# Creation and annihilation operators in 2D Dilaton Gravity matter fields

In these notes by Strominger sec 3.6 we are given the creation and annihilation operators $$a_w = -\frac{i}{2\pi}\int\frac{dz^-}{\sqrt{2w}}f(z^-)\overleftrightarrow{\partial}_-e^{iwz^-}$$ $${a_w}^\dagger = \frac{i}{2\pi}\int\frac{dz^-}{\sqrt{2w}}f(z^-)\overleftrightarrow{\partial}_-e^{-iwz^-}$$ where I assumed the notation means $$a_w = -\frac{i}{2\pi}\int\frac{dz^-}{\sqrt{2w}}\left[\partial_-f(z^-)-iwf(z^-)\right]e^{iwz^-}$$ $$a_{w'}^\dagger = \frac{i}{2\pi}\int\frac{dy^-}{\sqrt{2w'}}\left[\partial_-f(y^-)+iw'f(y^-)\right]e^{-iw'y^-}$$ The $$f$$ is a massless matter field with action $$\mathcal{S} = -\frac{1}{4\pi}\int{d^2x\sqrt{-g}(\nabla f)^2}$$

and the above operators obey $$\left[a_w,a_{w'}^\dagger\right]=\delta(w-w')$$

I am trying to prove the commutation relationship but I am running into problems.

My attempt $$\left[a_w,a_{w'}^\dagger\right]= \frac{1}{4\pi^2}\int{\frac{dz^-dy^-}{2\sqrt{w'w}}e^{i(wz^--w'y^-)}\left(\left[\partial_-f(z^-),\partial_-f(y^-)\right]+iw'\left[\partial_-f(z^-),f(y^-)\right]-iw\left[f(z^-),\partial_-f(y^-)\right]+w'w\left[f(z^-),f(y^-)\right]\right)}$$ Now I don't see how we can integrate $$y^-$$ out such that we are left with a single integral and then use the definition of the delta function to get the desired result.