What does the $e$ operation mean in cryptography research papers?

It's the pairing function. This bilinear map which takes as Input the set $mathbb{G}timeshat{mathbb{G}}$ (in the common case, eliptic curves), and output a group element in $mathbb{G}_T$ the group target.

In the symmetric case (Type 1), because $e$ is bilinear, you can deduce that $$e(g_1, g_n)^t= e(g, g)^{x_1x_nt}$$

with $x_1,x_n$ respectively be the discrete logarithms of $g_1, g_n$ in base $g$.

For more details check: https://en.wikipedia.org/wiki/Pairing-based_cryptography