Antigonus is a new world mission in the new Enkanomiya area in Genshin Impact. Enkanomiya is a new location introduced in version 2.4 and brings tons of new world missions like this to the table. Make sure you switch to Evernight before starting this mission.
The search for the world is found here. You will have to slide a little over the edge of the cliff to see who gave the mission. It's hard to see, so you may need to glide a couple of times.
Starting the missions will have you talking to the two ghosts pictured above. After you are done here, you will be sent to another location to find an NPC named Antei. Antei is located in the area below. If you have trouble finding it, you can use the teleport point, then walk north. You will see it in a crack below.
Antei will let you find the core of a Ruin Guard, then you will find an abandoned Ruin Guard. Just follow the indicated locations. You will have to defeat this Ruin Guard. After it is destroyed, you can then collect the book "Before the sun and the moon". (This book is also required for another world mission, Collection of Dragons and Snakes.)
Return to Antei and deliver the book. This is the conclusion of Antigonus' research. You can now turn in the Before the Sun and the Moon book for the Dragon and Snake Collection worldwide mission. You will receive 40 Primogems and some other rewards for your troubles.