Cardy states as idempotents of fusion ring in string field theory

TL;DR

This paper shows that boundary states in string field theory can be understood as idempotents in an algebraic structure called the fusion ring, linking boundary states to algebraic properties in conformal field theory.

Contribution

It establishes a connection between Cardy states and idempotents in the fusion ring within string field theory, providing a new algebraic perspective on boundary states.

Findings

Boundary states satisfy a universal idempotency relation.

The algebra of Ishibashi states reduces to a fusion ring or group ring.

Cardy states correspond to idempotents in this algebra.

Abstract

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 can be identified with a fusion ring for the rational conformal field theory and a group ring for the orbifold. This observation supports our previous observation that boundary states satisfy a universal idempotency relation under closed string star product.