Weak Separation, Positivity and Extremal Yangian Invariants

Understand

We classify all positive n-particle N^kMHV Yangian invariants in N=4 Yang-Mills theory with n=5k, which we call extremal because none exist for n>5k.

• We show that this problem is equivalent to that of enumerating plane cactus graphs with k pentagons.
• We use the known solution of that problem to provide an exact expression for the number of cyclic classes of such invariants for any k, and a simple rule for writing them down explicitly.
• As a byproduct, we provide an alternative (but equivalent) classification by showing that a product of k five-brackets with disjoint sets of indices is a positive Yangian invariant if and only if the sets are all weakly separated.