One-loop supergravity on AdS4 × S 7/ℤ k and comparison with ABJM theory

Journal of High Energy Physics, Nov 2016

The large-N limit of ABJM theory is holographically dual to M-theory on AdS4 × S 7/ℤ k . The 3-sphere partition function has been obtained via localization, and its leading behavior F ABJM (0)  ∼ k 1/2 N 3/2 is exactly reproduced in the dual theory by tree-level supergravity. We extend this comparison to the sub-leading \( \mathcal{O}\left({N}^0\right) \) order by computing the one-loop supergravity free energy as a function of k and comparing it with the ABJM result. Curiously, we find that the expressions do not match, with F SUGRA (1)  ∼ k 6, while F ABJM (1)  ∼ k 2. This suggests that the low-energy approximation Z M-theory = Z SUGRA breaks down at one-loop order.

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One-loop supergravity on AdS4 × S 7/ℤ k and comparison with ABJM theory

Received: September S7=Zk and James T. Liu 0 Wenli Zhao 0 leading behavior F A 0 JM 0 Open Access 0 c The Authors. 0 0 Michigan Center for Theoretical Physics, Randall Laboratory of Physics, University of Michigan The large-N limit of ABJM theory is holographically dual to M-theory on S7=Zk. The 3-sphere partition function has been obtained via localization, and its k1=2N 3=2 is exactly reproduced in the dual theory by tree-level supergravity. We extend this comparison to the sub-leading O(N 0) order by computing the one-loop supergravity free energy as a function of k and comparing it with the ABJM result. Curiously, we nd that the expressions do not match, with FS(U1)GRA k2. This suggests that the low-energy approximation ZM-theory = ZSUGRA breaks AdS-CFT Correspondence; M-Theory F A(1B)JM down at one-loop order. 1 Introduction 2 Kaluza-Klein spectrum on the S7=Zk orbifold 3 One-loop free energy of supergravity on AdS4 Asymptotic expansion of FS(U1)GRA for large k 4 Discussion S7=Zk A The q 0 mod k states in the Kaluza-Klein spectrum B Regulator dependence of the one-loop free energy C The polynomials c1(l; m) and c2(l; m) Introduction The AdS/CFT correspondence is a remarkable duality between large-N eld theories and gravity in the bulk. As such, it has passed many non-trivial tests at the leading order Weyl anomaly [1], which for IIB string theory on AdS5 X5 yields c = a = 4 vol(X5) at tree-level in the supergravity limit. This result has been extended to the O(1) level by performing a one-loop computation, where the states running in the loop come from the Kaluza-Klein spectrum on X5 [2{11]. An interesting feature of the one-loop contribution multiplets in the Kaluza-Klein tower. As such, this provides a connection between the holographic central charges and the superconformal index [12, 13]. While the Weyl anomaly is a feature of even-dimensional eld theories, similar holohas been to focus on the holographic entanglement entropy which can be de ned in arbitrary dimensions [14]. Alternatively, the 3-sphere free energy F has been conjectured to play the role of the a-anomaly in odd-dimensional CFTs [15]. In this paper, we extend the one-loop tests of AdS/CFT to the odd-dimensional case by examining the O(1) contributions to F . In particular, we compute the holographic one-loop ABJM sphere partition function in the M-theory limit and compare with the matrix model result. (CSM) theory with gauge group U(N )k U(N ) k [16]. It is conjectured to be the holographic dual of IIA string theory on AdS4 CP3 in the `t Hooft limit with nite and the dual of M-theory on AdS4 S7=Zk in the limit N ! 1 with k5 function has been computed from the matrix model, and takes the form [17]: ZABJM = C 31 eA(k)Ai C 3 1 + ZNon-Perturbative; in the IIA (i.e. planar) limit as the all-genus sum of the constant map contributions to the free-energy [18]: A(k) = FABJM = the M-theory limit by 1 Z 1 x sinh2 x we are mostly interested in [18]. In particular, when expanded for small k, it reproduces the perturbative series computed with the Fermi gas approach in [17]. The ABJM free energy can be expanded in the large-N limit with the result1 2 k1=2N 3=2 F A(1B)JM = N 1=2 + F A(1B)JM + O(N 1=2); The holographic ABJM free energy was computed in [19], and is given at leading order in FS(U0)GRA = 2 k1=2N 3=2: This precisely matches the leading term in the expansion of the matrix partition funcity, which would be given in powers of the 11-dimensional Newton constant, G11 N 3=2. Instead, it arises as a quantum correction in M-theory, and in particular from a shifted relation between ABJM and M-theory parameters resulting from the eight-derivative C3R4 term [20{23], as anticipated in [24]. Our present focus is on the O(1) contribution, F A(1B)JM, which is dual to the oneloop free-energy in M-theory. The log N term in (1.5) has been identi ed as a universal X7, [23]. It is likely that this term is 1Here we use the convention F = log Z. to a ect the zero mode counting, as this ought to be a robust feature of the low energy (and hence supergravity) limit. Although the non-zero modes do not contribute to the log N term in (1.5), they are the O(1) term given in (1.5). We will perform this computation in the M-theory limit, where the dual of ABJM theory in low energy limit is given by 11-dimensional supergravity on S7=Zk. On the ABJM theory side, the AdS/CFT dictionary at leading order gives the relation2 N = where L is the AdS4 radius and lp is the 11 dimensional Planck length. Under (1.7), the O(1) term, (1.5), then becomes F A(1B)JM = On the supergravity side, we regulate the one-loop determinants by working with a 4 + 7 dimensional split. We use spectral zeta function methods for determinants in AdS4 before schematically by ( (0) + c0) log L + a(k); is the volume cuto in the one-loop determinants, c0 is the zero mode contribution, a(k) is a term only dependent on k, and both 0(0) and (0) refer to the regulated qu (...truncated)


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James T. Liu, Wenli Zhao. One-loop supergravity on AdS4 × S 7/ℤ k and comparison with ABJM theory, Journal of High Energy Physics, 2016, pp. 99, Volume 2016, Issue 11, DOI: 10.1007/JHEP11(2016)099