Sphere partition functions and the Zamolodchikov metric

Journal of High Energy Physics, Nov 2014

We study the finite part of the sphere partition function of d-dimensional Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In odd dimensions, this quantity is physical and independent of the exactly marginal couplings. In even dimensions, this object is generally regularization scheme dependent and thus unphysical. However, in the presence of additional symmetries, the partition function of even-dimensional CFTs can become physical. For two-dimensional \( \mathcal{N}=\left(2,2\right) \) supersymmetric CFTs, the continuum partition function exists and computes the Kähler potential on the chiral and twisted chiral superconformal manifolds. We provide a new elementary proof of this result using Ward identities on the sphere. The Kähler transformation ambiguity is identified with a local term in the corresponding \( \mathcal{N}=\left(2,2\right) \) supergravity theory. We derive an analogous, new, result in the case of four-dimensional \( \mathcal{N}=2 \) supersymmetric CFTs: the S 4 partition function computes the Kähler potential on the superconformal manifold. Finally, we show that \( \mathcal{N}=1 \) supersymmetry in four dimensions and \( \mathcal{N}=\left(1,1\right) \) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions well-defined functions of the exactly marginal parameters.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP11%282014%29001.pdf

Sphere partition functions and the Zamolodchikov metric

Efrat Gerchkovitz 0 1 3 Jaume Gomis 0 1 2 Zohar Komargodski 0 1 3 Open Access 0 1 c The Authors. 0 1 0 Waterloo , Ontario, N2L 2Y5 , Canada 1 Rehovot 76100 , Israel 2 Perimeter Institute for Theoretical Physics 3 Weizmann Institute of Science We study the finite part of the sphere partition function of d-dimensional Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In odd dimensions, this quantity is physical and independent of the exactly marginal couplings. In even dimensions, this object is generally regularization scheme dependent and thus unphysical. However, in the presence of additional symmetries, the partition function of even-dimensional CFTs can become physical. For two-dimensional N = (2, 2) supersymmetric CFTs, the continuum partition function exists and computes the Kahler potential on the chiral and twisted chiral superconformal manifolds. We provide a new elementary proof of this result using Ward identities on the sphere. The Kahler transformation ambiguity is identified with a local term in the corresponding N = (2, 2) supergravity theory. We derive an analogous, new, result in the case of four-dimensional N = 2 supersymmetric CFTs: the S4 partition function computes the Kahler potential on the superconformal manifold. Finally, we show that N = 1 supersymmetry in four dimensions and N = (1, 1) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions well-defined functions of the exactly marginal parameters. - Conformal manifolds and sphere partition functions Sphere partition functions of superconformal field theories N = (2, 2) theories and supersymmetric two-spheres The S2 partition function An explicit regularization N = 1 field theories on S4 N = 2 field theories on S4 The Kahler potential from the four-sphere neighborhood of the reference conformal field theory p, there is a family of CFTs. We will refer to this family of CFTs as the conformal manifold, and denote it by S admits a natural Riemannian metric, the Zamolodchikov metric gij (p), given by [1] Conformal manifolds and sphere partition functions Suppose that in a d-dimensional CFT, denoted by p, there are exactly marginal operators 1 Z hOi(x)Oj (0)ipS = 1We choose a normalization that will make later formulae simpler. S and the operators Oi represent vector fields in S . Thus, under a change of variables Consider an observable O. Suppose we are interested in computing this observable as 1 Z It is very important that (1.4) is ambiguous. Indeed, even if we solve the reference we still need the integrated correlation functions in order to compute (1.4). Integrated correlation functions generically have ultraviolet divergences. They need to be regularized by introducing a cutoff and appropriate counterterms need to be added to the effective action so that the correlation functions are finite in the continuum limit. Therefore, computations of various observables via (1.4) may depend on the regularization scheme, and consequently be subject to ambiguities. Let us now analyze these ambiguities in some detail. The formalism where the couallows to easily classify all such ambiguities. Different regularization schemes simply differ by local terms of dimension equal or smaller than d in the effective action. In order to preserve Ward identities, one may need to fine-tune some of the counterterms. Let us briefly describe a few general pedagogical examples. The finite counterterm with Cij any symmetric tensor. This counterterm accounts for a power divergent piece Our primary interest in this paper lies in CFTs on curved spaces. A conformal field theory can be placed canonically on any conformally-flat manifold, in particular, on Sd (in on the conformal manifold S [2]. In even dimensions, the local counterterm follows from conformal Ward identities. Indeed, to leading order in the expansion (1.4) we get R ddxhL(0)Oi(y)Oj (x)i. This correlator is infrared finite but has an ultraviolet tuning the local term (1.6). Finally, another example of a counterterm is this case we can use the stereographic