Heterotic stringy corrections to metrics of toroidal orbifolds and their resolutions

We explicitly analyze $O({\alpha }^{\prime })$ corrections to heterotic supergravity on toroidal orbifolds and their resolutions, which play important roles in string phenomenology as well as moduli stabilization. Using a conformal factor ansatz that is valid only for four-dimensional geometries, we obtain a closed expression for the $O({\alpha }^{\prime })$ metric corrections in the case of several orbifold limits of K3, namely $T^4/{\mathbb{Z}}_n$ where $n=2$, 3, 4, 6. However, we find that nonstandard embedding requires the inclusion of five-branes on such orbifolds. We also numerically investigate the behavior around orbifold fixed points by considering the metric correction on the resolution of a ${\mathbb{C}}^2/{\mathbb{Z}}_2$ singularity. In this case, a nontrivial conformal factor can be obtained in nonstandard embedding even without five-branes. In the same manner, we generalize our analysis to study metric corrections on $T^6/{\mathbb{Z}}_3$ and its resolution described by a complex line bundle over ${\mathbb{CP}}^2$. Further prospects of utilizing these $O({\alpha }^{\prime })$ corrected metrics as a novel approach in obtaining realistic or semirealistic Yukawa couplings are discussed.

Figure 1

Numerical solution for the conformal factor ${\mathrm{\Omega }}^2$ on the resolution ${\mathcal{M}}^2$ on the $(|z|,|x|)$ plane, where the blowup parameter is chosen as $r=5$, 1, 0.1 from left to right. In all panels, the solid, dashed, dot-dashed, and dotted curves denote ${\mathrm{\Omega }}^2=1$, 0.99, 0.98, 0.95, respectively.

Figure 2

Numerical solution for the trace of the metric correction $h$ and background metric $g$ on the resolution ${\mathcal{M}}^3$ with blowup parameter $r=3$. $h$ and $g$ are drawn in red and black curves in all panels. The left panel shows the $(|z_1|,|z_2|)$ plane with $|x|=0$. Red (black) solid, dashed, dot-dashed curves denote $h=-0.1$, $-0.3$, $-0.5$, [$\mathrm{tr}(g)=140$, 80, 20] respectively. In the middle ($(|z_1|,|x|)$ plane with $|z_2|=0$) and right panels ($(|z_2|,|x|)$ plane with $|z_1|=0$), the red (black) solid, dashed, dot-dashed curves denote $h=-0.1$, $-0.3$, $-0.5$, [$\mathrm{tr}(g)=20$, 10, 5] respectively.

Figure 3

Numerical solution for the trace of the metric correction $h$ and background metric $g$ on the resolution ${\mathcal{M}}^3$ with blowup parameter $r=1$. $h$ and $g$ are drawn in red and black curves in all panels. The left panel shows the $(|z_1|,|z_2|)$ plane with $|x|=0$. Red (black) solid, dashed, dot-dashed curves denote $h=-0.1$, $-0.4$, $-0.8$, [$\mathrm{tr}(g)=300$, 150, 50] respectively. In the middle ($(|z_1|,|x|)$ plane with $|z_2|=0$) and right panels ($(|z_2|,|x|)$ plane with $|z_1|=0$), the red (black) solid, dashed, dot-dashed curves denote $h=-0.05$, $-0.1$, $-0.2$, [$\mathrm{tr}(g)=20$, 10, 5] respectively.