Non-Supersymmetric Vacua and Self-Adjoint Extensions

TL;DR

This paper studies how boundary conditions at singularities in string compactifications affect the stability and spectrum of lower-dimensional vacua, revealing that specific self-adjoint extensions can ensure stability and determine the presence of massless modes.

Contribution

It characterizes the role of self-adjoint extensions of Schrödinger operators in non-supersymmetric string vacua and applies this to orientifold models with the tadpole potential.

Findings

Identifies stable boundary conditions with a massless graviton in nine dimensions.

Finds a spectrum of massive and massless modes depending on boundary choices.

Demonstrates the importance of self-adjoint extensions for stability analysis.

Abstract

Internal intervals spanned by finite ranges of a conformal coordinate $z$ and terminating at a pair of singularities are a common feature of many string compactifications with broken supersymmetry. The squared masses emerging in lower-dimensional Minkowski spaces are then eigenvalues of Schr\"odinger-like operators, whose potentials have double poles at the ends of the intervals. For one-component systems, the possible self-adjoint extensions of Schr\"odinger operators are described by points in $AdS_3 \times S^1$, and those corresponding to independent boundary conditions at the ends of the intervals by points on the boundary of $AdS_3$. The perturbative stability of compactifications to Minkowski space time depends, in general, on these choices of self-adjoint extensions. We apply this setup to the orientifold vacua driven by the ``tadpole potential'' $V=T \ e^{\,\frac{3}{2}\,\phi}$ and find, in nine dimensions, a massive scalar spectrum, a unique choice of boundary conditions with stable tensor modes and a massless graviton, and a wide range of choices leading to massless and/or massive vector modes.