Generalized charges, part II: Non-invertible symmetries and the symmetry TFT

Consider a d <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>d</mml:mi> </mml:math> -dimensional quantum field theory (QFT) \mathfrak{T} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="fraktur"> <mml:mi>𝔗</mml:mi> </mml:mstyle> </mml:math> , with a generalized symmetry \mathcal{S} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒮</mml:mi> </mml:mstyle> </mml:math> , which may or may not be invertible. We study the action of \mathcal{S} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒮</mml:mi> </mml:mstyle> </mml:math> on generalized or q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>q</mml:mi> </mml:math> -charges, i.e. q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>q</mml:mi> </mml:math> -dimensional operators. The main result of this paper is that q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>q</mml:mi> </mml:math> -charges are characterized in terms of the topological defects of the Symmetry Topological Field Theory (SymTFT) of \mathcal{S} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒮</mml:mi> </mml:mstyle> </mml:math> , also known as the “Sandwich Construction”. The SymTFT is a (d+1) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> -dimensional topological field theory, which encodes the symmetry \mathcal{S} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒮</mml:mi> </mml:mstyle> </mml:math> and the physical theory in terms of its boundary conditions. Our proposal applies quite generally to any finite symmetry \mathcal{S} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒮</mml:mi> </mml:mstyle> </mml:math> , including non-invertible, categorical symmetries. Mathematically, the topological defects of the SymTFT form the Drinfeld Center of the symmetry category \mathcal{S} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>𝒮</mml:mi> </mml:mstyle> </mml:math> . Applied to invertible symmetries, we recover the result of Part I of this series of papers. After providing general arguments for the identification of q <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>q</mml:mi> </mml:math> -charges with the topological defects of the SymTFT, we develop this program in detail for QFTs in 2d (for general fusion category symmetries) and 3d (for fusion 2-category symmetries).