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Computing 1/N^2 corrections in AdS/CFT

Stringy corrections in AdS/CFT generally fall into the category of either \alpha' effects or string loop effects, corresponding to 1/\lambda and 1/N corrections, respectively, in the dual field theory. While \alpha'^3R^4 corrections have been well studied, at least in the context of N=4 super-Yang-Mills, less is known about the 1/N^2 corrections arising from closed string loops. In this paper, we consider AdS_5 x SE_5 compactifications of the IIB string, and compute the closed string loop correction to the anomaly coefficients a and c in the dual field theory. For T^{1,1} reductions, we find the string loop correction to yield c-a=1/24, which is the contribution to c-a of a free N=2 hypermultiplet. We also comment on reductions to lower dimensional AdS theories as well as the nature of T-duality with higher derivatives.


Introduction
While many important features of string theory may be investigated in its low energy limit, it is often desirable to go beyond supergravity and to examine distinguishing features that separate string theory from ordinary supergravity. Such stringy effects include Kaluza-Klein modes and non-perturbative states such as D-branes as well as string worldsheet effects arising from the string loop expansion and the α ′ expansion. In fact, the latter α ′ expansion, which is equivalent to a higher derivative expansion in the effective field theory, has attracted much recent attention for multiple reasons.
From a quantum gravity point of view, higher derivative corrections serves as a means of probing string theory at a fundamental level. This has been successfully applied to the study of stringy black holes and higher derivative effects on black hole entropy [1][2][3][4] (see e.g. [5,6] and references therein). Alternatively, higher derivative corrections also play an important role in AdS/CFT, showing up as finite coupling and in some cases 1/N effects in the dual field theory.
The string α ′ expansion naturally leads to a higher derivative expansion in the effective field theory, with each factor of α ′ accompanied by two additional derivatives. In most cases, the focus has been on the gravitation action, including α ′ R 2 terms in the heterotic effective action and α ′3 R 4 terms in the type II theories. However, the complete α ′ expansion involves not just Riemann terms but all fields of the theory. Such corrections in their entirety are rather complicated, and often only incomplete information is known. Nevertheless, the first higher order curvature terms are generally well established, and for non-flux backgrounds, they are often sufficient for most purposes.
The perturbative α ′3 R 4 corrections in type II string theory arise at both tree level and one-loop order. In the IIA case, the corrections take the schematic form The t 8 t 8 R 4 terms were first obtained by direct calculation of four graviton scattering at tree level [7] and one loop [8] order, while the ǫ 10 ǫ 10 R 4 terms are related to the eight-dimensional Euler density, and first arise at the level of five graviton scattering. The one-loop CP-odd term B ∧ X 8 is related to the five-brane anomaly, and computed in [9,10]. This is the IIA analog of the heterotic Green-Schwarz term, and can be computed directly through a parity violating five-point amplitude [11,12] or more abstractly through the elliptic genus [13]. Because of its topological nature, the B ∧ X 8 term in (1.1) provides a useful handle on the study of higher derivative corrections in dimensionally reduced IIA theories. While there are often technical difficulties involved for practical calculations, abstractly once the corrections arising from B ∧ X 8 are pinned down, many of the remain terms may be obtained by supersymmetry. As an example of this, consider the lift of IIA to eleven-dimensional supergravity. In this case, B ∧ X 8 lifts to C 3 ∧ X 8 , so that [10]  Compactifying to five dimensions on a Calabi-Yau three-fold gives N = 2 supergravity coupled to n v vector multiplets and n h hypermultiplets with n v = h (1,1) − 1 and n h = h (2,1) + 1 [14]. Focusing only on the vector multiplets, the compactification of (1.2) on CY 3 proceeds by expanding the Kähler form J in a basis of (1, 1) forms ω I on CY 3 . The Chern-Simons term then reduces in a straightforward manner (1.4) where c IJK are the triple intersection numbers. Similarly, the gravitational Chern-Simons term reduces as [15,16] M 11 where c 2I arises from the expansion of the second Chern class (1.6) The power of supersymmetry then enables us to deduce the entire five-dimensional N = 2 action for the vector multiplets in terms of the topological data c IJK and c 2I . In particular, at the R 2 level, the supersymmetric completion of A I ∧ Tr R 2 was obtained in [17] using superconformal tensor calculus and an off-shell formalism [18][19][20][21]. The resulting bosonic action has the form [17] The addition of these R 2 terms lead to corrections to the entropy of five-dimensional N = 2 black holes [22][23][24]. Furthermore, as the R 2 terms are related to the five-brane anomaly, many of these entropy results are in fact exact [2,3,25].

