One framework for Hastings-Haah floquet codes is via loops in the space of gapped Hamiltonians for the toric code (arxiv 2203.11137). This raised an interesting question about the homotopy type of moduli spaces of topological theories. I will discuss some progress towards a conjectured picture of the homotopy type of such moduli spaces in a joint work with Po-shen Hsin (arxiv 2211.16535).
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