Michael Atiyah and Isadore Singer associated two differently defined numbers to an elliptic operator on a manifold: the topological index and the analytical index. The index theorem asserts that the two are equal.
The original articles are
Michael Atiyah, Isadore Singer, The index of elliptic operators I, Ann. of Math. (2) 87 (1968) pp. 484–530; III, Ann. of Math. (2) 87 (1968) pp. 546–604; IV, Ann. of Math. (2) 93 (1971) pp. 119–138; V, Ann. of Math. (2) 93 (1971) pp. 139–149
A lighning review of the proof is on the last pages of
Traditional physics arguments along these lines include for instance