Seminar Talk

In the quantum theory of angular momentum, the Racah--Wigner coefficient, often known as the 6j symbol, is a numerical invariant assigned to tetrahedra with half-integer edge-lengths. One important property of the 6j symbol is a set of exotic symmetries discovered by Regge in the 50s. In conjunction with the asymptotics of the 6j symbol, Regge symmetries have some profound connections beyond quantum mechanics, such as special function theory, differential equations, and even provide new insight into elementary geometry of tetrahedra. In this talk, I will introduce a generalization of 6j symbol of arithmetic flavor, dubbed tetrahedral symbol. It turns out the tetrahedral symbol enjoys a much larger symmetry group, which also prompts even more questions. The highlight among these is a surprising new connection with certain spin group, which may be explained by the philosophy of arithmetic field theory. As an interesting spin-off, we also discovered new symmetries of special values of hypergeometric functions. Joint work with Akshay Venkatesh.