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)