First Article Accepted!

#paper #swiss-cheese #operads

My paper Swiss-Cheese operad and Drinfeld center has finally been accepted! It is going to appear in the Israel Journal of Mathematics. I’ve made the few corrections suggested by the referee (the new version is available on the arXiv), and I’m now waiting for the final proofs before the paper can be published.

If you haven’t read the paper yet, I wrote a post some time ago about the Voronov product of operads. My paper starts with the following observation, due to Voronov: the homology of the Swiss-Cheese operad \(\mathtt{SC}\) split as a “product” of the Gerstenhaber operad and the associative operad. It thus seems natural to think that the Swiss-Cheese operad itself could split as some kind of product. Unfortunately, this operad is not formal [Livernet, 2015], which makes matters more complex.

The first result of the paper is about an operad weakly equivalent to the fundamental groupoid of \(\mathtt{SC}\), the operad \(\mathtt{PaPB}\) of parenthesized permutations and braids. This is a combination of the operads of parenthesized permutations (corresponding to the associative part) and of parenthesized braids (corresponding to the Gerstenhaber part). I then prove that an algebra over \(\mathtt{PaPB}\) is given by the following data:

  • A monoidal category \(\mathsf{M}\);
  • A braided category \(\mathsf{N}\);
  • A braided functor \(F : \mathscr{Z}(\mathsf{M}) \to \mathsf{N}\) from the Drinfeld center of \(\mathsf{M}\) to \(\mathsf{N}\);

which fits in nicely with Voronov’s description of \(H_*(\mathtt{SC})\). The second result is inspired by Tamarkin’s proof of the formality of the little \(2\)-disks operad, and it is roughly speaking given by a “semi-direct product” of the operad of parenthesized permutations and the operad of parenthesized chord diagrams. The data of a Drinfeld associator is needed for the construction of this operad.

I knew that publishing took time in mathematics, but I hadn’t realized just how much. Indeed, between the moment I put the first version of the paper on the arXiv (July 2015) and today, about a year and a half elapsed. This is explained by several factors: I did not submit the paper to a journal immediately, reviews take a lot of time and editors have a lot on their plate, and the paper wasn’t accepted the first time I submitted it (I think I could have chosen the journal better that first time). But still, as an early-career mathematician, it seemed like forever! I’m rather proud of finally having achieved this first milestone.

PS: This week I’m at the Université Catholique de Louvain to give a talk at the algebraic topology seminar of the IRMP. I’m very grateful to Pascal Lambrechts for this invitation!