On fermionic representation of the framed topological vertex

The Gromov-Witten invariants of ℂ³ with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al. in the fermionic picture. We will propose a similar formula for the framed topological vertex and prove it in the case when there are one or two branes.