Comments on ABJM free energy on $S^{3}$ at large $N$ and perturbative expansions in M-theory and string theory

We compare large $N$ expansion of the localization result for the free energy $F$ in the 3d $\mathcal{N}=6$ superconformal $U(N)_k \times U(N)_{-k}$ Chern-Simons-matter theory to its AdS/CFT counterpart, i.e. to the perturbative expansion of M-theory partition function on AdS$_{4}\times S^{7}/\mathbb{Z}_{k}$ and to the weak string coupling expansion of type IIA effective action on AdS$_{4}\times {\rm CP}^3$. We show that the general form of the perturbative expansions of $F$ on the two sides of the AdS/CFT duality is indeed the same. Moreover, the transcendentality properties of the coefficients in the large $N$, large $k$ expansion of $F$ match those in the corresponding M-theory or string theory expansions. To shed light on the structure of the 1-loop M-theory partition function on AdS$_{4}\times S^{7}/\mathbb{Z}_{k}$ we use the expression for the 1-loop 4-graviton scattering amplitude in the 11d supergravity. We also use the known information about the transcendental coefficients of the leading curvature invariants in the low-energy effective action of type II string theory. Matching of the remaining rational factors in the coefficients requires a precise information about currently unknown RR field strength terms in the corresponding superinvariants.


Introduction
Localization [1] provides a remarkable source of information about supersymmetric gauge theories beyond the standard weak-coupling perturbation theory. In the context of AdS/CFT duality [2] this information may be used to learn about the structure of string theory or M-theory corrections to the tree level or supergravity order.
Here we shall focus on the 3d N " 6 supersymmetric U pN q kˆU pN q´k Chern-Simons-matter theory [3] in which the free energy F pN, kq on S 3 was computed by localization in [4,5,6,7] (see [8] for a review and further references). For large N and fixed k this theory is dual to M-theory on AdS 4ˆS 7 {Z k while for large N and large k with fixed λ " N k it is dual to the 10d type IIA string theory on AdS 4ˆC P 3 background. 1 Our aim will be to compare the large N expansion of F to its AdS/CFT counterpart, i.e. to perturbative expansion of the M-theory partition function on AdS 4ˆS 7 {Z k or weak string coupling expansion of string theory effective action on AdS 4ˆC P 3 . Related work appeared in [9,10] and [11,12,13,14,15] and also in [16,17,18]. In the M-theory the expansion parameter is the inverse of the effective dimensionless M2-brane tension [3] while the type IIA string coupling and effective string tension are T " 1 8π where L 11 and L are curvature scales in the 11d and 10d metrics.
We will show that the general structure of perturbative expansions of F on the two sides of the AdS/CFT duality is indeed the same. Moreover, the transcendentality properties of the coefficients in the large N , large k expansion of the localization expression for F match those in the corresponding M-theory or string theory expansions. In particular, we will focus on the N -independent Apkq part of F and show that the leading ζp3qk 2 term in its large k expansion corresponds to the ζp3q term in the 1-loop 11d graviton amplitude on M 10ˆS1 [19,20] or the tree-level ζp3qR 4 term in the 10d string theory effective action. 3 Also, we will find that the π-dependent factors in the coefficients of subleading 1 k n terms match those in the coefficients of the corresponding curvature invariants in the M-theory or string theory effective actions. 4 To match the remaining rational factors in the coefficients requires precise knowledge of the structure of the corresponding superinvariants (RR flux terms in them) and remains an open problem.
The order N 0 term in F should correspond to the order pT 2 q 0 or 1-loop correction in M-theory. A similar k-dependent factor in the 1/2 BPS Wilson loop expectation value in the ABJM theory was recently reproduced [22] as the 1-loop quantum M2-brane correction. 5 The Apkq term in the free energy should represent the contribution of quantum M2-brane states propagating in the loop. In addition to point-like M2 branes one may need to include also contributions of BPS M2 branes wrapping 2-cycles in CP 3 part of S 7 {Z k . As we shall discuss below, the structure of the Apkq function suggests a close analogy of the present case with the Calabi-Yau compactification one in [24,25,26]. 6 This paper is organized as follows. In section 2 we review the structure of the large N perturbative part of the free energy as found from localization in ABJM theory on S 3 . In section 3 we compare its large N , fixed k expansion to the perturbative expansion of the partition function or effective action in M-theory. In section 3 we discuss the large N , fixed λ expansion of F and show its correspondence with the perturbative expansion in type IIA string theory on AdS 4ˆC P 3 . Some basic relations and notation are summarized in Appendix A. In Appendix B we recall the matching of the leading large N term in F with the 11d supergravity action evaluated on the AdS 4ˆS 7 {Z k background. In appendix C we present the AdS 4ˆC P 3 values of the R 4 invariants 3 In addition to the M-theory and weakly coupled string theory limits one may consider a limit of large N with fixed N {k 5 that corresponds to the type IIA string at finite string coupling and thus interpolates between M-theory at strong coupling and perturbative string theory at weak coupling. Ref. [14] used that limit to compute R 4 terms at finite coupling. 4 This is similar to what was observed [21] in the discussion of the leading strong-coupling terms in the localization result for free energy in the orbifold N " 2 gauge theory at each order in the 1{N 2 expansion. These terms take the form of a series in λ 3{2 N 2 " g 2 s T and can be matched (up to rational coefficients) with the contributions coming from the D n R m terms (of lowest order in α 1 at each order in g 2 s ) in type IIB string effective action. 5 It was observed in [23] that this psin 2π k q´1 prefactor (where 1 k 2 " g 2 s 8πT as in (1.3)) in the Wilson loop expectation value effectively sums up the leading large T contributions at each order in g 2 s . In [22] this prefactor was derived as a 1-loop correction in the M2-brane world-volume theory and thus it was concluded that this 1-loop M2 brane correction effectively sums up all large tension terms at all orders in the weak string coupling expansion in the dual type IIA theory. 6 It would be interesting also to try to do a similar matching in the case of the topological indices (or special partition functions on S 2ˆS1 ) for which localization results were discussed in [17,18]. that appear in the tree level and one loop term in the type IIA string effective action. Appendix D contains a brief review of the structure of non-perturbative terms in the ABJM free energy.

Free energy of ABJM model in the large N expansion
Our starting point will be the localization result for the free energy of the ABJM theory on S 3 expanded at large N . We will consider both fixed k and large k perturbative expansions ignoring non-perturbative corrections.
The partition function of the ABJM theory on S 3 was first expressed in terms of a localization matrix model in [4]. It was later mapped to a lens space matrix model and solved in the planar limit in [5]. Higher genus 1{N corrections were computed in [6,7] by integrating the holomorphic anomaly equation. Neglecting non-perturbative corrections (reviewed in [27,8]) the resummed partition function was determined in [28]. The same result was later rederived by Fermi gas methods in [29] and tested numerically in [30] at finite N, k. The resulting perturbative partition function reads The presence of the function Apkq was first detected in [30] and incorporated into the Fermi gas formalism of [29] that provided its small k expansion. The large k expansion of Apkq was identified in [30] with a topological string "constant map" contribution [31]. 7 Ref. [30] proposed a resummed integral representation for Apkq (later improved in [34]) valid at both small and large k 8 (2. 2) The expansions of (2.2) in the two regimes may be determined as asymptotic series. For k ! 1 where B 2n are Bernoulli numbers. At large k one finds where q n are rational numbers expressed again in terms of the products of two Bernoulli numbers or even-argument zeta-function values ζp2nq " p´1q n`1 p2πq 2n 2p2nq! B 2n as The expansion (2.4) reproduces (see below) the dominant terms in λ " N k " 1 in the 1{N expansion of (2.1). For this reason the resummation proposal (2.2) is usually considered to be correct. 9 7 The lens space Chern-Simons matrix model partition function can be interpreted as a partition function of a large N dual of a topological string theory on a certain class of local Calabi-Yau geometries [32]. This is a generalization of the Gopakumar-Vafa duality [33]. 8 The specific values of Apkq at integer k are given in Eq. (3.14) of [34]. In particular, Ap1q "´ζ p3q 8π 2`1 4 log 2. 9 Numerical tests of (2.2) for intermediate values of k were also presented in [30].
Below we shall use (2.1),(2.2) as a starting point ignoring non-perturbative corrections (for some comments on them see Appendix D).
From the exact expression of F pN, kq we can work out its large N expansion at fixed k (2.6) One can then assume that k is large and isolate the leading terms in k order by order in large N F k"1 " ?
Here the ζp3q term came from the first term in (2.4). As follows from (2.1), for large k at each order in 1{N the relevant combination should be N´1 24 k and indeed one finds that (2.7) may be rewritten as In the 't Hooft expansion, i.e. the expansion in 1{N with fixed λ " N k , the resulting large N expression of F may be written as Since we isolated in (2.9) the 1{N factors in the combination λ N " 1 k , it follows from (2.1) that the functions f h pλq should naturally depend on the shifted coupling Explicitly, one finds the following expressions for f n [17] (A is Glaisher constant) where all p h and p h,s are rational coefficients.
As a further refinement, we may consider the λ " 1 expansion and isolate the leading powers of λ at each order in 1{N in (2.9). These special terms read (omitting log k " log N λ term in (2.9)) r F " F λ"1 " N 2´π (2.14) Here we kept few subleading large λ terms only in the first N 2 term. Comparing to (2.6),(2.4) we conclude that the coefficients p h are related to q h inĀ in (2.4) as (cf. (2.5)) The two expansions we have discussed (large N at fixed k and large N , large k with fixed λ " N k ) should correspond to the M-theory and type IIA string theory expansions. We shall discuss this connection in the next sections.

M-theory perturbative expansion
The large N , fixed k expansion of the ABJM theory should be dual to the perturbative expansion of M-theory on AdS 4ˆS 7 {Z k in which the curvature scale L 11 is small compared to the 11d Planck length P so that the effective dimensionless M2-brane tension T 2 is large (see Appendix A for our notation) while the parameter k of the 11d background (related to the radius of the 11d circle) is fixed. Indeed, since according to (A.9) L 6 " 32π 2 N k , (3.2) this limit is equivalent to the large N , fixed k expansion.
Thus the M-theory perturbative expansion should be in inverse powers of T 2 or in powers of L´3. Expressed in terms of L and k the large N expansion of the free energy F in (2.6) is indeed As was suggested in [35], the presence of the ş R 4 C 3 term in the 11d effective action [36] implies the following shift of the M2-brane charge N N Ñ N´1 24 pk´k´1q .
Expressing the localization result for F pN, kq in (2.6) in terms of this redefined parameter L and k we find a remarkable simplification of the k-dependent coefficients of the L 3 powers Thus the k-dependence of the L 9 , L 3 and L´3 terms becomes simply 1 k (though this does not apply to L´6 and higher order terms in the expansion).
