One of the key characters in the [thesis](https://arxiv.org/pdf/0807.4834.pdf) of Zwegers is the modular correction $\tilde\mu(u,v;\tau)=\mu(u,v;\tau)+\frac i2R(u-v;\tau)$ of the Lerch sum $\mu(u,v;\tau):=\frac{e^{\pi i u}}{\theta(v;\tau)}\sum_{n\in\mathbb Z}(-1)^n\frac{e^{\pi i(n^2+n)\tau}e^{2\pi inv}}{1-e^{2\pi in\tau}e^{2\pi inu}}$, where
$$
R(u;\tau)=\sum_{\nu\in\frac12+\mathbb Z}(-1)^{\nu-\frac12}\left\{\operatorname{sgn}(\nu)-E\left(\left(\nu+\frac{\Im(u)}{\Im(\tau)}\right)\sqrt{2\,\Im(\tau)}\right)\right\}e^{-\pi i\nu^2\tau}e^{-2\pi i\nu u},
$$
with $E(z)=2\int_0^ze^{-\pi t^2}dt$.

In an illuminating (for me at any rate) paper "[M. P. Appell's function and Vector Bundles
of Rank 2 on Elliptic Curves](https://pdfs.semanticscholar.org/4aef/7841fc86e1e2024c020094f75a1e6e9ef8f9.pdf)", Polishchuk explains that the function $\kappa_a(z;q)=\sum_{n\in\mathbb Z}\frac{q^{n^2/2}}{q^n-a}z^n$ can be uniquely characterized by the fact that $(\kappa_a(z;q),\theta(z;q))$ defines a section of the rank 2 vector bundle on the elliptic curve $\mathbb C^*/q^{\mathbb Z}$ given by the quotient $(z,v_1,v_2)\sim(qz,av_1+v_2,q^{-1/2}z^{-1}v_2)$ of $\mathbb C^*\times\mathbb C^2$.

What confuses me is this - since $\kappa$ can be viewed as a section of a bundle on the universal elliptic curve, it must have certain modularity properties; and it is clearly holomorphic (well, almost everywhere at any rate). On the other hand $\mu$, which is obviously closely related to it, although holomorphic, is not modular, while $\tilde\mu$, being modular, is not holomorphic. So seemingly $\tilde\mu$ must have a counterpart in the context considered by Polishchuk; but the latter does not mention any possible nonholomorphic modular corrections.

How exactly are $\mu$, $\tilde\mu$ and $\kappa$ related to each other?