Dark Energy in String Theory

We consider various candidates for Dark Energy, motivated by string theory. Several no-go theorems push de Sitter string vacua, with $w=-1$, to the limits of theoretical control, and all known examples depend on a delicate interplay between different string theoretic ingredients. On the other hand, runaway moduli directions are ubiquitous in string theory, and could plausibly source slow-roll quintessence. We consider various candidate supergravity potentials, motivated by string theory, including single-field K\"ahler potentials for bulk and local moduli, and leading superpotentials of the form $W = W_0 + A e^{-a \Phi}$ or $W = W_0 + A \Phi^p$. Conditions on the scalar potential imposed by supergravity are very restrictive, ruling out e.g. quintessence with $K=-n\ln(\Phi+\bar{\Phi})$ and $W = W_0+A \Phi^p$. Out of the examples considered, one can simultaneously satisfy $V>0$ and $\epsilon_V<1$ only for a deformation-like modulus with $K = k_0 + \frac{|\Phi|^{2n}}{k1}$ and a blow-up like modulus with $K=k_0 +\frac{(\Phi+\bar{\Phi})^{2n}}{k_1}$ when the leading order in the perturbative superpotential, $p$, is equal to $n$. We also review the scenario of Thermal Dark Energy, where thermal effects in a light hidden sector hold a scalar field up away from the minimum of its zero-temperature potential. This provides a viable model of Dark Energy with $w=-1$, consistent with known swampland conjectures, and motivates further early Thermal Dark Energy epochs with potentially observable consequences.