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It has been also expected that the Mtheory includes the\nelevendimensional supergravity (11d SUGRA) as a lowenergy limit. The 11d SUGRA consists\nof the graviton, gravitino and threeform gauge field. The threeform field in eleven dimensions\nelectrically (magnetically) couples to two(five)dimensional object. Such objects naturally appear\nas black brane solutions conserving a part of supersymmetries in the 11d SUGRA. On the\nanalogy of the relation between such solutions in the tendimensional supergravities and objects\nin the superstring theories as string, NS5brane and Dbranes, we can expect that the Mtheory\nhas fundamental two and fivedimensional objects. These objects are called as ``M2brane``\nand ``M5brane``, respectively. In this thesis, we focus on Physics of the multiple M2branes.\nAs well known, a lowenergy limit of parallel N Dpbranes is described by the (p+1)dimensional\nU(N) maximally supersymmetric YangMills theory. This U(N) gauge symmetry can be\nintuitively understood by the facts that open string includes spin1 massless boson in its spectrum\nand have an U(1) charge called as a ChanPaton factor. What is a lowenergy effective\ntheory of the parallel N M2branes? Unfortunately, we have not an established answer to this\nquestion yet as we will argue below.\nFrom the single M2brane analysis and implication of the AdS/CFT correspondence, we\nexpect that the low energy effective theory for $N$ M2branes has the following properties:\n(1) Three dimensional conformal symmetry, (2) N=8 supersymmetry, (3) SO(8) Rsymmetry,\nand so on.\nHowever, such a theory had not been found for long years. There are many reasons for this.\nOne of most serious obstacle is difficulty of quantization of supermembrane. This prevents us\nfrom finding spectrum and something like a ChanPaton factor for M2branes. Another difficult\nty is that it is not easy to construct gauge theory with conformal and high supersymmetry\nexcept for four dimensions. Since YangMills action is scale invariant only for four dimensionns,\nwe can use only ChernSimons term of vector multiplet and marginal term of chiral multiplet\nfor the construction. Indeed in 1990\u0027s, a maximal supersymmetric extension of Chern\nSimons theory had been N=3.\nIn 2008, Aharony, Bergman, Jafferis and Maldacena (ABJM) has proposed a U(N)xU(N)\ntheory with ChernSimons levels k and k coupled to bifundamental matters. This theory has\nN=8 supersymmetry for k=1,2 and N=6 supersymmetry for other values of k. It has been\nconjectured tobe dual to Mtheory on AdS_4 x S^7/Z_k for k\u003c\u003cN^{1/5}, and to type IIA\nsuperstring onAdS_4xCP^3 in the planar largeN limit with the \u0027t Hooft coupling constant λ=N/\nk kept fixed. From the viewpoint of quantum gravity, the ABJM theory is important since it\nmay provide us with a nonperturbative definition of type IIA superstring theory or Mtheory\non AdS_4 backgrounds since the theory is welldefined for finite N. One may draw a precise\nanalogy with the way maximally supersymmetric YangMills theories may provide us with non\nperturbative formulations of type IIA/IIB superstring theories on Dbrane backgrounds through\nthe gauge/gravity duality. In particular, the Mtheory limit is important given that Mtheory is\nnot defined even perturbatively, although there is a wellknown conjecture on its nonperturbative\nformulation in the infinite momentum frame in terms of matrix quantum mechanics. The\nplanar limit, which corresponds to type IIA superstring theory, has interest on its own since it\nmay allow us to perform more detailed tests of the gauge/gravity duality than in the case of\nAdS_5/CFT_4. In particular, we may hope to calculate the 1/N corrections to the planar limit,\nwhich enables us to test the gauge/gravity duality at the quantum string level, little of which\nis known so far.\nIn all these prospectives, one needs to study the ABJM theory in the strong coupling regime.\nAs in the case of QCD, it would be nice if one could study the ABJM theory on a lattice by\nMonte Carlo methods. This seems quite difficult, though, for the following three reasons.\nFirstly, the construction of the ChernSimons term on the lattice is not straightforward,\nalthough there is a proposal based on its connection to the parity anomaly. Secondly, the\nChernSimons term is purely imaginary in the Euclidean formulation, which causes a technical\nproblem known as the sign problem when one tries to apply the idea of importance sampling.\nThirdly, the lattice discretization necessarily breaks supersymmetry, and one needs to restore it\nin the continuum limit by finetuning the coupling constants of the supersymmetry breaking\nrelevant operators. This might, however, be overcome by the use of a nonlattice regularization\nof the ABJM theory based on the largeN reduction on S^3, which is shown to be useful in\nstudying the planar limit of the 4d N=4 super YangMills theory.\nIn this thesis, we show that the ABJM theory can be studied for arbitrary N at arbitrary cou\npling constant by applying a simple Monte Carlo method to the matrix model that can be deri\nved from the theory by using the localization technique. This opens up the possibility of\nprobing the quantum aspects of Mtheory and testing the AdS_4/CFT_3 duality at the quantum\nlevel. Here we calculate the free energy, and confirm the N^3/2 scaling in the Mtheory limit\npredicted from the gravity side. We also find that our results nicely interpolate the analytical\nformulae proposed previously in the Mtheory and type IIA regimes. Furthermore, we show\nthat some results obtained by the Fermi gas approach can be clearly understood from the\nconstant map contribution obtained by the genus expansion. The method can be easily\ngeneralized to the calculations of BPS operators and to other theories that reduce to matrix\nmodels. We also study the supersymmetric Wilson loops in the ABJM theory. 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Numerical test of AdS/CFT correspondence for M2branes
https://ir.soken.ac.jp/records/4067
https://ir.soken.ac.jp/records/40672ed657a5b863427c9199a8bfa7bb48af
名前 / ファイル  ライセンス  アクション 

