# Central charge and topological invariant of Calabi-Yau manifolds

## TL;DR

This paper explores the relationship between the central charge and topological invariants of Calabi-Yau manifolds within F-theory, using orbifold constructions, quiver diagrams, and BPS charge calculations.

## Contribution

It introduces a method to compute Euler characters and BPS central charges for orbifold Calabi-Yau threefolds using quiver diagrams and coherent sheaves.

## Key findings

- Euler character calculations for orbifold spaces
- BPS central charge expressions for $C^3/Z_3$ orbifold
- Characterization of fractional sheaves via Ramond-Ramond charges

## Abstract

F-theory, as a 12-dimensional theory that is a contender of the Theory of Everything, should be compactified into elliptically fibered threefolds or fourfolds of Calabi-Yau. Such manifolds have an elliptic curve as a fiber, and their bases may have singularities. We considered orbifold as simplest non-flat construction. Blow up modes of orbifold singularities can be considered as coordinates of complexified Kahler moduli space. Quiver diagrams are used for discribing D-branes near orbifold point. In this case it is possible to calculate Euler character defined through $\mbox{Ext}^i(A,B)$ groups and coherent sheaves $A, B$ over projective space, which are representations of orbifold space after blowing up procedure. These fractional sheaves are characterized by $Q_0$, $Q_2$ and $Q_4$ Ramon-Ramon charges, which have special type, calculated for $C^3/Z_3$ case. BPS central charge for $C^3/Z_3$ orbifold is calculated through Ramon-Ramon charges and Picard-Fuchs periods.