map). Since there are no infrared divergences, we can study the partition function ZSd . We can repeat this procedure at any point on the conformal manifold. This provides an interesting probe of the conformal manifold, ZSd (p). While infrared divergences are absent, ultraviolet divergences remain. As in our disd = 2n : log ZS2n = A1(i)(rUV )2n + A2(i)(rUV )2n2 + . . . + An(i)(rUV )2 d = 2n + 1 : log ZS2n+1 = B1(i)(rUV )2n+1 + B2(i)(rUV )2n1 where r is the radius of Sd. The power-law divergent terms correspond to counterterms of in even dimensions and in odd dimensions. Therefore, all the power-law divergent terms in (1.8) and (1.9) can be tuned to zero in the continuum limit. In even dimensions, the sphere partition function has a logarithmic dependence on the radius (see (1.8)), which cannot be canceled by a local counterterm. It is associated to the Weyl anomaly. The variation of the partition function under a Weyl transformation with in the Weyl anomaly vanish on the sphere, see [3]). However, this violates the Wess-Zumino with the usual A-type anomaly [3], which is therefore independent of exactly marginal deformations.2 This is necessary for its interpretation as a monotonic function under as it can be removed by the local counterterm3 In summary, we have shown that the only physical data in the continuum limit of ZS2n is the A-anomaly, which is independent of the exactly marginal parameters. 2We thank L. Di Pietro and A. Schwimmer for discussions. has a natural normalization via radial quantization. For odd dimensions, absent additional restrictions on the counterterms, we have seen above that all the Bi are ambiguous and can be tuned to zero (a logarithmic term is absent because one cannot write an appropriate local anomaly polynomial in odd dimensions.) only conceivable dimensionless counterterm would be a gravitational Chern-Simons term Moreover, it has to have an imaginary coefficient due to CPT symmetry. Hence, the choice of regularization scheme that preserves coordinate invariance.4 It measures the finite entanglement entropy across a S2n1 in IR2n,1 [11].5 . Start at an arbitrary point on the conformal manifold and expand to second order ddx ghOii + ddx g ddy ghOi(x)Oj (y)i . (1.14) Conformal Ward identities on the sphere imply that hOii = 0 . Since this is true at every point on the conformal manifold, we conclude that the continuum conformal field theory sphere partition function is independent of exactly marginal parameters. As we will see, this argument cannot be repeated in even dimensions because there are finite counterterms. Indeed, we will see examples where the sphere partition function does depend on exactly marginal parameters. We can understand this simplicity of the sphere partition function in odd dimensions from another point of view. The integrated two-point function of exactly marginal operators in the last term of (1.14) is ultraviolet divergent. The singularity arises from the domain where x y. The two-point function on the sphere can be obtained from the corresponding two-point function in flat space by a stereographic map. Hence, the integrated two-point function in (1.14) is proportional to ddyg 4The imaginary part is more subtle. Only its fractional part is well defined. See, for example, the discussions in [810]. We will not comment any further on the imaginary part of F2n+1. 5The entanglement entropy provides another way to see that the finite part in even dimensions is ambiguous while in odd dimensions it is physical. Indeed, it is straightforward to write finite local counterterms on the entangling surface of even-dimensional spheres, while in odd dimensions this is impossible. For example, absolute value renders it nonlocal, while without the absolute value symbol it is not consistent with the vacuum being a pure state. We thank S. Pufu for discussing this with us. where d(x, y) is the SO(d + 1) invariant distance on the sphere Sd. This can be regulated in a region where it converges, analytically continuing to the region of interest at the end, 0 for d = 2n + 1 . Therefore the integral vanishes in odd dimensions, and since we have shown that there are no finite counterterms in odd dimensions, the answer computed in any other choice of regularization scheme would yield the same result. The fact that the second term in the expansion (1.14) vanishes holds true around any point on the conformal manifold. This is consistent with the independence of the partition function of exactly marginal couplings. Note that the same regularization scheme yields a nonzero answer in even dimensions. This will be important below. In summary, we have shown that the (real part of the) finite part of ZS2n+1 in the continuum limit is physical and unambiguous, and is independent of the exactly marginal parameters of the conformal field theory. This is a necessary requirement for the finite part of ZS2n+1 to serve as a candidate monotonic function under renormalization group flows Sphere partition functions of superconformal field theories Our analysis of the ambiguities of the sphere partition function of a CFT followed from assuming that the partition function can be regulated in a diffeomorphism invariant way. We then classified all the diffeomorphism invariant counterterms and determined their influence on the sphere partition function. Starting in this section, we pursue an analogous analysis for superconformal field theories (SCFTs). Imposing that the sphere partition function of SCFTs can be regulated in a supersymmetric way, we show that the partition