Single versus multifield scalar potentials from string theory

TL;DR

This paper argues that string theory effective theories with scalar fields are typically multifield, with at least two non-compact directions, making single-field intuitions for the potential fragile in cosmological contexts. By distinguishing off-shell multifield analyses from on-shell trajectories, the authors show that in $V>0$ settings there exist controlled string-theory examples where an asymptotic direction yields $\frac{|\partial_{\hat{\varphi}} V|}{V} \to 0$, implying no universal lower bound on the single-field slope and highlighting the necessity of using $\nabla V$ as a robust diagnostic. They provide explicit off-shell multifield realizations across group-manifold, Calabi–Yau, Landau–Ginzburg, and DGKT compactifications, demonstrating how an asymptotically flat direction can emerge while keeping $V$ positive and corrections under control. The analysis also discusses negative potentials and on-shell considerations, arguing that gradient-based conditions like SdSC remain more reliable off-shell and that on-shell dynamics can, under gradient-flow assumptions, reduce to an effectively single-field description. Together, these results sharpen the connection between swampland constraints and multifield cosmology, with implications for inflation, quintessence, and the viability of de Sitter constructions in string theory.

Abstract

In this work, we investigate the properties of string effective theories with scalar field(s) and a scalar potential. We first claim that in most examples known, such theories are multifield, with at least 2 non-compact field directions; the few counter-examples appear to be very specific and isolated. Such a systematic multifield situation has important implications for cosmology. Characterising properties of the scalar potential $V$ is also more delicate in a multifield setting. We provide several examples of string effective theories with $V>0$, where the latter admits an asymptotically flat direction along an off-shell field trajectory: in other words, there exists a limit $\varphi \rightarrow \infty$ for which $\frac{|\partial_{\varphi} V|}{V} \rightarrow 0$. It is thus meaningless to look for a lower bound to this single field quantity in a multifield setting; the complete gradient $\nabla V$ is then better suited. Restricting to on-shell trajectories, this question remains open, especially when following the steepest descent or more generally a gradient flow evolution. Interestingly, single field statements in multifield theories seem less problematic for $V<0$.

Figures (8)

• Figure 1: Potentials $V(\hat{\lambda},\hat{\lambda}_{\bot})$ allowing for an off-shell trajectory $(\hat{\lambda},\hat{\lambda}_{\bot}=0)$, in red, such that the single slope ratio $\frac{|\partial_{\hat{\lambda}} V|}{V} \rightarrow 0$ at $\hat{\lambda}\rightarrow \infty$. The general form of these potentials will be discussed in Section \ref{['sec:finiteslope']}, and string theory realisations will be provided in Section \ref{['sec:examples']}.
• Figure 2: Scalar potential $V$ obtained with the de Sitter solution $s_{6666}^+ 4$, setting all axions and saxions to their critical point value except for one saxion: this gives the potential profile along that saxionic direction. The profile for $g_{11}$ and $g_{22}, g_{33}$ are very similar, those for $g_{55}$ and $g_{66}$ as well.
• Figure 3: $V(\phi,g_{11})$ (the other fields being set to their critical point value), with the curve $g_{11} = e^{-4\phi}$ (in red), which provides an asymptotically flat direction.
• Figure 4: $V(\hat{\tau}_{I}\,,\hat{U}_{I})$ for $f_1=f^0=h_0=1$, and the trajectory that goes asymptotically flat in red.
• Figure 5: $V(\hat{\tau}_{I}\,,\hat{U}_{I})$ with $f^1=h_0=1$, and the trajectory that goes asymptotically flat in red, for which we make the stabilising shift $\hat{\lambda}_{\perp} \rightarrow \hat{\lambda}_{\perp} +\frac{1}{2\sqrt{2}}\log{\frac{16}{h_{0}^2}}$.