<p>In our article [<a href="http://arxiv.org/abs/1511.05226">arxiv:1511.05226</a>], we studied the commutant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}'\subset \operatorname {Bim}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mo>′</mo> </msup> <mo>⊂</mo> <mo>Bim</mo> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a unitary fusion category <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation>, where <i>R</i> is a hyperfinite factor of type <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathrm II_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">I</mi> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathrm II_\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">I</mi> <msub> <mi>I</mi> <mi>∞</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, or <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathrm III_1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">I</mi> <mi>I</mi> <msub> <mi>I</mi> <mn>1</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>, and showed that it is a bicommutant category. In other recent work [<a href="http://arxiv.org/abs/1607.06041">arxiv:1607.06041</a>, <a href="http://arxiv.org/abs/2301.11114">arxiv:2301.11114</a>] we introduced the notion of a (unitary) anchored planar algebra in a (unitary) braided pivotal category <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation>, and showed that they classify (unitary) module tensor categories for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">D</mi> </math></EquationSource> </InlineEquation> equipped with a distinguished object. Here, we connect these two notions and show that finite depth objects of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {C}'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation> are classified by connected finite depth unitary anchored planar algebras in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {Z}(\mathcal {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Z</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. This extends the classification of finite depth objects of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\operatorname {Bim}(R)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Bim</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> by connected finite depth unitary planar algebras.</p>

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

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2026年3月9日

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André Henriques,
David Penneys,
James Tener

摘要

In our article [arxiv:1511.05226], we studied the commutant $\mathcal {C}'\subset \operatorname {Bim}(R)$ $C^{'} \subset Bim (R)$ of a unitary fusion category $\mathcal {C}$ $C$ , where R is a hyperfinite factor of type $\mathrm II_1$ $I I_{1}$ , $\mathrm II_\infty $ $I I_{\infty}$ , or $\mathrm III_1$ $I I I_{1}$ , 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 $\mathcal {D}$ $D$ , and showed that they classify (unitary) module tensor categories for $\mathcal {D}$ $D$ equipped with a distinguished object. Here, we connect these two notions and show that finite depth objects of $\mathcal {C}'$ $C^{'}$ are classified by connected finite depth unitary anchored planar algebras in $\mathcal {Z}(\mathcal {C})$ $Z (C)$ . This extends the classification of finite depth objects of $\operatorname {Bim}(R)$ $Bim (R)$ by connected finite depth unitary planar algebras.