Pis’ma v ZhETF, vol. 118, iss. 10, pp. 711 - 712
© 2023 November 25
Review on special geometry and mirror symmetry
for Calabi-Yau manifolds
(Mini-review)
A. Belavin+∗1), B. Eremin∗×◦1), 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
Submitted 11 October 2023
Resubmitted 11 October 2023
Accepted 16
October 2023
DOI: 10.31857/S1234567823220019, EDN: pihhac
Ten-dimensional Superstring theory unifies the
pression was found for the Kähler potential of the metric
Standard Model of the strong, electromagnetic, and
of special geometry for an arbitrary surface of Fermat
weak interactions with quantum gravity. Starting
type.
with 10-dimensional superstring theory, we can get a
Previously, Jockers et al. proposed a conjecture
4-dimensional theory with spacetime supersymmetry
(JKLMR conjecture), which establishes a connection be-
following the Kaluza-Klein idea by compactifying six
tween Kähler potentials in the moduli space of Kähler
of the ten dimensions.
structures of CY manifolds and the exact partition func-
For phenomenological reasons we need to do
tion of N = (2, 2) supersymmetric gauge linear sigma
this while maintaining N
= 1 Supersymmetry of
models (GLSM) on a two-dimensional sphere. We tested
4-dimensional spacetime. To achieve this, as Candelas,
the JKLMR conjecture for several simple examples. To
Horowitz, Strominger and Witten have shown, we must
do this, in [3, 4] we used our results for the explicit ex-
compactify six of the ten dimensions of to the so called
pression for the Kähler potential together with mirror
Calabi-Yau (CY) manifolds.
symmetry.
Another equivalent approach developed by D. Gep-
We also tested the mirror version of the JKLMR con-
ner is the compactification of 6 dimensions onto some
jecture using the example of the CY manifold X, which
N = 2 Superconformal Field theory with the central
does not belong to the Fermat class [5]. Moreover, for
charge c = 9. Each of these two equivalent approaches
the general class of Berglund-Hübsch families, a con-
has its own merits. Say, using exactly solvability of the
nection was found between GLSM for Y and the CY
Minimal models of N = 2 Superconformal Field The-
family X [6].
ory, it is possible to obtain the explicit solution of the
In the work [7], devoted to the study of CY mani-
considered models.
folds of the Berglund-Hübsch type, we discovered the
In a series of papers [1, 2], we proposed a new method
phenomenon of coincidence of the Kähler potential on
for calculating special geometry on the moduli space of
the moduli space of complex structures of two differ-
the complex structures of CY manifolds, defined as a
ent CY families. Two cases of such coincidences were
hypersurfaces in a weighted projective space. This ap-
considered. Each pair has two families of Calabi-Yau
proach is based on the connection between the coho-
manifolds with matching Hodge numbers. For both fam-
mology of CY manifolds and their periods, specified
ilies in each pair, using our method proposed in [1], the
by oscillatory integrals, with supersymmetric Landau-
Kähler potentials were found on the moduli space of
Ginzburg models. Knowing the periods allows us to cal-
complex structures. After this, the fact that the calcu-
culate the main components of the special geometry of
lated potentials coincide in each pair was checked. To do
the moduli space. Using this calculation, an explicit ex-
this for both families in each pair a so-called fan that
defines this CY family was constructed.
In the works [8, 9] we studied the Mirror symmetry
1)e-mail:
belavin@itp.ac.ru;
Eremin.ba@phystech.edu;
spark@itp.ac.ru
for various examples of CY manifolds. We checked the
Письма в ЖЭТФ том 118 вып. 9 - 10
2023
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