Calabi–Yau manifolds realizing symplectically rigid monodromy tuples

Abstract

We define an iterative construction that produces a family of elliptically fibered Calabi-Yau n-folds with section from a family of elliptic Calabi-Yau varieties of one dimension lower. Parallel to the geometric construction, we iteratively obtain for each family with a point of maximal unipotent monodromy, normalized to be at t = 0, its Picard-Fuchs operator and a closed-form expression for the period holomorphic at t = 0, through a generalization of the classical Euler transform for hypergeometric functions. In particular, our construction yields one-parameter families of elliptically fibered Calabi-Yau manifolds with section whose Picard-Fuchs operators realize all symplectically rigid Calabi-Yau differential operators with three regular singular points classified by Bogner and Reiter, but also non-rigid operators with four singular points.

Heun functions and diagonals of rational functions

“…Furthermore, we constructed in subsection 2.4, a rational function whose diagonal is given by a Heun function that has already ‡ Or, more generally monomial transformations. been identified as a "Period" of an extremal rational elliptic surface [21], and that has also emerged in the context of pullbacked 2 F 1 hypergeometric functions [25]. The emergence of squares of Heun functions for most of the diagonals of rational functions of this paper, suggests a "Period" of algebraic surfaces (possibly product of elliptic curves) interpretation.…”

“…The Heun function on the RHS of (5) happens to be a period of an extremal rational curve as can be seen in the work of Doran and Malmendier [21]. The diagonal (5) can also be written as a Hadamard product † of a simple algebraic function and a Heun function:…”

“…Note: The Heun function (63) resembles the Heun function associated with extreme rational surfaces [21,24] (φ denotes the golden number (1 + √ 5)/2):…”

Hodge Numbers from Picard-Fuchs Equations

“…Finally we note that, according to work of Doran and Malmendier [11], most of Doran's and Morgan's [12] 14 hypergeometric local systems can be obtained from hypergeometric local systems of rank 3. These rank 3 hypergeometric local systems underlie variations of Hodge structure corresponding to families of K3 surfaces, for which the Lyapunov exponents have been computed by Filip [14].…”

GLSMs, joins, and nonperturbatively-realized geometries

“…Concrete examples will be discussed in sections 2.4 and 2.5. (See also [15] for a discussion of Hadamard products in a slightly different context. )…”