http://swrc.ontoware.org/ontology#Thesis
Numerical test of AdS/CFT correspondence for M2-branes
en
本多 正純
ホンダ マサズミ
HONDA Masazumi
総研大甲第1590号
Numerical test of AdS/CFT correspondence for M2-branesM-theory is an eleven-dimensional theory, which has been proposed as a strong coupling limitof the type IIA superstring theory. It has been also expected that the M-theory includes theeleven-dimensional supergravity (11d SUGRA) as a low-energy limit. The 11d SUGRA consistsof the graviton, gravitino and three-form gauge field. The three-form field in eleven dimensionselectrically (magnetically) couples to two(five)-dimensional object. Such objects naturally appearas black brane solutions conserving a part of supersymmetries in the 11d SUGRA. On theanalogy of the relation between such solutions in the ten-dimensional supergravities and objectsin the superstring theories as string, NS5-brane and D-branes, we can expect that the M-theoryhas 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)-dimensionalU(N) maximally supersymmetric Yang-Mills theory. This U(N) gauge symmetry can beintuitively understood by the facts that open string includes spin-1 massless boson in its spectrumand have an U(1) charge called as a Chan-Paton factor. What is a low-energy effectivetheory of the parallel N M2-branes? Unfortunately, we have not an established answer to thisquestion yet as we will argue below.From the single M2-brane analysis and implication of the AdS/CFT correspondence, weexpect 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 usfrom 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 supersymmetryexcept 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 multipletfor 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 hasN=8 supersymmetry for k=1,2 and N=6 supersymmetry for other values of k. It has beenconjectured tobe dual to M-theory on AdS_4 x S^7/Z_k for k<<N^{1/5}, and to type IIAsuperstring 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 itmay provide us with a nonperturbative definition of type IIA superstring theory or M-theoryon AdS_4 back-grounds since the theory is well-defined for finite N. One may draw a preciseanalogy with the way maximally supersymmetric Yang-Mills theories may provide us with nonperturbative formulations of type IIA/IIB superstring theories on D-brane backgrounds throughthe gauge/gravity duality. In particular, the M-theory limit is important given that M-theory isnot defined even perturbatively, although there is a well-known conjecture on its nonperturbativeformulation in the infinite momentum frame in terms of matrix quantum mechanics. Theplanar limit, which corresponds to type IIA superstring theory, has interest on its own since itmay allow us to perform more detailed tests of the gauge/gravity duality than in the case ofAdS_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 whichis 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 byMonte 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, theChern-Simons term is purely imaginary in the Euclidean formulation, which causes a technicalproblem 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 itin the continuum limit by fine-tuning the coupling constants of the supersymmetry breakingrelevant operators. This might, however, be overcome by the use of a non-lattice regularizationof the ABJM theory based on the large-N reduction on S^3, which is shown to be useful instudying 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 coupling constant by applying a simple Monte Carlo method to the matrix model that can be derived from the theory by using the localization technique. This opens up the possibility ofprobing the quantum aspects of M-theory and testing the AdS_4/CFT_3 duality at the quantumlevel. Here we calculate the free energy, and confirm the N^3/2 scaling in the M-theory limitpredicted from the gravity side. We also find that our results nicely interpolate the analyticalformulae proposed previously in the M-theory and type IIA regimes. Furthermore, we showthat some results obtained by the Fermi gas approach can be clearly understood from theconstant map contribution obtained by the genus expansion. The method can be easilygeneralized to the calculations of BPS operators and to other theories that reduce to matrixmodels. We also study the super-symmetric Wilson loops in the ABJM theory. Our resultnicely interpolates the expressions at weak and strong coupling regions.