In this paper on page 4 the authors write:
We would like to apply such a construction to the tangent bundle of a free loopspace. [...]. However, such a splitting does exist in a neighborhood of the constant loops: $$M = LM^{\mathbb{T}}\subset LM$$ has normal bundle$$\nu(M\subset LM) = TM\otimes_{\mathbb{C}}(\mathbb{C}[q,q^{-1}]/\mathbb{C})$$ (at least, up to completions; and assuming things complex for convenience). Here small perturbations of a constant loop are identified with their Fourier expansions $$\sum_{n\in\mathbb{Z}}a_n q^n$$ with $q^n = e^{i\theta}$.
Here $LM$ is the free loop space of a manifold $M$ and $\mathbb{T}$ is the circle group.
I don't understand what the Fourier expansion of a (small perturbation of a constant) loop is. So my question is: What is the Fourier expansion of a (small perturbation of a constant) loop? In particular:
- In what space do the $a_n$ of the formula $\sum_{n\in\mathbb{Z}}a_n q^n$ live?
- How does addition work in the formula $\sum_{n\in\mathbb{Z}}a_n q^n$?
- How is this Fourier expansion constructed? (I'm okay with answers that refer to some usual version of Fourier expansion)
My idea would be something like: If $\gamma: S^1 \to M$ is a small perturbation of a constant loop, the image of $\gamma$ is probably contained in one chart of $M$. Because of this I can think of $\gamma$ as a map $\gamma: S^1 \to \mathbb{R}^k$ and from there I can probably find a Fourier expansion of $\gamma$ (I think it is possible to construct a Fourier expansion of such a map but I don't exactly know because I don't know much about Fourier expansions in general).
Lets assume a tangent vector at $\gamma$ is some equivalence class of a curve $\alpha:[0,1]\to LM$ with $\alpha(0) = \gamma$. We can write $\alpha$ as $\alpha: [0,1]\times S^1 \to M$, which I would call a perturbation of $\gamma$ (without a precise definition in my mind).
Now define $V:S^1 \to TM$ by $V(\theta) = \frac{\partial \alpha}{\partial t}(0,\theta) \in T_{\gamma(\theta)} M. $ That should motivate (maybe even proof) that $$T_\gamma LM \overset{\sim}{=} \Gamma(\gamma^*TM).$$
Now if $\gamma = p \in M$ is a constant loop, this simplifies to $$ T_p LM = LT_pM. $$ Your claim now is, that any $V \in LT_pM$ can be written as $$V(\theta) = \sum\limits_{n \in \mathbb{Z}} a_n q^n$$ where $a_n \in T_pM \otimes \mathbb{C}$ and $q = e^{2\pi i \theta}$. This comes down to do the usual fourier expansion in each component of $T_p M = \mathbb{R}^n$.
You will get $a_n = \int_0^1 V(\theta) e^{-2\pi i n \theta} d\theta \in T_pM\otimes\mathbb{C}$ and $a_{-n} = \bar{a}_n$ for all $n \in \mathbb{Z}$.