Shortening Anomalies in Supersymmetric Theories

We present new anomalies in two-dimensional ${\mathcal N} =(2, 2)$ superconformal theories. They obstruct the shortening conditions of chiral and twisted chiral multiplets at coincident points. This implies that marginal couplings cannot be promoted to background superfields in short representations. Therefore, standard results that follow from ${\mathcal N} =(2, 2)$ spurion analysis are invalidated. These anomalies appear only if supersymmetry is enhanced beyond ${\mathcal N} =(2, 2)$. These anomalies explain why the conformal manifolds of the K3 and $T^4$ sigma models are not K\"ahler and do not factorize into chiral and twisted chiral moduli spaces and why there are no ${\mathcal N} =(2, 2)$ gauged linear sigma models that cover these conformal manifolds. We also present these results from the point of view of the Riemann curvature of conformal manifolds.


Introduction
Conformal Field Theories (CFT's) often come in a continuous family labeled by their exactly marginal couplings. This family, known as the conformal manifold M, is endowed with a canonical metric, the Zamolodchikov metric [1]. The Zamolodchikov metric is determined by the two-point functions of the exactly marginal operators. The general Riemannian structure of conformal manifolds was first discussed in [2,3].
Such conformal manifolds appear in theories with extended symmetries, such as supersymmetry or current algebra (e.g., the c = 1 models), and in certain large N theories. 1 These conformal manifolds also play an important role in the AdS/CFT correspondence and on string worldsheets. In the former case, the conformal manifold of the boundary CFT maps to the space of vacua in the bulk Anti-de Sitter space (AdS). In the latter case, the conformal manifold of the worldsheet theory maps to the space of solutions of the equations of motion in spacetime.
One of the fundamental properties of the conformal manifold of two-dimensional N = (2, 2) superconformal field theories (SCFT's) is that it factorizes locally 2 into the product of two Kähler manifolds, (1.1) Coordinates of M c and M tc are the coupling constants of the exactly marginal operators constructed from the dimension ( 1 2 , 1 2 ) operators in the chiral and twisted chiral rings [7] of the SCFT. When the Virasoro central charge is c = 9 and the SCFT is realized as a non-linear sigma model with a Calabi-Yau threefold as its target space, the factorization (1.1) was proven in [8] by combining string theory worldsheet SCFT and target space arguments. 3 For other early papers on the N = (2, 2) conformal manifold from the target space and the worldsheet view points, see [9,10] and [11,8,12,13], respectively.
More recently, in [14] the Weyl anomaly on the conformal manifold of N = (2,2) theories was used to rederive the factorization (1.1). The argument in [14] applies to any N = (2, 2) SCFT, with no restriction on the Virasoro central charge. An assumption made 1 Some general arguments about when such families may exist can be found in [4]. 2 There are examples where the conformal manifold is modded out by a discrete symmetry, which prevents it from being a product globally [5,6]. We thank D. Morrison for a useful discussion about this point. 3 For SCFT's realizing Calabi-Yau compactifications, M c and M tc parameterize the moduli space of complex structure and complexified Kähler class.
in [14] was that the coupling constants parameterizing M c and M tc could be promoted to supersymmetric dimension (0, 0) background chiral and twisted chiral multiplets, in the spirit of [15]. Factorization of the conformal manifold (1.1) then followed from the classification of anomalies of the partition function under super-Weyl transformations.
Alternatively, one can also easily provide an argument for factorization (1.1) in the spirit of [16].
On the other hand, it is well-known that the conformal manifold of the n-dimensional torus T n SCFT is locally [17], O(n, n) O(n) × O(n) , (1.2) while the conformal manifold of the K3 SCFT is locally [2], (20) .
More generally, the conformal manifold of N = (4, 4) SCFT's is locally of the form [18] (see also [19]), O(4, n) O(4) × O(n) , (1.4) for some n. 4 These examples appear to be at odds with the factorization (1.1) of the conformal manifold of N = (2, 2) SCFT's proven in [14]. Indeed, with the exception of the n = 2 case in (1.2), these conformal manifolds do not factorize locally into a product of Kähler manifolds. In fact, they are not even Kähler manifolds. This is in spite of the fact that the T n SCFT with n even and the other SCFT's enjoy N = (2, 2) superconformal symmetry.
Another way of presenting this puzzle is the following. Normally, when a particular global symmetry of a theory implies some special properties, extending that symmetry does not ruin those properties. Here, we see a counter-example to that. Specifically, if we consider a theory with N = (4, 4) supersymmetry and view it as a special case of a N = (2, 2) theory, we might conclude that the conformal manifold should be Kähler and that it should factorize as in (1.1). This conclusion turns out to be wrong. Later we will discuss additional properties of N = (2, 2) theories that do not hold true in theories with extended supersymmetry.
