Effective actions in supersymmetric gauge theories: heat kernels for non-minimal operators

Abstract

We study the quantum dynamics of a system of n Abelian mathcal{N} = 1 vector multiplets coupled to frac{1}{2}nleft(n+1right) chiral multiplets which parametrise the Hermitian symmetric space Sp(2n, ℝ)/U(n). In the presence of supergravity, this model is super-Weyl invariant and possesses the maximal non-compact duality group Sp(2n, ℝ) at the classical level. These symmetries should be respected by the logarithmically divergent term (the “induced action”) of the effective action obtained by integrating out the vector multiplets. In computing the effective action, one has to deal with non-minimal operators for which the known heat kernel techniques are not directly applicable, even in flat (super)space. In this paper we develop a method to compute the induced action in Minkowski superspace. The induced action is derived in closed form and has a simple structure. It is a higher-derivative superconformal sigma model on Sp(2n, ℝ)/U(n). The obtained mathcal{N} = 1 results are generalised to the case of mathcal{N} = 2 local supersymmetry: a system of n Abelian mathcal{N} = 2 vector multiplets coupled to mathcal{N} = 2 chiral multiplets XI parametrising Sp(2n, ℝ)/U(n). The induced action is shown to be proportional to int {textrm{d}}^4x{textrm{d}}^4theta {textrm{d}}^4overline{theta}Emathfrak{K}left(X,overline{X}right) , where mathfrak{K}left(X,overline{X}right) is the Kähler potential for Sp(2n, ℝ)/U(n). We also apply our method to compute DeWitt’s a2 coefficients in some non-supersymmetric theories with non-minimal operators.