  We discuss how the background geometry can be traced from the one-loop
effective actions in nonsupersymmetric theories in the external abelian fields.
It is shown that upon the proper identification of the Schwinger parameter the
Heisenberg-Euler abelian effective action involves the integration over the
$AdS_3$, $S_3$ and $T^{*}S^3$ geometries, depending on the type of the external
field. The interpretation of the effective action in the sefdual field in terms
of the topological strings is found and the corresponding matrix model
description is suggested. It is shown that the low energy abelian MHV one-loop
amplitudes are expressed in terms of the type B topological string amplitudes
in mirror to $T^{*}S^3$ manifold. We also make some comments on the relation
between the imaginary part of the effective action and the branes in SU(2) as
well as on the geometry of the contours relevant for the path integral.