R 2 corrections and AdS/CFT
The supersymmetry analysis of [17] suggests that any R 2 correction to five-dimensional N = 2 supergravity has the form (1.7), with a precise relation between the coefficient of C 2 µνρσ and the gravitational Chern-Simons term A I ∧ Tr R 2 . Truncating to the pure supergravity sector and integrating out the auxiliary fields of the off-shell theory, the effective four-derivative action has the form [26] At this level, the theory is completely determined by two parameters: L, the AdS radius and α, the coefficient of the four-derivative correction terms. Note that we have chosen a non-canonical normalization for the graviphoton which however is natural in the context of IIB supergravity reduced on a Sasaki-Einstein manifold. As highlighted above, for Calabi-Yau compactifications of eleven-dimensional supergravity, α is given by the second Chern class of CY 3 . However, in an AdS/CFT setup, α also has a direct relation to the central charges of the dual gauge theory. This is perhaps best seen through the holographic Weyl anomaly [27], where the αC 2 µνρσ term in (1.8) shifts the leading supergravity result [28][29][30][31]. Anomaly matching then yields the AdS/CFT connection [26,32] where L is the AdS radius, and where a and c are the central charges of the dual N = 1 gauge theory. Large N theories with an AdS dual have leading behavior a = c ∼ N 2 /4 [27]. However, from (1.9) we see that 1/N and further subleading corrections will show up in the dual gravity theory as R 2 corrections parametrized by α. These R 2 corrections have received much recent attention in computations of the shear viscosity and consequences for the conjectured KSS bound η/s ≥ 1/4π for the ratio of the shear viscosity to the entropy density of the dual gauge theory plasma [33,34]. In particular, at linearized order, the ratio takes the form [26,32,[35][36][37] so that theories with c > a will violate the KSS bound.
While many examples of super-Yang Mills theories are known with c = a, we are mainly interested in theories admitting a dual string description. Several explicit examples have been constructed with, e.g., seven-branes and orientifolds, where c − a ∼ O(N ) [29,[38][39][40][41]. From the stringy point of view, the correction α in (1.8) arises from the effective theory of the branes at the singularities, and can be viewed as an open string effect that gives rise to a 1/N correction to the leading N 2 behavior of the central charges.
In this paper, we wish to examine the closed string and hence O(1) corrections to c − a by appropriate reduction of the higher derivative terms in the bulk effective action. Focusing on Sasaki-Einstein compactifications of the IIB string, we immediately run into a puzzle. Namely, as discussed above, the α correction term in (1.8) is easily related to the reduction of the C 3 ∧ X 8 term in elevendimensional supergravity or the corresponding B 2 ∧ X 8 term in IIA theory. However, it is well known that such a B 2 ∧ X 8 term is absent in the IIB case, as the (p, q) IIB fivebrane is non-chiral. Thus, in contrast with (1.1), the α ′3 corrections in IIB have the schematic form 11) where in fact the tree and loop terms combine with non-perturbative corrections into a modularinvariant form in terms of the IIB axi-dilaton. When reduced on SE 5 , it is clear that the above action will not yield a non-vanishing A ∧ Tr R 2 term of the form given in (1.8). Hence this suggests that α = 0, and therefore that all gauge theories dual to IIB string theory on AdS 5 × SE 5 will have a vanishing 1/N 2 correction to the central charges (but will still generically have 1/λ 3/2 corrections arising from R 4 terms in the dual theory). The above argument for the absence of 1/N 2 corrections, however, fails to fully take stringy effects into account. In particular, as we demonstrate below, the finite volume of SE 5 leads to a nonvanishing contribution to α which is not present in uncompactified IIB theory. One way to see this is to note that any Sasaki-Einstein manifold admits a preferred U(1) fibration over a four-dimensional where dA = 2J, with J the Kähler form on B. This isometry circle then allows us to relate IIB theory on AdS 5 × SE 5 to IIA theory on AdS 5 × B × S 1 via T-duality [42,43]. This circle furthermore allows us to reduce first from ten to nine dimensions, and then from nine to five. In nine dimensions, the B 2 ∧ X 8 term in IIA theory reduces to Under T-duality, we thus see that the compactified IIB theory necessarily has a similar term, however this time with A µ = g µ9 . Independent of the duality frame, this term reduces to five dimensions on the base B to give rise to a generically non-vanishing 1/N 2 correction parameterized by α in (1.8).
In fact, working at finite circle radius, we demonstrate that there are a large class of gravitational and mixed Chern-Simons terms of the form A ∧ X 8 which arise in string theory. While some of these have been identified previously, the full story appears to be as yet incomplete. We thus begin in Section 2 with a reexamination of such terms which arise from the one-loop CP-odd sector of type II string theory, paying particular attention to the requirements of T-duality invariance. Following this, in Section 3 we compute the O(1) corrections to c − a arising from the closed string sector of IIB theory on AdS 5 × SE 5 . Since these one-loop CP-odd terms are generic in string theory, we conclude in Section 4 with a discussion of similar corrections in AdS 4 and AdS 3 theories. We also comment on T-duality invariance in the presence of higher derivative terms in an Appendix.

One-loop CP-odd terms in string theory
Before considering the reduction to five dimensions, we review the origin of the B ∧ X 8 term in ten dimensions. The structure of this CP-odd term can be obtained from an explicit one-loop five-point computation following the procedure outlined in [11,12]. In the RNS formalism for the type II string, this parity violating term arises as a sum of two contributions, one from the odd-even and the other from the even-odd spin structure sector. Focusing on the odd-even sector, the one-loop amplitude may be set up with one vertex operator in the (−1, 0) picture and the remaining four in the (0, 0) picture along with a picture changing operator δ(β)ψ · ∂X in the left-moving sector. It is conventional to take the first vertex to be the antisymmetric tensor B µν and the remaining four to be gravitons. However, it should be noted that the NSNS fields h µν , B µν and φ all involve the same vertex operators, with the only difference being the nature of the polarization tensors ζ (i) . The five-point function of the vertex operators (2.1) vanishes unless all ten fermion zero modes are soaked up in the odd spin structure sector. This gives rise to an amplitude where the sum is over the three even spin structures. Upon integration of the vertex operators, this takes the form This integrand A(q) is only a function ofq = e −2πiτ , and computes the elliptic genus [13]. Proceeding either by direct computation or through the elliptic genus [9], we then see that the above amplitude gives rise to the B 2 ∧ X 8 term, as expected. Before proceeding, we note that the extra factor of 1/4τ 2 in (2.3) arises from the zero mode contraction which we will revisit below in the compact case. This factor is crucial in showing that the amplitude is a total worldsheet derivative, so that only the τ 2 → ∞ boundary term contributes when integrating τ over the fundamental domain [9,12].