function of SCFTs with various amounts of supersymmetry and in various dimensions have (not unexpectedly) a more restricted set of ambiguities. This makes the sphere partition function of such SCFTs interesting, rich observables: the sphere partition functions of some of these theories are known to compute the Kahler potential on the conformal manifold [14, 15]. We now outline the logic of our arguments. Consider a d-dimensional SCFT and place it on the sphere by stereographic projection. Such a SCFT is by definition invariant under the corresponding superconformal algebra. As we explained in section 1, however, the partition function of a CFT suffers from ultraviolet divergences that arise when exploring the dependence of the partition function on the conformal manifold S . We now assume that we can regulate the SCFT partition function while preserving the subalgebra of the superconformal algebra that closes into the super-isometries of the d-dimensional supersymmetric sphere, which projects out all conformal generators of Sd. This is the general supersymmetry algebra of a massive supersymmetric theory on the d-dimensional sphere. In the supersymmetric context, the local counterterms that parametrize the ambiguities of the partition function are now supergravity counterterms. In supersymmetric theories, the coordinates that parametrize the conformal manifold are the bottom component of an apropriate supersymmetry multiplet. Therefore we must consider the most sponding supergravity theory. The supergravity theory that must be considered is found by embedding the rigid supersymmetric d-dimensional sphere as a supersymmetric background of d-dimensional off-shell supergravity, in the framework put forward by [16]. By constructing marginal supergravity counterterms in a given supergravity theory, we can determine the ambiguities and the leftover physical content of the partition function of SCFTs on Sd. Let us briefly summarize our main results explained in the rest of the paper. In manifold of such theories is a Kahler manifold, and locally takes a direct product form Sc Stc, where Sc, and Stc are the chiral and twisted chiral manifolds, respectively. These theories can be placed on supersymmetric S2 in two different ways while preserving four supercharges [15, 1719]. For each choice there is a corresponding two-dimensional of [17, 19], where the corresponding supersymmetry algebras were labeled by SU(2|1)A and SU(2|1)B. Imposing that the SCFT partition function preserves SU(2|1)B, we find that there is a finite B-type supergravity counterterm [20] sure. A similar A-type supergravity counterterm can be constructed for a SCFT preserving S2 background they give rise to the following Kahler ambiguities in the partition function of counterterms, then the partition functions may differ by (2.2). This means that the two-sphere partition functions ZA and ZB are generally not functions, but rather sections: the transitions between different patches may involve holomorphic functions. Our explicit construction of these counterterms gives a microscopic realization of the Kahler ambiguities implied by the proof [15] that the exact two-sphere partition function [1719] computes the Kahler potential K on the conformal manifold Sc Stc [14, 15] ZSA2 = eKtc ZSB2 = eKc . For some followup work see e.g. [21, 22]. One well-known way in which the partition function can turn into a section is if there are t Hooft anomalies. Then, the partition function transforms nontrivially under gauge transformations. Ordinary derivatives of the partition function then yield gauge noninvariant quantities (which can be fixed by the Bardeen-Zumino procedure [23], or equivalently, anomaly inflow [24]). Here we encounter a reminiscent situation. The partition function becomes a nontrivial section. As a result, various derivatives of the partition function need to be supplemented with an appropriate connection that for holomorphic nice to understand if the analogy with ordinary anomalies is deeper than these superficial In section 3 we also give a new elementary proof of (2.3) that follows from supersymmetry Ward identities and therefore does not require the existence of a Lagrangian description or localization of the partition function. We also show that the two-sphere parthat changes the finite part of the partition function arbitrarily, similar to what we found for even-dimensional nonsupersymmetric CFTs. give rise to a Kahler manifold [25]. We argue that in this case the partition function on the four-sphere does not have a preferred, unambiguous nontrivial continuum limit. We show Evaluated on S4, this shifts the finite part of the partition function by an arbitrary function ways (one of them being supersymmetric localization and the other by using an explicit supersymmetric regularization of the second term in (1.14)) that of the moduli. ZS4 = eK/12 , where K is the Kahler potential on the space of exactly marginal deformations S. That is, the four-sphere partition function becomes physical if the counterterms are restricted partition function is again a section rather than a function, where the transition functions are holomorphic plus anti-holomorphic in the moduli. Note the sign difference of the exponent (2.5) with respect to (2.3). One can therefore, in principle, use the localization computation on S4 [26] to comcan also derive (2.5) by generalizing the argument of [15] (i.e. geometrically deforming the equations should be along the lines of [28]). One may also try to extend to four dimensions our supersymmetry