Review on special geometry and mirror symmetry for Calabi-Yau manifolds (Mini-review)

A. Belavin^{+* 1)}, B. Eremin^{*×°}, S. Parkhomenko^{+× 1)}

^{+}Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia

^{*}Kharkevich Institute for Information Transmission Problems, 127051 Moscow, Russia

^{×}Moscow Institute of Physics and Technology, 115184 Dolgoprudny, Russia

^{°}Skolkovo Institute of Science and Technology, 121205 Moscow, Russia

**Abstract**

Ten-dimensional Superstring theory unifies the Standard Model of the
strong, electromagnetic, and weak interactions with quantum gravity.
Starting with 10-dimensional superstring theory, we can get a 4-dimensional theory
with spacetime supersymmetry following the Kaluza-Klein
idea by compactifying six of the ten dimensions.
For phenomenological reasons we need to do this while maintaining N=1
Supersymmetry of 4-dimensional spacetime. To achieve this, as Candelas,
Horowitz, Strominger and Witten [Candelas:1985en] have shown, we
must compactify six of the ten dimensions of to the so called Calabi-Yau
manifolds.
Another equivalent approach developed by D. Gepner [Gepner1,
Gepner2] is the compactification of 6 dimensions onto some N=2
Superconformal Field theory with the central charge c=9. Each of these
two equivalent approaches has its own merits. Say, using exactly
solvability of the Minimal models of N=2 Superconformal Field Theory,
it is possible to obtain the explicit solution of the considered models.
In this article we review a series of our works on Calabi-Yau manifolds,
their mirror symmetry, special geometry on the moduli space of Calabi-Yau, the
connection of Calabi-Yau manifolds with N=2 superconformal minimal
models and with supersymmetric gauged linear sigma models.