Generalized and non-invertible symmetries

4 problems

• When do non-invertible symmetries forbid a featureless gapped phase?

  Partially resolved

  Some symmetries force a theory to have massless particles or a degenerate vacuum. A test telling which non-invertible symmetries do this exists only in two dimensions.

• Sharp distinction between Higgs and confining phases with fundamental matter

  Contested

  With quarks present, the usual test for confinement fails, and the confining phase and the Higgs phase (where a condensed field gives the force carriers mass) seem smoothly connected. Whether some generalized symmetry or order parameter still separates them is unknown.

• Observable consequences of non-invertible symmetries in the Standard Model

  Open

  Some symmetries of the Standard Model, thought to be destroyed by quantum effects, survive in a form with no undo. Which measurable predictions follow from them is unclear.

• Mathematical structure of finite generalized symmetries above two dimensions

  Partially resolved

  In two dimensions all finite symmetries, including those without an undo, are described by a known algebraic structure. In higher dimensions the right structure is conjectured but not established.