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Say there is a (non-holomorphic) real function $Z(\tau,\bar{\tau})$ which obeys the (non-holomorphic) modularity conditions of some weight $k$ $$Z(\tau+1,\bar{\tau}+1)=Z(\tau,\bar{\tau}),\qquad Z(-1/\tau,-1/\bar{\tau})=|\tau|^kZ(\tau,\bar{\tau})$$ Assuming that $Z$ can be represented as a sum of $N$ holomorphic squares $$Z(\tau,\bar{\tau})=\sum_{i=1}^N|f_i(\tau)|^2$$ can one say somethig about the functions $f_i(\tau)$? Of course they can be modular forms of weight $k/2$, consistent with the transformation properties of $Z$. The question is whether they can be anything else (I think they can't at least for $N=1$)?

The question is not very precise, but I hope it still makes sense. My practical situation is that I have some explicit real functions $Z$ which obey "real modularity", and I know they must be representable as sum of holomorphic squares. But this representation is not evident from the definition of $Z$. My hope was that modular forms could provide a convenient basis make this decomposition.

Afterthoughts.

I now believe that if functions $f_i$ are linearly independent then the action of $S$ and $T$ generators on them must be represented by some linear transformations with possible weight factors. $$f_i(\tau+1)=T_{ij}f_{j}(\tau),\qquad f_i(-1/\tau)=\tau^kS_{ij}f_{j}(\tau)$$ (This follows from an observation that if $\{f_i\}$ and $\{g_i\}$ are holomorphic and linearly independent, and $\sum |f_i(\tau)|^2=\sum |g_i(\tau)|^2$, then there must be a unitary transformation mapping $f_i$ to $g_i$.)

If $T=S=Id$ each $f_i$ is a modular form. The question then is what other representations of $PSL(2,\mathbb{Z})$ can occur, and what are the corresponding functions $f_i$? Alas, I do not know the answer even for $N=1$, namely how to describe the space of holomorphic functions (assuming nice properties at infinity etc) which satisfy $$|f(\tau+1)|=|f(\tau)|,\qquad |f(-1/\tau)|=|\tau^kf(\tau)|$$ I guess that by dividing this by a suitable power of the Dedekind Eta-function one can reduce this to $k=0$ case, and now maybe this is something to look for in a textbook. Anyway, any comments are appreciated.

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This situation occurs in two-dimensional conformal field theory for the case $k=0$ where the $f_i(\tau)$ are characters of a chiral algebra including the Virasoso algebra and transform under some N-dimensional representation $\rho: PSL(2,Z) \rightarrow GL(V)$ of the modular group.

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  • $\begingroup$ Well, this question is CFT-inspired, but not exactly. I think you can show in general that if $f_i$ are linearly independent, they must furnish some representation of $PSL(2,\mathbb{Z})$. In CFT these things come as characters. Now I ask a reverse question: having such a decomposition, how can the space of functions $f_i$ be described? I don't even know the answer when $N=1$, so perhaps I should've started with a simpler question, I will make the corresponding edits. $\endgroup$
    – Weather Report
    Commented Nov 3, 2019 at 20:46
  • $\begingroup$ For $N=1$ I am pretty sure the only possibility is that $f_1$ is a modular form with a character. For example the CFT of affine $E_8$ at level one has character $f=E_4/\eta^4$. This is not a modular function, invariant under $SL(2,Z)$, but it transforms by a phase which will cancel out in $|f|^2$. $\endgroup$
    – jah
    Commented Nov 3, 2019 at 23:30

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