d N = 2 off-shell supergravity that when evaluated on the supersymmetric S 4 background yields (1.2). This is the 4d counterpart of the Kähler ambiguity counterterm for 2d N = (2, 2) SCFTs constructed in [1].
The plan is as follows. In section 2 we use supersymmetry Ward identities to show that the S 4 partition function of 4d N = 2 SCFTs computes the Kähler potential in the conformal manifold. In section 3 we identify the off-shell 4d N = 2 Poincaré supergravity theory in which the S 4 is a supersymmetric background. In section 4 we construct the supergravity invariant in the relevant Poincaré supergravity theory that once evaluated on S 4 provides a first principles realization of the Kähler transformation (1.2).

Kähler Potential from S 4 Partition Function
An exactly marginal operator in a four dimensional N = 2 SCFT is a scalar operator of dimension four which is a superconformal descendant of a scalar chiral primary operator of U(1) R charge w = 2. An N = 2 SCFT can be deformed while preserving superconformal invariance by 1 1 The exactly marginal couplings τ I are holomorphic coordinates in the space of exactly marginal deformations, known as the conformal manifold. The canonical metric in the conformal manifold g IJ is the Zamolodchikov metric which in four dimensional N = 2 SCFTs is Kähler, that is An N = 2 SCFT can be canonically placed on S 4 by the stereographic projection. The N = 2 superconformal transformations on S 4 are parametrized by chiral conformal Killing spinors ǫ i and ǫ i of opposite chirality transforming as doublets of the SU(2) R R-symmetry, which obey 2 so that η i = 1 4 ∇ /ǫ i and η i = 1 4 ∇ /ǫ i . An exactly marginal operator in an N = 2 SCFT can be represented as the top component of a four dimensional N = 2 chiral multiplet of R-charge w = 2, whose bottom component realizes the parent chiral primary operator. The holomorphic coordinates on the conformal manifold can be promoted to supersymmetric background chiral superfields with vanishing R-charge w = 0. The N = 2 superconformal transformations of a chiral multiplet with R-charge w on S 4 are given by [10] (we use [11]): 3 1 We use the same conventions as in [1]. 2 Γ a denotes tangent space gamma matrices while γ m = e m a Γ a denotes curved space ones. 3 Throughout a barred spinor isλ = λ T C, where C is the charge conjugation matrix.
where in Euclidean signature F − ab is a self-dual rank-two tensor. Indeed, for w = 2, the integrated top component is superconformal invariant and we have the identification For w = 0, an arbitrary covariantly constant background value for the bottom component of the chiral multiplet 4 is superconformal invariant, and serves as the spurion field for the holomorphic coordinates on the conformal manifold We denote by A I the chiral multiplets to which the coordinates in the conformal manifold have been promoted. Consider now the SCFT partition function on S 4 as a function of the exactly marginal couplings Z S 4 (τ,τ ). The second derivative is the integrated connected two-point function of exactly marginal operators. This correlator is ultraviolet divergent, divergences arising when the operators collide. These ultraviolet divergences can be regularized by introducing a massive deformation. Regulating divergences in a supersymmetric manner leads us to consider the OSp(2|4) massive subalgebra of the N = 2 superconformal algebra on S 4 , which is the supersymmetry algebra of an arbitrary massive four dimensional N = 2 theory on S 4 . The OSp(2|4) massive subalgebra on S 4 is generated by supercharges that anticommute to the SO(5) isometries of S 4 and an SO(2) R ⊂ SU(2) R R-symmetry. Conformal generators and U(1) R are projected out. The OSp(2|4) transformations are generated by Killing spinors which obey where and τ jk p = (iσ 3 , −1, −iσ 1 ) = (τ p jk ) * , where σ p are the Pauli matrices. In stereographic coordinates, where ds 2 = 1 1+ x 2 4r 2 2 dx m dx m , we have The constant spinors χ j 0 parametrize the transformations of the eight supercharges in OSp(2|4). If these parameters are chiral the corresponding spinors generate an OSp(2|2) subalgebra OSp(2|4). The chiral components of these spinors χ j L and χ j vanish at the North and the South poles of the sphere respectively. If the parameters are further constrained by the corresponding spinors generate a further SU(1|1) subalgebra Our strategy is to first prove that the integrated top component of the chiral multiplet in (2.17) 5 P L and P R are the spinor chirality projectors: P 2 L = P L , P 2 R = P R and P L + P R = 1. The Killing spinors obey P L ǫ i = ǫ i and P R ǫ i = ǫ i .