It is natural to expect that the terms in the free energy that scale as 1 k may originate from local terms in the M-theory partition function or the effective action evaluated on the AdS 4ˆS 7 {Z k background. Indeed, as this background is homogeneous (and its curvature does not depend on k explicitly, apart from the dependence via L 11 or L) the integrals of curvature (and 4-form) invariants will be proportional to the factor of the radius a " 1 k of the 11d circle coming from the integration volume. Other terms that do not scale as 1 k may come from non-local contributions to the M-theory partition function.

Local terms
The L 9 term in (3.7) comes effectively from the 11d supergravity action S 0 " 1 and also that 2κ 2 11 " p2πq 8 9 P (see (A.1)) we conclude that the coefficient of the supergravity term should scale as 1 π 2 1 k L 9 matching the π´2-dependence of c 0 in (3.4). Similarly, the 1 k L 3 term in (3.7) should come from the local 1-loop R 4`¨¨¨t erm in the 11d effective action [38,19,20, 39] 10 (3.10) Here we isolated the factor of the M2 brane tension T 2 . This term may be viewed as the 1-loop 11d supergravity contribution Λ 3 R 4`¨¨¨t hat scales as κ 0 11 but is cubically divergent [40] leading to a finite term in (3.10) after assuming the M-theory UV cutoff " ´1 P . 11 Thus this local 1-loop R 4 term is the one that corresponds to the k´1 {2 N 1{2 term in F in (2.6). 10 Note that here our P (see Appendix A for notation) is related to 11 used in [20,39] as 3 11 " 2π 3 P so that the values of κ11 and M2-brane tension T2 are the same as in these papers. 11 This R 4 term should be a superpartner of the R 4 C3 term. The fact that accounting for the shift (3.6) removes the "non-local" kL 3 term in (3.3) may be viewed as a consequence of supersymmetry. Note also that in general higher loop supergravity contributions should scale as pκ 2 11 q L´1 " pT2q´3 pL´1q but in local terms extra factors of the M-theory UV cutoff " ´1 P may introduce extra positive powers of T2, see [20,41] and Eq. (3.11) below.
Let us recall that similar terms " N 3{2 and " N 1{2 appear in the finite temperature free energy of the world-volume theory of multiple M2 branes and have similar origins in the R [42] and R 4 [43] terms in the M-theory effective action.
To reproduce the value of the coefficient c 1 1 in (3.7),(3.8) one needs the information about the precise structure of the 4-form dependent terms in the R 4 superinvariant which is not yet known (cf. [39]). 12 Still, it is remarkable that the fact that the value of c 1 1 that comes from (3.10) on the AdS 4ˆS 7 {Z k background is rational as in (3.8) does follow from the values of b 1 in (3.10) and of the volume factor (3.9): all factors of π cancel out.
In general, on dimensional grounds, all local terms in the M-theory effective action should contain particular powers of the M2-brane tension, i.e. should be given by the sum of terms like [20] 13 where dots stand for other possible terms (depending also on F 4 ) that have the same mass dimension 6p`2. Explicitly, the S 1 in (3.3) corresponds to the p " 1 case of (3.11), the p " 2 case is 7 {Z k background (3.11) will scale as 1 k L 9´6p and may, in principle, match some of the subleading terms in in the free energy (3.7). Terms that do not scale as 1 k should come from non-local parts of the quantum M-theory effective action.

Terms corresponding to the 1-loop M-theory contribution
The terms 1 2 log L 3 πk and´Apkq in (3.3) which are of zeroth order in the effective M2-brane tension (3.1) should originate from the (UV finite part of) 1-loop contribution to the M-theory partition function. The logarithmic 3 2 log L term coming from 1 4 log 32N k " 1 2 log L 3 πk term in (2.6) was reproduced by a 1-loop computation in 11d supergravity in [9] as a universal contribution of the zero modes of the 11d supergravity fluctuation operators on AdS 4ˆX 7 background (with the dependence on P via L coming from normalization factor related to κ 11 ). The´1 2 logpπkq term should have a similar origin (being also related to the volume factor in the normalization of the supergravity modes).
The´Apkq term in (3.3) (see (2.6),(2.4)) should correspond to the L-independent part of the 1-loop contribution in M-theory on AdS 4ˆS 7 {Z k .
In general, the 1-loop M-theory partition function should be the contribution of virtual M2brane propagating in the loop but it is not clear how to define it precisely. In the case of a large amount of supersymmetry of the background one may conjecture that only special BPS states (e.g.corresponding to M2-branes wrapped on special 2-cycles of internal space) may be contributing to the 1-loop partition function, while contributions of non-BPS states may cancel due to extended supersymmetry of the background (cf. [24,25,26]).
One may start with the contribution of just point-like BPS states corresponding to the 11d supergravitons, i.e. approximate the M-theory 1-loop partition function by its 11d supergravity 12 While matching the overall coefficient c 1 1 is thus an open problem, in [16] the dependence of the coefficient of the similar N 1{2 " L 3 term on extra geometric parameters (like squashing of the S 3 ) in the localization result for the free energy F was reproduced from the effective 4d effective action with the supersymmetric R 2 terms that should originate from the 11d R 4 superinvariant compactification to 4d. 13 The special role of the terms (3.11) noted in [20] is that upon reduction to 10d they have perturbative dependence on the string coupling gs. Note that some of these terms may be interpreted as higher-loop corrections in 11d supergravity proportional to pκ 2 11 q L´1 Λ 3n " pT2q 3´3L`n (Λ " ´1 P is the 11d UV cutoff).
