20 research outputs found
Non-commutative deformation of Chern-Simons theory
The problem of the consistent definition of gauge theories living on the
non-commutative (NC) spaces with a non-constant NC parameter is
discussed. Working in the L formalism we specify the undeformed
theory, d abelian Chern-Simons, by setting the initial brackets.
The deformation is introduced by assigning the star commutator to the
bracket. For this initial set up we construct the corresponding L
structure which defines both the NC deformation of the abelian gauge
transformations and the field equations covariant under these transformations.
To compensate the violation of the Leibniz rule one needs the higher brackets
which are proportional to the derivatives of . Proceeding in the slowly
varying field approximation when the star commutator is approximated by the
Poisson bracket we derive the recurrence relations for the definition of these
brackets for arbitrary . For the particular case of -like NC
space we obtain an explicit all orders formulas for both NC gauge
transformations and NC deformation of Chern-Simons equations. The latter are
non-Lagrangian and are satisfied if the NC field strength vanishes everywhere.Comment: 33 pages, published version, exposition improved, new material
regarding the definition of the non-commutative field strength and the
treatment of the non-commutativity of general form adde
A novel approach to non-commutative gauge theory
We propose a field theoretical model defined on non-commutative space-time
with non-constant non-commutativity parameter , which satisfies two
main requirements: it is gauge invariant and reproduces in the commutative
limit, , the standard gauge theory. We work in the slowly
varying field approximation where higher derivatives terms in the star
commutator are neglected and the latter is approximated by the Poisson bracket,
. We derive an explicit expression for both the NC
deformation of Abelian gauge transformations which close the algebra
, and the NC field strength ,
covariant under these transformations, . NC
Chern-Simons equations are equivalent to the requirement that the NC field
strength, , should vanish identically. Such equations are
non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the
gauge invariant action, . As guiding example, the case of
-like non-commutativity, corresponding to rotationally invariant NC
space, is worked out in detail.Comment: 16 pages, no figures. Minor correction
Symplectic realisation of electric charge in fields of monopole distributions
We construct a symplectic realisation of the twisted Poisson structure on the
phase space of an electric charge in the background of an arbitrary smooth
magnetic monopole density in three dimensions. We use the extended phase space
variables to study the classical and quantum dynamics of charged particles in
arbitrary magnetic fields by constructing a suitable Hamiltonian that
reproduces the Lorentz force law for the physical degrees of freedom. In the
source-free case the auxiliary variables can be eliminated via Hamiltonian
reduction, while for non-zero monopole densities they are necessary for a
consistent formulation and are related to the extra degrees of freedom usually
required in the Hamiltonian description of dissipative systems. We obtain new
perspectives on the dynamics of dyons and motion in the field of a Dirac
monopole, which can be formulated without Dirac strings. We compare our
associative phase space formalism with the approach based on nonassociative
quantum mechanics, reproducing extended versions of the characteristic
translation group three-cocycles and minimal momentum space volumes, and prove
that the two approaches are formally equivalent. We also comment on the
implications of our symplectic realisation in the dual framework of
non-geometric string theory and double field theory.Comment: 39 pages, 1 figure; v2: references added; v3: clarifying comments and
references added; Final version to be published in Physical Review
Symplectic Groupoids and Poisson Electrodynamics
We develop a geometric approach to Poisson electrodynamics, that is, the
semi-classical limit of noncommutative gauge theory. Our framework is
based on an integrating symplectic groupoid for the underlying Poisson
brackets, which we interpret as the classical phase space of a point particle
on noncommutative spacetime. In this picture gauge fields arise as bisections
of the symplectic groupoid while gauge transformations are parameterized by
Lagrangian bisections. We provide a geometric construction of a gauge invariant
action functional which minimally couples a dynamical charged particle to a
background electromagnetic field. Our constructions are elucidated by several
explicit examples, demonstrating the appearances of curved and even compact
momentum spaces, the interplay between gauge transformations and spacetime
diffeomorphisms, as well as emergent gravity phenomena.Comment: 31 pages, 2 figures, 1 tabl
G<sub>2</sub>-structures and quantization of non-geometric M-theory backgrounds
We describe the quantization of a four-dimensional locally non-geometric
M-theory background dual to a twisted three-torus by deriving a phase space
star product for deformation quantization of quasi-Poisson brackets related to
the nonassociative algebra of octonions. The construction is based on a choice
of -structure which defines a nonassociative deformation of the addition
law on the seven-dimensional vector space of Fourier momenta. We demonstrate
explicitly that this star product reduces to that of the three-dimensional
parabolic constant -flux model in the contraction of M-theory to string
theory, and use it to derive quantum phase space uncertainty relations as well
as triproducts for the nonassociative geometry of the four-dimensional
configuration space. By extending the -structure to a -structure,
we propose a 3-algebra structure on the full eight-dimensional M2-brane phase
space which reduces to the quasi-Poisson algebra after imposing a particular
gauge constraint, and whose deformation quantisation simultaneously encompasses
both the phase space star products and the configuration space triproducts. We
demonstrate how these structures naturally fit in with previous occurences of
3-algebras in M-theory.Comment: 41 pages; v2: Final version published in JHE
Noncommutative Rd via closed star product
We consider linear star products on RdRd of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, Tr (f ⋆ g) = Tr (f · g). We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibniz rule holds true up to a total derivative. As a particular example we study the space R θ 3 with su(2)su(2) type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibniz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents
Nonassociative Weyl star products
Deformation quantization is a formal deformation of the algebra of smooth functions on some manifold. In the classical setting, the Poisson bracket serves as an initial conditions, while the associativity allows to proceed to higher orders. Some applications to string theory require deformation in the direction of a quasi-Poisson bracket (that does not satisfy the Jacobi identity). This initial condition is incompatible with associativity, it is quite unclear which restrictions can be imposed on the deformation. We show that for any quasi-Poisson bracket the deformation quantization exists and is essentially unique if one requires (weak) hermiticity and the Weyl condition. We also propose an iterative procedure that allows to compute the star product up to any desired order
Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
We formulate general definitions of semi-classical gauge transformations for
noncommutative gauge theories in general backgrounds of string theory, and give
novel explicit constructions using techniques based on symplectic embeddings of
almost Poisson structures. In the absence of fluxes the gauge symmetries close
a Poisson gauge algebra and their action is governed by a -algebra
which we construct explicitly from the symplectic embedding. In curved
backgrounds they close a field dependent gauge algebra governed by an
-algebra which is not a -algebra. Our technique produces
new all orders constructions which are significantly simpler compared to
previous approaches, and we illustrate its applicability in several examples of
interest in noncommutative field theory and gravity. We further show that our
symplectic embeddings naturally define a -structure on the exterior
algebra of differential forms on a generic almost Poisson manifold, which
generalizes earlier constructions of differential graded Poisson algebras, and
suggests a new approach to defining noncommutative gauge theories beyond the
gauge sector and the semi-classical limit based on -algebras.Comment: 60 pages; v2: introduction expanded, typos corrected; v3: minor
changes, references added; Final version published in Journal of Physics