Non-invertible anyon condensation and level-rank dualities

Abstract

<jats:p> We derive new dualities of topological quantum field theories in three spacetime dimensions that generalize the familiar level-rank dualities of Chern-Simons gauge theories. The key ingredient in these dualities is non-Abelian anyon condensation, which is a gauging operation for topological lines with non-group-like i.e. non-invertible fusion rules. We find that, generically, dualities involve such non-invertible anyon condensation and that this unifies a variety of exceptional phenomena in topological field theories and their associated boundary rational conformal field theories, including conformal embeddings, and Maverick cosets (those where standard algorithms for constructing a coset model fail.) We illustrate our discussion in a variety of isolated examples as well as new infinite series of dualities involving non-Abelian anyon condensation including: i) a new description of the parafermion theory as <jats:inline-formula> <jats:alternatives> <jats:tex-math>(SU(N)_{2} × Spin(N)_{-4})/\mathcal{A}_{N},</jats:tex-math> <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>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mrow> <mml:mi>−</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:msub> <mml:mi mathvariant="script">𝒜</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> ii) a new presentation of a series of points on the orbifold branch of <jats:inline-formula> <jats:alternatives> <jats:tex-math>c=1</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> conformal field theories as <jats:inline-formula> <jats:alternatives> <jats:tex-math>(Spin(2N)_{2} × Spin(N)_{-2} × Spin(N)_{-2})/\mathcal{B}_{N}</jats:tex-math> <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>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mrow> <mml:mi>−</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mrow> <mml:mi>−</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:msub> <mml:mi mathvariant="script">ℬ</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , and iii) a new dual form of <jats:inline-formula> <jats:alternatives> <jats:tex-math>SU(2)_{N}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> as <jats:inline-formula> <jats:alternatives> <jats:tex-math>(USp(2N)_{1} × SO(N)_{-4})/\mathcal{C}_{N}</jats:tex-math> <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>U</mml:mi> <mml:mi>S</mml:mi> <mml:mi>p</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mrow> <mml:mi>−</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:msub> <mml:mi mathvariant="script">𝒞</mml:mi> <mml:mi>N</mml:mi> </mml:msub> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> arising from conformal embeddings, where <jats:inline-formula> <jats:alternatives> <jats:tex-math>\mathcal{A}_{N}, \mathcal{B}_{N},</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub>