Gauss–Manin Connection in Disguise: Calabi–Yau Threefolds

Abstract

We describe a Lie Algebra on the moduli space of Calabi-Yau threefolds enhanced with differential forms and its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation. In particular, we describe algebraic topological string partition functions F alg g , g ≥ 1, which encode the polynomial structure of holomorphic and non-holomorphic topological string partition functions. Our approach is based on Grothendieck's algebraic de Rham cohomology and on the algebraic Gauss-Manin connection. In this way, we recover a result of Yamaguchi-Yau and Alim-Länge in an algebraic context. Our proofs use the fact that the special polynomial generators defined using the special geometry of deformation spaces of Calabi-Yau threefolds correspond to coordinates on such a moduli space. We discuss the mirror quintic as an example.

Calabi-Yau modular forms in limit: Elliptic fibrations

“…This is actually the case for some examples of such q-expansions, however, in general they transcend the world of modular and automorphic forms. The case of the mirror quintic is the most well-known one, and it is argued in [Mov15b, Mov15a,AMSY14] that there is a parallel modular form theory in this case. These are called Calabi-Yau modular forms.…”

Manifold Ways to Darboux-Halphen System

“…In particular, the Frobenius manifold presented in Section 6.3 is a case of modular Frobenius manifold where the prepotential is preserved under a inverse symmetry that acts as an S generator of the modular group SL(2, Z) in t 3 direction [22]. Such modularity is a desirable property that can establish a relation to Gauss-Manin connection in disguise and may be extended to the group of transformations of Calabi-Yau modular forms [2,25].…”

“…Moduli of charge 2 monopoles reduced by quotient by R 3 action 2. Note that here the four coordinates of the moduli are not spacetime directions, but internal parameters of a 2-monopole solution.…”

Modular Vector Fields Attached to Dwork Family: s l 2 ( C ) Lie algebra

“…The enhanced moduli space T and the relevant modular vector field R of this family have been discussed in [MN16]. These facts lead us to extend the Lie algebras discussed in [AMSY16] to the enhanced moduli space T arising from the Dwork family for any positive integer n. From now on, unless otherwise is stated, by mirror variety X we mean the Calabi-Yau n-fold X = X ψ , ψ ∈ P 1 \ {0, 1, ∞}, obtained by the desingularization of the quotient space of the Calabi-Yau varieties of the Dwork family (1.1) under a group action (see [MN16,§2]). By enhanced moduli space T = T n we mean the moduli space of the pairs (X, [α 1 , α 2 , .…”

“…On one hand, the mirror quintic 3-fold is the main example of [AMSY16] in which the authors describe a special Lie Algebra on the moduli space of the non-rigid compact Calabi-Yau threefolds enhanced with differential forms and discuss its relation to the Bershadsky-Cecotti-Ooguri-Vafa holomorphic anomaly equation (see also [Ali17]). On the other hand, the mirror quintic 3-fold is the particular case n = 3 of the family of the ndimensional Calabi-Yau varieties X = X ψ , ψ ∈ P 1 − {0, 1, ∞}, arising from the so-called Dwork family:…”