Hamiltonian approach and quantization of D = 3,$$ \mathcal{N}=1 $$ supersymmetric non-Abelian multiwave system

Abstract

We develop Hamiltonian formalism and quantize supersymmetric non-Abelian multiwave system (nAmW) in D=3 spacetime constructed as a simple counterpart of 11D multiple M-wave system. Its action can be obtained from massless superparticle one by putting on its worldline 1d dimensional reduction of the 3d SYM model in such a way that the new system still possesses local fermionic kappa-symmetry. The quantization results in a set of equation of supersymmetric field theory in an unusual space with su(N)-valued matrix coordinates. Their superpartners, the fermionic su(N)-valued matrices, cannot be split on coordinates and momenta in a covariant manner and hence are included as abstract operators acting on the state vector in the generic form of our D=3 Matrix model field equations. We discuss the Clifford superfield representation for the quantum state vector and in the simplest case of N = 2 elaborate it in a bit more detail. As a check of consistency, we show that the bosonic Matrix model field equations obtained by quantization of the purely bosonic limit of our D=3 nAmW system have a nontrivial solution.

3D supersymmetric nonlinear multiple D0-brane action and 4D counterpart of multiple M-wave system

“…To study this latter problem on a toy model, the D = 3 counterpart of mM0 action was constructed in [33], where it was called non-Abelian multiwave system (nAmW) and was also generalized for the case of curved AdS superspace. In [45] the quantization of D = 3 nAmW model was performed, which resulted in a system of relativistic field equations for the bosonic and fermionic fields on the space with additional non-commutative bosonic and non-anticommutative fermionic coordinates. It was noticed their that the structure of D = 4 counterpart of 3D nAmW quantum theory promises to be simpler and more transparent due to the intrinsic complex structure characteristic for the D = 4 N = 1 superspace (and for the N = 2 D = 3 one).…”

“…As far as the study of D = 3 and D = 4 supersymmetric systems is concerned, an interesting problem is the development of Hamiltonian approach and quantization of D = 4 N = 1 nAmW system as well as of D = 3 N = 2 mD0-brane. These should result in new interesting supersymmetric systems of relativistic wave equations in the (super)spaces with noncommutative coordinates which promise to be simpler than the result of quantization of D = 3 N = 1 nAmW system in [45] due to the complex structure characteristic for the both D = 4 N = 1 and D = 3 N = 2 superspaces.…”