Two-loop corrections to partition function of Pohlmeyer-reduced theory forAdS5×S5superstring

Abstract

Pohlmeyer reduction of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ superstring, involving the solution of Virasoro conditions in terms of coset current variables, leads to a set of equations of motion following from an action containing a bosonic $Sp(2,2)\ifmmode\times\else\texttimes\fi{}Sp(4)/[SU(2){]}^{4}$ gauged Wess-Zumino-Witten (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 ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ 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 $\frac{\sqrt{\ensuremath{\lambda}}}{2\ensuremath{\pi}}$. Its quantum part is $f(\ensuremath{\lambda})={a}_{1}+\frac{1}{\sqrt{\ensuremath{\lambda}}}{a}_{2}+\dots{}$, where the one-loop term is ${a}_{1}=\ensuremath{-}3\mathrm{ln}2$ and the two-loop term is the negative of the Catalan's constant, ${a}_{2}=\ensuremath{-}\mathrm{K}$. We find that the counterpart of $f(\ensuremath{\lambda})$ in the reduced theory is $f(k)={\mathrm{a}}_{1}+\frac{2}{k}{\mathrm{a}}_{2}+\dots{}$, where $k$ is the coupling of the reduced theory. Here the one-loop coefficient is the same as in string theory, ${\mathrm{a}}_{1}={a}_{1}$, while the two-loop one is ${\mathrm{a}}_{2}={a}_{2}\ensuremath{-}\frac{1}{4}({a}_{1}{)}^{2}$. Remarkably, the first Catalan's constant term here matches the string-theory result if we identify the two couplings as $k=2\sqrt{\ensuremath{\lambda}}$. Nevertheless, the presence of the additional $({a}_{1}{)}^{2}\ensuremath{\sim}(\mathrm{ln}2{)}^{2}$ term suggests that the relation between the two quantum partition functions (if any) is not a simple equality. Similar results are found in the case of the ${\mathrm{AdS}}_{3}\ifmmode\times\else\texttimes\fi{}{S}^{3}$ string theory where ${a}_{1}=\ensuremath{-}2\mathrm{ln}2$ and ${a}_{2}=0$, while in the corresponding reduced theory ${\mathrm{a}}_{1}={a}_{1}$ and ${\mathrm{a}}_{2}={a}_{2}\ensuremath{-}\frac{1}{4}({a}_{1}{)}^{2}$.