Squashing, Mass, and Holography for 3d Sphere Free Energy
SSquashing, Mass, and Holography for 3dSphere Free Energy
Shai M. Chester , Rohit R. Kalloor , and Adar Sharon Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, Israel
Abstract
We consider the sphere free energy F ( b ; m I ) in N = 6 ABJ(M) theory deformed by boththree real masses m I and the squashing parameter b , which has been computed in terms ofan N dimensional matrix model integral using supersymmetric localization. We show thatsetting m = i b − b − relates F ( b ; m I ) to the round sphere free energy, which implies infiniterelations between m I and b derivatives of F ( b ; m I ) evaluated at m I = 0 and b = 1. For N = 8 ABJ(M) theory, these relations fix all fourth order and some fifth order derivatives interms of derivatives of m , m , which were previously computed to all orders in 1 /N usingthe Fermi gas method. This allows us to compute ∂ b F | b =1 and ∂ b F | b =1 to all orders in1 /N , which we precisely match to a recent prediction to sub-leading order in 1 /N from theholographically dual AdS bulk theory. a r X i v : . [ h e p - t h ] F e b ontents N and holography 115 Conclusion 12A Details of squashed sphere calculation 15B The large k expansion 18 The free energy F ( b, m ) for a quantum field theory placed on the d dimensional squashedsphere S db and deformed by a mass m is one of the few quantities that can be computedexactly in interacting theories. For a rank N supersymmetric gauge theory, supersymmetriclocalization has been used to compute F ( b, m ) in terms of an N dimensional matrix modelintegral for 2d N = (2 ,
2) [1], 3d N = 2 [2, 3], 4d N = 2 [4, 5], and 5d N = 1 [6, 7] theories.The massless theory on the round sphere, i.e. m = 0 , b = 1, typically flows in the IR to aconformal field theory. At large N the CFT is often dual to weakly coupled supergravityon AdS d +1 [9], while turning on mass and squashing in the CFT corresponds to suitablydeforming the bulk away from AdS d +1 . One can then study the weakly coupled gravitytheory using the large N CFT either by directly comparing the deformed theories at finite m, b , or by using the small m, b expansion in the CFT to constrain correlation functions onflat space, which are then holographically dual to scattering in undeformed
AdS d +1 . Foreither method, it is crucial to know the explicit large N expansion of F ( b, m ), not just thematrix model integral given by localization.This work will focus on F ( b ; m I ) for the 3d N ≥ U ( N ) k × U ( N + M ) − k and Chern-Simons level k . Like any 3d N = 6 SCFT, ABJ(M) hasan SO (6) R symmetry and a U (1) global symmetry [12], so that from the N = 2 perspective This is generically the case for supersymmetric gauge theories with matrix degrees of freedom. Fortheories with vector degrees of freedom, the large N limit is holographically dual to weakly coupled higherspin theory on AdS d +1 , see [8] for a review. SO (4) × U (1) flavor symmetry. The theory can then be deformed bythree real masses m I , where m , m correspond to Cartans of the SO (4) and m to theCartan of the U (1). The free energy F ( b ; m I ) on the squashed sphere in the presence ofthese masses was computed using localization in terms of an N dimensional matrix modelintegral in [3]. For the massless round sphere F (1; 0), [13] showed that this matrix modelcould be understood as a free Fermi gas with a nontrivial potential, which allowed F (1; 0)to be explicitly computed to all orders in 1 /N . This Fermi gas method was then extendedto F (1; m , m ,
0) (or F (1; m , , m )) [14], and to F ( √
3; 0) (or F (1 / √
3; 0)) [15]. For moregeneral b, m I , however, the matrix model takes a more complicated form that is not amenableto this technique. In this work, we will use methods inspired from the Fermi gas approachto derive the exact relation F ( b ; m , m , i b − b − F (1; b − ( m + m ) + b ( m − m )2 , b − ( m + m ) − b ( m − m )2 , , (1.1)where the RHS is now related to the round sphere expression F (1; m , m ,
0) that was com-puted to all orders in 1 /N . We can then expand both sides around the massless round sphereto derive infinite constraints between m I , b derivatives at each order. For instance, we findthat all combinations of 4 derivatives of m I , b can be written in terms of the quantities ∂ m ± F , ∂ m + ∂ m − F , ∂ m ± F , ∂ m ∂ m F , ∂ m ± ∂ m ∓ F , ∂ m + ∂ m − F , (1.2)all evaluated at m I = 0 and b = 1, where m ± ≡ m ± m (or m ± ≡ m ± m ). The quantitiesin red have an odd number of m ± derivatives and are pure imaginary, the quantity ∂ m ∂ m F in green is generically complex, while the remaining quantities in black are always real. Forparity preserving ABJ(M) theories, which includes all N = 8 theories, ∂ m ± ∂ m ∓ F in factvanishes, while ∂ m + ∂ m − F vanishes for ABJM theory. The ABJ(M) theories have N = 8supersymmetry when k = 1 ,
2, in which case the SO (6) R × U (1) global symmetry is promotedto SO (8) R . As a result ∂ m ∂ m F is related to the other real non-vanishing quantities, whichare all written as derivatives of m ± , or equivalently as derivatives of F (1; m , m , F (1; m , m ,
