Gorsky, A.; Lysov, V. From effective actions to the background geometry. (English) Zbl 1207.81149 Nucl. Phys., B 718, No. 1-2, 293-318 (2005). Summary: We discuss how the background geometry can be traced from the one-loop effective actions in non-supersymmetric 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 \(\text{AdS}_{3}, S_{3}\) and \(T^{*}S^{3}\) geometries, depending on the type of the external field. The interpretation of the effective action in the selfdual 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. 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