要旨・審査要旨 (287.7 kB)


本文 (1.7 MB)

Item type  学位論文 / Thesis or Dissertation(1)  

公開日  20131115  
タイトル  
タイトル  Numerical test of AdS/CFT correspondence for M2branes  
タイトル  
言語  en  
タイトル  Numerical test of AdS/CFT correspondence for M2branes  
言語  
言語  eng  
資源タイプ  
資源タイプ識別子  http://purl.org/coar/resource_type/c_46ec  
資源タイプ  thesis  
著者名 
本多, 正純
× 本多, 正純 

フリガナ 
ホンダ, マサズミ
× ホンダ, マサズミ 

著者 
HONDA, Masazumi
× HONDA, Masazumi 

学位授与機関  
学位授与機関名  総合研究大学院大学  
学位名  
学位名  博士（理学）  
学位記番号  
内容記述タイプ  Other  
内容記述  総研大甲第1590号  
研究科  
値  高エネルギー加速器科学研究科  
専攻  
値  14 素粒子原子核専攻  
学位授与年月日  
学位授与年月日  20130322  
学位授与年度  
2012  
要旨  
内容記述タイプ  Other  
内容記述  Numerical test of AdS/CFT correspondence for M2branes Mtheory is an elevendimensional theory, which has been proposed as a strong coupling limit of the type IIA superstring theory. It has been also expected that the Mtheory includes the elevendimensional supergravity (11d SUGRA) as a lowenergy limit. The 11d SUGRA consists of the graviton, gravitino and threeform gauge field. The threeform 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 tendimensional supergravities and objects in the superstring theories as string, NS5brane and Dbranes, we can expect that the Mtheory has fundamental two and fivedimensional objects. These objects are called as ``M2brane`` and ``M5brane``, respectively. In this thesis, we focus on Physics of the multiple M2branes. As well known, a lowenergy limit of parallel N Dpbranes is described by the (p+1)dimensional U(N) maximally supersymmetric YangMills theory. This U(N) gauge symmetry can be intuitively understood by the facts that open string includes spin1 massless boson in its spectrum and have an U(1) charge called as a ChanPaton factor. What is a lowenergy effective theory of the parallel N M2branes? Unfortunately, we have not an established answer to this question yet as we will argue below. From the single M2brane analysis and implication of the AdS/CFT correspondence, we expect that the low energy effective theory for $N$ M2branes has the following properties: (1) Three dimensional conformal symmetry, (2) N=8 supersymmetry, (3) SO(8) Rsymmetry, 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 ChanPaton factor for M2branes. Another difficult ty is that it is not easy to construct gauge theory with conformal and high supersymmetry except for four dimensions. Since YangMills action is scale invariant only for four dimensionns, we can use only ChernSimons 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 ChernSimons levels k and k coupled to bifundamental 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 Mtheory on AdS_4 x S^7/Z_k for k<<N^{1/5}, and to type IIA superstring onAdS_4xCP^3 in the planar largeN 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 Mtheory on AdS_4 backgrounds since the theory is welldefined for finite N. One may draw a precise analogy with the way maximally supersymmetric YangMills theories may provide us with non perturbative formulations of type IIA/IIB superstring theories on Dbrane backgrounds through the gauge/gravity duality. In particular, the Mtheory limit is important given that Mtheory is not defined even perturbatively, although there is a wellknown 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 ChernSimons term on the lattice is not straightforward, although there is a proposal based on its connection to the parity anomaly. Secondly, the ChernSimons 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 finetuning the coupling constants of the supersymmetry breaking relevant operators. This might, however, be overcome by the use of a nonlattice regularization of the ABJM theory based on the largeN reduction on S^3, which is shown to be useful in studying the planar limit of the 4d N=4 super YangMills 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 Mtheory 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 Mtheory limit predicted from the gravity side. We also find that our results nicely interpolate the analytical formulae proposed previously in the Mtheory 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 supersymmetric Wilson loops in the ABJM theory. Our result nicely interpolates the expressions at weak and strong coupling regions. 

所蔵  
値  有 