On gauge transformation property of coordinate independent SO(9) vector states in SU(2) Matrix Theory

Abstract

We investigate coordinate independent SO(9) vector states in SU(2) Matrix theory. There are 36 vector states, and we determine what representations of SU(2) they are decomposed into. Among them we find a unique set of states transforming in adjoint representation. We show that this set of states can appear as the linear term in the coordinate matrices in Taylor expansion of zero energy bound state wavefunction around the origin i.e. it satisfies the condition of full supersymmetry.

Towards the ground state of the supermembrane

“…the coordinate independent one) for the SU(2) model has been constructed explicitly [17] and proven to be unique [18,19] which confirmed earlier symbolic results using Mathematica [20]. The 1st order term is now also available and turns out to be unique as well [21]. Because the zero-energy state |ψ satisfies (schematically) (∂ X + X 2 ) |ψ = 0, the 0th, the 1st and the 2nd order terms are crucial in finding the higher order terms by an appropriate recurrence equation.…”

“…For this reason the Taylor expansion approach initiated in [17] is a natural step forward. The 2nd order terms, determined in this paper, together with the 0th [17] and the 1st [21] order ones complete the initial conditions needed to solve the recurrence relation (2.6) for all higher terms. It is therefore a crucial step towards finding the ground state by this method.…”

“…The unique candidate for |φ which we denote here by |S , has been constructed in [17,20], and the unique candidate for |φ a i which satisfies (2.4) has also been constructed in [21]. Thus we have the starting points for the sequences m = 0 and 1.…”

“…† For the definition of states |ij a , |ijk a and |αi a we use conventions of [21] which differ from the conventions of [17]…”

“…For the 1st order term a similar expression is more complicated however as it turns out it can be written in an elegant form when using |S . One finds that [21]…”