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I recently read about the construction of closed quasi-periodic differential forms on elliptic curves (1-dim Calabi-Yaus) via the Kronecker-Eisenstein series. I now wonder if similar constructions are known for higher dimensional Calabi-Yaus.

Context following [1].

Let $X=\mathbb{C}/\Lambda_{(1,\tau)}$, where $\Lambda_{(1,\tau)}=\mathbb{Z}\oplus \tau\mathbb{Z}$, stands for our elliptic curve with periods 1 and $\tau$. Then the Kronecker-Eisenstein series is defined by

$$F(z,\alpha,\tau)=\theta_1'(0,\bar{q})\frac{\theta_1(z+\alpha,\bar{q})}{\theta_1(z,\bar{q})\theta_1(\alpha,\bar{q})},$$

where $\bar{q}=\exp(2\pi i\tau)$ and

$$\theta_1(z,\bar{q})=-i\sum_{n\in\mathbb{Z}}(-1)^n\bar{q}^{\frac{1}{2}(n+1/2)^2}\exp(i\pi(2n+1)z).$$

We now look at the expansion in $\alpha$ of the Kronecker-Eisenstein series -- i.e., at the ${\color{red}g^{(k)}}$ in

$$F(z,\alpha,\tau)=\theta_1'(0,\bar{q})\frac{\theta_1(z+\alpha,\bar{q})}{\theta_1(z,\bar{q})\theta_1(\alpha,\bar{q})}=\frac{1}{\alpha}\sum_{k=0}^\infty{\color{red}g^{(k)}(z|\tau)}\alpha^k,$$

where the first few are given by $$\text{e.g., :} \quad g^{(0)}=1, \quad g^{(1)}(z)=\frac{\theta_1(z)}{\theta_1(z)}=\frac{1}{z}+\mathcal{O}(z), \quad g^{(2)}(z)=\frac{\wp(z)-g^{(1)}(z)^2} {2}.$$

More generally, we can show that:

  1. ${\color{red}g^{(k)}}$'s only have simple poles on the lattice as a function of $z$

  2. ${\color{red}g^{(k)}}$'s are periodic in $z\to z+1$, but not in $z\to z+\tau$

  3. ${\color{red}g^{(k)}}$ transforms in a relatively simple way under the modular group $$g^{(k)}\left(\frac{z}{c\tau+d}\mid\frac{a\tau +b}{c\tau+d}\right)=(c\tau +d)^{k}\sum_{j=0}^k g^{(k-j)}(z|\tau)\frac{\left(\frac{2\pi i c z}{c\tau+d}\right)^j}{j!}.$$

Then, if we fix a finite set $\Sigma\subset \mathbb{C}$ of punctures to define closed forms $$\omega_\sigma^{(n)}(z)=g^{(n)}(z-\sigma)\text{d}z\quad \in \Omega_1(\mathbb{C}\setminus(\sigma+\Lambda_{(1,\tau)})),$$ for each $n\ge 0$ and $\sigma\in\Sigma$.

These are particularly convenient because they allow to construct elliptic multi-polylogs, which are iterated integrals $$\int_0^z \omega_{z_1}^{(n_1)}...\omega_{z_r}^{(n_r)}=\tilde{\Gamma}\left(\substack{n_1, n_2,...,n_k\\ z_1,z_2,...,z_k};z\right)=\int_0^zdzg^{(n_1)}(z-z_1,\tau)\tilde{\Gamma}\left(\substack{n_2,...,n_k\\z_2,...,z_k};z\right).$$

This kind of integral is nice because they define holomorphic and homotopy invariant functions. This is at the price of not being doubly periodic since they live on the cover $$\mathbb{C}\setminus \bigcup_{\sigma\in\Sigma}(\sigma+\Lambda_{(1,\tau)}),$$ of $X\setminus \Sigma$.

Question.

I there a known analogue of $F$ for higher dimensional Calaby-Yaus, say for example a K3 manifold? In particular, I would be interested in a function that would lead to a well-defined generalization of elliptic multi-polylogs.

As a first guess I thought that promoting the Jacobi theta to a Riemann–Siegel theta function would give something interesting, but that would be for generalizations on higher genus tori, not on higher dimensional Calabi-Yaus.

Any comment or reference would help.

[1] https://arxiv.org/abs/1110.6917?context=math

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