UCLA

In this manuscript, we detail three recent calculations

addressing the ultraviolet behavior of supersymmetric quantum

gravity. First, we revisit the classic calculation of the two-loop

pure gravity divergence. We argue that the $\frac{1}{\epsilon}$

divergence is regulator and duality dependent. In its place, we

propose that examining the running of the scaling parameter,

$\log \mu$, is a duality and regulator independent approach to

assessing gravity divergences in four dimensions. We explicitly

calculate the $\log \mu$ coefficient at two loops for gravity with

any particle content, explicitly verifying the divergences for pure

gravity, and the finiteness for supersymmetric gravities. Second,

we analyze fully-integrated $\NeqFour$ supergravity at one loop

using the double copy. We find that there are evanescent effects at

one loop that come directly from evanescent terms in

pure-Yang--Mills. Using this observation, we lay the groundwork for

deeper analysis of the $U(1)$ anomaly with respect to the observed

evanescent behavior. Finally, we tackle the long-standing question

of the critical dimension of $\NeqEight$ supergravity at five loops.

We construct an integrand using the generalized double copy, expand

the integrand in large loop momentum, and reduce the resulting

integrals using $\mathfrak{sl}(L)$ integration-by-parts relations.

This procedure yields a critical dimension of $d_c = 24/5$.