counterpart. To get an insight about the structure of the latter and to compare it with F in (3.7) we will be guided by the expression for the low-energy expansion of the 1-loop correction to the 4-graviton amplitude in 11d supergravity [20]. While there is no a priori reason why just the supergravity correction should be enough to capture the full M-theory result, we will show that it indeed reproduces the structure of the large k expansion of the corresponding term in F . 14 Our strategy will be as follows. We shall consider the expression for the 1-loop 4-graviton amplitude in 11d supergravity expanded near flat space with 11d circle of radius R 11 (found under a simplifying assumption that only 10d components of the 4 polarization tensors and external momenta are non-zero) following [20]. We shall then expand this amplitude in powers of momenta and extract its dependence on R 11 and 11d UV cutoff Λ " ´1 P . Finally, we will assume that it can be used to shed light on the structure of 11d supergravity 1-loop partition function on a curved background. Specifying to the case of the AdS 4ˆS 7 {Z k background we shall reproduce the structure of the L 3 , log L and Apkq terms in (3.7). Remarkably, we shall find the terms with the same transcendental coefficients ζp3q and π 2h´2 that appear in the large k expansion of Apkq in (2.4). 15 In order to match the remaining rational factors in the coefficients it appears that one is to include other contributions to the M-theory partition function on AdS 4ˆS 7 {Z k background. These are presumably of other (extended) BPS M2-brane states propagating in the loop. By analogy with the case of the Calabi-Yau compactification [25] we shall then discuss how one could try to modify the supergravity-based result in order to reproduce the double-Bernoulli structure of the coefficients in Apkq in (2.4),(2.5).
The 4-graviton amplitude may be written as (omitting polarization tensor and normalization factors including 10d volume and momentum delta-function) [44,19,20] A 4 ps, tq " A 4 ps, tq`psymm in s, t, uq, where s, t, u are the standard kinematic variables depending on 10d momenta, the sum is over 11d component of the virtual momentum and τ has dimension of length squared. The function P is given by 16 Focussing on the first term in the sum in (3.12) and expanding e´τ M in powers of momenta, or, equivalently, in powers of M we get The h " 0 term in (3.15) may be written (using Poisson resummation and H 0 " 1 6 ) as [19] Here the first term comes effectively from the n " 0 contribution and is thus the same as in the 1-loop contribution in 10d supergravity. The second term comes from the contribution of 11d supergravity states with non-zero 11d momentum (or, from the 10d string theory point of view, from the contribution of the massive D0-brane states in the loop [19]). The h " 1 contribution vanishes after integrating over ρ, in agreement with the absence of 1-loop logarithmic divergences in 11d theory (and also in the 1-loop 4-graviton amplitude in 10d supergravity [45]).
The remaining h ě 2 terms are UV finite. The n " 0 term in the sum in (3.16) with h ě 2 gives a non-analytic contribution (" s log s, etc.) to (3.15) which is independent of R 11 (and thus should be the same as the 1-loop amplitude in 10d supergravity) The contribution of the n ‰ 0 terms may be written as where C h is given by Let us now interpret (3.22) as providing an indication about the structure of the M-theory 1loop partition function on a curved background. Specializing to AdS 4ˆS 7 {Z k we will have the 11d radius R 11 Ñ 1 k L 11 and, just on dimensional grounds, the momentum variables s, t scaling as L´2 11 . Rescaling (3.22) by L 2 11 to get a dimensionless expression we would then get from (3.22) (L " L 11 P " L 11 Λ) Here u 0 " 2 3π w where w is the coefficient of proportionality in Λ 3 " w ´3 P so that matching the rational c 1 1 coefficient in (3.7) requires w " π. 17 The terms u 1 log L`u 2 come from sHp s t q term in (3.22). The coefficients d 1 h are related to d h in (3.21) by rescaling by some rational factors. Thus (3.24) has the same structure as the sum of the L 3 , log L and´Apkq terms in (3.7). The missing log k term should be coming from the 1-loop 11d supergravity zero mode normalization contribution mentioned above and is thus not expected to be captured by this qualitative argument based just on the structure of the 4-graviton amplitude.
Remarkably,F in (3.24) has exactly the same form as the leading ζp3qk 2 term plus the sum of the subleading 1 k 2h´2 terms in´Apkq in (2.4), (2.5). Furthermore, the transcendental factors of π match between the coefficients d h " d 1 h in (3.21) and q h (2.5) in Apkq. To match the rational coefficient of the ζp3q term in´Apkq we would need an extra factor of 1 8 , i.e.
The exact equality of the rational factors in d 1 h (that should differ from d h in (3.21) just by rational factors) and q h in (2.5) may be hard to expect a priori given the crude nature of the above relation between the 1-loop supergravity amplitude and the 1-loop partition function on a curved background. But a definite mismatch in powers of the Bernoulli number factors between (3.21) and in (2.5) suggests that some other contributions (in addition to 1-loop 11d supergravity one) may be missing.
