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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?

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