  Pohlmeyer reduction of AdS_5 x S^5 superstring leads to a set of equations of
motion following from an action containing a bosonic Sp(2,2) x Sp(4)/[SU(2)]^4
gauged WZW term, an integrable potential and a fermionic part coupling bosons
from the two factors. The original superstring and the reduced model are in
direct correspondence at the classical level but their relation at the quantum
level remains an open question. As was found earlier, the one-loop partition
functions of the two theories computed on the respective classical backgrounds
match; here we explore the fate of this relation at the two-loop level. We
consider the example of the reduced theory solution corresponding to the long
folded spinning string in AdS. The logarithm of the AdS_5 x S^5 superstring
partition function computed on the spinning string background is known to be
proportional to the universal scaling function which depends on the string
tension ~ \l^{1/2} where \l=\lambda is `t Hooft coupling. Its "quantum" part is
f(\l) = a1 + \l^{-1/2} a2 + ... where the one-loop term is a1 = - 3 \ln 2 and
the two-loop term is minus the Catalan's constant, a2=-K. We find that the
counterpart of f(\l) in the reduced theory is f'(k) =a1' + 2 k^{-1} a2' + ...,
where k is the coupling of the reduced theory. Here the one-loop coefficient is
the same as in the string theory, a1'= a1, while the two-loop one is a2' =a2 -
1/4 (a1)^2. Remarkably, the first Catalan's constant term here matches the
string theory result if we identify the two couplings as k= 2\l^{1/2}.
Nevertheless, the presence of the additional (a1)^2 ~ (\ln 2)^2 term implies
that a relation between the two quantum partition functions (if any) is not a
simple equality. Similar results are found in the case of AdS_3 x S^3
superstring theory where a1= - 2 \ln 2 and a2=0, while in the corresponding
reduced theory a1'=a1, a2'= a2- 1/4 (a1)^2.
