It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive expectation: It should make sense to talk about modular cycles too. However, when I skim the literature, I find nothing on such objects...
Let me clarify a bit more what I mean.
Let $\tau\in\mathbb{H}$ and let $f(\tau)$ be a holomorphic function in $\mathbb{H}\cup \{i\infty\}$. Then we can look for functions $f(\tau)$ that transform under a modular transformation in a certain way. Ultimately, this leads us to the well-known concept of modular forms of weight $k$, satisfying the following transformation law $$f\left(\tau\right)d\tau\to(c\tau+d)^{k-2}f(\tau)d\tau,$$ under $\tau\to\frac{a\tau+b}{c\tau+d}$, where $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\text{SL}(2,\mathbb{Z})$.
What I am wondering about is if there exists in a dual version of such a criterion for cycles, which would lead to a well-define notion of modular cycles. I expect such a criterion to exist naively due to de Rham duality.
Any guidance to some literature would be very appreciated!
Disclaimer: I am a physicist, so this is not my field of expertise and I may ask something totally trivial or something that doesn't make any sense.
