Line bundle twisted chiral de Rham complex and bound states of D-branes on toric manifolds

TL;DR

This paper computes elliptic genera of twisted chiral de Rham complexes on toric manifolds and Calabi-Yau hypersurfaces, revealing insights into D-brane bound states and open string boundary conditions.

Contribution

It provides explicit calculations of elliptic genera for various twisted chiral de Rham complexes on toric manifolds, connecting them to D-brane bound states and open string spectra.

Findings

Elliptic genus calculations for line bundle twisted complexes on toric manifolds and K3 hypersurfaces.

Identification of open string oscillator contributions with D-brane bound states.

Explicit boundary conditions and D-brane charges derived from elliptic genus data.

Abstract

In this note we calculate elliptic genus in various examples of twisted chiral de Rham complex on two dimensional toric compact manifolds and Calabi-Yau hypersurfaces in toric manifolds. At first the elliptic genus is calculated for the line bundle twisted chiral de Rham complex on a compact smooth toric manifold and K3 hypersurface in \mathbb{P}^{3} . Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in \mathbb{P}^{2} and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on \mathbb{P}^{2} twisted by SL(N) vector bundle with instanton number k is calculated. In all cases considered we find the infinite tower of open string oscillator contributions of the corresponding bound state of D-branes and identify directly the open string boundary conditions and D-brane charges.