In this note we resolve this tension. In a generic N = (2, 2) SCFT, the operator product expansion (OPE) between a chiral multiplet O and a twisted chiral multipletÕ 4 For a sigma model with a hyper-Kähler target space M , n = h 1,1 (M ).
of scaling dimensions ( 1 2 , 1 2 ) does not have poles. On the other hand, in all the counterexamples to factorization mentioned above, the OPE has a pole 5 where the symbol | on the left-hand side picks up the bottom component of each multiplet.
The operator J ++ appearing on the right-hand side must be both chiral and twisted chiral, and with scaling dimensions (1, 0) and R-charges (2, 0). J ++ is the bottom component of a special N = (2, 2) short multiplet, which we shall denote by J ++ . The existence of such a multiplet implies that the R-symmetry is enhanced and that the SCFT enjoys enlarged superconformal symmetry beyond N = (2, 2) supersymmetry. For example, in the K3 SCFT, the current J ++ enlarges the R-symmetry from U (1) to SU (2). 6 Let us imagine that we deform the SCFT with exactly marginal operators A powerful idea [15] is to promote λ andλ to background chiral and twisted chiral superfields. We will see that whenever the OPE between O andÕ is singular as in (1.5), then the couplings cannot be promoted to such background superfields due to an anomaly! We can either promote the λs to background chiral superfields or theλs to background twisted chiral superfields, but we cannot do both simultaneously. To our knowledge, this type of obstruction has not been discussed before.
Before we explain how this comes about let us explain the physics of promoting couplings to background fields. The effective action as a function of the background fields satisfies the required symmetries when the operators do not have contact terms that spoil those symmetries. For example, if a conserved current is not conserved at coincident points then the background effective action will not be gauge invariant. Another example is if in a CFT the operator equation T µ µ = 0 does not hold at coincident points, then the background effective action fails to depend just on the conformal class of the background 5 Throughout this note we study the theories in Euclidean space. However, we use Lorentzian signature notation with coordinates x ±± etc., which are complex conjugates of each other. The reason for using this notation is that we use complex conjugation notation on the charged chiral objects like the odd coordinates θ, as if we are in Lorentzian signature. 6 It is important that the current J ++ which enlarges the R-symmetry corresponds to a normalizable state in the SCFT.
metric. Similarly, in supersymmetric theories, for the effective action to depend on λ andλ as chiral and twisted chiral superfields, the operators O andÕ have to obey their defining not only at separated points (which holds by definition) but also at coincident points.
Loosely speaking, one could say that the equations (1.7) have to be obeyed off-shell. As with all anomalies, our shortening anomaly can be understood as the failure of the parti- We will show that the OPE (1.5) induces contact terms of the form and that it is impossible to tune both of these contact terms away (the notation will be explained in section 3.) Therefore, the effective action does not depend on the background coupling constants as if they were chiral and twisted chiral superfields. Some of the background couplings have to sit in long multiplets. We call this phenomenon a shortening anomaly. 7 The discussion above is reminiscent of the clash between conservation of the vector and axial current in theories with fermions, where by adjusting counter-terms either symmetry can be preserved, but not both simultaneously.
As is standard in such situations, where some operator equations are violated at coincident points, when we turn on nontrivial backgrounds for λ andλ then the contact 7 We would like to emphasize that the operators O,Õ remain short in the standard situation where the couplings are constant. Indeed, λ,λ have vanishing beta functions, since the operators O,Õ have no operators to combine with (see [20] and also earlier literature, e.g. [11]). In particular, (1.5) does not induce a beta function. The shortening anomaly is in the background fields, not in the operators. The operator equations are modified only in nontrivial configurations for the background superfields (see equation (1.9)).
terms (1.8) lead to nontrivial operator equations. Depending on which counter-terms we choose, one of the equations below has to be true: For constant couplings, where only the bottom components of λ andλ are turned on, the operators O andÕ remain short, as they should.
The obstruction to promoting both chiral and twisted chiral couplings to short multiplets (which exists only if supersymmetry is enhanced) invalidates the conclusions found in [14] for such theories. 8 We therefore conclude that factorization (1. This phenomenon is similar to a familiar situation in supergravity. The target space of N = 1 supergravity in 4d is known to be Kähler (see [21] for details). But N = 2 supergravity with hypermultiplets has a quaternionic target space, which is not Kähler (see [22] for a review of the scalar manifolds of supergravities in various dimensions).
