Defect Verlinde Formula for Edge Excitations in Topological Order

We revisit the problem of boundary excitations at a topological boundary or junction defects between topological boundaries in nonchiral bosonic topological orders in 2+1 dimensions. Based on physical considerations, we derive a formula that relates the fusion rules of the boundary excitations and the "half-linking" number between condensed anyons and confined boundary excitations. This formula is a direct analogue of the Verlinde formula. We also demonstrate how these half-linking numbers can be computed in explicit Abelian and non-Abelian examples. As a fundamental property of topological orders and their allowed boundaries, this should also find applications in the search for suitable platforms realizing quantum computing devices.