计算非费马-卡拉比丘特殊几何的模空间方法与新度量

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本文主要探讨的是在数学物理领域的一个重要课题，即"特殊几何形状在二模非费马·卡拉比丘（two-moduli non-Fermat Calabi-Yau）模空间上的应用与计算"。非费马·卡拉比丘模空间是Calabi-Yau复结构的一种特殊表示，Calabi-Yau manifolds在超弦理论和镜像对称性中扮演着核心角色。特殊Kähler几何是物理学中的一个关键概念，特别是在超引力理论中，它涉及到复结构的不变量和其相关的度量。文章指出，模空间实际上嵌入在一个特定的Frobenius流形之中，Frobenius流形是一种具有特殊性质的复代数结构，它的局部几何特性与特殊Kähler几何密切相关。Frobenius流形在量子力学、代数几何和数学物理中有广泛应用，尤其在理解复杂结构的变分原理时发挥重要作用。作者Konstantin Aleshkina和Alexander Belavin等人基于这种关系，提出了一个新颖的方法来计算Calabi-Yau复杂结构模空间上的特殊Kähler度量。他们利用Frobenius流形的结构特征，对之前未被解决的两个模参数的非费马模型进行了计算。这种方法不仅提升了我们对Calabi-Yau几何的理解，而且对于探索这些高维空间的动态行为以及它们在宇宙学和粒子物理中的潜在应用具有重要意义。文章于2018年发布在《Physics Letters B》期刊上，这表明了研究结果的学术价值和同行评审的质量。文章的接收、修订和接受过程也体现了研究的严谨性和科学交流的有效性。此外，Open Access的标签意味着该研究成果可以免费获取，促进了知识的传播和共享。总结来说，本文的核心内容是利用Frobenius流形的理论框架来解决特殊Kähler度量的计算问题，并且通过实例展示了在非费马·卡拉比丘模空间上的具体应用，这是现代数学物理研究中一个前沿且重要的进展。

Physics Letters B 776 (2018) 139–144

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Special geometry on the moduli space for the two-moduli non-Fermat

Calabi–Yau

Konstantin Aleshkin

a,b,∗

, Alexander Belavin

a,c,d

L.D. Landau Institute for Theoretical Physics, Akademika Semenova av. 1-A, Chernogolovka, 142432 Moscow region, Russia

International School of Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy

Department of Quantum Physics, Institute for Information Transmission Problems, Bolshoy Karetny per. 19, 127994 Moscow, Russia

Moscow Institute of Physics and Technology, Dolgoprudnyi, 141700 Moscow region, Russia

a r t i c l e i n f o a b s t r a c t

Article history:

Received

29 August 2017

Received

in revised form 10 October 2017

Accepted

15 November 2017

Available

online 21 November 2017

Editor:

M. Cveti

We clarify the recently proposed method for computing a special Kähler metric on a Calabi–Yau complex

structure moduli space using the fact that the moduli space is a subspace of a particular Frobenius

manifold. We use this method to compute a previously unknown special Kähler metric in a two-moduli

non-Fermat model.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

To ﬁnd the low-energy Lagrangian of string theory compacti-

ﬁed

on a Calabi–Yau (CY) manifold X, one must know the special

Kähler geometry on the moduli space of the CY manifold [1–4].

An alternative method for computing the Kähler potential in the

case where the CY manifold is given by a hypersurface W

(x) = 0

a weighted projective space was recently presented in [5]. In

this paper, we brieﬂy review this method and use it to compute

the special Kähler geometry on the moduli space of a two-moduli

non-Fermat threefold considered in [6]. This approach is based on

the fact that the moduli space of a CY manifold is a subspace in

the particular Frobenius manifold (FM) [7] that arises from the de-

formation

space of the singularity deﬁned by the LG superpotential

(x). In this paper, we discuss this FM and its connection with

the geometry of the CY moduli space in more detail.

is known, that an isolated singularity W

(x) deﬁnes an ob-

ject

called a Milnor ring R

, Endowed with an invariant pairing

this ring becomes a Frobenius algebra. Its elements deﬁne versal

deformations of the singularity. It is well known that the space

of deformations admits a FM structure. In our case, W

(x) is a

quasihomogeneous polynomial, which deﬁnes a CY manifold in

the weighted projective space. For our purposes of computation

of the CY moduli space geometry we actually need the Frobenius

subalgebra R

of R

, that is multiplicatively generated only by

polynomials of the same degree d as W

(x). These polynomials are

related to the complex structure deformations. The subalgebra R

Corresponding author.

E-mail

address: kaleshkin@sissa.it (K. Aleshkin).

consists of elements of degrees 0, d, 2d and 3d, and each homo-

geneous

component of this algebra is naturally a subgroup of the

respective middle Hodge cohomology H

3,0

(X ), H

2,1

(X ), H

1,2

(X )

and H

0,3

(X ). In our examples, R

is actually isomorphic to H

(X ).

The FM in which we are interested is precisely the submanifold of

the FM on the versal deformations of W

(X ) restricted to the de-

formations

by the elements of R

The

relation to the FM structure allows obtaining the follow-

ing

explicit expression [5] for the Kähler potential in terms of the

holomorphic FM metric η

μν

in the ﬂat coordinates and periods of

the holomorphic 3-form :

−K

= σ

(φ)η

μρ

−

(φ), M = T

−1

T . (1)

Here, φ is a parameter on the moduli space, and σ

(φ) are periods

of the holomorphic 3-form , deﬁned below as the integrals of 

over some special basis of cycles γ

in the homology group H

(X ),

which are deﬁned in terms of the connection with the FM.

The

constant nondegenerate complex matrix T connects the ba-

sis

(φ) with another basis ω

(φ) deﬁned in [8,6] as integrals of

 over another special basis of the cycles q

by the linear relation

(φ) = T

(φ). (2)

The explicit expression for the Kähler potential is obtained as fol-

lows.

The basis ω

is deﬁned as integrals over the homology cycles

with real coeﬃcients. These cycles have a well-deﬁned inter-

section

matrix C

μν

= q

∩ q

. The matrix M and formula (1) are

independent of the speciﬁc choice of ω

. The Kähler potential

K (φ) on the moduli space with the coordinates φ and the holo-

morphic

FM metric η

αβ

can be expressed in the terms of C

μν

, the

https://doi.org/10.1016/j.physletb.2017.11.030

0370-2693/

SCOAP

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