Exact Path Integral Methods in Supersymmetric $\text{AdS}_2\times \mathbf{S}^2$ Backgrounds

Abstract

We determine the exact functional determinants of charged, massive spin-0 and spin-$\frac12$ particles in $\text{AdS}_2\times \mathbf{S}^2$ backgrounds threaded by constant electric and magnetic fields. This is achieved using Schwinger proper-time formalism, which allows us to derive the full non-perturbative effective action in the 1-loop and constant background field approximations. We then specialize the computation to supersymmetric settings and we obtain the effective action for a 4d $\mathcal{N}=2$ BPS massive hypermultiplet in a supersymmetric $\text{AdS}_2\times \mathbf{S}^2$ spacetime. This setup can be seen to be equivalent to the near-horizon geometry of a BPS black hole which solves the attractor equations of 4d $\mathcal{N}=2$ supergravity. Our results provide a necessary intermediate step for the evaluation of the quantum-corrected black hole partition function. We also comment on the relation with the celebrated Gopakumar-Vafa integral representation.

Figures (9)

• Figure 1: Complex $\lambda$-plane for the variable parametrizing the continuous $\mathbb{H}^2$ principal series \ref{['eq:jppalseries']}. The dots denote the (simple) poles of the integrand in the heat kernel $\mathcal{K}^{(0)}_{\mathbb{H}^2}$ (cf. eq. \ref{['eq:heatkernelH2spin0continuous']}), whereas the crosses mark the branch points of the 1-loop amplitude \ref{['eq:logZphi']}, which occur when $(mR)^2+E_\lambda <0$. The contour $\mathcal{C}$ is chosen so as to enclose the poles responsible for the discrete series \ref{['eq:discreteSpectrumH2']}.
• Figure 2: Singularity structure of the density of (continuous) energy states for a charged spin-0 particle in $\mathbb{H}^2$ (left) and AdS$_2$ (right), when expressed as in eqs. \ref{['eq:heatkernelH2spin0continuous']} and \ref{['eq:heatkernelAdS2spin0complete']}. The $\mathbb{H}^2$ poles have been slightly separated from the imaginary axis for clarity. The red lines indicate the contour of integration $\mathcal{C}=- i (g-\frac{1}{2}+\epsilon) + \mathbb{R}$ (left) and its continuation $\mathcal{C}'= \frac{i}{2}-i\epsilon + \mathbb{R}$ (right) via \ref{['eq:H2->AdS2continuation']}.
• Figure 3: Singularity structure of the density of (continuous) energy states for a charged spin-$\frac{1}{2}$ particle in $\mathbb{H}^2$ (left) and AdS$_2$ (right), when expressed as in eqs. \ref{['eq:heatkernelH2spin12continuous']} and \ref{['eq:heatkernelAdS2spin12complete']}. The $\mathbb{H}^2$ poles have been slightly separated from the imaginary axis for clarity. The red lines indicate the contour of integration $\mathcal{C}=- i (g+\epsilon) + \mathbb{R}$ (left) as well as its analytic continuation $\mathcal{C}'= -i\epsilon + \mathbb{R}$ (right) via \ref{['eq:H2->AdS2continuation']}.
• Figure 4: Spectral density of eigenstates for a charged spin-less particle in AdS$_2$ with $ER_{\rm{A}}^2 =1$. The variable $\lambda$ parametrizes the different (continuous) $SU(1,1)$ representations and is non-negative.
• Figure 5: Spectral density of energy eigenstates for a charged spin-$\frac{1}{2}$ particle in AdS$_2$ with $ER_{\rm{A}}^2 =1$. Naïvely, the function (blue) presents a simple pole at $\lambda=e$, thus rendering the density negative definite for $\lambda <e$. However, upon taking the principal value (red) and separating the localized zero mode contribution at $\lambda=e$ (cf. discussion around eq. \ref{['eq:densitydistributionfermions']}), one effectively restores positivity in $\rho_F(\lambda)$.