Gauge theory on nonassociative spaces

Journal of Mathematical Physics, 2010

We show that several standard associative quantizations in mathematical physics can be expressed as cochain module-algebra twists in the spirit of Moyal products at least to O( 3 ), but to achieve this we twist not by a 2-cocycle but by a 2-cochain. This implies a hidden nonassociavitity not visible in the algebra itself but present in its deeper noncommutative differential geometry, a phenomenon first seen in our previous work on semiclassicalisation of differential structures. The quantisations are induced by a classical group covariance and include: enveloping algebras U (g) as quantisations of g * , a Fedosov-type quantisation of the sphere S 2 under a Lorentz group covariance, the Mackey quantisation of homogeneous spaces, and the standard quantum groups Cq [G]. We also consider the differential quantisation of R n for a given symplectic connection as part of our semiclassical analysis and we outline a proposal for the Dirac operator.

Fortschritte der Physik, 2018

The Einstein-Hilbert action in three dimensions and the transformation rules for the dreibein and spin connection can be naturally described in terms of gauge theory. In this spirit, we use covariant coordinates in noncommutative gauge theory in order to describe 3D gravity in the framework of noncommutative geometry. We consider 3D noncommutative spaces based on SU(2) and SU(1,1), as foliations of fuzzy 2-spheres and fuzzy 2-hyperboloids respectively. Then we construct a U(2)× U(2) and a GL(2,C) gauge theory on them, identifying the corresponding noncommutative vielbein and spin connection. We determine the transformations of the fields and an action in terms of a matrix model and discuss its relation to 3D gravity.

The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques is extended to Noncommutative and Nonassociative Spacetime coordinates associated with octonionic-valued coordinates and momenta. The octonionic metric G µν already encompasses the ordinary spacetime metric g µν , in addition to the Maxwell U (1) and SU (2) Yang-Mills fields such that implements the Kaluza-Klein Grand Unification program without introducing extra spacetime dimensions. The color group SU (3) is a subgroup of the exceptional G 2 group which is the automorphism group of the octonion algebra. It is shown that the flux of the SU (2) Yang-Mills field strength F µν through the areamomentum Σ µν in the internal isospin space yields corrections O(1/M 2 P lanck ) to the energy-momentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. The known Octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic string actions that should have a correspondence with ordinary string actions for strings moving in a curved Clifford-space target background associated with a Cl(3, 1) algebra.

A ternary gauge field theory is explicitly constructed based on a totally antisymmetric ternary-bracket structure associated with a 3-Lie algebra. It is shown that the ternary infinitesimal gauge transformations do obey the key closure relations [δ1, δ2] = δ3. Invariant actions for the 3-Lie algebra-valued gauge fields and scalar fields are displayed. We analyze and point out the difficulties in formulating a nonassociative Octonionic ternary gauge field theory based on a ternary-bracket associated with the octonion algebra and defined earlier by Yamazaki. It is shown that a Yang-Mills-like quadratic action is invariant under global (rigid) transformations involving the Yamazaki ternary octonionic bracket, and that there is closure of these global (rigid) transformations based on constant antisymmetric parameters Λ ab = -Λ ba . Promoting the latter parameters to spacetime dependent ones Λ ab (x µ ) allows to build an octonionic ternary gauge field theory when one imposes gauge covariant constraints on the latter gauge parameters leading to field-dependent gauge parameters and nonlinear gauge transformations. In this fashion one does not spoil the gauge invariance of the quadratic action under this restricted set of gauge transformations and which are tantamount to spacetime-dependent scalings (homothecy) of the gauge fields.

arXiv: High Energy Physics - Theory, 1999

In this thesis we will discuss various aspects of noncommutative geometry and compactified Little-String theories. First we will give an introduction to the use of noncommutative geometry in string theory. Thereafter we will present a proof of the connection between D-brane dynamics and noncommutative geometry. Then we will explain the concept of instantons in noncommutative gauge theories. The last chapters shift the focus to Little-String- and $(2,0)$-theories. We study compactifications of these theories on tori with twists. First we study the case of two coinciding branes in detail. Afterwards we study the case of an arbitrary number of coinciding branes. The main result here is that the moduli spaces of vacua for the twisted compactifications are equal to moduli spaces of instantons on a noncommutative torus. A special case of this is that a large class of gauge theories with $\SUSY{2}$ supersymmetry in D=4 or $\SUSY{4}$ in D=3 has moduli spaces which are moduli spaces of insta...