There is no contradiction in that because the N = 2 theory is not merely a special N = 1 supergravity theory because it has a multiplet including a graviphoton and a gravitino, which is not present in the generic N = 1 theory. The special N = (2, 2) multiplet J ++ that resolves our puzzle, which includes a conserved spin-3 2 current, plays a similar role to the graviphoton multiplet in this supergravity analog. For a related discussion see [23].
We can relate our discussion to supergravity more directly if we view our 2d models as worldsheet theories for strings and we study their spacetime description. The low energy description of string compactifications on Calabi-Yau threefolds, described by N = (2, 2) SCFT's on the worldsheet, is captured by four-dimensional N = 2 supergravity, whose scalar manifold indeed takes the factorized form (1.1). On the other hand, compactifications on manifolds leading to enhanced spacetime supersymmetry, such as T 6 or K3 ×T 2 , 8 The analysis in the spirit of [16] is also invalidated by this anomaly since an implicit assumption in such an analysis is that it is possible to preserve the shortening conditions of all the dimension ( 1 2 , 1 2 ) chiral and twisted chiral operators both at separate and coincident points. 9 Such an operator does not exist for the T 2 SCFT. And indeed, the T 2 conformal manifold (1.2) factorizes into a product of Kähler manifolds, locally given by SL(2)/U (1) × SL(2)/U (1). are instead described by N = 8 or N = 4 supergravity. The scalar manifold in these theories does not factorize. 10 As with all 't Hooft-like anomalies, our analysis leads to theorems about the nonexistence of certain renormalization group (RG) flows. If there exists a strictly N = (2,2) supersymmetric RG flow in which all the infrared marginal couplings are realized as chiral or twisted chiral couplings along the flow, then it is guaranteed that the corresponding coupling constants in the infrared are in short representations and hence there is no shortening anomaly. Therefore, the conformal manifold would have to factorize into a chiral Kähler manifold and a twisted chiral Kähler manifold. Therefore, one can immediately conclude that there cannot exist an N = (2, 2) RG flow that realizes the full conformal manifold of the K3 SCFT. Indeed, constructions of gauged linear sigma models (GLSM's) [24] which lead to subspaces of the K3 conformal manifold are known (see for example [25,18]), but it has never been possible to embed the K3 SCFT in a UV completion that covers the full conformal manifold. The same holds for T 4 but not for T 2 , which does not have a shortening anomaly, and indeed admits an N = (2, 2) GLSM representation that realizes the complete T 2 SCFT conformal manifold. 11 We now see that the obstruction for K3 and T 4 is due to an anomaly in the infrared that must be matched in the ultraviolet. 12 It is interesting to relate this discussion to recent developments concerning the S 2 partition functions of N = (2, 2) SCFT's [26][27][28][29]. There are two inequivalent ways to compactify such theories on S 2 and they compute, respectively, the Kähler potentials of the chiral and twisted chiral deformations [30,28] (see also [31,14]). However, this statement is not meaningful, if the total space does not factorize as in (1.1)! Therefore, when our anomaly is present, also this result about general N = (2, 2) SCFT's is invalidated.
(One can understand it again as being due to the failure of spurion analysis.) Because of the anomaly, there is no N = (2, 2) UV completion that would cover the full conformal manifold, therefore, what one can extract from the sphere partition function is at best the 10 The conformal manifold of the worldsheet SCFT is a subspace of the supergravity scalar manifold. 11 The GLSM is an N = (2, 2) U (1) gauge theory with chiral multiplets (X 1 , X 2 , X 3 , P ) of is a homogeneous polynomial of degree 3. The twisted chiral coupling is realized by the complexified FI parameter and the chiral coupling by the single complex parameter in G 3 (X). 12 Our analysis however does not rule out N = (4, 4) supersymmetric RG flows that would cover the full conformal manifold of sigma models on K3 or T 4 .
Kähler potential on some Kähler submanifolds of the conformal manifold. An alternative sphere compactification that utilizes the extended supersymmetry may exist and it may probe the full conformal manifold.
We also provide a complementary perspective on the anomalies by studying the Riemann curvature of the conformal manifold in N = (2, 2) SCFT's, extending the previous work [8,32,19].
Our aim is to determine to what extent these shortening conditions can be maintained as we explore the conformal manifold of an N = (2, 2) SCFT.
Monitoring the shortening conditions (2.1) under an exactly marginal perturbation leads us to analyze the contact terms in where Z I ≡ (x ±± I , θ ± I ,θ ± I ) are points in superspace. In this section we determine the operator product expansion (OPE) of a chiral and a twisted chiral multiplets, whose top components yield marginal operators, leaving the analysis of contact terms to the following section.
