Dualities between 2+1d fusion surface models from braided fusion categories

Fusion surface models generalize the concept of anyon chains to 2+1 dimensions, utilizing fusion 2-categories as their input. We investigate bond-algebraic dualities in these systems and show that distinct module tensor categories \mathcal{M} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>ℳ</mml:mi> </mml:mstyle> </mml:math> over the same braided fusion category \mathcal{B} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle mathvariant="script"> <mml:mi>ℬ</mml:mi> </mml:mstyle> </mml:math> give rise to dual lattice models. This extends the 1+1d result that dualities in anyon chains are classified by module categories over fusion categories. We analyze two concrete examples: (i) a \text{Rep}(S_3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mtext mathvariant="normal">Rep</mml:mtext> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> model with a constrained Hilbert space, dual to the spin- \tfrac{1}{2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:math> XXZ model on the honeycomb lattice, and (ii) a bilayer Kitaev honeycomb model, dual to a spin- \tfrac{1}{2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:math> model with XXZ and Ising interactions. Unlike regular \mathcal{M}=\mathcal{B} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>ℳ</mml:mi> </mml:mstyle> <mml:mo>=</mml:mo> <mml:mstyle mathvariant="script"> <mml:mi>ℬ</mml:mi> </mml:mstyle> </mml:mrow> </mml:math> fusion surface models, which conserve only 1-form symmetries, models constructed from \mathcal{M} ≠ \mathcal{B} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mstyle mathvariant="script"> <mml:mi>ℳ</mml:mi> </mml:mstyle> <mml:mo>≠</mml:mo> <mml:mstyle mathvariant="script"> <mml:mi>ℬ</mml:mi> </mml:mstyle> </mml:mrow> </mml:math> can exhibit both 1-form and 0-form symmetries, including non-invertible ones.