The supersymmetry transformation of the fermions in a chiral multiplet with R-charge w = 2 can be written as (2.5) 6 Using the SU(1|1) supersymmetry transformation δ obtained by imposing the constraints (2.13) and (2.15) on the Killing spinors, we get after multiplying (2.18a) by τ 2ij τ jk where we have used that for the SU(1|1) Killing spinors The terms proportional to F − ab and ∇ µ F − ab in (2.18a) (2.18b) also vanish. Their coefficients are anti-self-dual in the tangent space indices since where γ (r) is the product of r distinct gamma matrices. Since F − ab is self-dual in Euclidean signature, all the terms involving F − ab vanish. We can eliminate B 1 from (2.19b) by using (2.19a), which yields where, for brevity, we have defined (2.22) We now show that the sum of the terms in (2.21) involving A are a total derivative.
For any OSp(2|2) supersymmetry parameter χ j and any scalar quantity X we have Using this, the top component C of a chiral multiplet with w = 2 can be written locally as the sum of an SU(1|1) supersymmetry transformation δ and total derivatives This formula fails at the North pole, where χ 1 L = χ 2 L = 0 and Ξ diverges. Therefore the integrated top component is non-trivial in correlation functions, as it is not supersymmetry-exact globally, but the entire contribution localizes to the North pole, just as in the analysis of 2d N = (2, 2) SCFTs in [1]. 8 Let us consider the integrated correlator with an operator O obeying δO = 0 (2.25) We have divided S 4 into two-regions: a four-dimensional ball B 4 R of radius R around the North pole and its complement S 4 \B 4 R . In the R → 0 limit the ball contribution vanishes 9 and we are left with Using (2.24), which is valid in S 4 \B 4 R , and δΦ = 0, we can replace C by the last three terms in (2.24), which inside (2.26) can be written as an integral over the three-sphere S 3 R of radius R at the boundary of S 4 \B 4 R . For any OSp(2|2) Killing spinor χ j (2.14), we have that in the R → 0 limit Therefore, a simple scaling argument shows that the last term in (2.24) cannot compensate for the R 3 measure factor coming from S 3 R and gives a vanishing contribution in the R → 0 7 By using that ∇ m χ † L = − i 2r χ † R γ m . 8 We note that had we assumed that the partition function can be regulated while preserving full N = 2 superconformal invariance, we would have concluded that the partition function is independent of the moduli, as the top component C is globally superconformal-exact. 9 The R 4 measure factor suppresses the ball contribution in the R → 0 limit.
limit. Therefore, we have shown that in the presence of δ-closed operators (2.28) In the limit R → 0 we can replace the bottom component A(x) by its value at the North pole A(N), as higher order terms in the expansion in R vanish in the limit, and using Stoke's theorem andη is the unit vector towards the North pole of S 4 along the radial direction. 10 Going to spherical coordinates, where R is the radial coordinate, we find that The integration in (2.29) is over S 3 R , therefore and lim R→0 512A(N)r 6 (R 2 + 2r 2 ) This yields the desired formula The integrated top component C of a chiral multiplet is equivalent to inserting the bottom 10 The unit radial vector in cartesian coordinates is given byη a = − x a √ x 2 .

component A at the North pole. A very similar analysis yields
The integrated top componentC of an anti-chiral multiplet is equivalent to inserting the bottom componentĀ at the South pole.
We can now use (2.34) and (2.35) to express the derivative of the partition function in (2.8) as an unintegrated two-point function   The partition function of a field theory in a curved geometry can be ambiguous. These ambiguities are encoded in finite counterterms for the background fields that capture the background geometry and the parameters of the theory. When the partition function of a supersymmetric theory can be regulated in a diffeomorphism invariant and supersymmetric manner, the counterterms are supergravity invariants constructed out of the supergravity multiplet encoding the background geometry and the supersymmetry multiplets to which the other parameters of the theory can be promoted, since all parameters in a supersymmetric field theory can be promoted to background supermultiplets [12]. Constructing these supergravity invariants requires identifying first the supergravity theory in which the curved geometry over which the partition function is computed is a supersymmetric background. This can be analyzed in the framework of off-shell supergravity [13]. In this section we identify the four dimensional N = 2 off-shell Poincaré supergravity theory and the background fields in that supergravity multiplet that give rise to the OSp(2|4)-invariant four-sphere background geometry.
