# How can one derive Schwarzian derivative action as low energy effective field theory invariant under global $SL(2,\mathbb{R})$?

In a recent paper (page 47, below eq (4.173)) they make a passing claim that the Schwarzian derivative action can be derived using effective low energy field theory reasoning. I imagine they mean that if I want to construct a least derivative action which is invariant under global $SL(2,\mathbb{R})$ transformations of the coordinates, then I will end up with Schwarzian derivative. I was wondering if this has been worked out anywhere. Also, using the same approach, what are the higher derivative invariants that I can possibly construct as 'less relevant' terms.

The answer to this question is rather simple. Let us assume that our field is $$f(x)$$ and we want to find an effective low energy action invariant under the $$SL(2,\mathbb R)$$ transformations,$$f(x)\to\frac{a\, f(x)+b}{c\, f(x)+d}~,\tag{1}$$with $$a,b,c,d\in\mathbb R$$ and $$ad-bc=1$$. Let us look at these transformations one by one:
1. Translations: $$f(x)\to f(x)+b$$ imply that effective low energy action can only be a function of derivatives of the $$f(x)$$ field, $$f'(x),f''(x),f^{(3)}(x),\ldots$$.
2. Scaling: $$f(x)\to a\, f(x)$$ implies that the effective action is a function of ratios of the fields, $$\dfrac{f^{(n)}(x)}{f^{(m)}(x)}$$.
Since we are working with effective action we want an action with the least number of derivatives. Since $$\dfrac{f''(x)}{f'(x)}$$ is a total derivative our lowest derivative candidates are $$\dfrac{f'''(x)}{f'(x)}$$ and $$\left(\dfrac{f''(x)}{f'(x)}\right)^2$$. Let's assume our action is, $$\dfrac{f'''(x)}{f'(x)}-\mu \left(\dfrac{f''(x)}{f'(x)}\right)^2~.\tag{2}$$
1. Finally imposing invariance of the action under inversion: $$f(x)\to\dfrac1{f(x)}$$ one can easily fix the value of $$\mu=\frac32$$