Ward identity proof in section 3. It would also be very nice to counterterm (2.1), perhaps using the tools of [29, 30]. Finally, it would be interesting to geometric structure beyond Kahler (additional geometric structure is known to exist in In this paper we have confined ourselves to discussing conformal field theories and their sphere partition functions. However, some of our results can be relevant for the interpretation of partition functions of gapped theories as well. For example, if we study a massive theory with a gap at the scale M on a manifold with typical scale R much larger than the the scale associated to the gap R 1/M , then the partition function corresponding to local terms in the action for the background fields. Therefore, supergravity counterterms of the type discussed in this paper can be useful to understand the leading terms in the expansion of partition functions of massive supersymmetric theories. R-symmetry group. Exactly marginal operators are superconformal descendants of operators in the chiral and twisted chiral rings which carry charge (2, 0) and (0, 2) under U(1)V U(1)A. We can view the coupling constants for these operators as charge (0, 0) background chiral superfields and charge (0, 0) background twisted chiral superfields, rethe metric in the chiral and twisted chiral directions.6 and the Zamolodchikov metric takes a factorized form Sc Stc: locally it is the sum of Such conformal field theories can be placed on S2 by a stereographic transformation. Killing spinors and , which satisfy 6One can derive this as follows. Regardless of supersymmetry, in any two-dimensional CFT, promoting where we omitted the usual A-type anomaly, as we have already discussed it in the introduction. Ckl is some symmetric tensor that depends on the exactly marginal parameters. Ckl is proportional to the Zamolodchikov metric on the space of CFTs. This is because the momentum space two-point function of exactly marginal operators looks like p2 log(p2), which clearly contains a rescaling anomaly. Such a nonlinear sigma model (3.1) is supersymmetrized as usual, by replacing the two-derivative term with a Kahler function for the background superfields. Hence, the total space of exactly marginal couplings is Kahler. Since the top component of the product of a chiral field with twisted chiral fields is a total derivative, this leads to a factorized Zamolodchikov metric. the R-symmetry group are as follows U(1)V U(1)A 1 1 1 1 The equations (3.2) have an eight-complex-dimensional space of solutions on S2. These furnish four doublets of the SU(2) isometry group of the two-sphere. The superconformal algebra on S2 can be linearly realized on supermultiplets. For our A superconformal invariant can be constructed from the top component of a chiral multiplet of Weyl weight 1 (rV = 2) I = d2xg F , since in this case F = im (m), and the variation integrates to zero. The twisted chiral multiplet transformations are given by [15] I = d2xg G is a superconformal invariant, since G = im ( Pm) im (P+m). The superconformal invariants (3.5)(3.7) represent the exactly marginal operators that are descendants of operators in the chiral and twisted chiral rings. The anti-chiral and antitwisted-chiral transformations and invariants are constructed similarly. The superconformal superalgebra just described admits interesting massive subalgetheories can be placed on S2 while respecting these subalgebras [15, 1719]. A regularization scheme can be thought of as a deformation of the theory by some massive sector. Hence, in order to understand the possible ways of regularizing the partition function ZS2 while preserving some supersymmetry, we need to study these massive subalgebras in detail. We will mainly focus on the subalgebras that preserve four supercharges, but will also pay some attention to one subalgebra that preserves two supercharges. sphere. They are generated by supercharges in the superconformal algebra that close into the SU(2) isometry of the two-sphere and a U(1) subgroup of the U(1)V U(1)A superconformal R-symmetry. The algebras that preserve U(1)V and U(1)A have been denoted by SU(2|1)A and SU(2|1)B in [17, 19]. They are mapped to each other by the One can also study massive subalgebras preserving only two supercharges. We do not carry out an exhaustive analysis of this case, but only discuss one example: the massive into the SU(2) isometries of S2, thus breaking the R-symmetry group completely. This supersymmetry algebra is obtained by restricting the conformal Killing spinors on the two-sphere (3.2) to obey so that = + + and = + + satisfy In stereographic coordinates, where ds2 = dimensional space of solutions is given by tion of (3.5) and (3.7) to these conformal Killing spinors. This supersymmetry algebra is obtained by restricting the conformal Killing spinors on the two-sphere (3.2) to obey m = m = = m = m = The SU(2|1)B supersymmetry transformations are the restriction of (3.5) and (3.7) to these conformal Killing spinors. OSp(1|2): massive N = (1, 1). The superconformal N = (1, 1) superalgebra is = two supercharges. The chiral and twisted chiral multiplets become reducible, each breaking up into two N = (1, 1) real scalar multiplets. The S2 partition function the formal expansion (1.4) contains divergences, we need to decide which symmetries our regulator preserves. Below we will analyze the consequences of the assumption that the superconformal algebra discussed above. Let us begin by assuming that the physics at coincident points is SU(2|1)A invariant. That means