  Spacetime Virasoro and affine Lie algebras for strings propagating in AdS3
are known to all orders in $\alpha'$. The central extension of such algebras is
a string vertex, whose expectation value can depend on the number of long
strings present in the background but should be otherwise state-independent. In
hep-th/0106004, on the other hand, a state-dependent expectation value was
found. Another puzzling feature of the theory is lack of cluster decomposition
property in certain connected correlators. This note shows that both problems
can be removed by defining the free energy of the spacetime boundary conformal
field theory as the Legendre transform of the formula proposed in the
literature. This corresponds to pass from a canonical ensemble, where the
number of fundamental strings that create the background can fluctuate, to a
microcanonical one, where it is fixed.