A conceptual way of constructing off-shell Poincaré supergravity theories is to start with off-shell conformal supergravity and partially gauge fix the conformal symmetries down to Poincaré by adding compensating supermultiplets. Different choices of compensating multiplets give rise to different off-shell Poincaré supergravity theories, with different sets of auxiliary fields. 11 The Poincaré supersymmetry transformations of the gauge fixed theory are constructed by combining the Poincaré supersymmetry transformations in conformal supergravity with field dependent superconformal transformations that are needed to preserve the gauge choice. 12 Our starting point is four dimensional N = 2 conformal supergravity [15] (we refer to [14] for more details). Off-shell N = 2 superconformal transformations are realized on the Weyl multiplet, whose independent fields are The fields e a m , b m , V j m i , A R m , ψ i m are the gauge fields for translations, dilatations, SU(2) R , U(1) R and Poincaré supersymmetry generators in the N = 2 superconformal algebra. The Weyl multiplet is completed by the bosonic auxiliary fields T − ab and D, and the fermionic auxiliary field χ i . In Euclidean signature T − ab is a self-dual rank-two tensor. The embedding of the OSp(2|4)-invariant S 4 in conformal supergravity appeared in [16] [17].
Four dimensional N = 2 Poincaré supergravity [18] contains a graviphoton gauge field A m . This field is furnished in the conformal approach by coupling an abelian vector multiplet to the Weyl multiplet [19] [20]. An N = 2 vector multiplet, also known as a restricted chiral multiplet, is an N = 2 chiral multiplet (2.5) with w = 1 subject to constraints, and consists of bosonic: X, A m , Y ij In order to preserve this gauge, supersymmetry transformations must be accompanied by a compensating special conformal transformation, which acts nontrivially on b m . Fortunately, all elementary fields in conformal supergravity and all fields in N = 2 matter multiplets transform trivially under special conformal transformations, and therefore the supersymmetry transformations of these fields are not modified by the gauge choice (3.3). Dilatations and U(1) R are gauge fixed by setting [15] X = µ , (3.4) where µ is an arbitrary mass scale, while [15] Ω i = 0 (3.5) fixes the special conformal supersymmetry transformations. Under supersymmetry [14] δX = 1 2 where δ ≡ δ ǫ + δ η , and (ǫ i , ǫ i ) and (η i , η i ) parametrize the Poincaré and conformal supersymmetry transformations. F ab is the superconformal covariant field strength (see equation (20.77) in [14]) and is the superconformal covariant derivative acting on the scalar field X. In order to preserve the gauge choice (3.4)(3.5), we must accompany the Poincaré supersymmetry transformations δ ǫ with a field dependent compensating conformal supersymmetry transformation δ η with parameter 13 Different Poincaré supergravity theories depend on the choice of a second multiplet which gauge fixes the remaining SU(2) R symmetry. Three choices for this compensating multiplet have been considered in the literature (see [21]): a non-linear multiplet, a hypermultiplet and a tensor multiplet. We now demonstrate that the OSp(2|4)-invariant S 4 is a supersymmetric background of the N = 2 Poincaré supergravity theory constructed with a tensor multiplet (and not with the non-linear or hypermultiplet).
Consider the off-shell N = 2 Poincaré supergravity multiplet constructed by coupling a vector multiplet and a tensor multiplet to the Weyl multiplet. An N = 2 tensor multiplet [21] bosonic : L ij , G, E mn fermionic : φ i (3.10) consists of a triplet of real scalars L ij = L ji , a tensor gauge field E mn , a complex scalar G and a doublet of spinors φ i . The SU(2) R symmetry can be gauge fixed by setting which breaks SU(2) R down to SO(2) R . The supersymmetry transformation [11] δL ij = ǫ (i φ j) + ε ik ε jl ǫ (k φ l) (3.12) implies that to preserve (3.11), we must accompany the Poincaré supersymmetry transformation δ ǫ with a compensating SU(2) R transformation δ SU (2) R (Λ k j ) with parameter 14 In summary, this off-shell Poincaré supergravity multiplet constructed by gauge fixing a Weyl, vector and tensor multiplet completes the on-shell multiplet e a m , ψ i m , A m with bosonic auxiliary fields and fermionic auxiliary fields χ i , φ i . The Poincaré supersymmetry transformations in this N = 2 Poincaré supergravity theory are given by the following combination of superconformal transformations with η in (3.9) and Λ k j in (3.13). In this N = 2 Poincaré supergravity theory the supersymmetric backgrounds where the background values of all fermions vanish are solutions to the following equations with η in (3.9), since Λ k j = 0 vanish on bosonic backgrounds. The explicit form of these transformations are [19] [20][21] (we use [11])

16)
14 The parameter is determined only up to an SO(2) R transformation.
with η in (3.9). D is superconformal covariant derivative and R ab (T ) and R ab (U i j ) are covariant curvatures for U(1) R and SU(2) R .
