Topological Quantum Field Theory: Help reading a paper

[nateHI]

https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf

I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space $Y$ and a spacetime $X$ the author talks about the associated Quantum Hilbert Spaces $E(Y)$ and $E(\partial X)$.

The Simple Question: Elements of $E(Y)$ are just $L^2$ functions $f:Y\to \mathbb{R}$ right?

The Sanity Check: If $X$ is the spacetime and $Y$ the space then $Y$ is the boundary of $X$. So why the different notations when talking about the Hilbert space $E(Y)$ itself and the other $E(\partial X)$ when talking about path integrals?

 

[nateHI]

You can disregard this question. I figured it out.