Bounds on field range for slowly varying positive potentials
Damian van de Heisteeg, Cumrun Vafa, Max Wiesner, David H. Wu
Abstract
A bstract In the context of quantum gravitational systems, we place bounds on regions in field space with slowly varying positive potentials. Using the fact that $$ V<{\Lambda}_s^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> <mml:mo><</mml:mo> <mml:msubsup> <mml:mi>Λ</mml:mi> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> , where Λ s ( ϕ ) is the species scale, and the emergent string conjecture, we show this places a bound on the maximum diameter of such regions in field space: ∆ ϕ ≤ a log(1/ V ) + b in Planck units, where a ≤ $$ \sqrt{\left(d-1\right)\left(d-2\right)} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mrow> <mml:mfenced> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfenced> <mml:mfenced> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:msqrt> </mml:math> , and b is an 𝒪(1) number and expected to be negative. The coefficient of the logarithmic term has previously been derived using TCC, providing further confirmation. For type II string flux compactifications on Calabi-Yau threefolds, using the recent results on the moduli dependence of the species scale, we can check the above relation and determine the constant b , which we verify is 𝒪(1) and negative in all the examples we studied.