The lattice path operad and Hochschild cochains

We introduce two coloured operads in sets – the lattice path operad and a cyclic extension of it – closely related to iterated loop spaces and to universal operations on cochains. As main application we present a formal construction of an $E_2$-action (resp. framed $E_2$-action) on the Hochschild cochain complex of an associative (resp. symmetric Frobenius) algebra.

💡 Research Summary

The paper introduces two new coloured operads in the category of sets: the lattice path operad (LPO) and its cyclic extension (CLPO). Both operads are built from combinatorial data of lattice paths in an $n$‑dimensional cube, where the colour $n$ records the number of input slots. An $n$‑ary operation in LPO is a collection of lattice paths that start at the origin, end at the opposite corner, and move by unit steps in one coordinate direction at a time. Composition in LPO is defined by concatenating compatible paths and then re‑indexing the resulting steps; the construction respects the orthogonal decomposition of the cube and yields a well‑behaved operadic structure.

The cyclic version CLPO removes the distinction between inputs and output by arranging the $n$ slots on a circle. Paths are now required to respect this cyclic ordering, and composition glues paths along the circle while preserving the cyclic symmetry. This cyclic operad models framed double loop spaces $\Omega^2_{\mathrm{fr}}X$, i.e. double loop spaces equipped with a framing of the tangent bundle, and therefore captures the additional rotational symmetry present in many two‑dimensional topological quantum field theories.

The central application of these operads is to the Hochschild cochain complex $C^\bullet(A,A)$ of an associative algebra $A$. Classical results (Gerstenhaber, Deligne’s conjecture) assert that $C^\bullet(A,A)$ carries an $E_2$‑algebra structure, but existing constructions typically rely on intricate homotopy transfer arguments, operadic bar‑cobar resolutions, or model‑category techniques. By interpreting each lattice path cell as a multi‑linear operation on cochains, the authors obtain a concrete, “formal” $E_2$‑action: for a given lattice path $\gamma$ with $n$ inputs, one defines an operation \