Counting SO(9) × SU(2) representations in coordinate independent state space of SU(2) matrix theory

Abstract

We consider decomposition of coordinate independent states into SO(9)×SU(2) representations in SU(2) Matrix theory. To see what and how many representations appear in the decomposition, we compute the character, which is given by a trace over the coordinate independent states, and decompose it into the sum of products of SO(9) and SU(2) characters.

Towards the ground state of the supermembrane

“…Performing the Taylor expansion of the ground state about X = 0, the 0th order term (i.e. 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].…”

“…coordinate independent states φ a 1 ...an i 1 ...in , which are constructed by acting creation operators made of θ a α on the vacuum for those operators, play an important role. In our case of SU(2) gauge group, classification of the coordinate independent states by representations has been given in [19].…”

“…For SU(2), t, a, and s correspond to spin 0, 1, and 2 representations respectively. For SO(9), t, a, and s correspond to Dynkin labels [0000], [0100], and [2000] in Table 1 in [19] respectively. Then the table tells us that in the coordinate independent state space, (i) there is only one (t, t) representation (and therefore proportional to |S ).…”

“…ii ab is in (s, t) representation. However there is no (s, t) representation in the table in [19]. This means φ ′…”

Hamiltonian truncation study of supersymmetric quantum mechanics: S-matrix and metastable states

“…However, there are only 3 SO(9) × SU (2) invariants formed out of the bosonic coordinates, which will play the role of r,r variables in the N = 4 SQM analyzed in this paper. Furthermore, the 2 24 component fermionic wave function reduces to a few hundred irreducible representations of SO(9) × SU (2) [45,46], and the direct diagonalization of a truncated Hamiltonian could be manageable when restricted to a sector with fixed SO(9) angular momentum. We hope to report on this in the near future.…”