What is the importance of studying degeneration on $M_g$

Question

Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$.

It seems to be important in physics to study the degeneration of certain functions on $\overline{M_g}$ as you approach its boundary.

For which functions on $M_g$ is it interesting to physicists to study their degeneration as you approach the boundary?

I know of theta functions, Green functions and delta invariants. Are there others?

As in my other question, here is an article by Wentworth which I find interesting.

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Comments

This old question of mine over at MathOverflow might be relevant: http://mathoverflow.net/questions/1312/gromov-witten-theory-and-compactifications-of-the-moduli-of-curves

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