Harvard Mathematical Picture Language Seminar: 4D Hex and Baby Yang-Mills

The 2D O(n) model is a very close analog of 4D lattice Yang-Mills. The quest to understand the former (mass gap, scaling limit, etc.) has been colloquially called the “baby Yang-Mills problem” or the “half million-dollar-problem.” Now, several recent OpenAI papers on the 2D O(n) model (one of which comes with Lean code) claim to have solved much of this problem. But how exactly is this relevant to 4D Yang-Mills.

I will review and explain how the 2D model derives special properties from the identity 1+1=2 while the 4D model derives special properties from the identity 2+2=4. The 2D model is related to random loop ensembles, while the 4D model is related to random surface ensembles. I will discuss the new proofs and the challenges involved in constructing analogous proofs in 4D. As a warmup, try playing a few games of 2D Hex (players compete to make spanning paths on a 2D board) and 4D Hex (players compete to make spanning surfaces on a 4D board) and think about what makes the second game harder to analyze.

Recorded seminars can be viewed on the Mathematical Picture Language YouTube channel.