0) was computed to all orders in 1 /N in [14], we thus have all orders in 1 /N expressions for all combinations of 4 derivatives of m I , b in N = 8 ABJ(M) theories. We can All m I , b derivatives of F that are considered in this work are assumed to be evaluated at m I = 0 and b = 1. For ABJ theory with unequal rank, generically ∂ m + ∂ m − F is nonzero, and is related to a choice ofbackground Chern-Simons level [16, 17]. ∂ b F in terms of m ± derivatives, sothat they too can be computed to all orders in 1 /N .We can then compare these all orders in 1 /N results for F ( b, m I ) to the holographicdual of U ( N ) k × U ( N + M ) − k ABJ(M) theory, which for large N and fixed M, k is dualto weakly coupled M-theory on
AdS × S / Z k , while for large N, k and fixed
M, λ ≡ N/k and then large λ is dual to weakly coupled Type IIA string theory on AdS × CP . Thefirst way we do this is to directly compare F ( b, m I ) to the renormalized on-shell action inthe AdS theory dual to these deformations. The leading large N term corresponds to theaction of N = 8 gauged supergravity on AdS [18] evaluated on solutions to the equationsof motion that preserve the suitable symmetry group of the CFT deformation, which for m (cid:54) = 0 or b (cid:54) = 1 breaks the amount of supersymmetry to N = 2 while preserving certainabelian flavor groups. These solutions were matched to F ( b, m I ) at leading order in large N for nonzero m I in [19] and nonzero b in [20]. The sub-leading 1 /N corrections to F ( b, m I )correspond to higher derivative corrections to supergravity evaluated on the correspondingsolution. The first higher derivative corrections, i.e. the four derivative terms, were recentlyderived in [21] for any minimal N = 2 gauged supergravity on AdS in terms of two theorydependent coefficients. These coefficients were then fixed for the ABJ(M) M-theory dualat finite k using the large N results for F (1 ,
0) and the coefficient c T of the stress tensortwo-point function, which was computed using ∂ m ± F in [22]. The free energy could then becomputed on any asymptotically AdS solution to sub-leading order in 1 /N , which for thecase of squashing gave [21]: F ( b ; 0) = π √ k (cid:34)(cid:18) b + 1 b (cid:19) (cid:18) N + ( 1 k − k
16 ) N (cid:19) − k N (cid:35) + O ( N ) . (1.3)We will match this gravity prediction for the sub-leading N terms to our all orders in1 /N expression for ∂ b F and ∂ b F in N = 8 ABJ(M) theory, i.e. for k = 1 ,
2. The furthersub-leading powers of 1 /N in our result will allow the coefficients of future higher derivativecorrections to supergravity to be similarly fixed. Note that once these higher derivativeterms are known, they can be used to compute gravity quantities on any asymptotically AdS solution, not just that corresponding to squashing, and can even be used to computethermodynamic quantities like higher derivative corrections to the the black hole entropy [21],which are much more difficult to compute directly from CFT using holography. Note that the finite value of M does not appear to any perturbative order in these holographic descrip-tions. AdS theory using F ( b, m ) is using the relationbetween the small m, b expansion of F ( b, m ) and integrated correlators of the stress tensormultiplet correlator, which is dual to scattering of gravitons on AdS . In particular, sinceboth m I and b couple to operators in the stress tensor multiplet for 3d N = 6 SCFTs, itshould be possible to relate n derivatives of F ( b ; m I ) evaluated at m = 0 , b = 1 to correlatorsof n stress tensor multiplet operators integrated on S [17]. These integrated constraints werederived for ∂ m ± F , ∂ m + ∂ m − F , and ∂ m ± F for N = 8 SCFTs in [22, 23] and for N = 6 SCFTsin [24]. The stress tensor multiplet four point function in ABJ(M) can then be constrainedin the large N limit using analyticity, crossing symmetry, and the superconformal wardidentities in terms of just a few terms at each order [25, 26], whose coefficients can then befixed using the integrated constraints and the large N expressions for derivatives of F ( b ; m I ).One can then take the flat space limit of this holographic correlator as in [27] and compareto the dual quantum gravity S-matrix in flat space, where 1 /N corrections correspond tohigher derivative corrections to supergravity. This program was carried out to sub-leadingorder in 1 /N for the M-theory limit in [23, 26], and the Type IIA limit in [24]. To go tofurther orders, one needs to both derive the integrated constraints for the remaining massand squashing derivatives, as well as the large N expansions of the localization expressions.This paper completes the latter task for all such fourth order derivatives for N = 8 ABJ(M),while for N = 6 ABJ(M) a large N expansion is still needed for ∂ m ∂ m F .The rest of this paper is organized as follows. In Section 2, we review the matrix modelexpression of F ( b, m ) for U ( N ) k × U ( N + M ) − k ABJ(M) theory, as well as previous all ordersin 1 /N results from the Fermi Gas method. In Section 3, we derive the exact relation (1.1)between mass and squashing, and use it to show that all