We consider dimensional reduction of the Bagger-Lambert-Gustavsson theory to a zero-dimensional 3-Lie algebra model and construct various stable solutions corresponding to quantized Nambu-Poisson manifolds. A recently proposed Higgs mechanism reduces this model to the IKKT matrix model. We find that in the strong coupling limit, our solutions correspond to ordinary noncommutative spaces arising as stable solutions in the IKKT model with D-brane backgrounds. In particular, this happens for S^3, R^3 and five-dimensional Neveu-Schwarz Hpp-waves. We expand our model around these backgrounds and find effective noncommutative field theories with complicated interactions involving higher-derivative terms. We also describe the relation of our reduced model to a cubic supermatrix model based on an osp(1|32) supersymmetry algebra.
We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras, as well as the approach based on harmonic analysi…
We consider Nambu-Poisson 3-algebras on three dimensional manifolds $ {\cal M}_{3} $, such as the Euclidean 3-space $R^{3}$, the 3-sphere $S^{3}$ as well as the 3-torus $T^{3}$. We demonstrate that in the Clebsch-Monge gauge, the Lie algebra of volume preserving diffeomorphisms $SDiff({\cal M}_{3})$ is identical to the Nambu-Poisson algebra on ${\cal M}_{3}$. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation of $ SDiff({\cal M}_{3})$. We propose a quantization prescription for the Nambu-Poisson algebra which provides us with the correct classical limit.…
We propose an extension of n-ary Nambu-Poisson bracket to superspace R^{n|m} and construct by means of superdeterminant a family of Nambu-Poisson algebras of even degree functions, where the parameter of this family is an invertible transformation of Grassmann coordinates in superspace R^{n|m}. We prove in the case of the superspaces R^{n|1} and R^{n|2} that our n-ary bracket, defined with the help of superdeterminant, satisfies the conditions for n-ary Nambu-Poisson bracket, i.e. it is totally skew-symmetric and it satisfies the Leibniz rule and the Filippov-Jacobi identity (fundamental ident…
We generalize noncommutative gauge theory using Nambu-Poisson structures to obtain a new type of gauge theory with higher brackets and gauge fields. The approach is based on covariant coordinates and higher versions of the Seiberg-Witten map. We construct a covariant Nambu-Poisson gauge theory action, give its first order expansion in the Nambu-Poisson tensor and relate it to a Nambu-Poisson matrix model.
We review the AKSZ construction as applied to the topological open membranes and Poisson sigma models. We describe a generalization to open topological p-branes and Nambu-Poisson sigma models.
Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundam…
It is shown that the algebra of diffeomorphism-invariant charges of the Nambu-Goto string cannot be quantized in the framework of canonical quantization. The argument is shown to be independent of the dimension of the underlying Minkowski space.
We analyze BPS equations for string-like configurations derived from the M5-brane worldvolume action with a Nambu-Poisson structure constructed in arXiv:0804.3629, arXiv:0805.2898. We solve the BPS equations up to the first order in the parameter $g$ which characterizes the strength of the Nambu-Poisson bracket. We compare our solutions to previously constructed BPS string solitons in the conventional description of M5-brane in a constant three-form background via Seiberg-Witten map, and find agreement.
Recently it was proposed that the Bagger-Lambert-Gustavsson theory with Nambu-Poisson structure describes an M5-brane in a three-form flux background. In this paper we investigate the superalgebra associated with this theory. We derive the central charges corresponding to M5-brane solitons in 3-form backgrounds. We also show that double dimensional reduction of the superalgebra gives rise to the Poisson bracket terms of a non-commutative D4-brane superalgebra. We provide interpretations of the D4-brane charges in terms of spacetime intersections.
We show that there exists a cut-off version of Nambu-Poisson bracket which defines a finite dimensional Lie 3-algebra. The algebra still satisfies the fundamental identity and thus produces N=8 supersymmetric BLG type equation of motion for multiple M2 branes. By counting the number of the moduli and the degree of freedom, we derive an entropy formula which scales as N^{3/2} as expected for the multiple M2 branes.
We continue our investigation on the Nambu-Poisson description of M5-brane in a large constant C-field background (NP M5-brane theory) constructed in Refs.[1, 2]. In this paper, the low energy limit where the NP M5-brane theory is applicable is clarified. The background independence of the NP M5-brane theory is made manifest using the variables in the BLG model of multiple M2-branes. An all order solution to the Seiberg-Witten map is also constructed.
