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 $\mathbb{F}_2$ 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 towards extending these spectral 2-representations to integer coefficients, we also work in the $\mathfrak{gl}_2$ setting and lift the Blanchet-Khovanov algebra to a multifunctor into a multicategory version of Sarkar-Scaduto-Stoffregen's signed Burnside category.