Numerical test of AdS/CFT correspondence for M2-branes
M-theory is an eleven-dimensional theory, which has been proposed as a strong coupling limit
of the type IIA superstring theory. It has been also expected that the M-theory includes the
eleven-dimensional supergravity (11d SUGRA) as a low-energy limit. The 11d SUGRA consists
of the graviton, gravitino and three-form gauge field. The three-form field in eleven dimensions
electrically (magnetically) couples to two(five)-dimensional object. Such objects naturally appear
as black brane solutions conserving a part of supersymmetries in the 11d SUGRA. On the
analogy of the relation between such solutions in the ten-dimensional supergravities and objects
in the superstring theories as string, NS5-brane and D-branes, we can expect that the M-theory
has fundamental two- and five-dimensional objects. These objects are called as ``M2-brane``
and ``M5-brane``, respectively. In this thesis, we focus on Physics of the multiple M2-branes.
As well known, a low-energy limit of parallel N Dp-branes is described by the (p+1)-dimensional
U(N) maximally supersymmetric Yang-Mills theory. This U(N) gauge symmetry can be
intuitively understood by the facts that open string includes spin-1 massless boson in its spectrum
and have an U(1) charge called as a Chan-Paton factor. What is a low-energy effective
theory of the parallel N M2-branes? Unfortunately, we have not an established answer to this
question yet as we will argue below.
From the single M2-brane analysis and implication of the AdS/CFT correspondence, we
expect that the low energy effective theory for $N$ M2-branes has the following properties:
(1) Three dimensional conformal symmetry, (2) N=8 supersymmetry, (3) SO(8) R-symmetry,
and so on.
However, such a theory had not been found for long years. There are many reasons for this.
One of most serious obstacle is difficulty of quantization of supermembrane. This prevents us
from finding spectrum and something like a Chan-Paton factor for M2-branes. Another difficult
-ty is that it is not easy to construct gauge theory with conformal and high supersymmetry
except for four dimensions. Since Yang-Mills action is scale invariant only for four dimensionns,
we can use only Chern-Simons term of vector multiplet and marginal term of chiral multiplet
for the construction. Indeed in 1990's, a maximal supersymmetric extension of Chern-
Simons theory had been N=3.
In 2008, Aharony, Bergman, Jafferis and Maldacena (ABJM) has proposed a U(N)xU(N)
theory with Chern-Simons levels k and -k coupled to bi-fundamental matters. This theory has
N=8 supersymmetry for k=1,2 and N=6 supersymmetry for other values of k. It has been
conjectured tobe dual to M-theory on AdS_4 x S^7/Z_k for k<<N^{1/5}, and to type IIA
superstring onAdS_4xCP^3 in the planar large-N limit with the 't Hooft coupling constant λ=N/
k kept fixed. From the viewpoint of quantum gravity, the ABJM theory is important since it
may provide us with a nonperturbative definition of type IIA superstring theory or M-theory
on AdS_4 back-grounds since the theory is well-defined for finite N. One may draw a precise
analogy with the way maximally supersymmetric Yang-Mills theories may provide us with non
perturbative formulations of type IIA/IIB superstring theories on D-brane backgrounds through
the gauge/gravity duality. In particular, the M-theory limit is important given that M-theory is
not defined even perturbatively, although there is a well-known conjecture on its nonperturbative
formulation in the infinite momentum frame in terms of matrix quantum mechanics. The
planar limit, which corresponds to type IIA superstring theory, has interest on its own since it
may allow us to perform more detailed tests of the gauge/gravity duality than in the case of
AdS_5/CFT_4. In particular, we may hope to calculate the 1/N corrections to the planar limit,
which enables us to test the gauge/gravity duality at the quantum string level, little of which
is known so far.
In all these prospectives, one needs to study the ABJM theory in the strong coupling regime.
As in the case of QCD, it would be nice if one could study the ABJM theory on a lattice by
Monte Carlo methods. This seems quite difficult, though, for the following three reasons.
Firstly, the construction of the Chern-Simons term on the lattice is not straightforward,
although there is a proposal based on its connection to the parity anomaly. Secondly, the
Chern-Simons term is purely imaginary in the Euclidean formulation, which causes a technical
problem known as the sign problem when one tries to apply the idea of importance sampling.
Thirdly, the lattice discretization necessarily breaks supersymmetry, and one needs to restore it
in the continuum limit by fine-tuning the coupling constants of the supersymmetry breaking
relevant operators. This might, however, be overcome by the use of a non-lattice regularization
of the ABJM theory based on the large-N reduction on S^3, which is shown to be useful in
studying the planar limit of the 4d N=4 super Yang-Mills theory.
In this thesis, we show that the ABJM theory can be studied for arbitrary N at arbitrary cou
pling constant by applying a simple Monte Carlo method to the matrix model that can be deri
ved from the theory by using the localization technique. This opens up the possibility of
probing the quantum aspects of M-theory and testing the AdS_4/CFT_3 duality at the quantum
level. Here we calculate the free energy, and confirm the N^3/2 scaling in the M-theory limit
predicted from the gravity side. We also find that our results nicely interpolate the analytical
formulae proposed previously in the M-theory and type IIA regimes. Furthermore, we show
that some results obtained by the Fermi gas approach can be clearly understood from the
constant map contribution obtained by the genus expansion. The method can be easily
generalized to the calculations of BPS operators and to other theories that reduce to matrix
models. We also study the super-symmetric Wilson loops in the ABJM theory. Our result
nicely interpolates the expressions at weak and strong coupling regions.
Numerical test of AdS/CFT correspondence for M2-branes
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