Brane dynamics in M-theory M理論におけるブレーンのダイナミクス
Brane dynamics in M-theory
We discuss recent developments in M-theory and low energy effective theories of M-theory branes. This thesis consists of three parts. In part I, we briefly review the foundations of M-theory. Especially we take a look at M2-brane and M5-brane solutions in 11-dim supergravity and their implications to dual CFTs through the AdS/CFT correspondence. In part II, we summarizes the recent progress on the low energy effective theories of M2-branes and M5-branes with particular emphasis on the role of the Lie 3-algebra. In part III, we provide more details about M-theory branes and its reduction to D-branes in string theory. Part III is the main part of this thesis and is based on the author's works. In order to understand the nonperturbative aspects of superstring theory, it is essential to investigate the dynamics of M-theory and its branes. In modern perspective, it is known that there are two types of description about M2-branes. One is the Bagger-Lambert-Gustavsson (BLG) theory based on a novel gauge structure, Lie 3-algebra and the other is the Aharony-Bergman-Jafferis-Maldacena (ABJM) theory based on two Chern-Simons theories with four bifundamental matter multiplets.In this thesis, we mainly consider the relationships between BLG theory and ABJM theory. There are two types of Lie 3-algebras classified by the metric of generators, namely Euclidean and Lorentzian. We first explain the general reduction of the Lorentzian-BLG (L-BLG) theory to D2-brane theory and confirm that the L-BLG theory can be regarded as a reformulation of D2-brane theory. However, such a formulation of the L-BLG theory in terms of ordinary gauge theory enables us to connect this theory to the ABJM theory.Then we see that the 3d N=8 BLG theory based on the Lorentzian type 3-algebra can be derived by taking a certain scaling limit of 3d N=6 U(N)k×U(N)－k ABJM theory whose moduli space is SymN(C4/Zk). The scaling limit which can be interpreted as the Inonu-Wigner contraction is to scale the trace part of the bifundamental fields and the axial combination of the two gauge fields. Simultaneously we scale the Chern-Simons level. In this scaling limit, M2-branes are located far from the origin of C4/Zk compared to their fluctuations and Zk identification becomes a circle identification.Furthermore, we show that the BLG theory with two pairs of negative norm generators is derived from the scaling limit of an orbifolded ABJM theory. The BLG theory with many Lorentzian pairs is known to be reduced to the Dp-brane theory via the Higgs mechanism, so our scaling procedure can be used to derive Dp-branes from M2-branes. We also investigate the scaling limits of various quiver Chern-Simons theories obtained from different orbifolding actions. Remarkably, in the case of N=2 quiver CS theories, the resulting D3-brane action covers a larger region in the parameter space of the complex structure moduli than the N=4 quiver CS theories. How the SL(2,Z) duality transformation is realized in the resultant D3-brane theory is also discussed. Moreover, we explain the recent progress on the application of Lie 3-algebra to M5-branes. For M5-branes, its nonabelian action has not been discovered due to the lack of understanding about consistent coupling between arbitrary number of tensor multiplets and Yang-Mills multiplets. Recently, however, it was suggested that the equations of motion of M5-branes can be constructed by using Lie 3-algebra. We describe its consistency with the known string dualities and confirm that the proposed system has to be modified to realize the dynamics of multiple M5-branes. We also comment about type IIA/IIB NS5-brane and Kaluza-Klein monopoles by taking various compactification cycles. Because both longitudinal and transverse directions to the worldvolume can be compactified in the proposed model, we can realize these systems. This situation is entirely different from the case of BLG theory. Realization of the moduli parameters in the U-duality group is also discussed.