By using the recently proposed prescription arXiv:0804.3629 for obtaining the $M5$ brane action from multiple $M2$ branes action in BLG theory, we examine such transition when 11 Dimensional background antisymmetric fluxes couple to the $M2$ brane world volume. Such couplings was suggested in arXiv:0805.3427 where it was used the fact that various fields in BLG theory are valued in a Lie 3-algebra. We argue that this action and promoting it by Nambu-Poisson bracket gives the expected coupling of fluxes with $M5$ brane at least at weak coupling limit. We also study some other aspects of the act…
We introduce a Nambu-Poisson bracket in the geometrical description of the D=11 M5-brane. This procedure allows us, under some assumptions, to eliminate the local degrees of freedom of the antisymmetric field in the M5-brane Hamiltonian and to express it as a D=11 p-brane theory invariant under symplectomorphisms. A regularization of the M5-brane in terms of a multi D1-brane theory invariant under the $SU(N)\times SU(N)$ group in the limit when $N\to\infty$ is proposed. Also, a regularization for the D=10 D4-brane in terms of a multi D0-brane theory is suggested.
A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher order graded Leibniz rule). The $(n+1)$-ary bracket fulfills the properties of the Nambu bracket including the ``fundamental identity'', thus leading to the Nambu…
The BLG model for multiple M2-branes motivates an M5-brane theory with a novel gauge symmetry defined by the Nambu-Poisson structure. This Nambu-Poisson gauge symmetry for an M5-brane in large C-field background can be matched, on double dimension reduction, with the Poisson limit of the noncommutative gauge symmetry for a D4-brane in B-field background. Naively, one expects that there should exist a certain deformation of the Nambu-Poisson structure to match with the full noncommutative gauge symmetry including higher order terms. However, We prove the no-go theorem that there is no way to de…
Nonlinear generalization of the Dirac equation extending the standard paradigm of nonlinear Hamiltonians is discussed. ``Faster-than-light telegraphs" are absent for all theories formulated within the new framework. A new metric for infinite dimensional Lie algebras associated with Lie-Poisson dynamics is introduced.
In analogy to Nambu's generalization of mechanics, we present a generalization of Poisson sigma models to higher dimensional world volumes. We find corresponding generalizations of string sigma models and open-closed string relations for background fields to the case of p-branes. As an application, we discuss a DBI-type effective action for open membranes.
The characterization of the Nambu-Poisson n-tensors as a subfamily of the Generalized-Poisson ones recently introduced (and here extended to the odd order case) is discussed. The homology and cohomology complexes of both structures are compared, and some physical considerations are made.
In this note, we gauge the rigid vectorial supersymmetry of the two-dimensional Poisson sigma model presented in arXiv:1503.05625. We show that the consistency of the construction does not impose any further constraints on the differential Poisson algebra geometry than those required for the ungauged model. We conclude by proposing that the gauged model provides a first-quantized framework for higher spin gravity.
We study Kelvin modes and translational zero modes excited along a quantized vortex and relativistic global string in superfluids and a relativistic field theory, respectively, by constructing the low-energy effective theory of these modes. We find that they become exact gapless Nambu-Goldstone modes only in a system with infinite volume limit. On the other hand, in a system with the finite volume, we find an imaginary massive gap causing the tachyonic instability above some critical wavelength in the relativistic theory. We also find in the non-relativistic theory that Kelvin modes with wavel…
We propose the following receipt to obtain the quantization of the Poisson submanifold $N$ defined by the equations $f_i=0$ (where $f_i$ are Casimirs) from the known quantization of the manifold $M$: one should consider factor algebra of the quantized functions on $M$ by the images of $D(f_i)$, where $D: Fun(M) \to Fun(M)\otimes \CC[\hbar]$ is Duflo-Kirillov-Kontsevich map. We conjecture that this algebra is isomorphic to quantization of $Fun(N)$ with Poisson structure inherited from $M$. Analogous conjecture concerning the Hamiltonian reduction saying that "deformation quantization commutes w…
We consider D1-string in a constant R-R 3-form flux background and analyze its low energy limit. The leading order low energy theory has reparametrization symmetry and is a generalization of an earlier work by Takhtajan. We show that the dynamical evolution of the theory takes a generalized Hamiltonian form in terms of a Nambu bracket. This description is formulated in terms of reparametrization invariant quantities and requires no fixing of the reparametrization symmetry. We also show that a Nambu- Poisson (p + 2)-bracket arises naturally in the reparametrization invariant description of the …
The Nambu-Goto string in a 3-dimensional (3D) Minkowski spacetime is quantized preserving Lorentz invariance and parity. The spectrum of massive states contains anyons. An ambiguity in the ground state energy is resolved by the 3D N=1 Green-Schwarz superstring, which has massless ground states describing a dilaton and dilatino, and first-excited states of spin 1/4.