One may wonder whether to match the full expression for the Apkq term in F in (3.7) one needs to include contributions of other M2-brane BPS states to the M-theory 1-loop partition function. 18 By analogy with a discussion in [24,25] one may conjecture that this may lead to a modification of the measure in the proper-time integral in (3.19) like where µ is a mass parameter (that may be related to L´1 11 in the present context, so that µR 11 " 1 k ). Then an extra factor of the Bernoulli numbers required to match d h in (3.21) with q h in (2.5) may come from the expansion To see at the heuristic level how that may work out we may start with the localization expression forĀpkq in (2.4),(2.5) that has the following integral representation [30] Apkq " 17 This is indeed the right identification as follows from the discussion in [19,20] or the comparison with the coefficient of the corresponding R 4 term in 11d effective action (3.10). In the present notation w " 1 2 π, i.e. Λ 3 " 1 2 π ´3 P . 18 In particular, one may consider contributions of M2-branes wrapped on 2-cycles of CP 3 part of S 7 {Z k (which, in the perturbative 10d string limit, are related to the type IIA string world-sheet instantons [46,47] but here play a role of massive modes propagating in the loop). These may be the analogs of M2-branes wrapped on 2-cycles of CY space in [25]. Note also that the field strength of the RR 1-form A in the S 7 {Z k metric (A.10) may be playing the role of the graviphoton strength in the discussion of [26].
One may also rewrite the full expression for Apkq in (2.4) as 19 Apkq " In (3.29) we are assuming that the evaluation of the singular terms (corresponding to the last two terms in the bracket in (3.28)) is done using a suitable regularization. 20 Here k " L 11 R 11 and we may redefine t Ñ pL 11 q´1τ to put the integral into a similar form as in (3.26).
Eq. (3.29) closely resembles the expression in [25,26] used to reproduce the coefficients of the special protected R 2 F 2h´2 terms [48] in the 4d effective action of type II string (compactified on a CY space) from a conjectured 1-loop M-theory correction coming from M2-brane BPS states. Indeed, we may compare the summand in (3.29) with the 4d effective action of a charged scalar of mass m (representing an M2-brane wrapped on a 2-cycle in CY space ) in a constant self-dual gauge field background Here F is the gauge field strength and the UV divergent term is assumed to be subtracted out. Specializing to a BPS state with m " e and rescaling t one gets the integrand as 1 sinh 2 t 1 e´2 mF´1 t 1 . Accounting for multiple wrappings corresponds to m Ñ nm and summing over n so that we get (3.31) This matches (3.29) if 2mF´1 is identified with k. 21 Then the coefficients in the 1 k expansion of (3.29) are directly related to the coefficients in the expansion of (3.31) in powers of F. In the present case 1 k " R 11 L 11 scales as the square root of the effective curvature of AdS 4ˆS 7 {Z k and is thus analogous to F. 22

Type IIA string perturbative expansion
Let us now compare the expansion of the localization result for free energy in the 't Hooft limit (2.9) with the perturbative expansion of the effective action in type IIA string theory in AdS 4ˆC P 3 background.
Let us start with the free energy expanded in large N and large k with fixed λ " N k (2.9) and then expanded further in large λ (see (2.13),(2.14)). Expressing F in terms of the type IIA string parameters g s and T in (A.14),(A.15) we will attempt to match the result to the perturbative 19 Note that using 1 sinh 2 t " 4 ř 8 n"1 n e´2 nt one may also write Apkq " 4 ř 8 n,m"1 n ş 8 0 dt t e´p mk`2nqt . 20 For instance, with an analytic regularization like dt t Ñ dt t 1`ε one has ş 8 The log k and k 2 terms here agree with (2.4). The pole term (plus regularization-dependent transcendental constants) should be discarded as part of the regularization prescription. 21 This matching is not totally unexpected given that both functions were noticed to be related to the topological string amplitudes (cf. [49,48] and [6,50]). 22 The difference between (3.29) or (3.31) and the attempted modification (3.26) is that the sum over the 11d KK modes in (3.26) involves n 2 rather than n, but in going from (3.30) to (3.31) this is taken care of by a rescaling of t. For this to be possible requires the measure in (3.26) to depend on n. low-energy or α 1 expansion of type IIA string effective action in the corresponding AdS 4ˆC P 3 background (A.12) order by order in small g s .

Using the original relations between the parameters (A.14),(A.15) [3]
N " 4 ? 2π T 5{2 g´1 s , λ " 2T 2 , T " 1 8π where L is the scale in 10d metric in (A.12) we get from (2.13) Here the first term L 8 α 14 scales as the contribution of the R term in the type IIA effective action while the third L 2 α 1 term -as the contribution of the R 4 term. The second L 4 α 12 contribution could come from the 1 g 2 s α 12 ş d 10 x ? G R 3 term in tree-level string effective action but such term is absent in type IIA 10d string theory (on supersymmetry grounds).
This problematic term is eliminated if one takes into account the shift of N in (3.5),(A. 18) implying that the relations between the gauge theory pN, λq and string theory pg s , T q parameters take the modified form (A.20),(A. 19). Note that if we shift N in (3.5) as N Ñ N´k 24`a 24 k´1 with a " 1 then this term will not be eliminated. 23 As a result, (2.13) then gives (cf. (3.7),(3.8)) where we kept only the leading at large tension (large λ or small α 1 L 2 ) contribution at each order in g 2 s expansion (apart from the ζp3qg´2 s term). Since according to (A.16) the q h coefficients in (4.4) are the same as (2.5),(2.15) appearing in the large k expansion of Apkq term in (2.6) or in (3.7).

Transcendentality structure of the coefficients
Similarly to the discussion in section 3.1 above, the first term in (4.3) originates from the supergravity part 1 g 2 s α 14 ş d 10 x ?´G pR`¨¨¨q of the tree-level 10d superstring effective action evaluated on the AdS 4ˆC P 3 background (see (B.2)).