that we are allowed to use SU(2|1)A Ward identities, even at coincident points. With this assumption, the first observation we make is that the chiral multiplet invariant (3.5) is SU(2|1)A-exact. By taking = 0 in (3.5) we have that F = fore, the S2 partition function is independent of the chiral couplings constants. already shows that the two-sphere partition function is not completely ambiguous. The independence of the chiral couplings implies that there is no local SU(2|1)A supergravity counterterm that could reduce on S2 to a function of the bottom components of the chiral The twisted chiral multiplet invariant (3.7) is more interesting. It is not exact with respect to the SU(2|1)A subalgebra of the superconformal algebra. Indeed, in SU(2|1)A the SU(2|1)A Killing spinor equation (3.8). The SU(2|1)A-invariant partition function may thus depend on the twisted chiral moduli. Even though the top component of the twisted chiral multiplet G is not exact, its integral is almost so. From (3.7) we see that If + had been nowhere vanishing, then the integrated top component would have been + = = d2xg G = 4rY () = 4rY (S) . We note that the insertion of the bottom component of a twisted chiral multiplet Y is invariant under this supersymmetry transformation if the operator is inserted at the North Pole of the two-sphere (and if also vanishes at the North Pole). We can focus on the physics near the point where the spinor vanishes as follows. and z = 0), after some simple algebra we arrive at This expression is only valid away from the North Pole. We rewrite the supersymmetric invariant as + 2iz d2xg G = d2xg G + d2xg G , where DN = {x|x2 R2} and DS = {x|x2 R2}. Using the expression (3.17), which is regular in DS, we get d2xg G = d2xg z ishes and the entire contribution from DS comes from the second, total derivative term. Ultimately we will take the limit that R 0, so that the contribution from DN is vanishingly small. We thus need to evaluate the total derivative term in (3.19). Using polar coordinates, Stokes theorem, and the identities ||+||2 = 4r2 1 + 4zrz2 ||0|| d2xg G = 2r limR0 1 + 4Rr22 = 4rY (0) = 4rY (N ) . An analogous analysis for the top component of the anti-twisted chiral multiplet yields Therefore, the integrated twisted chiral (anti-twisted chiral) multiplet top component inside a correlation function inserts the bottom component of the twisted chiral (anti-twisted chiral) multiplet at the North (South) Pole of the two-sphere. We will now use these formulae to prove that ZSA2 = exp(Ktc). Differentiating the partition function twice with respect to twisted chiral moduli we get ij log ZSA2 = d2xg Gi(x) d2yg Gj (y)i . Using (3.21), (3.22) we arrive at In completing the proof we have used a supersymmetry Ward identity relating the two-point ambiguity which will be discussed in detail in the next subsection, ZSA2 = eKtc . ZSB2 = eKc . If we assume that the physics at coincident points is SU(2|1)B invariant, the analysis is similar to what we have done above, essentially only exchanging the role of chiral and twisted chiral multiplets. The twisted chiral invariant (3.7) is SU(2|1)B-exact. The chiral invariant (3.5) is not SU(2|1)B-exact, again, because one cannot construct it as a variation of a fermion without encountering some zero of a Killing spinor. Repeating the steps above ij ln ZSB2 = (4r)2hi(N )j (S)i = (2r)4hFi(N )F j (S)i = ij Kc . Thus, up to a holomorphic ambiguity which we will discuss below, OSp(1|2) described in the previous subsection. The exactly marginal operators are given by superconformal invariant is (3.5), but with F now being real. Let us now consider the sphere partition function regulated in an OSp(1|2) invariant way. One can attempt to repeat the analysis performed for SU(2|1)A and SU(2|1)B. However, we find that the integrated twopoint function of the top component of a real scalar multiplet does not reduce to the analysis. While it is still true that the top component is not supersymmetry exact (it is of derivation of (3.24) we used in a crucial way that the insertion of the bottom component of involving an arbitrary function of the moduli, which also sit in bottom components of SCFTs on S2 is scheme dependent and thus ambiguous. We will reach a very similar Before we turn to the holomorphic ambiguity afflicting the partition functions ZSA2,B, let us make some comments. We discussed what happens if the theory is regulated in a manner that preserves SU(2|1)A,B. But what if we could find a regulator that preserves a bigger symmetry group, i.e. the full superconformal group which has eight supercharges on S2? Clearly, such a regulator cannot exist. The reason is simply that if it had existed, using two different subgroups of the superconformal symmetry group, we would have arrived at two different results (3.25), (3.27). In fact, if all the eight supercharges are available to us, then one can formally prove that the chiral invariant (3.5) and the twisted chiral invariant (3.7) are both superconformal exact. Hence, one would be led to the conclusion that the partition function is independent of the exactly marginal parameters. The fact that group cannot be preserved. This is reminiscent of an anomaly. The Kahler potential admits the following well-known ambiguity supergravity counterterm that when evaluated on the two-sphere captures the Kahler ambiguity (3.28). For example, if we describe the same conformal field theory in two different duality frames, the partition functions may differ by some holomorphic (plus antiholomorphic) function of the