Matrix models and the gravitational interaction by Azuma, T
By Azuma, T
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Example text
76) Utilizing these relations, we can discern that A′ , as well as H′ is the representations of osp(1|32, R) and also that the algebra A′ is a representation of osp(1|32, R) super Lie algebra by the commutation relation (2)[H, A′ ] ∈ A′ for H ∈ H and A′ ∈ A′ . This commutation relation states that A′ remain in the super Lie algebra A′ after the infinitesimal translation by the elements H ∈ H. In this sense, we can understand that A′ is another representation of osp(1|32, R). The introduction of these two representations of osp(1|32, R) teaches us the relationship of osp(1|32, R) and u(1|16, 16) super Lie algebras.
91) Wigner-In¨ on¨ u contraction and supersymmetry In considering the relation to the IIB matrix model, we need to reduce the model to the ten dimensions. However, this matrix model has a grave difference from the osp(1|32, R) nongauged supermatrix. 88) is no longer based on the commutator for the gl(N |R) matrices, it is not invariant under the inhomogeneous supersymmetry, namely the translation of the fermion. 88) accommodates 32 + 32 = 64 supercharges. This leads us to speculate that this model may also have the two-fold supersymmetry structure of the IIB-like supersymmetry.
12 In the S 2 × S 2 × S 2 fuzzy-sphere background, the 2-form field Cij is Cij = ǫijk Bk . 135) Therefore, the total energy is 9 ES 2 = = SS 2 = −64µ µ=1 3 T r(Bµ B µ ) = −3 × 64µ −12µ3 N (N − 1)(N + 1). 136) This result shows that the S 2 × S 2 × S 2 fuzzy-sphere classical solution has a lower energy compared to the trivial commutative solution and hence a higher probability. Other curved-space solutions and the fuzzy 8-sphere So far, we have considered the simplest curved-space solution S 2 × S 2 × S 2 .