# Axion RG flows and the holographic dynamics of instanton densities

## TL;DR

This paper explores axion-driven holographic RG flows in Einstein-Axion-Dilaton theories, revealing constraints on IR fixed points, the importance of regularity conditions, and the finite range of axion sources, with implications for axion swampland conjectures.

## Contribution

It provides analytical solutions and a systematic expansion for axion holographic flows, establishing the connection between IR regularity, axion source bounds, and the landscape of saddle points.

## Key findings

- Non-trivial axion profiles exclude IR conformal fixed points.
- Regularity conditions impose finite bounds on UV axion sources.
- Numerical examples confirm the finite number of saddle points.

## Abstract

Axionic holographic RG flow solutions are studied in the context of general Einstein-Axion-Dilaton theories. A non-trivial axion profile is dual to the (non-perturbative) running of the $\theta$-term for the corresponding instanton density operator. It is shown that a non-trivial axion solution is incompatible with a non-trivial (holographic) IR conformal fixed point. Imposing a suitable axion regularity condition allows to select the IR geometry in a unique way. The solutions are found analytically in the asymptotic UV and IR regimes, and it is shown that in those regimes the axion backreaction is always negligible. The axion backreaction may become important in the intermediate region of the bulk. To make contact with the axion probe limit solutions, a systematic expansion of the solution is developed. Several concrete examples are worked out numerically. It is shown that the regularity condition always implies a finite allowed range for the axion source parameter in the UV. This translates into the existence of a finite (but large) number of saddle-points in the large $N_c$ limit. This ties in well with axion-swampland conjectures.