In supersymmetry, it is often useful to employ spurion analysis. E.g., the coupling constants of chiral operators O are promoted to background chiral multiplets [15]. This procedure makes sense only if the operator equationD ± O = 0 is respected also at coincident points (loosely speaking, we can say that O is chiral off-shell). This is because when we write the partition function depending on some background fields, by taking derivatives with respect to the background fields, we can probe the correlation functions of the corresponding operators both at separated and at coincident points. By definition, operators equations are always obeyed at separated points, but they may fail at coincident points. The famous (continuous) 't Hooft anomalies arise when a conservation equation is not obeyed at coincident points. As a consequence, the partition function depending on the associated background fields does not obey the naively expected equations (in the famous case of the chiral anomaly, the partition function is not gauge invariant). Similarly, we can couple twisted chiral background fields to twisted chiral operators as long as the corresponding shortening conditions are valid off-shell.
A general method that guarantees that some conservation equation is obeyed also at coincident points is to construct a regularization obeying the conservation equation. This automatically tunes the contact terms in the infrared to zero. Such a regularization may not exist, if there is a genuine anomaly in the conservation equation. In the next section we will establish the existence of some contact terms that violate the shortening conditions (2.1) at coincident points. Therefore, the standard arguments relying on the selection rules of background superfields are not valid. This also means that, when this occurs, N = (2, 2) RG flows that contain all the infrared chiral and twisted chiral couplings cannot exist.
We will show that such a subtlety in the spurion analysis takes place in an N = (2, 2) SCFT in two dimensions when it has an operator of dimensions (1, 0) and R-charges (2, 0). Unitarity implies that this operator, which we denote by J ++ , is a conserved current, obeying ∂ −− J ++ = 0. Since J ++ carries a non-vanishing R-charge, its existence implies that the supersymmetry is enhanced.
where · · · encode the contributions of non-chiral operators and descendants. For simplicity, we are suppressing some coefficients that we will make more explicit later. In particular, when the N = (4, 4) SCFT is realized as a non-linear sigma model whose target space is a hyper-Kähler manifold M , there is a unique holomorphic 2-form Ω ∈ H 2,0 (M ), and the extra R-current is expressed as J ++ = Ω ab ψ a + ψ b + . Chiral and twisted chiral operators are expressed as, where k ∈ H 1,1 (M ), g ab is the Kähler metric on M , and ψ ± are fermions. The OPE (2.5) More generally, one can show that the existence of J ++ of dimensions (1, 0) and Rcharges (2, 0) alone is sufficient for the pairing (2.4) and therefore the OPE (2.5) follows without assuming the small N = 4 superconformal symmetry. Indeed, since J ++ does not commute with the U (1) + × U (1) − R-symmetry in an N = (2, 2) SCFT, the R-symmetry group must be larger than just U (1) + × U (1) − and the dimension ( 1 2 , 1 2 ) chiral and twisted chiral operators must furnish a representation of it. Therefore, J ++ cannot act trivially and hence some pairing as in (2.4) and (2.5) must be present.
Unitarity and supersymmetry further imply that the extra R-current J ++ is the bottom component of a very short multiplet, which we denote by J ++ . It obeys For convenience, we summarize the R-charges of various objects used in the following: Our final goal is to monitor the shortening conditions (2.1) as we explore the conformal manifold while preserving N = (2, 2) supersymmetry. This prompts us to determine the OPE of the supermultiplets O,Õ and J ++ in N = (2, 2) superspace, which makes supersymmetry manifest. Introducing coordinates (x, θ,θ) (suppressing the +, − indices) we define and are chiral/anti-chiral respectively, which we denote also as c/c. In this language O is (c, c),Õ is (c,c) and J ++ is simultaneously (c, c) and (c,c).
Supertranslation invariance implies that correlators depend on the position of operators through z ij and zī j . The coordinates we have defined are rather convenient. Indeed, if the i-th operator is chiral, the correlators depend on z ij only, while if it is anti-chiral they depend on zī j .
Our correlator of interest is: where Z I ≡ (x ±± I , θ ± I ,θ ± I ). The shortening conditions, together with supertranslational and rotational invariance imply that the correlator (2.13) depends on z ++ 13 , z ++ 23 and z −− 12 . A subtlety in SCFT's that must be taken into account is the existence of a superconformal invariant X(Z 1 , Z 2 , Z 3 ) constructed out of three points in superspace [33]. Superconformal invariance and nilpotency of X(Z 1 , Z 2 , Z 3 ) imply that the most general correlator consistent with superconformal invariance is given by Imposing that O(Z 1 ) is chiral forces a = 0, as X(Z 1 , Z 2 , Z 3 ) depends onθ 1 [33]. We therefore conclude that the correlator at separated points is (2.16) Moreover, using that we obtain our desired OPE: (2.18) The superspace correlators we have constructed obey, by construction, the shortening conditions (2.14) at separated points. Our next task is to study the shortening conditions at coincident points (2.3) and determine whether counter-terms can be adjusted so that the shortening conditions for chirals and twisted chirals (2.1) can be both simultaneously obeyed.

