Fermions and algebraic closure
When
3 – 4 p.m., April 9, 2025
Where
<nolink> Math 402
Title: Fermions and algebraic closure
Abstract: What makes Cardano's ring C of complex numbers special? This is answered by Hilbert's "Nullstellensatz": over C, you can solve any algebraically-consistent system of polynomial equations. It turns out that Dirac's category sVec of super vector spaces is special in the same way: Deligne's "Existence of super fibre functors" can be understood as the statement that, over sVec, you can solve any algebraically-consistent system of "categorified polynomial equations". These statements have physical content: quantum mechanical operators have eigenvalues in C, not R; quantum-mechanical particles have internal state spaces in sVec, not Vec. I will describe how the sequence C, sVec, ... continues, with strings and branes in place of operators and particles. Time permitting, I will describe the \infty-categorical Galois group, which is closely related to the stable piecewise-linear group. This is joint work in progress with David Reutter.