moduli. This means that the partition function is a section on the conformal manifold. We proceed to construct this local supergravity counterterm explicitly. of the U(1) R-symmetry that is gauged (vector or axial), and are mapped into each other under the Z2 mirror automorphism. The rigid limits of these two supergravities correspond to preserving SU(2|1)A or SU(2|1)B on the two-sphere. We begin with the SU(2|1)B theory. The ambiguity (3.28) depends on the couplings to the exactly marginal operators from the chiral ring. They sit in background chiral multiplets of vanishing U(1)A R-charge. The supergravity counterterm supersymmetrizes the product of the Ricci scalar curvature R with the sum of an arbitrary holomorphic function and an anti-holomorphic function of chiral multiplets d2xg R The relevant local counter term is obtained from the SU(2|1)B supergravity action7 density superspace measure and R is a chiral superfield whose bottom component is a complex scalar auxiliary field that belongs to the SU(2|1)B supergravity multiplet. The spacetime scalar curvature R sits in the top component of the superfield R. When (3.30) is evaluated in the SU(2|1)B supersymmetric two-sphere background and the superfields establishes the Kahler ambiguity for the SU(2|1)B theory. Similarly, for the SU(2|1)A theory one has to consider twisted chiral density superspace measure. F is a twisted chiral superfield that contains as its lowest component a complex scalar auxiliary field that belongs to the SU(2|1)Asupergravity multiplet and the scalar curvature R in its top component. The rigid limit of this coupling evaluated on supersymmetric backgrounds was recently considered in [37]. This supergravity counterterm evaluated in the SU(2|1)A supersymmetric two-sphere back Hence, the finite piece of the two-sphere partition function has a holomorphic plus an anti-holomorphic ambiguity. The pieces that cannot be shifted away by a holomorphic and an anti-holomorphic function of the exactly marginal parameters are physical and calculable, as long as our regularization scheme respects SU(2|1)A or SU(2|1)B, as explained in the previous subsection. contains the graviton, the gravitino and a real auxiliary field B. The superspace description of this N = (1, 1) supergravity counterterm is 7We thank N. Seiberg for a discussion.,KetovES component [38]. This supergravity counterterm evaluated in the OSp(1|2) supersymmetric two-sphere background yields the marginal counterterm (3.32). Therefore, the sphere function of nonsupersymmetric CFTs, for which the S2 partition function does not have a preferred nontrivial continuum limit. As we shall see in section 4, we will reach the same An explicit regularization Let us for a moment forget about supersymmetry and examine a little more closely the proportional to the double integral ddxg Oi(x) ddyg Oj (y)i . hOi(x)Oj (y)iSd = d(x, y) = |x y| The two-point function on Sd can be obtained from the one in flat space by a Weyl transsufficiently large and positive the integral converges and evaluates to hRSd ddxg Oi(x) RSd ddyg Oj (y)i = gij vol(Sd)vol(Sd1) (d/2)(d/2+ ) . 2d+1 ( ) Setting d = 2, in the 0 limit we get that d2xg Oi(x) In the case without any supersymmetry, this procedure of regularizing (3.34) by continuing the dimension of the operator is just one of many possible regularization schemes and since the answer is not renormalization scheme invariant, there is no particularly special meaning see that (3.38) gives an incorrect answer in some situations! Indeed, we have shown above that, if we preserve SU(2|1)A, the partition function is independent of the chiral couplings. However (3.38) does not distinguish the chiral from the twisted chiral couplings. The resolution is, of course, that the procedure of analytically continuing the dimension of chiral operators is inconsistent with SU(2|1)A. This is because if one changes the dimension of a chiral operator in the superpotential, one necessarily breaks U(1)V and therefore SU(2|1)A. However, this procedure is perfectly SU(2|1)A invariant for the twisted chiral operators, and since we have already explained that the answer is unique, (3.38) must give the right result in this case. Indeed, (3.38) is completely consistent with the formula derived in (3.24), which implies (3.25). The mirror symmetric statement is that this regularization scheme preserves SU(2|1)B for chiral operators while it breaks it for twisted chiral operators, and therefore leads to (3.27). We will use this derivation via analytic continuation in the conformal dimension of the exactly marginal operators again in section 5, when we discuss four-dimensional N = 2 SCFTs. = 1 field theories on S4 superconformal field theories on their exactly-marginal couplings. The exactly-marginal conformal theories [39]. These theories have a U(1)R R-symmetry. The exactly marginal ral ring; they are the top component of a chiral multiplet with R-charge 2 (and Weyl weight 3). We note that not all such operators are exactly marginal in four-dimensional superconformal field theories with a normalizable vacuum state, where superconformal descendant operators of the type described in the previous section are necessarily exactly sphere. The superconformal transformations are generated by a Dirac conformal Killing spinor on S4, which obeys Under superconformal transformations the components of a chiral multiplet of Weyl weight w (and R-charge 23 w) transform as [41] with the projectors PL = 12 (1 + ) and PR = 12 (1 ), so that PL,R = L,R. T C, and C is the charge conjugation matrix. We have also defined chiral spinors of Weyl weight w = 3 I = d4xgF . 