In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.

What is G'?

I know there is concrete description in terms of pairs (M,f) with M in G and f a function (see Lemma 2.14 of Huybrechts http://arxiv.org/pdf/1111.1745v2.pdf). But I'd like to know if G' is diffeomorphic to some other manifold.

At some point I convinced myself that G was actually diffeomorphic to $\mathbb C^* \times \mathbb C$ and therefore $G'$ was $\mathbb C^2$. But I've never seen this written down, so I'm getting suspicious.

EDIT: For those who have voted to close: I realize this question was very elementary. However, none of the resources on stability conditions I know of state this fact explicitly, which is quite frustrating. I think it would be very useful for people looking for a quick answer to have a place where this fact appears unambiguously.

ispossible that someone might be happy about it some day. $\endgroup$ – Helene Sigloch Apr 29 '16 at 12:31preciselythe point of this site: to allow people to put questions far from their expertise in front of those who have such expertise. $\endgroup$ – Charles Rezk Apr 29 '16 at 14:02