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Nambu proposed an extension of dynamical system through the introduction of a new bracket (Nambu bracket) in 1973. This article is a short review of the developments after his paper. Some emphasis are put on a viewpoint that the Nambu bracket naturally describes extended objects which appear in M-theory and the fluid dynamics. The latter part of the paper is devoted to a review of the studies on the Nambu bracket (Lie 3-algebra) in Bagger-Lambert-Gustavsson theory of multiple M2-branes. This paper is a contribution to the proceedings of Nambu memorial symposium (Osaka City University, September 29, 2015).
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The derivation of the self-dual relations for the two-form gauge field in the Nambu-bracket description of M5-brane in a constant C-field background initiated in arXiv:0907.4596 is completed by including contributions from all the fields in the M5-brane action. The result is used to examine Seiberg-Witten map of the BPS conditions for the string solitons, up to the first order in the expansion by the parameter $g$ which characterizes the strength of the interactions through the Nambu-bracket.
We construct zero-temperature solutions of supergravity theories in five and four dimensions which interpolate between two copies of anti-de Sitter space, one of which preserves an abelian gauge symmetry while the other breaks it. These domain wall solutions can be lifted to solutions of type IIB string theory and eleven-dimensional supergravity. They describe quantum critical behavior and emergent relativistic conformal symmetry in a superfluid or superconducting state of a strongly coupled dual gauge theory. We include computations of frequency-dependent conductivities which exhibit power law scaling in the infrared, with exponents determined by irrelevant perturbations to the symmetry-breaking anti-de Sitter background.
We study M-theory compactification on ${mathbb{T}^7/ mathbb{Z}_2^3}$ in the presence of a seven-flux, metric fluxes and KK monopoles. The effective four-dimensional supergravity has seven chiral multiplets whose couplings are specified by the $G_2$-structure of the internal manifold. We supplement the corresponding superpotential by a KKLT type non-perturbative exponential contribution for all, or for some of the seven moduli, and find a discrete set of supersymmetric Minkowski minima. We also study type IIA and type IIB string theory compactified on ${mathbb{T}^6/ mathbb{Z}_2^2}$. In type IIA, we use a six-flux, geometric fluxes and non-perturbative exponents. In type IIB theory, we use F and H fluxes, and non-geometric Q and P fluxes, corresponding to consistently gauged supergravity with certain embedding tensor components, emph{without non-perturbative exponents}. Also in these situations, we produce discrete Minkowski minima. Finally, to construct dS vacua starting from these Minkowski progenitors, we follow the procedure of mass production of dS vacua.
In the quest for mathematical foundations of M-theory, the Hypothesis H that fluxes are quantized in Cohomotopy theory, implies, on flat but possibly singular spacetimes, that M-brane charges locally organize into equivariant homotopy groups of spheres. Here we show how this leads to a correspondence between phenomena conjectured in M-theory and fundamental mathematical concepts/results in stable homotopy, generalized cohomology and Cobordism theory Mf: Stems of homotopy groups correspond to charges of probe p-branes near black b-branes; stabilization within a stem is the boundary-bulk transition; the Adams d-invariant measures G4-flux; trivialization of the d-invariant corresponds to H3-flux; refined Toda brackets measure H3-flux; the refined Adams e-invariant sees the H3-charge lattice; vanishing Adams e-invariant implies consistent global C3-fields; Conner-Floyds e-invariant is H3-flux seen in the Green-Schwarz mechanism; the Hopf invariant is the M2-brane Page charge (G7-flux); the Pontrjagin-Thom theorem associates the polarized brane worldvolumes sourcing all these charges. Cobordism in the third stable stem witnesses spontaneous KK-compactification on K3-surfaces; the order of the third stable stem implies 24 NS5/D7-branes in M/F-theory on K3. Quaternionic orientations correspond to unit H3-fluxes near M2-branes; complex orientations lift these unit H3-fluxes to heterotic M-theory with heterotic line bundles. In fact, we find quaternionic/complex Ravenel-orientations bounded in dimension; and we find the bound to be 10, as befits spacetime dimension 10+1.
The dissertation consists of two parts. The first presents an account of the effective worldvolume description of $N$ coincident M2-branes ending on an M5-brane in M-theory. It reviews Basu and Harveys recent description of the worldvolume theory of the M2-branes in terms of a Bogomolnyi equation, and its solution via a fuzzy (three-) funnel. Tests of the consistency of this picture are then performed and many of the issues with it are addressed. This is followed by a discussion of how a refinement of the fuzzy three-sphere algebra used can lead to the correct $N^{3/2}$ scaling of degrees of freedom for this system. A reduction of this Basu-Harvey picture to the D1-string picture of the D1-D3 intersection is then performed via constructing a reduction of the fuzzy-three sphere to the fuzzy two-sphere. The second part of the dissertation describes how a holomorphic factorisation argument can be used to demonstrate quantum equivalence of the doubled formalism of string theory with the standard formalism by deriving the partition function, including instanton and oscillator sectors.
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