8The origin of this difference is as follows: as explained in [40], a marginal operator can cease to be to eat our marginal operator. Therefore, it has to be exactly marginal. the following conformal Killing spinors on S4 This represents a general marginal operator in the superconformal field theory. From now on, we will only discuss the exactly marginal ones. The parameters multiplying these operators, the coordinates in the conformal manifold, are realized as the lowest component of background chiral multiplets with vanishing R-charge. which corresponds to choosing m = The supersymmetry transformation generated by these spinors anti-commute to give the SO(5) isometry of the four-sphere, hence, projecting out the conformal and R-symmetry theories on S4 preserving (4.4). The action of this subalgebra on the chiral multiplet (4.3) is of course induced from the action of the full superconformal group, restricted to the One can attempt to repeat our analysis in two dimensions and aim to write the invariant (4.3) as an OSp(1|4) supersymmetry variation of a fermion modulo a total derivative and a zero. However, one quickly encounters a geometrical obstruction. One finds an addicommutator cannot be put to zero on S4 (loosely speaking, the OSp(1|4) algebra contains only complexified Killing vectors not commuting with their complex conjugate). Therefore, the invariant (4.3) is not OSp(1|4)-exact (even if one ignores zeroes of spinors and total derivatives). In fact, we now prove that the dependence of the sphere partition therefore analogous to the situation in non-supersymmetric theories. real function of the chiral and anti-chiral multiplets. This counterterm once evaluated on the OSp(1|4) invariant four-sphere is non-vanishing and explicitly demonstrates that the dependence on the moduli of the partition function is ambiguous. Denoting chiral fields 2 Z 3 The supergravity multiplet contains, in addition to the graviton and the gravitino, two measure and is the superfield that contains the square root of the determinant of the metric in its bottom component. The chiral superfield R has the auxiliary field M as its lowest component. After setting the gravitino to zero, we get: R = 6 M 6 2 R + Manifolds that preserve four supercharges obey the integrability conditions [16]: 2 R bb 2M M = 0 , metric sphere (in addition, we have set the gravitino to zero). a little algebra, we then obtain that (4.6) evaluates to 1 Z on the supersymmetric sphere. Thus, (4.6) supersymmetrizes the finite counterterm (4.10) for an arbitrary function of the exactly marginal parameters. This renders the sphere partition function of four-dimensional N = 1 SCFTs ambiguous. = 2 field theories on S4 superconformal field theories. These theories have an SU(2)R U(1)R R-symmetry. An exactly marginal operator in such a theory is realized as a superconformal descendant of the bottom component of a chiral multiplet of U(1)R R-charge 2 (and Weyl weight 2). superconformal transformations are parametrized by an SU(2)R R-symmetry doublet of left chiral conformal Killing spinors i and right chiral conformal Killing spinors i m weight w are given by [43] = /(A i) + 2 Bij 2 Bij i = 41 ab/ Fab i 2 / Bij jk 1 is invariant, where C is the top component of the multiplet (5.2). of Weyl weight 0. Note that Weyl weight 0 chiral superfields are irreducible [44], thus, in We are interested in unraveling the physical content of the sphere partition function of sphere S4. We describe it below in detail. We will prove that the S4 partition function of the Kahler potential on the conformal manifold The OSp(2|4) supersymmetry transformations are generated by conformal Killing spinors j and j obeying where 1jk = i3jk = (1jk), and 3 = spinors (5.1) with 0 1 ZS4 = eK/12 . m j = m j = I = d4xg C C = ij i/ j + 2wij ij = m(ij imj ) + (2w 4)ij ij , such a multiplet of Weyl weight 2. Indeed, it follows from (5.2) that (1 + w) Bij jk k + (1 w) ab Fabi (1 + w) Bij jk k + (3 w) ab Fabi , 9 These correspond to conformal Killing 9We introduce this notation in order to follow the notations of [41]. We can diagonalize equations (5.5) by defining and bj = j , where PL/R = 12 (1 ). on the fields of a chiral multiplet gives a representation of the N = 2 superconformal algebra on the fields With the choice of spinors (5.5), it is easy to prove that the vector field produced by two superconformal transformations i 2 2im 1 is a Killing vector on S4. Moreover, with the choice of spinors (5.5) the parameters associated to local dilatations and U(1)R R-symmetry vanish, while the SU(2)R R-symmetry is broken down to SO(2)R i12i + 1i2i i21i 2i1i = 0 , i12i 1i2i i21i + 2i1i = 0 , a = i( i12j + 1j 2i + i21j 2j 1i)aij = 1 = symmetry algebra, whose supercharges close into an SO(5) isometry and an SO(2)R R Using the OSp(2|4) supersymmetry transformations of a chiral multiplet (5.2) we expect that the top component C of the multiplet with Weyl weight 2 can be written as three consecutive supersymmetry transformations of a linear combination of fermions in the multiplet modulo a zero and a total derivative. One could then repeat the argument of section 3 and arrive at (5.4). Instead, here we follow a closely related strategy, extending to four dimensions the localization proof of (5.4) presented in [15]. In addition, we derive (5.4) by an explicit supersymmetric regularization. The two derivations agree. The Kahler potential from the four-sphere First, we employ the localization computation [26] of the S4 partition function of Laon vector multiplets with gauge group G and hypermultiplets transforming in a representation R of G. The partition function can be computed by localizing the functional integral with respect to a supercharge in OSp(2|4). For our analysis, the details of the hypermultiplets, which vanish on the localization saddle points [26], are not important. Therefore, we can focus on the N = 2 vector multiplets. ponent of an N = 2 chiral and anti-chiral multiplet of Weyl weight 2 Ai = S = d4xg X where Ci and Ci are the top components of the composite chiral and anti-chiral multiplets. The exactly marginal parameters are the complexified gauge couplings 2 . 