Supersymmetric Contact Terms and the Shortening Anomaly
The OPE (2.18) may lead to some contact terms (2.3). 14 To understand these we need to study the superspace derivatives (the derivatives with respect to the second argument follow from these) (3.1) Our strategy to compute (3.1) is to extend the well-known formula, ∂ ++ 1 to Green's function in superspace. This line of inquiry makes manifest an inherent ambiguity in defining the derivatives (3.1) in superspace. In order to define these derivatives we need to specify the behavior of Green's function in superspace for the left-movers, and different choices yield different answers. This ambiguity is a manifestation of the fact that the pole 1/z −− 12 in N = (2, 2) superspace is too singular to yield an unambiguous distribution. 15 In fact, we will find an ambiguity of θ + 12θ + 12 δ (2) (z 12 ) in defining 1/z −− 12 . This is analogous to the well-known ambiguity in the distribution 1/x on the real line x ∈ R by the addition of δ(x). 16 As a result, the derivatives (3.1) suffer from certain ambiguities.
However, we will find that, whichever choice we make for these ambiguities, we cannot remove the contact terms for both chiral and twisted chiral operators simultaneously.
We can compute (3.1) starting from the Green's functions in superspace 14 A related discussion appeared in [34], where it clarified the need for a contact term, which had been found earlier in [35,36]. A more modern discussion of that problem appeared in [37]. 15 On the other hand, the pole 1/x −− on the 2-plane of (x ++ , x −− ) can be extended to an to a particular choice of regularization. 16 1/x is characterized by the fact that x n · 1/x = x n−1 for all positive integers n. Therefore, any ambiguity in 1/x must be annihilated by multiplication by x n , and the only distribution with this property is proportional to δ(x) (See, for example, theorem 9 in [38]). Indeed, the principal value P (1/x) and 1/(x ± iǫ) differ by distributions proportional to δ(x).

or alternativelyD
(1)   Symmetric violation of Ward identities −π π (3.8) We note that there is no choice of a that preserves simultaneously the chiral and twisted chiral Ward identity. Even though the derivatives are subject to some ambiguities, we now proceed to unambiguously establish our shortening anomaly. Anomalies arise when we have an operator that should satisfy an operator equation, such as (2.1), and one finds that such an equation is only correct at separated points while at coincident points there are various contact terms. 18 Establishing the anomaly amounts to showing that these contact terms cannot be removed by redefining the scheme since scheme redefinitions change the theory by contact terms.
We consider N = (2, 2) supersymmetric contact terms since we assume that the conformal manifold can be explored while maintaining N = (2, 2) supersymmetry. Therefore, the most general OPE between a chiral and a twisted chiral superfield in a theory with an enhanced R-symmetry, now allowing for supersymmetric contact terms, is given by where r is a scheme dependent constant, which can be shifted by changing the scheme.
In order to establish our shortening anomaly we must show that it is not possible to tune the contact term r such that the shortening conditions for a chiral and a twisted chiral multiplet can be maintained simultaneously. Since the contact term in (3.9) is annihilated byD (1) − and D  However, since a + b = 1, it is impossible to solve both equations. This implies that we cannot simultaneously preserve the chiral and twisted chiral shortening conditions along 18 By turning on suitable background fields the ambiguities in the contact terms can be described as an ambiguity in adding local counter-terms constructed out of the operators in the theory and the background fields. When such background fields are present, the problem with contact terms can be uplifted to problems at separated points. 19 The formulaD the conformal manifold. By tuning the contact term, we can either preserve the chiral or the twisted chiral constraint, but not both. This is our shortening anomaly.
The fact that we cannot preserve both constraints simultaneously is analogous to the situation in two dimensions with vector and axial anomalies [39], where contact terms cannot remove both anomalies simultaneously. For a complementary derivation of our shortening anomaly in the cohomological approach based on the Wess-Zumino consistency conditions, see Appendix A.