derivatives up to fourth order as wellas ∂ b F can be written in terms of the invariants shown in (1.2). In Section 4 we use theserelations as well as the previously derived all orders in 1 /N expressions for F (1; m , m , /N expressions for ∂ b F and ∂ b F , which we will match to the gravityprediction in (1.3). We end with a discussion of our results and future directions in Section 5.Details of our calculations are given in various Appendices, and an attached Mathematica notebook includes our result for c T in the large k weak coupling expansion to O ( k − ). We begin by reviewing ABJ(M) theory and localization results for F ( b ; m I ), including theall orders in 1 /N results from the Fermi Gas method for the round sphere with only m ± N = 2 language, ABJM theory consists of vector multiplets for each U ( N ) k × U ( N + M ) − k gauge group, as well as four chiral multiplets Z A , W A for A = 1 , SU (2) R × SU (2) R × U (1) flavor symmetry as shown in Table1. Seiberg duality relates different ABJ(M) theories as U ( N ) k × U ( N + M ) − k ←→ U ( N ) − k × U ( N + | k | − M ) k , (2.1)which implies that M ≤ | k | . Parity then sends k → − k , so the M = 0 theories can be seen tobe parity invariant from the Lagrangian, while Seiberg duality implies that the k = M, M theories must be parity invariant on the quantum level.field U ( N ) × U ( N + M ) SU (2) × SU (2) U (1) Z A ( N , N ) ( , ) 1 W A ( N , N ) ( , ) − N = 2 language.The partition function Z ( b ; m I ) = e − F ( b ; m I ) on a squashed sphere with squashing param-eter b and deformed by masses m I for the chiral fields can then be computed by assemblingthe standard N = 2 ingredients, as reviewed in [28], to get up to an overall m I , b independentconstant: Z ( b ; m I ) = e iπ k ( b − b − ) M ( M − (cid:90) d N + M µ d N νN ! ( N + M )! e iπk [ (cid:80) i ν i − (cid:80) a µ a ] (cid:89) a>b πb ( µ a − µ b )] sinh (cid:2) πb − ( µ a − µ b ) (cid:3) × (cid:89) i>j πb ( ν i − ν j )] sinh (cid:2) πb − ( ν i − ν j ) (cid:3) (cid:89) i,a (cid:20) s b (cid:18) iQ − ( µ a − ν i + m + m + m (cid:19) × s b (cid:18) iQ − ( µ a − ν i + m − m − m (cid:19) s b (cid:18) iQ − ( − µ a + ν i + − m − m + m (cid:19) × s b (cid:18) iQ − ( − µ a + ν i + − m + m − m (cid:19)(cid:21) , (2.2)where Q = b + b , the µ a , ν i correspond to the Cartans of the two gauge fields with a =1 , . . . , N + M and i = 1 , . . . , N , and each chiral field contributes a factor with masses m I determined by the charge assignments in Table 1, so that m corresponds to U (1) and m + m , m − m correspond to the Cartans of each factor in SU (2) × SU (2), respectively.5he functions s b ( x ) are reviewed in Appendix A. The phase factor was computed for generic N = 2 supersymmetric gauge theories in [29] in terms of the topological anomaly, whichwas given for ABJ(M) theory in [16, 30]. If we restrict to the round sphere with b = 1, andset m (or m ) zero, then the partition function can be simplified using identities in A andwritten in terms of m ± = m ± m (or m ± = m ± m ) as Z ( m + , m − ) = (cid:90) d N + M µd N νN ! ( N + M )! e iπk [ (cid:80) i ν i − (cid:80) a µ a ] × (cid:81) a
11 + m + 11 + m − (cid:21) − k
12 + k (cid:18) − Mk (cid:19) ,A = A [ k (1 + im + )] + A [ k (1 − im + )] + A [ k (1 + im − )] + A [ k (1 − im − )]4 , (2.5)6here the constant map function A is given by [33] A ( k ) = 2 ζ (3) π k (cid:18) − k (cid:19) + k π (cid:90) ∞ dx xe kx − (cid:0) − e − x (cid:1) = − ζ (3)8 π k + 2 ζ (cid:48) ( −
1) + log (cid:2) πk (cid:3) ∞ (cid:88) g =0 (cid:18) πik (cid:19) g − g B g B g − (4 g )(2 g − g − , (2.6)and in the second line we wrote A in the large k expansion [33]. Note that the all orders in1 /N formula only depends on M via the parameter B .A useful parameterization of ABJ(M) is given by the coefficient c T of the two-pointfunction of canonically normalized stress-tensors: (cid:104) T µν ( (cid:126)x ) T ρσ (0) (cid:105) = c T
64 ( P µρ P νσ + P νρ P µσ − P µν P ρσ ) 116 π (cid:126)x , P µν ≡ η µν ∇ − ∂ µ ∂ ν . (2.7)This quantity is related to the AdS Planck length, and so is a more natural expansionparameter in the holographic large N limit than N itself. We can compute c T in terms of Z ( m − , m + ) as [34] c T = 64 π ∂ m ± F , (2.8)so that it can be written to all orders in 1 /N using (2.5). It turns out that any other quantitycomputed by taking m I , b derivatives of F ( b ; m I ), when expanded at large c T in either theM-theory or Type IIA limits, becomes independent of M . In this sense these limits are blindto parity, which as discussed depends on the value of M . One can also check that only evennumbers of m ± derivatives are nonzero in this limit, and that these quantities are alwaysreal. We will now derive the relation between mass and squashing shown in (1.1). We will thenuse this result to show that all quartic order m I , b derivatives of F ( b ; m I ) can be written interms of the invariants (1.2), and that for N = 8 ABJ(M) all of these invariants are knownto all orders in 1 /N from previous Fermi gas results. For N = 6 ABJ(M), no such all ordersresult is known yet for ∂ m ∂ m F . We will verify all these relations at both small finite M, N and in the large k weak coupling expansion. Note that in general all m I , b derivatives areevaluated at m I = 0 and b = 1. 7e start by setting m = i b − b − in (2.2) and using properties of s b ( x ) given in AppendixA to write the partition function purely in terms of trigonometric functions: Z ( b ; m ,m , i b − b − e iπ k ( b − b − ) M ( M − (cid:90) d N + M µ d N νN ! ( N + M )! e iπk [ (cid:80) i ν i − (cid:80) a µ a ] × (cid:89) a>b πb ( µ a − µ b )] sinh (cid:2) πb − ( µ a − µ b ) (cid:3) (cid:89) i>j πb ( ν i − ν j )] sinh (cid:2) πb − ( ν i − ν j ) (cid:3) × (cid:89) i,a
