# Difference between $\tilde{\textrm{Diff}}_+(S^1)$ and ${\textrm{Diff}}_+(S^1)$

In this paper, where Liouville theory is being studied on a strip, after equation 2.3 it is mentioned that the conformal transformations of the strip are given by the same chiral and anti-chiral coordinate transformations, $$x\to\xi(x),\ \bar x\to\xi(\bar x)$$. But the associated group is identified as $$\tilde{\textrm{Diff}}_+(S^1)$$ which is the cover of $${\textrm{Diff}}_+(S^1)$$.

Does the subscript '$$+$$' denote orientation preserving transformations?

What is the difference between the two groups and what conformal transformations are included in the covering group but not the original group?