<p>We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-D fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-D topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on 4-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto–Kapustin and Tata–Kobayashi–Bulmash–Barkeshli, together with a “bosonization conjecture” which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5344_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="94" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}/4^T\times \mathbb {Z}/2^f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <msup> <mn>4</mn> <mi>T</mi> </msup> <mo>×</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <msup> <mn>2</mn> <mi>f</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> time-reversal symmetry with symmetry algebra <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5344_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {T}}^2 = (-1)^FC$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="script">T</mi> </mrow> <mn>2</mn> </msup> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>F</mi> </msup> <mi>C</mi> </mrow> </math></EquationSource> </InlineEquation>, which many fermionic topological orders enjoy, including the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5344_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{U}(1)_5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>U</mtext> <msub> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>5</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> spin Chern–Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5344_Article_IEq4.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {Z}/4^{Tf}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">/</mo> <msup> <mn>4</mn> <mrow> <mi mathvariant="italic">Tf</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> time-reversal symmetry with symmetry algebra <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5344_Article_IEq5.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal T^2 = (-1)^F$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="script">T</mi> <mn>2</mn> </msup> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>F</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, and other symmetry groups in the 10-fold way involving Lie group symmetries.</p>

Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders

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2025年7月2日

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Arun Debray,
Weicheng Ye,
Matthew Yu

摘要

We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-D fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-D topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on 4-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto–Kapustin and Tata–Kobayashi–Bulmash–Barkeshli, together with a “bosonization conjecture” which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is $\mathbb {Z}/4^T\times \mathbb {Z}/2^f$ $Z / 4^{T} \times Z / 2^{f}$ time-reversal symmetry with symmetry algebra ${\mathcal {T}}^2 = (-1)^FC$ $T^{2} = {(- 1)}^{F} C$ , which many fermionic topological orders enjoy, including the $\textrm{U}(1)_5$ $U {(1)}_{5}$ spin Chern–Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including $\mathbb {Z}/4^{Tf}$ $Z / 4^{Tf}$ time-reversal symmetry with symmetry algebra $\mathcal T^2 = (-1)^F$ $T^{2} = {(- 1)}^{F}$ , and other symmetry groups in the 10-fold way involving Lie group symmetries.