12 cosh( πb − ( µ a − ν i + m + )) 12 cosh( πb ( ν i − µ a + m − )) , (3.1)where recall that m ± = m ± m . In Appendix A, we then perform the standard Fermi gassteps of writing the products of trigonometric functions as Cauchy determinants, introducingauxiliary variables so that the µ, ν factorize into gaussian integrals, and finally performingthese integrals and rewriting the Cauchy determinant back into the standard form. Theresult is Z ( b ; m , m , i b − b − Z ( b − m + , bm − ) , (3.2)where on the RHS we wrote the simplified round sphere partition function defined in (2.4),and note that the b dependent phase that appeared in (2.2) is precisely cancelled, so thatthe RHS of (3.2) depends on b only through a rescaling of the masses. We can then simplyrewrite m ± in terms of m , m to get (1.1). This entire calculation can also be performedwith the roles of m and m switched with the same result on the RHS of (3.2), which isexpected since these masses both correspond to the Cartans of the SO (4) part of the flavorsymmetry.We can now expand both sides of (3.2) around m I = 0 and b = 1 to derive relationsbetween derivatives of F ( b ; m I ). The first nonzero relation appears at quadratic order andrelates ∂ b F = 2 ∂ m ± F + 2 ∂ m + ∂ m − F , (3.3)where we used the fact that various single derivatives of b and m I identically vanish. Fromthe explicit single variable partition function for Z ( m + , m − ) as given in (2.4), we see that¯ Z ( m + , m − ) = ( − MN Z ( − m + , m − ) . (3.4)8his implies that any odd number of derivatives of m + is pure imaginary, such as ∂ m + ∂ m − F .We thus conclude that Re ∂ b F = 2 ∂ m ± F , (3.5)where recall that ∂ m ± F is manifestly real. This relation is expected from the general resultsof [17, 35], which showed that the real part of two derivatives of any parameter that couplesto the stress tensor multiplet should be related to c T , where the precise relation in our casewas given in (2.8).At cubic order, we similarly find the nonzero relations i∂ m ∂ m ∂ m F = − ∂ m ± F , ∂ b F = − ∂ m ± F − ∂ m + ∂ m − F . (3.6)Since conformal symmetry fixes both three point and two point functions of the stress tensorto be proportional to c T [36], we therefore expect that all three derivative terms can bewritten as linear combinations of two derivative terms. The factor of i on the LHS of thefirst relation follows from the fact that the real cubic casimir invariant for the SO (4) × U (1)flavor symmetry (from the N = 2 perspective) is im m m .At quartic order, the full list of nonzero relations is ∂ m , F = 2 ∂ m ± F + 6 ∂ m + ∂ m − F + 8 ∂ m ± ∂ m ∓ F ,∂ m F = 2 ∂ m ± F + 6 ∂ m + ∂ m − F − ∂ m ± ∂ m ∓ F ,∂ m ∂ m , F = 2 ∂ m ± F − ∂ m + ∂ m − F ,∂ b ∂ m , F = 8 ∂ m ± F + ∂ m ∂ m F ,∂ b ∂ m F = 8 ∂ m ± F + 2 ∂ m ± F − ∂ m + ∂ m − F ,∂ b F = 78 ∂ m ± F − ∂ m ± F − ∂ m + ∂ m − F + 6 ∂ m ∂ m F − ∂ m ± ∂ m ∓ F − ∂ m + ∂ m − F , (3.7)which are all written in terms of the six invariants in (1.2). As discussed above, the twoinvariants ∂ m + ∂ m − F and ∂ m ± ∂ m ∓ F are both pure imaginary, since they involve an oddnumber of derivatives of m + , and they vanish for ABJM theory with equal rank. The oneinvariant that cannot generically be written as derivatives of m ± is ∂ m ∂ m F . For N = 8ABJ(M), however, the flavor group SO (4) × U (1) is enhanced to SO (6), which implies that9 ∂ m ∂ m F ∂ m ∂ m F − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i − . − . i
10 -44.9409 − . − . i Table 2: Explicit free energy derivatives for the U (1) k × U (2) − k ABJ theory for k = 1 , . . . m I = 0 and b = 1. ∂ m ∂ m F must be related to the other mass derivatives as [23] N = 8 : ∂ m I ∂ m J F = 2 ∂ m + F − ∂ m + ∂ m − F for I (cid:54) = J . (3.8)In this case, we find that ∂ b ∂ m , F = ∂ b ∂ m F in (3.7) as expected.At quintic and higher order of b, m I derivatives, the relation (3.2) is not sufficient to writeall derivatives in terms of just m ± derivatives even for N = 8. For instance, at quintic order ∂ m ∂ m ∂ m F cannot be further simplified. Nevertheless, at quintic order for N = 8 we canuse (3.2) and (3.8) to write ∂ b F as ∂ b F = − ∂ m ± F − ∂ m ± F + 180 ∂ m + ∂ m − F . (3.9)At higher order a >
5, we can no longer write ∂ ab F in terms of just m ± derivatives.We checked all the relations discussed in this section for U ( N + M ) k × U ( N ) − k at finite M, N, k as well as in the large k weak coupling expansion. For instance, in Table 2 we show ∂ m ∂ m F and ∂ m ∂ m F for the U (1) k × U (2) − k theory for k = 1 , . . .
