# Relation between the fundamental group and projective representations of a Lie group

For concreteness let's focus on $$SO(3)$$. A projective representation is such that, $$U(g_1) U(g_2) = e^{i\theta(g_1, g_2)} U(g_1g_2), \ \ g_1, g_2 \in SO(3).$$ What I'm struggling to fully appreciate is the relation between the phase factor $$\theta(g_1, g_2)$$ and the fundamental group $$\pi_1(SO(3)) = \mathbb Z_2$$. Reading this answer here, there seems to be a connection between $$\pi_1(SO(3)) = \mathbb Z_2$$ and the kind of projective representations $$SO(3)$$ can have. I'm not asking how to construct $$\pi_1(SO(3))$$, but rather how knowing $$\pi_1(SO(3))$$ we can know something about $$\theta(g_1, g_2)$$. The relationship seems to suggest the following, which are also my questions about this:

1. Can we write $$\theta(g_1, g_2)$$ as the integration of some quantity over a line connecting $$g_1$$, and $$g_2$$? Or more formally $$\theta(g_1, g_2) = {\int_{\gamma} A}$$, such that $$\gamma$$ is a path connecting $$g_1$$, and $$g_2$$.
2. If the above is true, then we would need to show that the integral is invariant under a smooth change of $$\gamma$$, or that $$\text{curl} \ A = 0$$. How to show this?

I have seen arguments about how projective representations of $$SO(3)$$ has to correspond to linear representations of the universal cover $$SU(2)$$, a statement that I'm also not sure how to prove.

• That's a question in algebraic/differential topology better suited for the Math.SE. Commented May 5, 2021 at 14:40
• I was fearing that if I posted in Math.SE I'll get a super abstract answer that I would not be able to follow. I was hoping for an answer that someone with a physics background (and not very hardcore on the math) could understand (if any exists). Commented May 5, 2021 at 14:50
• It's more a cohomology issue than a homotopy issue. Commented May 5, 2021 at 15:03
• Since it's a self-answered Q&A of mine, I don't want to mod-close this question as duplicate, but physics.stackexchange.com/q/203944/50583 has probably everything you need to know. Commented May 5, 2021 at 15:37
• I think the link with the fundamental group (as in the post linked by OP) is still to be made, so it's not a duplicate Commented May 5, 2021 at 15:48