MODULI DEPENDENCE OF STRING LOOP CORRECTIONS TO GAUGE COUPLING-CONSTANTS

We consider one-loop corrections DELTA-a to inverse gauge couplings g(a)-2 in supersymmetric vacua of the heterotic string. The form of these corrections plays an important role in scenarios for dynamical supersymmetry breaking in string theory. Specifically, we calculate the exact functional dependence of DELTA-a(U) on any untwisted modulus field U of an orbifold vacuum; it has the universal form DELTA-a(U,UBAR) = A(a) . log(\eta(U)\4 . Im U) + const., where A(a) are easily computable rational constants. The dependence is nontrivial (A(a) not-equal 0) only if some sectors of the orbifold Hilbert space have precisely N = 2 space-time supersymmetry. The expression for DELTA-a has an expected invariance under modular transformations of U, since these are symmetries of the orbifold vacuum state. However, DELTA-a is not the real part of a holomorphic function, in seeming contradiction with the existence of a supersymmetric effective lagrangian. The apparent paradox is an infrared problem, and can occur not just in string theory but in renormalizable supersymetric field theories as well. We show how the paradox is resolved in the field theory case and argue that the same resolution applies also to the string theory case.