Pis’ma v ZhETF, vol. 110, iss. 11, pp. 727 - 728
© 2019 December 10
GLSM for Berglund-Hübsch Type Calabi-Yau manifolds
K.Aleshkin+1), A.Belavin∗×◦1)
+Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA
∗L. D. Landau Institute for Theoretical Physics, 142432 Chernogolovka, Russia
×Moscow Institute of Physics and Technology, 141700 Dolgoprudnyi, Russia
◦Institute for Information Transmission Problems, 127051 Moscow, Russia
Submitted 31 October 2019
Resubmitted 31 October 2019
Accepted 5 November 2019
DOI: 10.1134/S0370274X19230024
In this note we briefly present the results of our com-
Kähler moduli space of the infrared limiting model of
putation of special Kähler geometry for polynomial de-
the GLSM in the Calabi-Yau case.
formations of Berglund-Hübsch type Calabi-Yau man-
Mirror symmetry connects quantum corrected
ifolds. We also build mirror symmetric Gauge Linear
Kähler moduli space of one model with complex
Sigma Model and check that its partition function com-
structures moduli space of another. For all Calabi-Yau
puted by Supersymmetric localization coincides with ex-
threefolds given by invertible singularities we con-
ponent of the Kähler potential of the special metric.
struct the corresponding mirror symmetric GLSM and
Special Kähler geometry is the geometrical structure
compute their partition functions. Then we explicitly
underlying particular coupling constants in superstring
check that they coincide with exponents of Kähler
compactifications. Knowledge of this metric is impor-
potentials of the metrics on the deformation spaces of
tant for phenomenological questions arising in super-
the invertible singularities.
string theories.
∑N ∏N
Namely, we use Batyrev approach to the mirror sym-
A polynomial W0(x1, . . . , xN ) =
i=1
j=1
xMijj in
metry. The idea is as follows. Let Calabi-Yau threefold
N variables is called an invertible singularity if M is an
X be defined as a hypersurface in a weighted projec-
invertible matrix with positive integer coefficients.
tive space P4(k
and given by zero locus of the
1,...,k5)
We explicitely describe the Kähler geometry on the
polynomial W(x, φ). Exponents of the monomials in
∑5
∏5
space of polynomial deformations of W0(x) which cor-
the deformed polynomial W (x, φ) =
i=1
j=1
xMijj +
responds to the space of deformations of complex struc-
∏5
∑h
+
=
l=1
φlj=1 xsljj define the finite set Va, a
tures on the Calabi-Yau manifold X0. The exponent of
= 1, . . ., N and thus define Batyrev’s polytope ΔX.
the Kähler potential of the Weil-Peterson metric on the
Knowing the set of vectors Va we can construct a fan,
deformation space is expressed in terms of the periods
which defines another toric variety. Calabi-Yau mani-
the periods σa(φ) as follows:
fold Y , the mirror to X, can be defined as a hyper-
surface in this variety given by the zero locus of the
e-K(φ
φ) =
(1)
(
)
homogeneous polynomial WY . Using the fan we also
∑
∏
Γ
(aj + 1)M-1
ji
find the corresponding GLSM with its gauge group and
=
(-1)|a|/d
(
) |σa(φ)|2.
Γ
1 - (aj + 1)M-1
the charges Qal of its chiral multiplets. The charges
a=0,2d-1
i≤5
ji
appear as coefficients of linear relations between the
The partition function for general Gauge Linear
vectors of the fan and set the weights of the toric
Sigma Models (GLSM) was computed exactly a few
variety.
years ago. The conjecture of Jockers et al. states that
The mirror symmetric GLSM has h + 5 chiral
this partition function of GLSM coincides with the ex-
superfields fields {Φa}h+5a=1 and h U(1) vector super-
ponent of the Kähler potential of the metric on the
fields {Vl}hl=1 acting on Φa with the charge matrix
{Qal}a≤h+5,l≤h. In our matrix Qal = sljM-1ja if ≤ a ≤ 5
1)aleshkin@math.columbia.edu; belavin@itp.ac.ru
and Qal = -δa-5,l if a > 5.
Письма в ЖЭТФ том 110 вып. 11 - 12
2019
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