Fractional Branes on a Non-compact Orbifold

Fractional branes on the non-compact orbifold ${\text{C}}^3\mathrm{/}{\mathbb{Z}}_5$ are studied. First, the boundary state description of the fractional branes are obtained. The open-string Witten index calculated using these states reproduces the adjacency matrix of the quiver of ${\mathbb{Z}}_5$. Then, using the toric crepant resolution of the orbifold ${\text{C}}^3\mathrm{/}{\mathbb{Z}}_5$ and invoking the local mirror principle, B-type branes wrapped on the holomorphic cycles of the resolution are studied. The boundary states corresponding to the five fractional branes are identified as bound states of BPS D-branes wrapping the 0-, 2- and 4-cycles in the exceptional divisor of the resolution of ${\text{C}}^3\mathrm{/}{\mathbb{Z}}_5$.