Type: Article
Publication Date: 2014-07-09
Citations: 14
DOI: https://doi.org/10.1515/crelle-2014-0037
Abstract We provide a general method for finding all natural operations on the Hochschild complex of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℰ</m:mi></m:math> ${\mathcal{E}}$ -algebras, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℰ</m:mi></m:math> ${\mathcal{E}}$ is any algebraic structure encoded in a prop with multiplication, as for example the prop of Frobenius, commutative or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>A</m:mi><m:mi>∞</m:mi></m:msub></m:math> ${A_{\infty}}$ -algebras. We show that the chain complex of all such natural operations is approximated by a certain chain complex of formal operations , for which we provide an explicit model that we can calculate in a number of cases. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℰ</m:mi></m:math> ${\mathcal{E}}$ encodes the structure of open topological conformal field theories, we identify this last chain complex, up quasi-isomorphism, with the moduli space of Riemann surfaces with boundaries, thus establishing that the operations constructed by Costello and Kontsevich–Soibelman via different methods identify with all formal operations. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℰ</m:mi></m:math> ${\mathcal{E}}$ encodes open topological quantum field theories (or symmetric Frobenius algebras) our chain complex identifies with Sullivan diagrams, thus showing that operations constructed by Tradler–Zeinalian, again by different methods, account for all formal operations. As an illustration of the last result we exhibit two infinite families of non-trivial operations and use these to produce non-trivial higher string topology operations, which had so far been elusive.