Phycisists working on QFT care about representations of certain Lie groups (e.g. SU(2n), SO(2n+1), E8, etc.) as they describe physical symmetries. It turns out that fully understanding the representation theory of these groups is a horrifyingly difficult task: in fact, as most of these fall under the class of so-called (quasi-)split reductive groups, they are objects of interest for people in the Langlands Programme, which is a very large research programme within pure mathematics.

There are also many objects in the theory of Lie algebras, such as Kac-Moody algebras, Yangians,the Heisenberg algebra, or vertex operator algebras, which originated from QFT. Mathematicians care about these things because they exhibit interesting combinatorial behaviours, and also because they too pertain to the Langlands Programme (although a bit more tangentially, via what's known as the Quantum Local Langlands Programme).

People working in PDEs obviously care about the many equations coming from physics.

Algebraic geometers also care a bit about QFT stuff, thanks to the emergence of topics like mirror symmetry and algebro-geometric objects such as Calabi-Yau manifolds.