basically unknown even among string theorists, in what follows we will summarize the basics of these developments, following the beautiful thesis [@15].

Automorphisms of Gauge – and Symmetry Transformations {#s3.1}
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The first step is to introduce an equivalent set of constraints called the two Virasoro constraints \[3.1\] V\_(u):=\_[S\^1]{}dx u (CD)\_[S\^1]{}dx \_ Y\^\_ Y\^\_where $u$ is a smearing function and \[3.2\] Y\^\_:=\^\_X\^ Here and in what follows a prime denotes derivation with respect to $x$ while a dot denotes derivation with respect to $t$. The advantage of (\[3.1\]) over (\[2.3\]) is that the constraint algebra simplifies to \[3.3\] {V\_(u),V\_(v)}=0, {V\_(u),V\_(v)}=V\_(s(u,v)) where $s(u,v)=u' v- u v'$. Thus the constraint algebra can be displayed as the direct sum of two algebras each of which is isomorphic to the Lie algebra of the diffeomorphism group Diff$(S^1)$ of the circle. These two algebras are, however, not the same as yet a third copy of diff$(S^1)$ generated by $D=V_+ + V_-$ which generates diffeomorphisms of the circle for all phase space functions while $V_\pm$ do that only for functions of $Y_\pm$. Hence, there are three different diffeomorphism groups of the circle at play which have to be cleanly distinguished.

The important functions $Y_\pm(f):=\int_{S^1}\;dx\; f_\mu\; Y^\mu_\pm$ 