Abstract:
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.
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