# Some comments on symmetric orbifolds of K3

## TL;DR

This paper classifies symmetries and twining genera in symmetric orbifolds of K3 superconformal field theories, revealing surprising identities between modular forms and analyzing the structure of their moduli space.

## Contribution

It provides a partial classification of symmetry groups and twining genera in these models, and discusses properties and puzzles of their moduli space.

## Key findings

- Classification of symmetry groups under certain assumptions
- Partial classification of twining genera and their identities
- Analysis of the moduli space properties and dualities

## Abstract

We consider two dimensional $\mathcal{N}=(4,4)$ superconformal field theories in the moduli space of symmetric orbifolds of K3. We complete a classification of the discrete groups of symmetries of these models, conditional to a series of assumptions and with certain restrictions. Furthermore, we provide a partial classification of the set of twining genera, encoding the action of a discrete symmetry $g$ on a space of supersymmetric states in these models. These results suggest the existence of a number of surprising identities between seemingly different Borcherds products, representing Siegel modular forms of degree two and level $N>1$. We also provide a critical review of various properties of the moduli space of these superconformal field theories, including the groups of dualities, the set of singular models and the locus of symmetric orbifold points, and describe some puzzles related to our (lack of) understanding of these properties.