All possible graded Poisson-Lie structures on the external algebra of $SL(2)$ are described. We prove that differential Poisson-Lie structures prolonging the Sklyanin brackets do not exist on $SL(2)$. There are two and only two graded Poisson-Lie structures on $SL (2)$ and neither of them can be obtained by a reduction of graded Poisson-Lie structures on the external algebra of $GL(2)$. Both of them can be quantized and as a result we get a new graded algebra of quantum right-invariant forms on $SL_q(2)$ with three generators.
We study the classical current algebra for principial chiral model defined on two dimensional world-sheet with general metric. We develop the Hamiltonian formalism and determine the form of the Poisson brackets between currents. Then we determine the Poisson bracket for Lax connection and we show that this Possion bracket does not depend on the world-sheet metric. We also study the Nambu-Gotto form of this model. We prove an existence of the Lax connection and determine their Poisson bracket.
The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual, and their Heisenberg double in a way which extends both the theory of coadjoint orbits and the classical Fourier transform. We also describe the twisted Heisenberg double which is relevant for the study of nontrivial deformations of the quantized universal enveloping algebras.
We give a selfcontained introduction to the theory of quantum groups according to Drinfeld highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After having defined Poisson-Lie groups we study their relation to Lie-bi algebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of $sl_2$ is explicitly carried out. Next we show how quantum groups are rela…
We study metric-compatible Poisson structures in the semi-classical limit of noncommutative emergent gravity. Space-time is realized as quantized symplectic submanifold embedded in R^D, whose effective metric depends on the embedding as well as on the Poisson structure. We study solutions of the equations of motion for the Poisson structure, focusing on a natural class of solutions such that the effective metric coincides with the embedding metric. This leads to i-(anti-) self-dual complexified Poisson structures in four space-time dimensions with Lorentzian signature. Solutions on manifolds w…
The Poisson algebra $\mathfrak h$ of invariants of the Nambu-Goto string, which was first introduced by K. Pohlmeyer in 1982, is described using the Shuffle Hopf algebra. In particular, an underlying auxiliary Lie algebra is reformulated in terms of the image of the first Eulerian idempotent of the Shuffle Hopf algebra. This facilitates the comparison of different approaches to the quantization of $\mathfrak h$.
Regularizing volume preserving diffeomorphism (VPD) is equivalent to a long standing problem, namely regularizing Nambu-Poisson bracket. In this paper, as a first step to regularizing VPD, we find general complete independent basis of VPD algebra. Especially, we find complete independent basis that give simple structure constants, where three area preserving diffeomorphism (APD) algebras are manifest. This implies that an algebra that regularizes VPD algebra should include three u(N) Lie algebras.
We show that an action of a supermembrane in an eleven-dimensional spacetime with a semi-light-cone gauge can be written only with Nambu-Poisson bracket and an invariant symmetric bilinear form under an approximation. Thus, the action under the conditions is manifestly covariant under volume preserving diffeomorphism even when the world-volume metric is flat. Next, we propose two 3-algebraic models of M-theory which are obtained as a second quantization of an action that is equivalent to the supermembrane action under the approximation. The second quantization is defined by replacing Nambu-Poi…
Recently an action based on Lie 3-algebras was proposed to describe M2-branes. We study the case of infinite dimensional Lie 3-algebras based on the Nambu-Poisson structure of three dimensional manifolds. We show that the model contains self-dual 2-form gauge fields in 6 dimensions, and the result may be interpreted as the M5-brane world-volume action.
1-loop vacuum energies of (fuzzy) spacetimes from a supersymmetric reduced model with Filippov 3-algebra are discussed. A_{2,2} algebra, Nambu-Poisson algebra in flat spacetime, and a Lorentzian 3-algebra are examined as 3-algebras.