Furthermore, as mentioned above, the closed string vertex operators (2.1) encode h µν , B µν and φ through the polarization tensors ζ (i) . Thus, at this linear order, it is clear that the B 2 ∧ X 8 term incorporates not just R 4 but also the full set of NSNS fields according to While this expression is linearized in H 3 and φ, we expect it to have a non-linear completion, so that R µνρσ becomes the curvature of the connection with torsionω = ω + H. This completion is also needed in order to obtain a T-duality invariant combination of g µ9 and B µ9 under circle reduction. We discuss this point further in the Appendix. The simple replacement of R(ω) byR(ω), however, cannot be the entire story, as the type II string receives contributions from both the odd-even and even-odd spin structures. These contributions are essentially identical except for the important fact that B µν , being antisymmetric, has opposite worldsheet parity from h µν and φ. Noting that the IIA and IIB amplitudes differ by a relative sign in the flip between odd-even and even-odd spin structures (because of the differing GSO projection), we finally obtain the result for the CP-odd sector and In particular, the B 2 ∧ X 8 (R) term is projected out in the IIB case by worldsheet parity. However, terms of the form B 2 ∧ R(∇H) 3 and B 2 ∧ R 3 (∇H) survive.

Reduction to D = 9 and T-duality
In the above, we have demonstrated the existence of a one-loop CP-odd correction to IIB supergravity given by (2.8). However, this term by itself has no effect on the holographic c − a computation, as H 3 vanishes in the AdS 5 × SE 5 background. Instead, new terms will show up when IIB string theory is compactified on a circle to nine dimensions. After all, from a IIA point of view, the familiar B 2 ∧ X 8 (R) term in (2.7) may be reduced on a circle in the x 9 direction to give A 1 ∧ X 8 + B 2 ∧ X 7 where A µ = B µ9 and X 7 is the circle reduction of X 8 . Focusing on the first term, we note that T-dualizing to a IIB frame yields Thus T-duality guarantees that IIB theory on a circle necessarily includes a one-loop A 1 ∧ X 8 term. Of course, this IIB CP-odd term does not lift to ten dimensions, as it would schematically lift to g 2 ∧ X 8 (where g 2 = 1 2 g µν dx µ ∧ dx ν ), which however vanishes because of the symmetry of the metric.
To directly see what is happening in nine dimensions, it is instructive to return to the five-point one-loop amplitude. Assuming a circle of radius R and the first vertex in (2.1) to have a leg on the circle (so that ζ µ9 ), we end up with a zero mode contraction Here where n and w correspond to momentum and winding on the circle, and the expectation is with respect to the partition function Z = Tr q (α ′ /4)p 2 This replaces the contraction (2.4) in the non-compact case that was used to obtain the additional factor of 1/τ 2 in (2.3).
To see the effect of this circle compactification, we first consider the large radius limit, R → ∞. In this case, only the zero winding sector contributes, and we obtain up to exponentially suppressed corrections. As expected, this directly reduces to the non-compact zero-mode contraction (2.4). On the other hand, in the small radius limit, R → 0, only the zero momentum sector contributes, and we have instead The difference in sign is apparent from the opposite sign of the winding term in p R .
Combining the odd-even and even-odd spin structure sectors, we see that the IIA CP-odd amplitude is proportional to the factor while the IIB amplitude has the opposite behavior Since T-duality relates large and small radius compactifications of IIA and IIB theory, this result explicitly demonstrates the T-duality covariance of B 2 ∧ X 8 in nine dimensions with B 2 on the circle. In fact, the extension of T-duality to the full reduction of B 2 ∧X 8 necessitates the use of the curvature with torsion (2.6) and the fact that B 2 ∧X 8 (R) is a top form, so that T-duality will always flip between even and odd terms in B 2 in (2.7) and (2.8).
While the expressions (2.14) and (2.15) are appropriate in the large and small radii limits, we are of course interested in corrections arising at a finite radius. With the supergravity limit in mind, we work with R finite and larger than √ α ′ . As in the R → ∞ limit, only the zero winding sector contributes. However, approximating the momentum sum by an integral is only valid for τ 2 R 2 /α ′ . For τ 2 R 2 /α ′ , the zero mode contraction (2.12) becomes exponentially suppressed. Hence the radius provides a natural cutoff The implication of this is that when the zero mode contraction takes place on the x 9 circle the integral over the fundamental domain in (2.3) is cut off at τ 2 ∼ R 2 /α ′ . So long as R 2 ≫ α ′ , the boundary contribution to the amplitude is still evaluated at large τ 2 and is hence dominated by theq 0 term in the elliptic genus A(q). Integrating dτ 2 /τ 2 2 up to a cutoff of R 2 /α ′ then demonstrates that the amplitude A picks up a finite radius correction factor of 1 − α ′ /R 2 compared to the non-compact result.
