q-Deformed quaternions and su(2) instantons

Abstract

We have recently introduced the notion of a q-quaternion bialgebra and shown its strict link with the SO_q(4)-covariant quantum Euclidean space R_q^4. Adopting the available differential geometric tools on the latter and the quaternion language we have formulated and found solutions of the (anti)selfduality equation [instantons and multi-instantons] of a would-be deformed su(2) Yang-Mills theory on this quantum space. The solutions depend on some noncommuting parameters, indicating that the moduli space of a complete theory should be a noncommutative manifold. We summarize these results and add an explicit comparison between the two SO_q(4)-covariant differential calculi on R_q^4 and the two 4-dimensional bicovariant differential calculi on the bi- (resp. Hopf) algebras M_q(2),GL_q(2),SU_q(2), showing that they essentially coincide.

arXiv commentLatex file, 18 pages

References in corpus (13)

• Instantons on noncommutative R^4, and (2,0) superconformal six dimensional theory
• Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations
• Construction of non-Abelian gauge theories on noncommutative spaces
• Differential Geometry on Linear Quantum Groups
• q-Euclidean space and quantum group wick rotation by twisting
• Noncommutative Instantons on the 4-Sphere from Quantum Groups
• Reality in the Differential Calculus on q-euclidean Spaces
• The $S{O}_q(N,\mathbf{R})$-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ${\mathbf{R}}_q^N$ and its Hilbert Space Structure
• Instantons on the Quantum 4-Spheres S^4_q
• Quantum group covariant (anti)symmetrizers, epsilon-tensors, vielbein, Hodge map and Laplacian
• Integration on quantum Euclidean space and sphere in $N$ dimensions
• Noncommutative Moduli for Multi-Instantons
• On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N

Cited by in corpus (1)

• Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$