Spectral 2-actions, foams, and frames in the spectrification of Khovanov arc algebras

Leveraging skew Howe duality, we show that Lawson–Lipshitz–Sarkar’s spectrification of Khovanov’s arc algebra gives rise to 2-representations of categorified quantum groups over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper F 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">F</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbb {F}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that we call spectral 2-representations. These spectral 2-representations take values in the homotopy category of spectral bimodules over spectral categories. We view this as a step toward a higher representation theoretic interpretation of spectral enhancements in link homology. A technical innovation in our work is a streamlined approach to spectrifying arc algebras, using a set of canonical cobordisms that we call frames, that may be of independent interest. As a step toward extending these spectral 2-representations to integer coefficients, we also work in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g German l Subscript 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak {gl}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> setting and lift the Blanchet–Khovanov algebra to a multifunctor into a multicategory version of Sarkar–Scaduto–Stoffregen’s signed Burnside category.