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 $\frac{1}{2}n(n+1)$ chiral multiplets which parametrise the Hermitian symmetric space $\mathsf{Sp}(2n,\mathbb{R})/\mathsf{U}(n)$. In the presence of supergravity, this model is super-Weyl invariant and possesses the maximal non-compact duality group $\mathsf{Sp}(2n,\mathbb{R})$ 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 $\mathsf{Sp}(2n,\mathbb{R})/\mathsf{U}(n)$. 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 $X^I$ parametrising $\mathsf{Sp}(2n,\mathbb{R})/\mathsf{U}(n)$. The induced action is shown to be proportional to $\int {\mathrm{d}}^{4}x{\mathrm{d}}^4\theta {\mathrm{d}}^4\bar{\theta }E\mathfrak{K}(X,\bar{X})$, where $\mathfrak{K}(X,\bar{X})$ is the Kähler potential for $\mathsf{Sp}(2n,\mathbb{R})/\mathsf{U}(n)$. We also apply our method to compute DeWitt's $a_2$ coefficients in some non-supersymmetric theories with non-minimal operators.

38 pages

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Cited by in corpus (4)

• Schwinger--DeWitt expansion for the heat kernel of nonminimal operators in causal theories
• Commutator technique for the heat kernel of minimal higher derivative operators
• Heat kernel of non-minimal second-order operators
• Heat kernel approach to the one-loop effective action for nonlinear electrodynamics