The OSp(2|4)-supersymmetric S 4 background is described by the following Killing spinor equations (2.9) From (3.16) we find that S 4 is a supersymmetric background of this supergravity theory with the following non-vanishing background fields turned on

The Kähler ambiguity Supergravity Counterterm
In this section we construct the N = 2 Poincaré supergravity invariant constructed out of the supergravity multiplet and the w = 0 chiral multiplets A I (see below (2.7)) which when evaluated on the OSp(2|4)-supersymmetric background (4.6) realizes the Kähler ambiguity (1.2).
Our approach is to construct a superconformal invariant constructed out of the Weyl multiplet, the compensating vector multiplet Φ, the compensating tensor multiplet and the chiral multiplets A I , the supermultiplets to which the coordinates in the conformal manifold τ I have been promoted. This invariant, when evaluated on the Poincaré gauge fixing choice described in the previous section yields an invariant in the associated N = 2 Poincaré supergravity theory. We first recall some facts about the construction of superconformal invariants.
Consider an abstract chiral multiplet (2.5) with w = 2, which we denote byÂ, coupled to the Weyl multiplet (3.1). The following superconformal invariant can be constructed from such a chiral multiplet [15] whereĈ andÂ denote the top and bottom components of the multipletÂ. The coupling of the chiral multiplet to the Weyl multiplet is responsible for the appearance of the terms 15 A similar analysis for the Poincaré supergravity theories constructed with a compensating non-linear multiplet and hypermultiplet demonstrates that the background fields that yield the S 4 Killing spinor equations are incompatible with the vanishing of the supersymmetry variations of the fermions in these multiplets. Therefore, S 4 is not a supersymmetric background of these supergravity theories. afterĈ in (4.1). The product of two chiral multiplets with R-charge w 1 and w 2 yields another chiral multiplet of R-charge w 1 + w 2 . Therefore, superconformal invariants can be constructed from products of chiral multiplets with total R-charge w = 2.
Consider now the compensating vector multiplet that appears in the construction of N = 2 Poincaré supergravity, which we denote by Φ. It is important to note that an N = 2 vector multiplet is a chiral multiplet with w = 1 subject to reducibility constraints [22], which express the last two components of the chiral multiplet in terms of the previous ones. It is also known as a restricted chiral multiplet. The components of a chiral multiplet (2.5) are given in terms of the fields in the abelian vector multiplet (3.2) by where F ab is the superconformal covariant field strength. Expressing a vector multiplet as a w = 1 chiral multiplet provides a way of constructing a superconformal invariant out of Φ using (4.1). We now write down the supergravity counterterm responsible for the Kähler ambiguity (1.2). It is the superconformal invariant (4.1) constructed from the w = 2 composite chiral multiplet where F is an arbitrary holomorphic function of the w = 0 chiral multiplets A I describing the coordinates in the conformal manifold. 16 The associated N = 2 Poincaré supergravity invariant is I[Φ 2 F (A I )] (4.4) with We now want to evaluate the N = 2 Poincaré supergravity invariant (4.4) on the 16 Another natural guess for the Kähler counterterm can be constructed from the w = 2 chiral multiplet W ab W ab F (A I ), where W ab is a chiral multiplet that encodes the covariant Weyl multiplet (3.1). However, upon evaluating these terms on the background (4.6) they all vanish, as these terms involve the Weyl tensor, which vanishes on S 4 . Supergravity couplings involving W 2 have been considered in the literature [23]. For other higher derivative invariants in N = 2 supergravity see e.g. [24][25] [26][27] [28].
The product of two chiral multiplets with bosonic components (A, B ij , F − ab , C) and (a, b ij , f − ab , c) yields a new chiral multiplet with bosonic components (setting all fermions to zero, as they vanish on the OSp(2|4)-invariant background (4.6)) Aa, Ab ij + aB ij , Af − ab + aF − ab , Ac + aC − Therefore, on the supersymmetric background (4.6) and finally C| Φ 2 F (A I ) = 12µ 2 r 2 F (τ I ) . (4.9) Using that µ is a fiducial scale introduced in gauge fixing to Poincaré supergravity, so that we can write µ = a/r, we have find that the invariant (4.4) evaluates to I[Φ 2 F (A I )] = d 4 x √ g C| Φ 2 F (A I ) = 32π 2 a 2 F (τ I ) .  This provides a microscopic realization of Kähler ambiguities in these SCFTs.