In addition, it is important to note that the discrete momentum sum for the partition function on a circle will affect the amplitude even if the zero mode contraction is in a non-compact dimension, as in (2.4). In particular, the bosonic zero mode contribution in (2.11) takes the form (2.17) Since this contribution no longer falls off as 1/ √ τ 2 as τ 2 → ∞, the integral of the elliptic genus A(q) is enhanced by a factor of 1 + α ′ /R 2 whenever the zero mode contraction ∂X µ ∂X ν is over a non-compact dimension. Combining these two finite radius corrections, we may now refine the above expressions (2.14) and (2.15) for the CP-odd amplitudes. In the large radius limit, the IIA amplitude is proportional to where A µ = g µ9 . What this indicates is that the α ′3 corrections to the effective supergravity actions contain the following CP-odd terms in nine dimensions 1 where of course the one-form potentials correspond to B µ9 and g µ9 for IIA and IIB, respectively.
Here we have explicitly written out the nine-dimensional Newton's constant, and furthermore these expressions are valid in the corresponding large radii limits. (For completeness, we note that there are other CP-odd term as well, such as those associated with B 2 ∧ X 7 . However, they vanish for the AdS 5 × SE 5 reduction and hence will not contribute to the c − a computation.) It is perhaps worth mentioning that the nine-dimensional CP-odd terms in (2.20) are related by naïve T-duality where R IIB = α ′ /R IIA . Thus we could have immediately written down the A 1 ∧ X 8 term for IIB theory compactified on a circle based on the existence of the corresponding well-known term in IIA theory. However, T-duality takes a large radius IIB theory into a corresponding small radius IIA limit, in which case the IIA supergravity reduction is not necessarily to be trusted so that a full string calculation is warranted. Nevertheless, it is reassuring to see that the string amplitude calculation and the supergravity reduction are in perfect agreement.
3 Actual computation for AdS 5 × SE 5 We now proceed to examine the O(1) contribution to c − a for N = 1 theories dual to IIB theory on AdS 5 × SE 5 . Our strategy is to take advantage of the fact that any Sasaki-Einstein metric may be given in terms of a U(1) fibration over a Kähler-Einstein base B. This allows us to proceed in two steps: first reduce to nine dimensions on S 1 , and then further reduce down to five dimensions on the base B. By taking the intermediate step of working in nine dimensions, we may then straightforwardly evaluate the nine-dimensional CP-odd term (2.20) to obtain the O(1) contribution to c − a.
At the two-derivative level, the full non-linear reduction of the bosonic sector of IIB theory on SE 5 was carried out in [44]. The dimensionally reduced fields (g µν , A µ ) comprise the bosonic components of the N = 2 supergraviton multiplet in five dimensions, and are related to the ten-dimensional fields according to Here we have written the Sasaki-Einstein metric as a U(1) bundle over B The resulting five-dimensional action is that of gauged N = 2 supergravity Here vol(SE 5 ) is the dimensionless volume of SE 5 and 2κ 2 10 = (2π) 7 α ′4 . The ansatz (3.1) corresponds to the reduction of IIB theory on a circle of constant radius L. From a nine-dimensional point of view, the fields are ds 2 9 = g µν dx µ dx ν + L 2 ds 2 (B), whereF 2 is the field strength of the Kaluza-Klein gauge field g µ9 . This allows us to directly compute the one-loop CP-odd term (2.20) which arises after circle compactification of IIB theory. Since H 3 vanishes for this background, we may take the familiar X 8 given in (1.3). Noting that the irreducible Tr R 4 term does not contribute for the direct product nine-dimensional metric (3.5), we obtain Comparing this expression with the general four-derivative action (1.8), and using the five-dimensional Newton's constant relation (3.4) allows us to extract the effective four-derivative coefficient We now use the AdS/CFT relation 4πg s N = L 4 /α ′2 (which is appropriate for the dual quiver gauge theories arising from a stack of D3-branes at the tip of the cone over SE 5 ) and the holographic anomaly matching relations (1.9) to write where vol(S 1 ) is the dimensionless volume of the U(1) circle. The a anomaly expression is familiar 2 , while the c − a difference picks up a calculable contribution from the closed string sector. Before proceeding, it is important to keep in mind that Tr R 2 in the expression for c − a is composed out of the pullback of the ten-dimensional curvature onto the base B. In particular, since the compactification manifold is a fibered space, the curvature of the U(1) bundle dA = 2J will contribute as well the curvature of the base B. In other words, the ten-dimensional Tr R 2 will reduce to the nine-dimensional TrR 2 plus terms involving the Kaluza-Klein field g µ9 through its field-strength F 2 . If it were not for the latter terms, then we would simply obtain where σ(B) = B p 1 /3 is the signature of the base B. This, however, is not the complete story, as the Kaluza-Klein gauge field is non-trivial on the base as well.

