Line bundle twisted chiral de Rham complex, chiral Riemann–Roch formula and $D$-branes on toric manifolds

I represent the results of the elliptic genus calculations in various examples of twisted chiral de Rham complex on one- and two-dimensional toric compact manifolds. The explicit calculations are made for line bundle twisted chiral de Rham complex on $\mathbb{P}^{1}$, $\mathbb{P}^{2}$ and Hirzebruch surface. Based on these results I propose the elliptic genus expression of the bundle twisted chiral de Rham complex for general smooth compact two dimensional toric manifold. The expression resembles Riemann–Roch formula and coincides with the later in certain limit. I interprete the result in terms of infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of $D$-branes.