The contact terms we encountered lead to an operatorial violation of the shortening equations upon deforming the theory by background superfield sources for the exactly marginal operators (as in (2.2)). The shortening conditions becomē This implies that promoting the couplings of dimension ( 1 2 , 1 2 ) chiral and twisted chiral operators to full-fledged background chiral and twisted chiral superfields is impossible whenever the theory includes the operator J ++ . We emphasize that the violations in (3.12) depend on the fermionic components of the multiplet to which the couplings have been promoted. The result in [8] follows from the vanishing of the mixed chiral and twisted chiral curvature components, which implies that the holonomy group is the direct product of two commuting subgroups, which in turn implies factorization (1.1). In this section we establish the following result about any N = (2, 2) SCFT by investigating the curvature on its conformal manifold explicitly: the conformal manifold of an N = (2, 2) SCFT does not factorize if and only if the SCFT has a conserved current with R-charges (2, 0).

Curvature of Conformal Manifold and Factorization
We now proceed to compute the mixed components of the Riemann tensor of M using CFT techniques, in the spirit of [3,41,19,42]. The tangent bundle T M comes equipped with a metric compatible connection, whose curvature we would like to find. The curvature of this connection is determined by a certain four-point function of the exactly marginal operators, which we denote by U i . The formula for the curvature can be written as [42] 20 The symbol 'RV' stands for the prescription where we cut out small discs around the fixed operators and remove the power-law divergent terms. 21 The subscript c stands for the connected correlator, which is defined by subtracting the three possible disconnected contributions to the four-point function. As shown in [41], an explicit formula for the curvature can be given in terms of the CFT data: spectrum of operators and OPE coefficients.
Extending the N = (2, 2) superconformal Ward identities first introduced in [8] and further exploited in [19] we can show that the only non-trivial mixed chiral and twisted chiral moduli four-point function that needs to be studied is where we have denoted the exactly marginal operators constructed from operators in the chiral and twisted chiral ring by 20 Note that our definition of the curvature includes an overall normalization of 4 compared to that of [42]. This is the convention in which the special geometry relation takes the standard form.
In addition, we define as usual We are now ready to compute the curvature using (4.1). It follows from (4.4) that the independent components of the Riemann curvature tensor are R iklj and R ilkj . The connected component prescription in (4.1) can be extended to both sides of (4.4), and therefore, by pulling the operator located at x to infinity we arrive at Integrating by parts in (4.1) and remembering that we are integrating over the complex plane with disks around the punctures removed, the answer reduces to contour integrals around the punctures. In order to get a non-zero contribution to the curvature, the function whereJ ++ has R-charges (−2, 0). Using we find that indeed near y = 1 the function g(y) has a simple pole A very similar analysis of the behavior of the function near y = 0, where now the relevant OPE is demonstrates that there is no contribution to the curvature from the contour integral around y = 0.
In conclusion, we have shown that there is a non-trivial component of the mixed curvature tensor between chiral and twisted chiral moduli in an N = (2, 2) SCFT if and only if the SCFT has a current with R-charges (2, 0). In such a SCFT, the Riemann curvature is given by 11) and the conformal manifold M no longer factorizes.

N = (4, 4) Conformal Manifolds
The small N = 4 superconformal algebra, which has an SU (2) R R-symmetry (see e.g. [19]), is an important example of extended supersymmetry. In this section, we use the formula (4.1) to compute the curvature on the conformal manifold of N = (4, 4) theories and give a purely field-theoretic derivation that the local geometry of the coset is (1.4).
We also study the bundle of N = 4 supercurrents over the conformal manifold and show that there is no consistent choice of an N = 2 subalgebra even over a local coordinate patch. This gives a geometric perspective on our shortening anomaly. Let us introduce some notation first: the left-moving N = 4 supercurrents are denoted by S αA +++ , and the SU (2) R currents by J (αβ) ++ , where α and A are doublet indices for the R-symmetry SU (2) R and the outer automorphism SU (2) out of the N = 4 superconformal algebra, respectively.

Riemannian Curvature
In the N = 4 notation above, the formula (4.1) for the Riemann curvature is expressed as By choosing an N = 2 subalgebra generated by S 11 +++ and S 22 +++ , marginal operators for (anti-)chiral multiples are F i11 (F i22 ), while those for (anti-)twisted chiral multiples are As shown in the previous section, there are non-zero curvature components in mixed chiral and twisted chiral directions, which in the N = 4 notation can be expressed as where k in the normalization factor is related to the Virasoro central charge by c = 6k.