The Kaluza-Klein reduction of Tr R 2
In order to evaluate the contribution of the Kaluza-Klein gauge field to Tr R 2 , we take the explicit circle reduction ds 2 10 = e α e α + e 2ϕ (dψ +Ã) 2 . (3.10) The resulting spin connections are ω αβ =ω αβ − 1 2 e ϕF αβ e 9 , ω α9 = − 1 2 e ϕF α β e β − ∂ α ϕe 9 , (3.11) whereω αβ is the nine-dimensional spin connection computed from the nine-dimensional metric ds 2 9 = e α e α . The curvature two-forms are then As indicated in (3.5), for the U(1) fibered SE 5 , the nine-dimensional graviphoton field strength is a sum of two terms: the Kähler form of B and the five-dimensional graviphoton. Since we are primarily interested in extracting the coefficient of A ∧ Tr R 2 in five dimensions, we ignore the graviphoton contribution. In this case,F 2 = 2J is covariantly constant on B. (Note that we have explicitly scaled out the AdS radius factor L.) Furthermore, the U(1) circle has constant radius, so we set ϕ = 0. The curvature two-forms then simplify as Since B is a Kähler manifold, we may use the identitiesR ab J ac J bd =R cd andR ab J ab = 2ρ (where ρ is the Ricci form) to simplify the second term above. In this case we have

Reduction on S 5
Given the above result, we now present a few examples where c − a may be directly computed. We start with IIB theory on AdS 5 × S 5 , which yields the familiar duality to N = 4 super-Yang-Mills theory. In this case, the five-sphere may be written as U (1) bundled over CP 2 . The signature of CP 2 is σ(CP 2 ) = 1, while its volume is π 2 /2. As a result, the two terms cancel in (3.17), and we verify that c = a in N = 4 super-Yang-Mills. Alternatively, we may compute the Riemann curvature from the Fubini-Study metric and directly show that TrR 2 = −24J 2 . This demonstrates that it is not just the integrated expression in (3.17), but also the integrand itself that vanishes everywhere on CP 2 . The result that c − a remains unshifted by string loop contributions is of course consistent with the expectation that while decoupling of the center of mass U(1) takes U(N ) to SU(N ), this shift affects both a and c identically. Thus c remains identified with a, with c = a = (N 2 −1)/4. Note that, since the integrand of (3.17) is trivial, this string loop correction will continue to vanish for orbifolds of S 5 , such as S 5 /Z 3 . Since this model is dual to N = 1, SU(N ) 3 gauge theory, it suggests that the O(1) contribution to c − a ought to be c − a = 3/16, corresponding to three decoupled N = 1 vectors from the U(1) factors. Although the string loop calculation does not give such a shift, this could be accounted for by a second contribution to c − a arising from quantum corrections from the states in the Kaluza-Klein tower [45,46]. In particular, the Kaluza-Klein modes that need to be considered are those arising from the massless fields in ten dimensions compactified on S 5 /Z 3 , and it is precisely these modes that have not been captured by the string loop calculation. While the contribution of complete N = 8 supergravity multiplets would not shift c − a, the orbifolding breaks this to N = 2, in which case their contributions would no longer be expected to vanish. It of course remains to be seen whether a direct computation along the lines of [45,46] will reproduce the predicted value of c − a = 3/16.

Reduction on T 1,1
The next simplest case to consider is IIB theory on AdS 5 × T 1,1 [47]. Since the base of T 1,1 is S 2 × S 2 , we simply have TrR 2 = 0. (The vanishing of the signature can also be understood from the existence of both a self-dual and an anti-self dual harmonic two-form.) In this case we are left with the volume term vol(B) = (4π/6) 2 = 4π 2 /9. Substituting this into (3.17) then gives the T 1,1 result where we also used the fact that the volume of the U(1) fiber is 4π/3. Curiously, this is the contribution to c − a for a free N = 2 hypermultiplet or negative that of a vector multiplet 3 . It would be interesting to see how this result may arise from the perspective of the dual gauge theory. Generalizing the idea of the S 5 /Z 3 orbifold, we focus on N = 1, SU(N ) quiver gauge theories. Since we expect c − a to count the number of decoupled N = 1 vectors, the natural prediction would be c − a = (# of nodes in the quiver) 16 .

(3.19)
This gives c − a = 1/8 for the conifold gauge theory, which however does not agree with (3.18). While it is possible that we have lost a factor of three in the string loop computation, we instead suggest as above that there is a second contribution to c − a from the Kaluza-Klein tower so that c − a = 1 24 string loop + 1 12 supergravity loop = 1 8 , (3.20) where the supergravity loop contribution arises from the Kaluza-Klein tower in five dimensions. While it would be interesting to perform such a calculation, in practice the non-trivial Kaluza-Klein spectroscopy on T 1,1 [48] would appear to make this a challenge.

Reduction on Y p,q
We now turn to reductions on the Sasaki-Einstein manifold Y p,q , which are dual to a large family of N = 1 superconformal quiver gauge theories [49][50][51]. On the IIB supergravity side, the Sasaki-Einstein manifold Y p,q has topology S 2 × S 3 . The metric in canonical form (1.12) is given by ds 2 = 1 − cy 6 (dθ 2 + sin 2 θdφ 2 ) + dy 2 w(y)q(y) + w(y)q(y) 36 (dβ + c cos θdφ) 2 The parameters a and c are to be chosen to avoid conical singularities at the poles y 1 ≤ y ≤ y 2 where y 1 and y 2 are the two smallest roots of the cubic a − 3y 2 + 2cy 3 = 0. Note that c = 0 corresponds to T 1,1 , while if c = 0 we can rescale the coordinates to set c = 1.