10, as computed fromthe explicit partition function in (2.2). As expected, for k = 1 , N = 8,these quantities are identical and real. For all higher k , when the theory is only N = 6,these quantities are distinct and ∂ m ∂ m F is complex. Finally, in Appendix B we show that ∂ m ∂ m F and ∂ m ∂ m F differ explicitly in the large k weak coupling expansion, which isautomatically N = 6. We also computed c T to O ( k − ) using an efficient algorithm for theweak coupling expansion, which improves the O ( k − ) result of [37].10 Large N and holography In the previous section, we showed that for N = 8 ABJ(M) theory, we can relate ∂ b F and ∂ b F to derivatives of F ( m − , m + ) using (3.7) and (3.8). As reviewed in Section 2, thisquantity was computed to all orders in 1 /N using Fermi gas methods, which implies thatwe can also compute ∂ b F and ∂ b F to all orders in 1 /N . For the U ( N ) k × U ( N ) − k theorywith finite k = 1 ,
2, in which case we have N = 8 supersymmetry, we find ∂ b F =10 √ kπN / − π ( k − √ √ k √ N + 39 k A (cid:48)(cid:48) ( k ) − k A (cid:48)(cid:48)(cid:48)(cid:48) ( k ) − π (5 ( k − k + 704)384 √ k / √ N + O ( N − ) ,∂ b F = − √ kπN / + 15 π ( k − √ √ k √ N − k A (cid:48)(cid:48) ( k ) + 50 k A (cid:48)(cid:48)(cid:48)(cid:48) ( k ) + 30 − π ( k − k + 448)64 √ k / √ N + O ( N − ) , (4.1)while it is straightforward to compute higher orders in 1 /N . The constant map A wasdefined in (2.6), and its derivatives can be computed exactly for any integer value of k . Inparticular, for the k = 1 , A (cid:48)(cid:48) (1) = 16 + π , A (cid:48)(cid:48) (2) = 124 , A (cid:48)(cid:48)(cid:48)(cid:48) (1) = 1 + 4 π − π , A (cid:48)(cid:48)(cid:48)(cid:48) (2) = 116 + π . (4.2)The leading and sub-leading terms in (4.1) exactly match ∂ b F and ∂ b F as computed fromthe bulk prediction (1.3). The bulk prediction is in fact for any k , not just the k = 1 , N = 8 supersymmetry that we could compute here. This implies that (4.1) must hold forany value of k up to O ( N ). As discussed above, in the large N limit it is more natural toexpand quantities in terms of c T than N . Using the large N expansion for c T given by (2.8)and (2.5), we find that ∂ b F and ∂ b F can be expanded to all orders in 1 /c T as1 c T ∂ b F = 15 π c T + 24 k A (cid:48)(cid:48) ( k ) − k A (cid:48)(cid:48)(cid:48)(cid:48) ( k ) + 6 c T + 4 (cid:18) πk (cid:19) / c T + O ( c T ) , c T ∂ b F = − π c T + − k A (cid:48)(cid:48) ( k ) + 50 k A (cid:48)(cid:48)(cid:48)(cid:48) ( k ) − c T − (cid:18) πk (cid:19) / c T + O ( c T ) . (4.3)11ere, the 1 /c T corresponds to the tree level supergravity correction, the 1 /c T correspondsto the 1-loop supergravity correction, and the 1 /c T term corresponds to the tree level D R correction. Curiously, the 1 /c T correction, which would correspond to the R correction,vanishes.Finally, we can also consider the limit of large N, k at fixed λ ≡ N/k and then large λ ,which is dual to weakly coupled Type IIA string theory on AdS × CP . Using the large k expansion of A ( k ) given on the second line of (2.6), we find that1 c T ∂ b F = 1 c T (cid:20) π − ζ (3)32 √ πλ + O ( λ − ) (cid:21) + 1 c T (cid:20) ζ (3) π λ − ζ (3)2 √ π λ + O ( λ − ) (cid:21) + O ( c − T ) , c T ∂ b F = 1 c T (cid:20) − π
16 + 45 ζ (3)16 √ πλ + O ( λ − ) (cid:21) + 1 c T (cid:20) − − ζ (3) π λ + 75 ζ (3) √ π λ + O ( λ − ) (cid:21) + O ( c − T ) . (4.4)Here, the c − T term corresponds to the tree level supergravity correction, the c − T λ − termcorresponds to tree level R , while the various c − T terms correspond to 1-loop corrections.Unlike the M-theory expansion in (4.3), we find that the R correction no longer vanishes.This result can also be compared to future bulk calculations in this background. The main result of this work is the exact relation between the mass and squashing deformedsphere free energy F ( b ; m I ) given in (1.1) for all N = 6 ABJ(M) theories. This relationimplies infinite relations between derivatives of F ( b ; m I ) evaluated at m = 0 , b = 1, suchas the fact that all four derivatives can be written in terms of the six quantities listed in(1.2). For the N = 8 ABJ(M) theories, these relations allowed us to compute ∂ b F ( b ; m I )and ∂ b F ( b ; m I ) to all orders in 1 /N , which at sub-leading order match the prediction givenin (1.3) from M-theory compactified on AdS × S / Z k and expanded to leading order beyondthe supergravity limit [21]. Our results provide constraints at further orders in 1 /N that willallow more higher derivative corrections to supergravity to be derived following the programoutlined in [21].It is instructive to compare the results of this work for F ( b ; m I ) in ABJ(M) to similarresults in 4d N = 4 SYM. The free energy F ( b ; m ; τ ) in this theory was computed using12ocalization in [4, 5] in terms of an N dimensional matrix model integral that depends on thecomplexified gauge coupling τ , a single mass m , and the squashing b , all of