Title:The 10d Uplift of the GPPZ Solution

Abstract:We present the uplift of the GPPZ solution of the five-dimensional maximal supergravity to ten dimensions. The five dimensional solution involves two real scalar fields, with one of them encoding holographically the (norm of the complex) supersymmetric ${\mathcal N}=1$ mass deformation and the other the real part of the gaugino condensate. We embed this solution in a consistent truncation of $D=5$ maximal supergravity which involves two complex scalars dual to the complex mass deformations and the complex gaugino condensate, and a $U(1)$ gauge field dual to the $U(1)_R$ current, and uplift it to ten dimensions. The ten dimensional solution is completely explicit, with all fields given in terms of elementary functions. The metric and the axion-dilaton agree with those of a partial uplift of the GPPZ flow by Pilch and Warner. We analyze the asymptotics and the singularity structure of the ten dimensional solution. The uplifted solution is singular, but the singularity is milder than that of the five dimensional solution, and there is conformal frame in which the metric is only singular at one point of $S^5$. We compare the asymptotics of the $10d$ solution with that of the Polchinski-Strassler and Freedman-Minahan solutions, and find agreement with Freedman-Minahan and disagreement with Polchinski-Strassler. In particular, we infer that while the Polchinski-Strassler $10d$ fields satisfy the correct boundary conditions, they do not solve the field equations near the boundary.