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

Abstract

AbstractWe study the quantum dynamics of a system ofnAbelian$$ \mathcal{N} $$N= 1 vector multiplets coupled to$$ \frac{1}{2}n\left(n+1\right) $$12nn+1chiral 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$$ \mathcal{N} $$N= 1 results are generalised to the case of$$ \mathcal{N} $$N= 2 local supersymmetry: a system ofnAbelian$$ \mathcal{N} $$N= 2 vector multiplets coupled to$$ \mathcal{N} $$N= 2 chiral multipletsXIparametrising Sp(2n, ℝ)/U(n). The induced action is shown to be proportional to$$ \int {\textrm{d}}^4x{\textrm{d}}^4\theta {\textrm{d}}^4\overline{\theta}E\mathfrak{K}\left(X,\overline{X}\right) $$∫d4xd4θd4θ¯EKXX¯, where$$ \mathfrak{K}\left(X,\overline{X}\right) $$KXX¯is the Kähler potential for Sp(2n, ℝ)/U(n). We also apply our method to compute DeWitt’sa2coefficients in some non-supersymmetric theories with non-minimal operators.