What is the difference between the first Chern class and integral first Chern class?

Question

I'm reading a paper which says that the first Chern class of a manifold is $0$, but the integral first Chern class is not $0$.

What is the difference between the two? Does it have to do with taking integer coefficients?

Answer 1

This is strange terminology; does it come from a physics paper or something? I think a topologist would think of the first Chern class $c_1 \in H^2(-, \mathbb{Z})$ as being integral by default.

I would guess that whoever wrote this is considering "the first Chern class" to refer to the real first Chern class $c_1 \in H^2(-, \mathbb{R})$, which has the advantage of being definable via Chern-Weil theory in terms of the curvature of a connection. There's a natural map $H^2(-, \mathbb{Z}) \to H^2(-, \mathbb{R})$ sending the first Chern class to the real first Chern class, whose kernel is (under mild hypotheses) the torsion subgroup of $H^2(-, \mathbb{Z})$, so if the real first Chern class vanishes this means the (integral) first Chern class is torsion.