Abelian matrix models in two loops

We perform a two-loop calculation of the effective Lagrangian for the low--energy modes of the quantum mechanical system obtained by dimensional reduction from 4D, N = 1 supersymmetric QED. The bosonic part of the Lagrangian describes the motion over moduli space of vector potentials A_i endowed with a nontrivial conformally flat metric. We determined the coefficient of the two-loop correction to the metric, which is proportional to 1/A^6. For the matrix model obtained from Abelian 4D, N = 2 theory, this correction vanishes, as it should.