By using the Bianchi identity, we can also determine 23 On the other hand, the curvatures in the purely chiral directions are controlled by the special geometry relation for c = 9 and its generalization for other c [10,12,13,43], which we re-derive in Appendix B using the formula (4.1). In the N = 4 notation, it takes the form where C J ik are the chiral ring coefficients. In Appendix D, we use the N = (4, 4) superconformal symmetry to find a relation between the curvature components (5.5), (5.6), and (5.7) R i11;j22;k11;ℓ22 + R i21;j12;k11;ℓ22 + R i21;j22;k11;ℓ12 = 0 . (5.8) This allows us to determine Comparing this with (5.7), we obtain as a by-product the following constraint on the chiral ring of any N = (4, 4) SCFT Combining these results, the full Riemannian curvature is

Supercurrent Bundle
Let us turn to the curvature of the bundle of the N = 4 supercurrents S αA +++ , 25 In Appendix C, we determine the connected four-point function using N = (4, 4) Ward identities to be S αA (5.14) 24 A curious observation is that (5.10) implies that there is a uniform bound on chiral ring coefficients (squared) associated with the R-charge (1, 1) chiral primaries. The bound takes the schematic form C 2 < 2 − 1/k. Note that this is very different from the N = (2, 2) case, where the chiral ring coefficients can blow up, say, as at the conifold point. 25 The k dependent normalization is due to the two-point function S αA The y-integral in (5.13) can then be performed R iCĊ;jDḊ;αA;βB = − 1 2k η ij ǫĊḊǫ αβ (ǫ AC ǫ BD + ǫ AD ǫ BC ) , (5.15) and R iCĊ;jDḊ;αȦ;βḂ = − 1 2k η ij ǫ CD ǫαβ(ǫȦĊǫḂḊ + ǫȦḊǫḂĊ) . (5.16) The nontrivial SU (2) out × SU (2) out holonomies shown in these curvatures means that it is not possible to choose an N = 2 subalgebra consistently, even on a local patch of M.
Since the tangent bundle T M is a tensor product of the left and right supercurrent bundles and the bundle of weight 1 2 , 1 2 chiral primaries, the curvature tensor for the supercurrents computed here can be combined with the curvatures for the chiral primaries computed in [19] to reproduce the Riemann curvature (5.11) on M.
Although M is not Kähler, it has Kähler sub-manifolds. In fact, the maximal Kähler sub-manifold of a quaternionic-Kähler manifold is middle-dimensional [44]. In our case, the maximal Kähler submanifold of M is locally .

Appendix A. The Wess-Zumino Perspective
Anomalies arise when we have some operator that should satisfy an operator equation, e.g. ∂ µ j µ = 0 or T µ µ = 0, but then one finds that such an equation is only correct at separated points while at coincident points there are various contact terms. The essence is to show that these contact terms cannot be removed by redefining the scheme. Indeed, scheme redefinitions change the theory by various contact terms and so we need to demonstrate that the anomaly is invariant under scheme redefinitions. A convenient way to establish it is to introduce background fields for the various operators. Then scheme redefinitions correspond to adding new local terms to the action, which depend on these background fields and also, possibly, on the operators in the theory.
We would like to examine the operator [Q + , O|], which is normally zero if O is chiral.
To this end we couple a background field to this operator. A standard procedure is to couple the superfield O to a background field in the superpotential but then we do not have a source for the redundant operator. Therefore we will couple O to a source in the Kähler potential. We add a corresponding term for a twisted chiral superfield O: The general principles that we reviewed above tell us that for infinitesimal χ ± , ψ ± , should be a local functional of the sources and operators in the theory i.e.
with L local some local function of the couplings and operators. The right hand side in (A.5) is restricted by demanding that it is supersymmetric and also by demanding that it obeys the Wess-Zumino consistency conditions [45].
Let us assume that the partition function is invariant under the ψ ± transformations, namely,Õ obeys the twisted chiral shortening conditions at both separated and coincident points. We can then write the variation under χ − as follows (the formula for the variation under χ + is analogous) We now proceed to prove that (A.6) is cohomologically non-trivial. If (A.6) were cohomologically trivial then one could add a local term to log Z[A, B] such that the right hand side of (A.6) would vanish while retaining supersymmetry and invariance under ψ ± transformations. It is clear (by integration by parts and using (2.7)) that the χ − variation of the local term could cancel the right hand side of (A.6). However, (A.7) spoils the invariance of the partition function under ψ ± transformations. One can easily verify that indeed the right hand side of (A.6) is physical as long as we insist on supersymmetry and invariance under ψ ± transformations.
To summarize let us make some comments 1. Suppose we always insist on preserving N = (2, 2) supersymmetry. Then, if κ in (A.6) is nonzero, then it turns out that we may not be able to respect both (A.3) and (A.4).
In other words, we cannot embed the coupling constants of chiral and twisted chiral operators into short multiplets. At least some of the couplings have to be in longer multiplets.
