Classification of Finite Depth Objects in Bicommutant Categories via Anchored Planar Algebras

Abstract In our article [ arxiv:1511.05226 ], we studied the commutant $$\mathcal {C}'\subset \operatorname {Bim}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>⊂</mml:mo> <mml:mo>Bim</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of a unitary fusion category $$\mathcal {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math> , where R is a hyperfinite factor of type $$\mathrm II_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> , $$\mathrm II_\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:mrow> </mml:math> , or $$\mathrm III_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>I</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> </mml:math> , and showed that it is a bicommutant category. In other recent work [ arxiv:1607.06041 , arxiv:2301.11114 ] we introduced the notion of a (unitary) anchored planar algebra in a (unitary) braided pivotal category $$\mathcal {D}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> , and showed that they classify (unitary) module tensor categories for $$\mathcal {D}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> equipped with a distinguished object. Here, we connect these two notions and show that finite depth objects of $$\mathcal {C}'$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> </mml:math> are classified by connected finite depth unitary anchored planar algebras in $$\mathcal {Z}(\mathcal {C})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . This extends the classification of finite depth objects of $$\operatorname {Bim}(R)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>Bim</mml:mo> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> by connected finite depth unitary planar algebras.