The second L 2 α 1 term in (4.3) has the structure that corresponds to the contribution of the sum of the tree level ? kN`... [11,12,14] which is naturally related to the definition of the Newton's constant in the gravitational dual (the coefficient in the graviton kinetic term). This then leads [12,14] to the expected higher derivative terms avoiding spurious terms like R 3 (we thank S. Chester for pointing this out). Also, an alternative shift of N in relation to 4d Newton's constant was used in [17] and shown to lead to simplification of perturbative expansion. In general, the relation between string/M-theory parameters and dual gauge theory ones is effectively a scheme choice required to make the duality manifest; unambiguous relations are found only when expressing one observable in terms of the others.
The factors of π in the coefficients match perfectly after we account for π 5 coming from the volume of AdS 4ˆC P 3 (see (B.3)). As discussed in more detail in the next subsection, fixing the remaining rational coefficients requires the information about the RR field strength dependent terms in the corresponding superinvariants which is not available at the moment.
The higher order h ě 2 terms in (4.4) may originate from local terms in type IIA effective action of the form (note that R " L´2 and the L 10 factor comes from the 10d volume) Here L h may contain several terms of the same dimension (depending on curvature and other fields) required on supersymmetry grounds. The structure of these invariants is not known but as the relative coefficients of the terms in (4.6) should be rational, we may get some information about their transcendentality properties by looking at the particular terms D 2h R 4 . Like for the tree-level and 1-loop R 4 terms, the coefficients of these terms may, in principle, be fixed using the type II string 4-graviton scattering amplitude.
In fact, one may follow [20] and conjecture that in the perturbative string theory limit (g s ! 1) the structure of the 11d supergravity amplitude (3.22) implies the presence of special 10d-local terms g 2h´2 s α 1h´1 ş d 10 x ?´G D 2h R 4 in the type IIA string effective action. These should correspond to local s h terms in (3.23), i.e. should have the coefficients proportional to d h in (3.21) or p2πq 2h (after including an overall normalization factor " π 2 as implied by (3.17)) and should thus match the π-dependence of the q h coefficients in (4.4). As was already pointed out in the previous section, the matching of the rational factors (proportional to B 2h B 2h´2 in (2.5),(2.15) instead of just B 2h´2 in d h in (3.21)) implies the need to account also for the contributions of other terms of the same dimension in the corresponding superinvariants in (4.6).
The same conclusion about the structure of the relevant coefficients can be reached also from the leading D 2n R 4 terms in the effective action reconstructed directly from the type II 4-graviton 10d superstring amplitudes. This applies also to the type IIB effective action (for a related discussion in connection with free energy of N " 2 4d gauge theory models see [21]). The leading D 2n R 4 terms are the same (at least at 1-loop and 2-loop orders) in both type IIA and type IIB theories [41,51]. In the type IIB case one finds The functions f 0 , f 1 , f 2 contain a finite number of perturbative contributions plus non-perturbative Ope´1 {g 2 s q corrections that we shall omit (see, e.g., [41,52]) 24 The leading α 1 terms at each order in g 2 s in (4.7) correspond to the last perturbative terms in f 0 , f 1 , f 2 in (4.9) and their coefficients are expected to be protected by supersymmetry.
This suggests that the coefficients of the terms g 2h´2 s α 1h´1 ş d 10 x ?´G D 2h R 4 with h ě 2 we are interested in are proportional to ζp2hq " p´1q h`1 p2πq 2h B 2h 2p2hq! " p2πq 2h . This is the same conclusion 24 Here f3 " 1 64 ζp9qg´2 s`k0 ζp3q logp´α 1 D 2 q`Opg 2 s q and may contain an infinite series of terms in g 2 s (though their presence appears to remain an open question). The logarithmic term is associated with a non-local term p 16 log p 2 in the 4-graviton amplitude on a flat background. that follows from the above conjectured relation to the 11d supergravity 1-loop amplitude. Once again, to match the remaining rational factors in the coefficients against those in the free energy (2.15) would require the precise knowledge of the superinvariants that have the same dimension as D 2h R 4 terms.

Contributions from tree level and 1-loop R 4 invariants
To illustrate this point let us go back to the discussion of the contribution of the tree-level and 1-loop R 4 terms in type IIA theory. They may be written as (assuming the dilaton is constant and ignoring dependence on B 2 field, see, e.g., [39] for a review) J 0 "t 8 t 8 RRRR`1 8 ε 10 ε 10 RRRR`¨¨¨, J 1 " t 8 t 8 RRRR´1 8 ε 10 ε 10 RRRR`¨¨¨, (4.11) Dots in J 0 and J 1 stand for other terms of the same dimension depending on RR fields. 25 In type IIB theory J 1 is replaced by J 0 so that ζp3q g 2 s`π 2 3 is the total coefficient of the R 4 terms as was already indicated in (4.8). In type IIA theory J 0 and J 1 should correspond to separate superinvariants.
Explicitly, the contribution of the R 4 terms is then To compare to the corresponding L 2 α 1 term in the free energy (4.3) we are to evaluate (4.13) on the AdS 4ˆC P 3 background. 26 As was already mentioned above, since the background is homogeneous with R " F 2 4 " F 2 2 " L´2 and the volume of the AdS 4ˆC P 3 given by (B.3), i.e. ż d 10 x ?´GˇˇˇA dS 4ˆC P 3 " 4π 2 3 p 1 2 Lq 4¨π 3 6 L 6 " π 5 3 2¨23 L 10 , (4.14) the coefficient of the ζp3q term in (4.13) scales as (4.4) thus requires an extra rational factor 3 3¨213 that should presumably come from the curvature contractions and other terms in J 0 , J 1 depending on fluxes.