which couple tooperators in the N = 4 stress tensor multiplet. In [38], it was found that all four derivativesof these three parameters can be written in terms of the three invariants ∂ m F ( b ; m ; τ ) , ∂ m ∂ b F ( b ; m ; τ ) , c , (5.1)where c is the conformal anomaly and the coefficient of the canonically normalized stresstensor two-point function. Recall that for N = 8 ABJ(M) theory, we also found that allfour derivatives of F ( b ; m I ) could be written in terms of the three quantities shown in blackin (1.2), where the similarity to 4d becomes even tighter once we use Table 3.7 to exchange ∂ m + ∂ m − F for ∂ m ± ∂ b F , and we note that the 3d analog of c is c T , which is proportional to thethird invariant ∂ m ± F . The fact that there are just three independent quartic derivatives formaximally supersymmetric theories in both 3d and 4d is in some sense expected, as in bothcases the unprotected D R term in the large N expansion of the stress tensor correlatorcan be fixed in terms of four coefficients [26, 39], so if there were four independent quarticderivatives then one could have derived an unprotected quantity from protected localizationconstraints. Another similarity between 3d and 4d is that in [40] it was shown that for aspecial value of the mass F ( b ; m ; τ ) obeys F ( b ; i b − b − τ ) = F (1; 0; τ ) , (5.2)i.e. it becomes independent of the squashing and mass, just as in (1.1) we showed that theexact same relation between m and b made F ( b ; m I ) equivalent to the round sphere freeenergy with m = 0, although in 3d the dependence on m and b is now captured by m and m . It would be interesting to find a deeper geometric explanation for this simplification inall theories where m, b both couple to the stress tensor multiplet.One application of our results that we did not explore in this work is the relation between n derivatives of m I , b of F ( b ; m I ) and correlators of n stress tensor multiplets. For N = 8ABJM theory, ∂ m ± F and ∂ m + ∂ m − F were related in [23] to integrated constraints on thestress tensor four point function, which were used to derive the large N expansion up toorder D R in the bulk language. One further constraint is needed to fix the D R term, which is the highest order protected term, and it is possible that the integrated constraint This term could equivalently be fixed using the flat space limit relation to the M-theory S-matrix, whichhas been computed to order D R . ∂ b ∂ m F will be sufficient to fix this term.For N = 6 ABJ(M), from the list of independent quartic derivatives in (1.2), we expectthat there will now be six total independent constraints. Recall that ∂ m + ∂ m − F and ∂ m ± ∂ m ∓ F are known to vanish in the Fermi gas expression that describes both the M-theory and TypeIIA string theory limits, so we expect just four constraints in these cases. The integratedconstraints for ∂ m ± F and ∂ m + ∂ m − F were derived in [24] and used to fix the stress tensorcorrelator in both the M-theory and Type IIA limit to order R . To fix the correlatorto order D R just from CFT results, one would need six constraints, which is probablymore than are even in principle independent. On the other hand, if one uses the knownType IIA amplitude in the flat space limit to fix two of these constraints, then just fourmore constraints are required, which matches the four invariants we found in this work. Tocomplete this program, one would need to derive the large N expansion of ∂ m ∂ m F , whichremains unknown for N = 6 ABJ(M). It would be nice if the Fermi gas method for n -bodyoperators as initiated in [41] could be used to compute this quantity. At strong coupling, onecould also try to compute them at large N and finite λ ≡ N/k using topological recursionas was done for Wilson loops and the free energy in [42]. Topological recursion for theABJM matrix model is quite complicated, however, especially for the multi-body operatorswe consider, so one could instead try to guess the large N and finite λ result from the small λ , i.e. large k , weak coupling expansion, which can be computed to very large order usingthe algorithm introduced in Appendix B of this work. A first step would be guessing thefinite λ resummation for c T , which we computed to O ( k − ) in this work.One final interesting limit of N = 6 ABJ(M) that we have not yet considered is the large M, k limit at fixed ˜ λ = M/k and N , which is holographically dual to weakly coupled N = 6higher spin theory [43]. Unlike the M-theory and string theory limits, this limit is sensitiveto the value of M , and thus to parity. The parity violating quartic invariants shown in redin (1.2) can also be computed in this limit following [44], and could potentially be used toconstrain the correlator. This limit was recently considered in the context of the 3d N = 6numerical bootstrap in [32], and will be further discussed in upcoming work. Acknowledgments
