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

Abstract

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 F 2 \mathbb {F}_2 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 g l 2 \mathfrak {gl}_2 setting and lift the Blanchet–Khovanov algebra to a multifunctor into a multicategory version of Sarkar–Scaduto–Stoffregen’s signed Burnside category.