Department of

September 2013 October 2013 November 2013 Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 1 2 3 4 5 1 2 8 9 10 11 12 13 14 6 7 8 9 10 11 12 3 4 5 6 7 8 9 15 16 17 18 19 20 21 13 14 15 16 17 18 19 10 11 12 13 14 15 16 22 23 24 25 26 27 28 20 21 22 23 24 25 26 17 18 19 20 21 22 23 29 30 27 28 29 30 31 24 25 26 27 28 29 30

Tuesday, October 15, 2013

**Abstract:** Let $E_n$ be the little disks operad. It is well known that for $n>1$, the rational homology of $E_n$ is $P_n$, the $(n-1)$ shifted Poisson operad. More generally, for all $n$, $E_n$ has a filtration whose associated graded operad is $P_n$. Dunn's additivity theorem states that the Boardman-Vogt tensor product of $E_k$ and $E_l$ is $E_{k+l}$. We will show that this equivalence is compatible with the filtration and, time permitting, explain the generalization of this statement to factorization algebras. This fact has a number of remarkable consequences. An immediate corollary is the formality theorem for $n>2$ by induction starting with formality for $E_2$. Furthermore, the factorization algebra version of the result provides a local-to-global version of the BV-AKSZ formalism in quantum field theory, and sheds new light on the problem of quantization.