Topological field theory (TFT) is a major client of higher-dimensional category theory. For $(\infty,
2)$-categories specifically, this specializes to two-dimensional TFT. One significant research area in this field
is taking physics ideas, making them mathematically rigorous, and then using the resulting mathematical objects to
prove interesting theorems internal to mathematics.
For example, string theorists originally predicted a form of duality now known as “mirror symmetry,”
coming out of a more general physics phenomenon called T-duality. The context is that given $X$ a Calabi-Yau
$3$-fold, physicists studied a two-dimensional supersymmetric quantum field theory called "the sigma model with
target $X$" and found that it admits two twists, called the A and B models, which are topological field theories
(in the physicists' sense). Mirror symmetry predicts that given $X$, there is another CY3 called the mirror
of $X$ and denoted $X^\vee$, such that the A model with target $X$ is equivalent to the B model with target
$X^\vee$, and vice versa.
Now, because these are topological field theories, it should be possible to make sense of them mathematically:
the A and B model should admit descriptions as symmetric monoidal functors from an $(\infty, 2)$-category of
bordisms in dimension 2 to some target $(\infty, 2)$-category. This has amounted to a lot of hard work by many
teams of researchers, but large parts of this story have been translated into mathematics, and the
higher-categorical language is important here!