A factor of the same order does come from the Weyl-tensor dependent part of J 0 : 27 25 Terms with Ricci tensor can be expressed in terms of flux-dependent terms using equations of motion (or field redefinitions). In J0, J1 we use Minkowski signature so that ε10ε10 "´10! and after reduction to 8 spatial dimensions εmn...εmn... Ñ´2ε8ε8. t8 is the 10-dimensional extension of the 8-dimensional light-cone gauge involving G µν (see, e.g., [53]). Explicitly, 26 It is curious to note that if we did not apply the redefinition in (A.18) then the value of the coefficient b1 in 3 q as in (4.2) and is thus exactly proportional to the coefficient of the second term in (4.13). 27 Note thatJ0 vanishes in the case of undeformed AdS5ˆS 5 background (implying, in particular, no correction to the radius or free energy, cf. e.g. [54,43,55]) but does not vanish on the AdS4ˆCP 3 one. (4.16) The difference from the required 3 3¨213 factor may be attributed to the contributions of other Ricci tensor dependent terms in t 8 t 8 RRRR and ε 10 ε 10 RRRR (discussed in Appendix C) and other RR field strength dependent terms in the invariants J 0 , J 1 (cf. [56]).

Acknowledgments
We thank S. Chester and S. Giombi for very useful discussions and comments on the draft. MB was supported by the INFN grant GSS (Gauge Theories, Strings and Supergravity). AAT was supported by the STFC grant ST/T000791/1 and also acknowledges A. Sfondrini for the hospitality at the Filicudi workshop on Integrability in lower-supersymmetry systems in June 2023.

A Notation and basic relations
Here we review the relations between M-theory and type IIA string theory parameters in general and also in the specific case of the AdS 4ˆS 7 {Z k background when they are expressed in terms of of N and k of ABJM theory [3].
The action of the 11d supergravity is where our normalization of 11d Planck length P here is the same as in, e.g., [3,57]. The M2-brane tension is then [42] T 2 " 1 p2πq 2 3 P . (A.2) Assuming compact x 11 direction the 11d metric may be written as where, upon reduction to 10d, ds 2 10 will be the string frame metric and φ the dilaton. The constant part of the dilaton is related to string coupling as g s " e φ , so that (A.1) reduces to the standard 10d type IIA supergravity action 28 with The M2 brane wrapped on the x 11 circle gives the fundamental string action with the standard tension 29 2π R 11 T 2 " T 1 , T 1 " 1 2πα 1 .
(A.5) 28 It reads 1 2κ 2 10 ş d 10 x ?´G re´2φpR`...q`...s whereφ is non-constant part of the dilaton, with constant part included in κ10. 29 In relating M2-brane action and the fundamental string action by this double dimensional reduction the dilaton factors cancel [58].
From (A.4) and (A.5), we then learn that in the above notation 30 For a constant φ the effective radius of the 11-th direction is (as in [59]) Let us now specialize to the AdS 4ˆS 7 space supported by the 4-form flux withN units of charge (which is the near-horizon limit of the background sourced by multiple M2-branes [60]) The flux quantization condition implies that Considering Z k quotient of S 7 we get [3] ds 2 and thus R 11 " g 2{3 s R 11 " Here L 11 and k are the parameters of the 11d M-theory background.
Upon dimensional reduction we then get the metric and parameters of 10d string theory Expressed in terms of the dual gauge-theory parameters N and k the string coupling and the effective dimensionless string tension are T " L 2 AdS 4 T 1 " The M-theory perturbative expansion corresponds to large curvature scale or large effective M2 brane tension for fixed parameter k of the background i.e. to the large N limit with fixed k. The 10d string perturbative expansion corresponds to g s ! 1 for fixed T , i.e. to the the 't Hooft expansion in the large N , large k limit with fixed λ " N k . As was argued in [35], the presence of the M-theory correction R 4 C 3 implies the shift which modifies the relation between L 11 and N in (A.11). This leads also to a modification of the expressions for the 10d string parameters g s and T in (A.14),(A.15) , (A. 19) or, equivalently, of how N and λ are expressed in terms of them: Note that the useful relation (A.16) remains unchanged.

B Supergravity contribution to the free energy
To find the leading term in the M-theory effective action one is to evaluate the 11d supergravity action (A.1) on the AdS 4ˆS 7 {Z k background. There is a subtlety here: as this is an "electric" solution, the sign of the flux F 2 term is to be reversed when evaluating the on-shell value of the action (this is also equivalent to adding a particular boundary term). This then gives the same value of the on-shell action as found from the effective 4d action having AdS 4 as its solution (see [61] and refs. there for related discussions). 31 The latter approach was used in [6].
Analogous remark applies to computing the on-shell value of the type IIA action. Explicitly, starting from the IIA supergravity solution and compactifying on CP 3 we get an effective 4d Einstein action with a cosmological constant that admits the AdS 4 solution with the radius L AdS 4 " 1 2 L (cf. (A.12)). Then (using the negative overall sign for the action corresponding to the Euclidean signature) Here κ 10 is given in (A.4) and on the AdS 4 solution R "´12L´2 AdS 4 so that We used that the volumes for the unit-radius spaces are 32 and also that L AdS 4 " 1 2 L and (A.13),(A.15),(A.14). Thus (B.2) matches the first term in F in (2.14) [6].
