Bound on scalar potential slopes in asymptotic field space
In plain words
A conjecture says that wherever string theory is weakly coupled, the vacuum energy must slope steeply downhill. Whether this holds, and how steep the slope must be, is unknown.
Precise statement
The refined de Sitter conjecture (2018) states that in asymptotic, weakly coupled regions of moduli space, $\mid \operatorname{grad} V \mid \ge c V/M_{P}$ or $\mathrm{min}\ \mathrm{eigenvalue}(\operatorname{grad}\ \operatorname{grad} V) \le -c' V/M_P^{2}$ with c, c' of order 1. Determine whether this holds in all asymptotic regions of string moduli spaces and find the optimal value of c in d dimensions.
What would settle it
A proof over all asymptotic limits of string compactifications with the sharp constant, or an explicit asymptotic counterexample.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Candidate sharp values are $c = 2/\sqrt{(d-1)(d-2)}$ from the trans-Planckian censorship conjecture (Bedroya, Vafa 2019) and $c = 2/\sqrt{d-2}$ (Rudelius 2021), tested in many but not all asymptotic limits (2021-2025).
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