Enriched string-net models and their excitations
David Green, Peter Huston, Kyle Kawagoe, David Penneys, Anup Poudel, Sean Sanford
Abstract
Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:math> representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:math>-enriched unitary fusion category <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi></mml:mrow></mml:math> on a 2D boundary of the 3D Walker-Wang model associated to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:math>. That article claimed that the boundary excitations were given by the enriched center/Müger centralizer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Z</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:math> in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Z</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math>.In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math>, and we construct bulk-to-boundary hopping operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">&#x2192;</mml:mo><mml:msup><mml:mi>Z</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math> reflecting how the UMTC of boundary excitations <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Z</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">X</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math> is symmetric-braided enriched in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:math>.This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>6</mml:mn><mml:mi>j</mml:mi></mml:math> symbol viewpoint.