Line bundle twisted chiral de Rham complex, chiral Riemann-Roch formula and D-branes on toric manifolds
I present 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 ℙ1, ℙ2 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 interpret 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.