1 Z Calculating the second derivative with respect to the marginal couplings we get ij log ZS4 = d4xg Ci(x) d4yg Cj (y)i = (32r2)2hAi(N )Aj (S)i. In the final step, we have used supersymmetric localization. To relate this to the Zamolodchikov metric we can use the supersymmetry transformations (5.2) to relate the two-point function of the bottom components (5.15) to the 1 ric. The result is hCi(N )Cj (S)i = (2r)8 gij . Combining all the factors we find Finally, we need to relate the two-point function hCi(N )Cj (S)i to the Zamolodchikov metThis shows (2.5) ij log ZS4 = 12 gij = 12 ij K . ZS4 = eK/12 . We can now compare this result to another derivation. In section 3, equation (3.37), we have evaluated the integrated two-point function using regularization by analytic continuation in the scaling dimension. As we have explained in section 3, this regularization does not always work. One needs to make sure that it preserves the massive supersymmetry 11The argument below can be carried without referring to a specific microscopic realization. This requires Plugging d = 4 in (3.37) and taking the limit 0, we find precisely (5.16) (after correctly normalizing the operators, as in (1.1)). This therefore provides a derivation of (5.16) that does not depend on localization. i.e. the gauge coupling. The S4 partition function depends on the masses of the fields in the adjoint hypermultiplet, and those need to be tuned such that the four-sphere partition function is that of the conformally coupled theory. For the correct, conformal choice of the mass parameter [45], the instanton contributions to the four-sphere partition function manifestation of a fact that we have explained in detail in section 3; the partition function is a section rather than a function. This Kahler transformation should be understood as a local supergravity counterterm, similar to the ones we have found in section 3. We leave We are very grateful to C. Closset, L. Di Pietro, N. Doroud, T. Dumitrescu, B. Le Floch, D. Gaiotto, S. Lee, J. Maldacena, V. Narovlansky, J. Polchinski, A. Schwimmer, A. Van Proeyen for useful discussions. We especially thank N. Seiberg for helpful discussions at various stages of the project. JG and ZK are grateful to the KITP for its warm hospitality during the initial stages of this project, which was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. EG and ZK thank the Perimeter Institute for its very kind hospitality during the course of this project. Research at the Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI. J.G. also acknowledges further support from an NSERC Discovery Grant and from an ERA grant by the Province of Ontario. ZK is supported by the ERC STG grant number 335182, by the Israel Science Foundation under grant number 884/11, by the United States-Israel Binational Science Foundation (BSF) under grant number 2010/629, and by the I-CORE Program of the Planning and Budgeting Committee and by the Israel Science Foundation under grant number 1937/12. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Theory, JETP Lett. 43 (1986) 730 [INSPIRE]. Lett. B 220 (1989) 153 [INSPIRE]. Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE]. JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE]. 069 [arXiv:1112.4538] [INSPIRE]. (1989) 351 [INSPIRE]. (2012) 053 [arXiv:1205.4142] [INSPIRE]. Unitarity and F-Maximization in Three-Dimensional Superconformal Theories, JHEP 10 Chern-Simons Contact Terms in Three Dimensions, JHEP 09 (2012) 091 [arXiv:1206.5218] [INSPIRE]. entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE]. Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE]. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE]. [arXiv:1208.6244] [INSPIRE]. JHEP 04 (2013) 019 [arXiv:1210.6022] [INSPIRE]. 06 (2011) 114 [arXiv:1105.0689] [INSPIRE]. [15] J. Gomis and S. Lee, Exact Kahler Potential from Gauge Theory and Mirror Symmetry, [16] G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP Gauge Theories, JHEP 05 (2013) 093 [arXiv:1206.2606] [INSPIRE]. vortices, arXiv:1206.2356 [INSPIRE]. complex moduli, JHEP 1312 (2013) 99 [arXiv:1309.2305] [INSPIRE]. Function, JHEP 01 (2013) 142 [arXiv:1211.0019] [INSPIRE]. and equivariant Gromov-Witten invariants, arXiv:1307.5997 [INSPIRE]. Domain Walls, Nucl. Phys. B 250 (1985) 427 [INSPIRE]. Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE]. B 303 (1988) 286 [INSPIRE]. F-theory, arXiv:1404.7645 [INSPIRE]. [35] H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE]. Phys. B 324 (1989) 427 [INSPIRE]. JHEP 07 (2014) 075 [arXiv:1404.2636] [INSPIRE]. http://itf.fys.kuleuven.be/toine/LectParis.pdf.


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP11%282014%29001.pdf

Efrat Gerchkovitz, Jaume Gomis, Zohar Komargodski. Sphere partition functions and the Zamolodchikov metric, Journal of High Energy Physics, 2014, 1, DOI: 10.1007/JHEP11(2014)001