Cardy states as idempotents of fusion ring in string field theory

@article{Kishimoto2004CardySA,
  title={Cardy states as idempotents of fusion ring in string field theory},
  author={I. Kishimoto and Y. Matsuo},
  journal={Physics Letters B},
  year={2004},
  volume={590},
  pages={303-308}
}
With some assumptions, the algebra between Ishibashi states in string field theory can be reduced to a commutative ring. From this viewpoint, Cardy states can be identified with its idempotents. The algebra is identified with the fusion ring for the rational conformal field theory and the group ring for the orbifold. This observation supports our previous observation that the boundary states satisfy a universal idempotency relation under the closed string star product. 
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