2. We can view the χ and ψ transformations as analogous to U (1) A and U (1) V transformations in 2d electrodynamics. If we preserve one we must give up on the other, but choosing which one to preserve is at our discretion. Therefore, the situation is very similar to the way the usual chiral 't Hooft anomalies arise [39].
Note that from equation (A.6) we can immediately write the anomaly in operatorial formalism. This is because χ − couples toD + O and so we find But since our partition function is invariant under ψ ± transformations and hence depends only onλ =D + D − B, which is a standard twisted chiral background field, we can also writeD Hence in a "fermionic background" for the twisted chiral coupling, the operator O ceases to be chiral.
It is now straightforward to make contact with the analysis in the bulk of the paper.
Our discussion in this appendix has shown that there may be an anomaly with coefficient κ and that it would manifest itself as (A.9). Comparing with (3.12) we thus see that this coefficient is nonzero whenever the OPE coefficient in (3.9) is nonzero. Furthermore, the analysis in this appendix sheds light on the choices we could make in (3.11). Indeed, we

Appendix B. Special Geometry Relation
It is well known that for an N = (2, 2) SCFT, the Riemannian curvature for marginal deformations generated by chiral primary fields is determined in terms of the Zamolodchikov metric and the chiral ring coefficients [10,12,13,43]. For c = 9, this is known as the special geometry relation. Below we will give a simple derivation of this relation using (4.1) (our derivation is valid for any central charge c).
Following [8], we can use N = 2 Ward identities to express the integrand of (4.1) as a total derivative, The curvature is then computed by integration by parts, with non-vanishing contributions coming from the origin (y = 0). 27 Therefore, we need to study the order-one terms in the . We can act with the derivatives on the prefactor |y − w| 2 and thus reduce the problem to studying the y → 0 limit of O i (1)Ōj(y)O k (∞)Ōl(0) c . There is a contribution from 27 A priori, there can be boundary contributions from contours around y = 1 and ∞ as well.
However careful analysis of the integrand shows that such contributions are absent for this particular four-point function.
the unit operator in the t and u channel as well as a contribution from the (2, 2) chiral primaries, O i (1)Ōj(y)O k (∞)Ōl(0) c = g ij g kl + g il g kj −CjlJ g IJ C ik I , (B.2) where g IJ is the metric associated with the R-charge (2, 2) chiral primaries.

Appendix C. Four-Point Function involving Supercurrents
Let us focus on the connected four-point function Since the four-point function is holomorphic in z and w, it is determined completely by the poles in the OPE between the supercurrents and other insertions. We first consider the singularities in x, and denote the polar terms in (x − w), (x − y), (x − z) by I 1 , I 2 , I 3 respectively.

(C.2)
we have where we have dropped the disconnected pieces and also the terms involving the SU (2) R currents since F i are SU (2) R singlets. It is then easy to obtain using the OPE between T and F i Similarly looking at the OPE between S αA +++ and F CĊ i S αA +++ (z)F BḂ (w) = we have Using again (C.5) and also S αA +++ (z)O ββ (w) = δ α β 2(z ++ − w ++ ) Q γA + O γβ (w) + . . . , (C.7) we get Finally from the OPE between S αA +++ and F DḊ j we get and following the same steps as above we obtain (C.10) Putting together (C.4), (C.8) and (C.10) while taking the limit w → ∞, we arrive at (C.11) Appendix D. SU (2) out Selection Rules In this appendix we prove that F i11 (0) F k11 (y) F ℓ22 (1) F j22 (∞) + F i21 (0) F k11 (y) F ℓ12 (1) F j22 (∞) + F i21 (0) F k11 (y) F ℓ22 (1) F j12 (∞) = 0, (D. 1) follows from N = (4, 4) superconformal Ward identities. More generally, the four-point function of the exactly marginal operators F iAȦ F jBḂ F kCĊ F ℓDḊ respects SU (2) out Ward identities. 28 28 The SU (2) out is the outer automorphism group of the N = 4 superconformal algebra, but not a symmetry of the full SCFT. However, the operator spectrum of the theory can still be organized into SU (2) out representations by postulating that the N = 4 superconformal primaries transform as singlets. Here we see that certain correlation functions respect the SU (2) out invariance. This is analogous to the bonus U (1) Y symmetry in the 4d N = 4 super Yang-Mills theory [46]. More recently, the outer automorphism group figured in the study of the K3 SCFT from spacetime arguments in [47].
For notational simplicity, let us focus on the left-moving side of (D.1). Using N = 4 Ward identities and (D.4) Moreover, from (C.5) we can derive