Operations on derived moduli spaces of branes

Abstract

The main theme of this work is the study of the operations that naturally exist on moduli spaces of maps $Map(S,X)$, also called the space of branes of $X$ with respect $S$. These operations will be constructed as operations on the (quasi-coherent) derived category $\D(Map(S,X))$, in the particular case where $S$ has some close relations with an operad $\OO$. More precisely, for an $\s$-operad $\OO$ and an algebraic variety $X$ (or more generally a derived algebraic stack), satisfying some natural conditions, we prove that $\OO$ acts on the object $\OO(2)$ by mean cospans. This universal action is used to prove that $\OO$ acts on the derived category of the space of maps $Map(\OO(2),X)$, which will call the brane operations. We apply the existence of these operations, as well as their naturality in $\OO$, in order to propose a sketch for a proof of the \emph{higher formality conjecture}, a far reaching extension of Konstevich's formality's theorem. By doing so we present a positive answer to a conjecture of Kapustin (see \cite[p. 14]{kap}), relating polyvector fields on a variety $X$ and deformations of the mono/"i dal derived category $\D(X)$.