In order to compute c − a given by (3.17), we take the natural vielbein basis dθ, e 2 = 1 − cy 6 sin θdφ, dy, e 4 = w(y)q(y) 6 (dβ + c cos θdφ). (3.23) For the U(1) fibration, we have (3.24) and it is easy to verify that dA = 2J where J = e 1 ∧ e 2 + e 3 ∧ e 4 . Given the above metric, we may directly compute TrR 2 on the base is the volume form on the base. In order to proceed, we need to integrate TrR 2 + 24J ∧ J on the base B. However, a difficulty arises in that for generic values of p and q the manifold Y p,q is irregular. This means that while the above expressions are valid locally the base B is ill defined as a base manifold. At best, for appropriate values of p and q the Sasaki-Einstein space is quasi-regular, and the base is then an orbifold. Although the base B may be ill defined, the five-dimensional Sasaki-Einstein manifold itself is smooth and free of curvature singularities. Thus instead of computing the signature and volume of B separately, we directly evaluate the five-dimensional quantity Integrating this over the entire SE 5 then gives ℓ(y 2 − y 1 )(1 + c(y 2 + y 1 )), (3.28) where ℓ = P 1 /p = P 2 /q is related to the period of the U(1) fiber in the notation of [49]. As a result, we find 2π/3 vol(S 1 ) 2 ℓ(y 2 − y 1 )(1 + c(y 2 + y 1 )). (3.29) Relating ℓ and the roots y 1 and y 2 to the Chern numbers p and q finally gives For p and q chosen appropriately, the square root becomes rational, and the base B is an orbifold. In this case, orbits of the U(1) fiber close, and we may take vol(S 1 ) = 2π/3, corresponding to 2π periodicity of the ψ ′ circle. However, for irregular Y p,q the orbits do not close, and this suggests that vol(S 1 ) should be taken to be infinite, in which case c−a would vanish. This difference in behavior for quasi-regular versus irregular Sasaki-Einstein manifolds appears rather unusual, and merits further investigation. Based on the dual quiver gauge theory with 2p gauge groups, we may expect from (3.19) that c − a = p/8. If this is the case, then the additional contribution from the Kaluza-Klein tower would have to compensate for the rather unwieldy function of p and q appearing in (3.30).

Graviphoton backgrounds and lower dimensional AdS reductions
While we have mainly focused on AdS 5 reductions of IIB supergravity, similar features arise when examining AdS 4 reductions of eleven dimensional supergravity on Sasaki-Einstein seven-folds given by a non-trivial circle fibration over a six-dimensional Kähler-Einstein base. In particular, we demonstrate that the reduction of C 3 ∧ X 8 gives rise to four-dimensional couplings of the N = 2 graviphoton T with the curvature tensor of the form and further compute its coefficient. So far, we had not been concerned with nontrivial ten-dimensional graviphoton backgrounds, since they do not give rise to AdS 5 compactifications. However, for AdS 4 reductions of eleven-dimensional supergravity on SE 7 , the ten-dimensional graviphoton is important, and some field redefinitions may be required in order to write down a consistent (one-loop) ten-dimensional action. To see how this arises, we first look at the reduction of C 3 ∧ X 8 to ten dimensions on a non-trivially fibered circle.
It is convenient to work on a twelve-dimensional manifold Y 12 whose boundary is the elevendimensional spacetime X 11 = ∂Y 11 . We are interested in the case where X 11 is a circle fibration over ten-dimensional spacetime M 10 : U (1)→X π → M . (In turn, M is a boundary to an eleven-dimensional manifold Y 11 ). The isometry is generated by a vector field v and the dual global connection one form is denoted by e: ı v e = 1 , d e = π * T , where T is the graviphoton field strength. Now consider the circle reduction: where the tilde eight-form is a polynomial of ten (rather than eleven)-dimensional curvatures and the graviphoton T . Every eleven-dimensional quantity respects the isometry, i.e. has a vanishing Lie derivative with respect to the vector v: L v (.) = (dı v + ı v d)(.) = 0. This means in particular that closed forms upon reduction yield closed forms of lesser rank. For now let us ignore the sources and take dG 4 = 0. Then L v G = 0 allows to write where F 4 and H 3 are ten-dimensional RR and NSNS fluxes respectively (ı v G = −H 3 ). The closure of G leads to a pair of ten-dimensional equations: Similarly, whereX 7 = −ı v X 8 . Note that all quantities with tildes are polynomials in the (ten-dimensional) curvature R and the graviphoton T :X n =X n (R 11 ) =X n (R 10 , T ). The closure of X 8 leads to: Moreover locally X 8 = dX . One can showX 7 = −ı v X 8 is not only closed, but is also exact:X 7 = dX 6 (R, T ). Note that the horizontal eight-formX 8 − T ∧X 6 is closed. We shall give explicit expressions for the polynomials with tildes shortly.
For now, let us get back to the reduction of the one-loop term (4.3): Note that the RR four-form appearing in (4.8) satisfies the correct Bianchi identity, dF 4 − H 3 ∧ T = 0, while B 2 comes wedged with a closed eight-form, hence ensuring that the one-loop term (in the absence of fivebrane sources) is invariant under the NSNS gauge transformation δB 2 → dΛ 1 . While further reductions are not important for our purposes, clearly we can extend this discussion to the case of multiple (commuting) isometries of the eleven-dimensional background. Let us discuss the reduction to nine dimensions. Starting from eleven dimensions, we should consider now a pair of isometries generated by v 1 and v 2 . Here we concentrate on B 1 = ı v 1 ı v 2 C 3 couplings. Even though the SL(2) doublets are not important for our purposes, and (in IIA language) we shall set the pair of graviphotons to zero, let us have a look at the complete set of lower dimensional descendants of C 3 ∧ X 8 . Due to isometries we can write where H 2 = dB 1 . One can check: Similarly for X 8 : where nowX n =X n (R 11 ) =X n (R 9 , T 1 , T 2 ). The closure of X 8 leads to: Let us introduce quantities: 13) and note that due to ǫ ij T i ∧ T j = 0 14) The result of the reduction of C 3 ∧ X 8 is then where F 4 and F i 3 satisfy the Bianchi identities (4.10). Once again, in absence of fivebranes, the action is invariant under δB 1 → dΛ 0 .
