# Complex dimensions and their observability

## TL;DR

This paper explores how complex-valued spacetime dimensions emerge from discrete scale invariance in quantum gravity models, potentially leaving observable signatures like log oscillations in the cosmic microwave background.

## Contribution

It introduces the concept of complex dimensions arising from discrete scale symmetry and discusses their potential observable effects in cosmology.

## Key findings

- Complex dimensions can emerge in quantum gravity models with discrete scale invariance.
- Observable signatures include log oscillations in the cosmic microwave background.
- Discrete scale invariance blurs the traditional UV-IR separation in physics.

## Abstract

We show that the dimension of spacetime becomes complex-valued when its short-scale geometry is invariant under a discrete scaling symmetry. This characteristic can generically arise in quantum gravities, for instance, in those based on combinatorial or multifractal structures or as the partial breaking of continuous dilation symmetry in any conformal-invariant theory. With its infinite scale hierarchy, discrete scale invariance overlaps with the traditional separation between ultraviolet and infrared physics and it can leave an all-range observable imprint, such as a pattern of log oscillations and sharp features in the cosmic microwave background primordial power spectrum.