The $\mathcal{N} = 2$ Prepotential and the Sphere Free Energy

TL;DR

This paper explores the relationship between the sphere free energy of 3D $ =2$ superconformal theories and the prepotential of their 4D holographic duals, providing explicit verifications in supergravity models.

Contribution

It proposes a conjecture linking boundary free energy to bulk prepotential and verifies it through explicit calculations in various supergravity examples.

Findings

Confirmed the proportionality between sphere free energy and prepotential in multiple models

Derived explicit formulas connecting boundary and bulk quantities

Supported the conjecture with consistent supergravity computations

Abstract

We study the mass-deformed sphere free energy of three-dimensional $\mathcal{N} = 2$ superconformal field theories with holographic duals. Building on previous observations, we conjecture a proportionality relation between the sphere free energy on the boundary and the prepotential of the four-dimensional $\mathcal{N} = 2$ supergravity theory in the bulk. We verify this formula by explicit computation in several examples of supergravity theories with vector multiplets and hypermultiplets.