Polyvector-valued gauge field theories in noncommutative Clifford spaces are presented. They are based on noncommutative (but associative) star products that require the use of the Baker-Campbell-Hausdorff formula. Using these star products allows the construction of actions for noncommutative p-branes (branes moving in noncommutative spaces). Noncommutative Clifford-space gravity as a poly-vector-valued gauge theory of twisted diffeomorphisms in Clifford-spaces would require quantum Hopf algebraic deformations of Clifford algebras. We proceed with the study of n-ary algebras and find an important relationship among the n-ary commutators of noncommuting spacetime coordinates [X 1 , X 2 , ......, X n ] with the poly-vector valued coordinates X 123...n in noncommutative Clifford spaces given by [X 1 , X 2 , ......, X n ] = n! X 123...n . The large N limit of n-ary commutators of n hyper-matrices Xi 1 i 2 ....in leads to Eguchi-Schild p-brane actions for p+1 = n. A noncomutative n-ary • product of n functions is constructed which is a generalization of the binary star product * of two functions and is associated with the deformation quantization of n-ary structures and deformations of the Nambu-Poisson brackets. * Dedicated to the loving memory of Adam Bowers 1 algebra of polyvector-valued coordinates in C-spaces to a noncommutative one which allows to construct a star product among the polyvector-valued coordinates (based on the Baker-Campbell-Hausdorff formula). Once the star product is properly defined one may proceed with the construction of polyvector-valued gauge field theories in noncommutative Clifford spaces and whose gauge group is also based on the Clifford algebra. Clifford algebras are deeply related and essential tools in many aspects in Physics. The Extended Relativity theory in Clifford-spaces ( C-spaces ) is a natural extension of the ordinary Relativity theory [3] whose generalized polyvectorvalued coordinates are Clifford-valued quantities which incorporate lines, areas, volumes, hyper-volumes.... degrees of freedom associated with the collective particle, string, membrane, p-brane,... dynamics of p-loops (closed p-branes) in D-dimensional target spacetime backgrounds.

Nuclear Physics B, 2014

We consider homogeneous spaces SU(4)/U(3) and Sp(2)/Sp(1)×U(1), both representing the complex projective space CP 3 , as well as the natural twistor spaces SU(4)/U(2)×U(1) and Sp(2)/U(1)×U(1) fibred over the above cosets with CP 2 and CP 1 fibres, respectively. We describe the relation between Hermitian Yang-Mills connections (instantons) on complex vector bundles over CP 3 and holomorphic bundles over the associated twistor spaces.

Nuclear Physics B, 1988

We study twistorial mechanics of particles and super-particles in six dimensions. To this end we formulate (in a general division algebra framework) a twistor theory in D = 6 based on quaternionic numbers, and prove the equivalence between this version of particle dynamics and the ordinary one. The super-twistors define a covariant and gauge invariant concept of a super world-line and allow us to write an action for the supersymmetric particle that is not plagued by the content of second class constraints that prevents a covariant quantization in the space-time picture. The notion and geometry of projective twistor space, and its connection to Minkowski space, are examined and shown to directly generalize the results in D = 3,4. Quantization is performed and analytic quaternionic eigenfunctions and integrations are discussed. We also draw some conclusions on the possible generalization to ten dimensions.

Zeitschrift für Naturforschung A

We have recently proposed a new matrix dynamics at the Planck scale, building on the theory of trace dynamics and Connes noncommutative geometry program. This is a Lagrangian dynamics in which the matrix degrees of freedom are made from Grassmann numbers, and the Lagrangian is trace of a matrix polynomial. Matrices made from even grade elements of the Grassmann algebra are called bosonic, and those made from odd grade elements are called fermionic—together they describe an ‘aikyon’. The Lagrangian of the theory is invariant under global unitary transformations and describes gravity and Yang–Mills fields coupled to fermions. In the present article, we provide a basic definition of spin angular momentum in this matrix dynamics and introduce a bosonic(fermionic) configuration variable conjugate to the spin of a boson(fermion). We then show that at energies below Planck scale, where the matrix dynamics reduces to quantum theory, fermions have half-integer spin (in multiples of Planck’s ...

Proceedings of Corfu Summer Institute 2021 "School and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2021), 2022

In the present article first we review the construction of three-and four-dimensional fuzzy spaces and subsequently we present the construction of fuzzy gravity theories on these noncommutative spaces following the gauge theoretic formulation of gravity.