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

S.E. Parkhomenko

doi:10.1016/j.nuclphysb.2014.02.003

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 P3. Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in P2 and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on P2 twisted by SL(N) vector bundle with instanton number k is calculated. In all the cases considered we find the infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of D-branes.

Highlights

It has been proposed by J.Harvey and G.Moore [1] that sheaves can be used to model Dbranes on large-radius Calabi-Yau manifolds
Soon after the significant application of chiral de Rham complex in the String Theory has been represented in the beautiful paper of Borisov [5] where the chiral de Rham complex construction has been given for each pair of dual reflexive polytopes defining toric CY manifold
As a result of the elliptic genus calculation we find the infinite tower of open string oscillator contributions coming from the bound states of k D0-branes and N D4-branes which is in agreement with the conjecture of Witten [16] on the relation between the instantons and D-branes

Summary

Introduction

It has been proposed by J.Harvey and G.Moore [1] that sheaves can be used to model Dbranes on large-radius Calabi-Yau manifolds. The elliptic genus calculation is made for the line bundle twisted chiral de Rham complex on K3 hypersurface embedded in P3 In this case we extract the corresponding open string oscillator contributions coming from bound state of. We are not giving the proof, the structure of the formulas (34), (41), (43) obviously confirms the assumption if we read them as a D0 and D2- branes or D2-brane and D4-brane binded together to form the bound state due to tachyon condensation [24] (see [2], [3] and references therein) so that the open string states are given by the cohomology of line bundle twisted chiral de Rham complex. Elliptic genus of chiral de Rham complex twisted by a sheaf localized on points

Let us consider for example the sheaf

Conclusion