From now on, we shall consider only ten-dimensional theories with a single non-trivial graviphoton. A quick comment about anomalies is in order. Indeed the term B 2 ∧ [X 8 − T ∧X 6 ] (just like its elevendimensional ancestor) is not invariant under ten-dimensional diffeomorphisms. For that matter, even the closed eight-form is not;X 8 − T ∧X 6 is invariant only under the combined action of tendimensional diffeomorphisms and graviphoton U(1) transformations. In the presence of NS5-branes, dH = η(W 6 ֒→ M 10 ), the variation will produce a complicated expression restricted to the fivebrane worldvolume. Note that from the other sideX 8 is simply a sum of the usual (closed) X 8 polynomial and a part that depends on the graviphoton,X 8 = X 8 (T M 10 ) + X 8 (R, T ). Hence in a trivial graviphoton background, T = 0, we recover the usual ten-dimensional anomaly inflow 4 . Of course, eleven-dimensional anomaly cancellation requires contributions from three sources -the fivebrane anomaly, the variations from the bulk C 3 ∧ X 8 (T M 11 ) and the (modified) Chern-Simons term C 3 ∧ G 4 ∧ G 4 [52]. The prediction of this argument is that the reduction of the latter should yield counterparts to (4.16) 5 .

Higher derivative couplings in AdS 4
Before turning to the reduction, it is instructive now to look at explicit expressions. Since the single trace part of X 8 does not contribute to the reductions, it is sufficient to look at X 4 = p 1 (T M 11 ) and X 4 andX 3 = dX 2 arising from the reduction: 4 With an abuse of notation, X(T MD) refers to forms constructed of polynomials in D-dimensional curvatures. Quantities with tildeX are polynomials in R and T . 5 A circumstantial argument in favor of this is given by recalling that the contribution from C3 ∧ G4 ∧ G4 is important when the normal bundle is not trivial. It is not hard to check that when the normal bundle N is trivial and M10 = W6×N , the lift to eleven dimensions is provided by fibering the M-theory circle over N . In this situation the graviphoton field strength T is a horizontal form on T and hence pulls back to zero on W6. In other words, when the normal bundle is trivial, the coupling B2 ∧ [(X8 − X8(T M10)) − T ∧X6] is invariant.
where T = 1 2 T ab e a ∧ e b and T a = T a b e b and the covariant derivative ∇ is taken with respect to the Levi-Civita connection. (All the curvatures here are ten-dimensional, and a, b, c, . . . are tendimensional tangent space indices.) One can now computẽ 18) and see that it is invariant. HenceX 3 is cohomologically trivial. While we have reduced C 3 ∧ X 8 from eleven dimensions, as highlighted in (2.7), the string loop amplitude necessarily involves the curvature of the connection with torsion (2.6). In particular, we need to make the replacement R ⇒R insideX in order to account for H contributions to the NSNS part of the couplings. However, these contributions are not important in the SE 7 reduction since we backgrounds have vanishing H.
We are now ready to discuss the graviphoton couplings in AdS 4 . From the original elevendimensional point of view, the Sasaki-Einstein reduction takes the form [53] The resulting four-dimensional action corresponds to minimal gauged N = 2 supergravity with AdS radius L/2 We are of course interested in the one-loop correction to (4.20). In the absence of the NSNS B field, the relevant term in (4.8) is M 10 F 4 ∧X 6 . Taking into account the constant factors in (2.7), we find Focusing on the N = 2 graviphoton, F 2 , we pick out the component where we have also used (4.21).
Making use of (4.18), and taking the AdS 4 graviphoton to be T = F 2 gives * 4 F 2 ∧X 2 = 1 8π 2 As a result, the four-dimensional action (4.20) picks up a correction at the four derivative level where the ellipsis denotes terms that we have not focused on. See [54] for a recent discussion of these terms.
The coefficient α may be extracted from (4.23) where we have made use of the six-dimensional version of (3.16). Finally, if the AdS 4 geometry arose from a stack of N M2-branes probing a C 4 /Z k singularity, we may use the relation L 6 = 32π 2 g 2 s kN α ′3 to write α = λ N 2 where λ = N/k [55]. This integral vanishes on CP 3 (the base for S 7 ), but is generally non-zero.
Note that in addition to (4.25), there are four-derivative order terms that are in general moduli dependent. Indeed, for B = b i ω i where ω i ∈ H 2 (B), the reduction yields where i.e. where B = CP 3 , the second cohomology is one-dimensional and ω = J. Moreover in this case b is constant, and the resulting AdS 4 Tr R 2 correction is non-dynamical.

Reducing to AdS 3
We conclude with a brief discussion of reductions to AdS 3 . In this case, it is convenient to pass via six dimensions. The lowest order in derivatives one-loop contributions are well-studied in the context of IIA/heterotic duality with 16 supercharges (see e.g. [9,10]), and can be collected into a Chern-Simons like term where F T = (T, G 2 , F I 2 ) with T as above denoting the IIA RR one-form (graviphoton), while G 2 and F I 2 descend from the RR 3-form (see [10] for details). The intersection matrix is given by L = [σ 1 ⊕ d IJ ], where d IJ is in turn the intersection matrix of the internal space K and I, J = 1, ..., h 2 (K). For theories with 16 supercharges K is a K3 surface, and d IJ has signature (3,19). We shall mostly ignore these modes and concentrate on T and G 2 , since these are terms that also survive the truncation to theories with lower supersymmetry.