We thank Ofer Aharony, Itamar Yaakov, Zohar Komargodski, Silviu Pufu, Damon Binder,Marcos Marino, Yifan Wang, Alba Grassi, Masazumi Honda, Erez Urbach and Ohad Mam- Recall that for N = 8 ABJ(M) this quantity was related by symmetry to derivatives of m + and m − which are known to all orders in 1 /N . A Details of squashed sphere calculation
Let us start by reviewing the properties of the double sine function s b ( x ) (for reviews see forinstance [15, 45]). This function is defined as s b ( x ) = exp (cid:20) − iπ x − iπ (cid:0) b + b − (cid:1) + (cid:90) R + i dt t e − itx sinh ( bt ) sinh ( t/b ) (cid:21) , (A.1)where the integration contour evades the the pole at t = 0 by going into the upper half–plane.This function obeys several identities:1. s − b ( x ) = s b ( − x ).2. s b − ( x ) = s b ( x ).3. s b (cid:0) ib − σ (cid:1) s b (cid:0) ib + σ (cid:1) =
12 cosh( πbσ ) .The last identity is what we used to get (2.3) and (3.1).Next, we will show how (3.1) is related to (2.3) as in (3.2), by adapting the usual Fermigas steps for ABJ as discussed in [30, 46]. We start from a slightly modified version of (3.1): Z ≡ Z (cid:18) b ; b m + − b − m − , b m + + b − m − , i ( b − b − )2 (cid:19) = N (cid:90) d M + N µ d N ν e iπ k ( b − b − ) M ( M − e − ik π ( (cid:80) j µ j − (cid:80) l ν l ) × (cid:32)(cid:89) j 12 cosh (cid:0) b (cid:0) µ r − ν s (cid:1) − πm − (cid:1) 12 cosh (cid:0) b − (cid:0) µ r − ν s (cid:1) + πm + (cid:1) (cid:35) , (A.2)where for convenience we changed variables ( µ, ν ) → ( µ, ν ) / (2 π ) relative to (3.1), and defined15he numerical constant N = 1(2 π ) M +2 N ( M + N ) ! N ! . (A.3)Our goal is to show that Z is independent of b . Once this is done, plugging in the right valuesof m ± will give (3.2). Our first step will be to use the Cauchy determinant formula [14, 46]to turn the integrand into a product of two determinants. From here, a clever change ofintegration variables will let us replace one of the determinants with the product of thediagonal elements of the corresponding matrix. We will then express the integrand as aFourier transform; as is routinely done in Fermi gas derivations. This allows us do the µ, ν integrals (these are simple Gaussian integrals at this stage), which gives a simple b -dependentphase factor that exactly cancels a similar factor (A.2), thus making Z independent of b .Since the object of our derivation is to exhibit the b -independence of (A.2), we do not needto keep track of any b -independent factors that are produced along the way. Hence, we willemploy a series of normalization factors N i that soak up all such b -independent factors.We now begin the calculation by using the Cauchy determinant formula as given in [14,46]to turn (A.2) into Z = N (cid:90) d M + N µ d N ν e − ik π ( (cid:80) j µ j − (cid:80) l ν l ) − M (cid:80) r ( Qµ r + πm + − πm − )+ M (cid:80) s Qν s + iπ k ( b − b − ) M ( M − (cid:34) det (cid:32) Θ N,l 12 cosh b − ( µ j − ν l )+ πm + + Θ l,N +1 e ( M + N − l + )( b − µ j + πm + ) (cid:33)(cid:35)(cid:34) det (cid:32) Θ N,s 12 cosh b ( µ r − ν s ) − πm − + Θ s,N +1 e ( M + N − s + ) ( bµ r − πm − ) (cid:33)(cid:35) , (A.4)where Θ r,s = Θ ( r − s ) is the step function, and the indices ( j, l, r, s ) run from 1 to N + M .Now we use the following identity (similar to the one in Appendix A of [47]): (cid:90) d M + N µ d N ν det ( f (( µ j , ν l )) × det ( f (( µ j , ν l ))= ( M + N ) ! (cid:90) d M + N µ d N ν (cid:32) M + N (cid:89) j =1 f (( µ j , ν j ) (cid:33) × det ( f (( µ r , ν s ))(A.5)16o get: Z = N (cid:88) σ ∈ S M + N ( − σ (cid:90) d M + N µ d N ν e − ik π ( (cid:80) j µ j − (cid:80) l ν l ) − M (cid:80) r ( Qµ r + πm + − πm − )+ M (cid:80) s Qν s + iπ k ( b − b − ) M ( M − (cid:32) N (cid:89) j =1 12 cosh b − ( µ j − ν j )+ πm + (cid:33) (cid:32) M + N (cid:89) l = N +1 e ( M + N − l + )( b − µ l + πm + ) (cid:33) N (cid:89) r =1 12 cosh b ( µ σ ( r ) − ν r ) − πm − (cid:32) M + N (cid:89) s = N +1 e ( M + N − s + )( bµ σ ( s ) − πm − ) (cid:33) , (A.6)where we’ve also expanded the remaining determinant term. We now rewrite the integralsas Fourier transforms: Z = N (cid:90) d M + N µ d N ν d N p d M + N q e − ik π ( (cid:80) j µ j − (cid:80) l ν l ) − M (cid:80) r ( Qµ r + πm + − πm − )+ M (cid:80) s Qν s + iπ k ( b − b − ) M ( M − N (cid:89) j =1 e i pj ( b − ( µj − νj ) + πm + ) π p j (cid:32) M + N (cid:89) l = N +1 e ( M + N − l + )( b − µ l + πm + ) (cid:33)(cid:32) M + N (cid:89) r = N +1 δ (cid:18) πi (cid:18) M + N − r + 12 (cid:19) + q r (cid:19) e − i qrm − π π (cid:33) N (cid:89) s =1 e − i qs ( πm − + bνs ) π q s (cid:88) σ ∈ S M + N ( − σ M + N (cid:89) t =1 e i qt ( bµσ ( t ) ) π . (A.7)The µ, ν integrals are now easy to do and generate some mass-dependent phases, whichare absorbed into the definition of N . The resulting expression may be massaged into thefollowing form: Z = (cid:32) M + N (cid:89) j = N +1 e i πk ( − πi ( M + N − j +1 / b − + iπMQ ) (cid:33) e iπ k ( b − b − ) M ( M − × N (cid:88) σ ∈ S M + N ( − σ (cid:90) d M + N q d N p (cid:32) M + N (cid:89) l = N +1 e i πk ( − iπq σ ( l ) ( M + N − l +1 / q l b +2 iπMQbq l ) (cid:33) N (cid:89) r =1 e i pr π (cid:18) qσ ( r ) − qrk + πm + (cid:19) p r (cid:32) N (cid:89) s =1 e − i qsm − q s (cid:33) (cid:32) M + N (cid:89) t = N +1 δ (cid:18) πi (cid:18) M + N − t + 12 (cid:19) + q t (cid:19)(cid:33) . (A.8)17e now observe that the b -dependent terms under the integral sign are functions of only ofthose q j ’s with j ∈ { N + 1 , ..., M + N } . These may be pulled out of the integral with thehelp of the delta functions – leaving behind an integral that is completely independent of b (and hence absorbed into the definition of N ): Z = N (cid:32) M + N (cid:89) j = N +1 e i πk (cid:16) ( − πi ( M + N − j +1 / b − + iπMQ ) − π ( M + N − j +1 / b +4 π MQb ( M + N − j +1 / (cid:17) (cid:33) × e iπ k ( b − b − ) M ( M − = N e − iπk ( b + b − ) M ( M − ) × e iπ k ( b − b − ) M ( M − = N , (A.9)where note in the second line that the b -dependent phase in (A.2) has cancelled. We havethus shown that Z = Z (cid:18) b ; b m + − b − m − , b m + + b − m − , i ( b − b − ) (cid:19) is independent of b , so we arenow free to plug in b = 1 to get: Z (cid:18) b ; b m + − b − m − , b m + + b − m − , i ( b − b − )2 (cid:19) = Z (cid:18) m + − m − , m + + m − , (cid:19) = Z (1; m , m , . (A.10)This result is valid for any value of m + and m − , so we can set m + → b − m + and m − → bm − to get (3.2). B The large k expansion In this appendix we compute some observables in ABJ(M) in perturbation theory in the CSlevel k . The computation follows standard procedure, for a recent example see [37].Our starting point is the partition function deformed by real masses on the squashedsphere (2.2). Taking derivatives with respect to masses and the squashing parameter, andsetting them to zero, we can define observables in the matrix model. In general, this proce-dure should lead to some expectation value in the matrix model of the form (cid:104)O(cid:105) = (cid:90) d N νd N + M µN ! ( N + M )! e iπk ( (cid:80) i ν i − (cid:80) a µ a ) Z − loop ( ν i , µ a ) O ( ν i , µ a ) . (B.1)18here Z − loop ( ν i , µ b ) = (cid:81) i 20s an example, we compute c T to order k − . We can find the operator O by usingequation (2.8): c T = 64 π ∂ m + F = − π (cid:32) Z (cid:48)(cid:48) Z − (cid:18) Z (cid:48) Z (cid:19) (cid:33) , (B.12)where primes denote derivatives by m + = m + m . The derivatives are given by Z (cid:48) = − (cid:90) d N νd N + M µN ! ( N + M )! e iπk ( (cid:80) i ν i − (cid:80) a µ a ) Z − loop ( ν i , µ a ) π (cid:88) a,i tanh π ( ν i − µ a ) , (B.13) Z (cid:48)(cid:48) = (cid:90) d N νd N + M µN ! ( N + M )! e iπk ( (cid:80) i ν i − (cid:80) a µ a ) Z − loop ( ν i , µ a ) × π (cid:32)(cid:88) a,i tanh ( π ( ν i − µ a )) (cid:33) − (cid:88) a,i ( π ( ν i − µ a )) . (B.14)Note that Z (cid:48) vanishes since it is odd under ν, µ → − ν, µ , and so it is enough to compute Z (cid:48)(cid:48) .The corresponding operator O can be read off from (B.14): O ( ν, µ ) = π (cid:32)(cid:88) a,i tanh ( π ( ν i − µ a )) (cid:33) − (cid:88) a,i ( π ( ν i − µ a )) . (B.15)Following the algorithm above, we computed c T to O ( k − ). Due to the length of theexpression, we give it in an attached Mathematica file.We will next be interested in computing ∂ m ∂ m F from equation (1.2) in perturbationtheory. Specifically, we would like to compare it to another independent quantity, ∂ m ∂ m F ,and to show that they are not the same in perturbation theory in large k (where we have N = 6 SUSY). Explicitly, ∂ m ∂ m F is given by ∂ m ∂ m F = − ∂ m ∂ m ZZ + ∂ m ZZ ∂ m ZZ . (B.16)Using the relation (1.1), we find that we can write the first term as ∂ m ∂ ZZ = 4 ∂ m − ∂ b ZZ − (cid:0) ∂ m + − ∂ m + ∂ m − + 16 ∂ m + (cid:1) ZZ , (B.17)while we find ∂ m ZZ = ∂ m ZZ = 2 ∂ m + Z + 2 ∂ m + ∂ m − ZZ . (B.18) the ratio of two numbers. To find the numerator, we start by inserting n into the box in the top-left corner, 21e thus have to compute the derivatives ∂ m − ∂ b Z , ∂ m + Z , ∂ m + ∂ m − Z , ∂ m + Z , ∂ m + ∂ m − Z . First,we write the operators O ( x i , y a ) corresponding to each term: ∂ m − ∂ b Z → (cid:88) a,i π sech ( g s r ai ) (cid:0) g s r ai tanh( g s r ai ) − (cid:0) g s r ai + π + 4 (cid:1) + 3 (cid:0) g s r ai + π (cid:1) sech ( g s r ai ) (cid:1) + π (cid:88) b,j tanh ( g s r bj ) (cid:88) a,i sech ( g s r ai ) (cid:0) g s r ai − (cid:0) g s r ai − cosh (2 g s r ai ) + π − (cid:1) tanh ( g s r ai ) (cid:1) + O b O + (0) (B.19) ∂ m + Z → O (0) − ∂ m O + ( m ) π (cid:88) a,i tanh ( πr ai ) + ∂ m O + ( m ) (B.20) ∂ m + ∂ m − Z → O (0) (B.21) ∂ m + Z → O + (0) (B.22) ∂ m + ∂ m − Z → − π (cid:88) a,i tanh ( g s r ai ) (B.23) where r ai ≡ x a − y i . Here O + ( m ) = π (cid:32)(cid:88) a,i tanh (cid:16) g s r ai + πm (cid:17)(cid:33) − (cid:88) a,i (cid:0) g s r ai + πm (cid:1) , (B.24) O b = (cid:88) a
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