The same value is found by starting with the 11d supergravity solution and again evaluating the effective 4d action on the AdS 4ˆS 7 {Z k background where we used (A.11). This matches the first term in the large N expansion of free energy (2.6). Note that if one directly evaluates the 11d action (A.1) on the AdS 4ˆS 7 {Z k solution (without inverting the sign of the F 2 term) one gets´1 2 of the value in (B.4). 33 C Values of R 4 invariants on AdS 4ˆC P 3 In (4.16) we presented the value of the Weyl-tensor part (4.15) of the J 0 invariant in (4.11) on AdS 4ˆC P 3 background. Here we shall present the values of the full curvature-dependent parts of the invariants J 0 and J 1 keeping also the Ricci tensor dependent contributions.
Evaluating these two invariants on AdS 4ˆC P 3 with the metric (A.12) introducing for generality γ " p L CP 3 L AdS 4 q 2 as the ratio of the squares of the radii we find 36 For γ " 4 corresponding to the metric in (A.12) we get Thus if we would keep only these curvature-dependent terms in J 0 , J 1 in (4.10) we would get from (4.13) using (4.14) This is of the same order as just the Weyl-tensor part contribution in (4.16) but does not match the precise rational coefficients in the L 2 α 1 term in r F in (4.3). This suggests that it is important to include also the contributions of the RR field strength dependent terms in J 0 and J 1 to get the matching.
As an aside, let us note that E 8 has an interpretation of an Euler density in 8 dimensions. In general, for a d dimensional space M d with Euclidean signature 37 a 2n´1 a 2n , (C.7) which vanishes for 2n ą d. For example, for a sphere S d we have R ab ce " 1 r 2 d δ ab ce , and thus E 2n pS d q " 2 n d! pd´2nq!`1 r 2 d˘n . For a product manifold M mˆSn , with d " m`n, we have [65] E 2p pM mˆSn q " rm{2s ÿ t"0ˆp t˙n ! pn´2pp´tqq!´1 r 2 n¯p´t 2 t´p E 2t pM m q . (C.8) This is a special case of the general relation 38 E 2p pM mˆMn q " rm{2s ÿ t"0ˆp t˙E 2t pM m qE 2pp´tq pM n q , (C.9) implying, in particular, that E 8 pM 4ˆM6 q " 4E 2 pM 4 q E 6 pM 6 q`6E 4 pM 4 q E 4 pM 6 q. Indeed, one can check that E 8 pAdS 4ˆC P 3 q " 4E 2 pAdS 4 q E 6 pCP 3 q`6E 4 pAdS 4 q E 4 pCP 3 q, (C. 10) in agreement with the value of E 8 in (C.4). 36 In particular, R kl R kl " p384`36γ 2 qL´4, R klmn R klmn " p384`24γ 2 qL´4, R ab R bc R cd Rca " p24576`324γ 4 qL´4. 37 Recall that in d dimensions εi 1¨¨¨in ε j 1¨¨¨jn " δ j 1¨¨¨jn i 1¨¨¨in " ř σ p´1q σ δ j 1 iσ 1¨¨¨δ jn iσ n , δ j 1¨¨¨js j s`1¨¨¨jp i 1¨¨¨is i s`1¨¨¨ip " pd´sq! pd´pq! δ j 1¨¨¨js i 1¨¨¨is . 38 We use this opportunity to point out a misprint in eq.(4.1) in [39]: the coefficient of the second term in E8pM 4ˆM7 q " 4E2pM 4 q E6pM 7 q`6E4pM 4 q E4pM 7 q is 6 not 12. The value of this coefficient was not, actually used in [39].

D Non-perturbative corrections to the ABJM free energy
Here, for completeness, we recall some facts about non-perturbative corrections to free energy of the ABJM theory.
In M-theory one may expect non-perturbative contributions to the M2-brane partition function related to membranes wrapping a 3-cycle C 3 of 11d space and thus producing a factor " expp´T 2 volpC 3 qq where T 2 is M2-brane tension in (A.2). If C 3 wraps 11d circle then this contribution may be interpreted as the 10d fundamental string instanton with T 2 volpC 3 q Ñ T 1 volpC 2 q where T 1 is the string tension (cf. (A.5)) and C 2 is a 2-cycle in 10d space on which the string worldsheet is wrapped. If C 3 lies in 10d subspace then the corresponding contribution is that of the D2-brane instanton.
In the Fermi gas approach of [29] the exact localization expression for the ABJM partition function is expressed in terms of the grand potential Jpµ, kq of a non-trivial fermionic system as where the two sums may be interpreted as accounting for the contributions of the two types of instantons mentioned above. Isolating the terms with " 0 and n " 0 we may write J np pµ, kq " J I pµ, kq`J II pµ, kq`δJ np pµ, kq . (D.7) Terms with both ą 0 and n ą 0 in (D.6) (or "bound state" corrections) given by δJ np pµ, kq were discussed in [66]. Here J I pµ, kq is given by J I pµ, kq " where d n may be determined using that the ABJM matrix integral is dual to the partition function of topological string theory on P 1ˆP1 . J II pµ, kq has the following structure for µ " 1 J II pµ, kq " where the expansion of the coefficients a , b , c for small k follows from the WKB expansion of the Fermi gas representation [29] a pkq " 1 k ř 8 m"0 a ,m k 2m , etc. Conjectures for the closed form of some of these coefficients were suggested in [67] and a unifying picture were all p , nq terms in (D.6) arise from a refined topological string representation was presented in [68]. The saddle point evaluation of (D. Recently, the prefactor of the leading worldsheet instanton correction to the free energy [6] was directly computed on the 10d string theory side in [47].