Our discussion makes it clear that there are two types of higher-derivative (but still one-loop) corrections to this coupling. Indeed in ten dimensions we have both new terms involving ∇H and H 2 and terms involving T . While along internal directions these vanish, they should appear in the six-dimensional effective theory. After integration by parts, the Chern-Simons term (4.29) takes the form The complete coupling should have O(4, 20) symmetry and hence the modified curvature terms should be written in terms of F and not simply T . However we do not try to impose this here, and just write down the terms that involve the graviphoton (and the ellipsis stands for the rest). The reduction to AdS 3 can now be readily performed, and yields a correction term of the form

A T-duality and higher-order terms in the effective action
We shall present here a very brief (and incomplete) discussion of the action of T-duality on the corrections to the effective action. Since the terms in the effective action built solely out of curvature cannot be invariant under T-duality, the corrections discussed in Section 2 can also be seen as completions required to make higher curvature terms invariant. One way of introducing a correction with torsion is to consider the Courant bracket-a generalization of the Lie bracket acting on vector fields, which however acts on sections of the so called generalized tangent bundle E. The latter locally is a product of tangent and cotangent bundles to the manifold M (dim(M ) = d): Sections of E are called generalized vectors. Locally they can be written as X = x + ξ where x ∈ T M and ξ ∈ T * M . In going from one coordinate patch U α to another U β , we have to first make the usual patching of vectors and one-forms, and then give a further patching describing how T * M is fibered over T M in E. The choice of B gives a canonical identification of E with T ⊕ T * , and the ordinary Courant bracket on E gets identified with the twisted Courant bracket on T ⊕ T * . The generalized connection is defined by analogy to the ordinary connection and is an operator where W is some vector bundle which carries a representation of O(d, d). We can now think of D as D = ∂ + Ω, where the ordinary derivative ∂ simply gives a term in the T * M part of E and nothing in the T M part. Thus one defines the derivative D, acting on a generalized vector X. The generalized connection is now defined by the Courant bracket: where [x, y] Lie is the usual Lie bracket between vectors and L x is the Lie derivative. We are interested in a case when the string background admits an isometry, hence both the metric on M and H are annihilated by the Lie derivative of some vector v, L v g = L v H = 0. We shall use the setup similar to that of Section 4. In particular we may use H = π * H 3 + (π * H 2 ) ∧ e, where ı v e = 1 and de = π * F . We may also decompose the sections of T M ⊕ T * M into horizontal and vertical components, x −→ x + f v and ρ −→ ρ + φe correspondingly, and consider the Courant bracket [(x + f v; ρ + φe), (y + gv; λ + ωe)] (H 3 ,H 2 ) = [(x; ρ), (y; λ)] H 3 + 0 + (L x g − L y f ) v; ξ base + (L x ω − L y φ)e , (A.4) where the first term is the Courant bracket on the base of the circle fibration and with η = 0 1 1 0 . One can now readily check that an O(1, 1) transformation leaves ξ base invariant and is an automorphism of the bracket (A.4), provided that X is an O(1, 1) matrix X T ηX = η. Hence the connection ω + H defined by (A.4) is T-duality invariant. Finally, by flipping the sign of H, we may remark that the connection ω − H is T-duality anti-invariant. This can be generalized for the case of multiple commuting isometries v I , provided ı v I ı v J H = 0 for any two vectors v I and v J . In general, for n isometries, O(n, n) transformations are not an automorphism of the Courant bracket, and hence one cannot construct an O(n, n) invariant generalized (twisted) connection. More details on generalized connection can be found in [56][57][58].
One may also use the Courant bracket to define a curvature operator that will be tensorial when restricted to integrable maximally isotropic subbundles of E. For our purpose, it suffices to look at the curvatureR written in (2.6) in the linearized approximation. Once more,R + =R(ω + H) is T-duality invariant, whileR − =R(ω − H) does not transform particularly nicely under T-duality. However writing locally X 4n (R ± ) = dX (0) 4n−1 (ω ± H), we recall that the latter contain only odd powers of the connection, and hence X 8 (R + ) and X 8 (R − ) are respectively even and odd under T-duality.
With all this in mind, the CP-odd corrections (2.7), (2.8) and (2.20) can be summarized as the T-duality invariant combination where the curvature expressions are constructed out of the original ten-dimensional fields. From the IIA point of view, we have introduced β = B inv + e∧ ı v B, with B inv = (1− e∧ ı v )B − 1 2 ı v g ∧ ı v B being the component of the B-field invariant under T-duality. We have also introduced γ = (α ′ /R 2 )e ∧ ı v g which vanishes in the absence of isometries and is suppressed in the large radius IIA ten-dimensional limit. Hence in ten dimensions the formula reproduces the known CP-odd one-loop terms. Upon reduction on a circle, (A.7) correctly reproduces the nine dimensional couplings. Note that T-duality exchanges the roles of ı v B and ı v g, so that the former becomes associated with γ and the latter with β from the IIB point of view. Of course, this cannot be the complete story, as it ought to be possible to express the CP-even corrections in a T-duality invariant manner as well. However, this looks somewhat more involved, as the circle reduction ofR ± appears rather unenlightening.