Operations on derived moduli spaces of branes

TL;DR

This paper develops operations on moduli spaces of maps called branes, using derived categories and operads, and applies these to propose a proof for the higher formality conjecture, extending Kontsevich's theorem.

Contribution

It introduces a universal operad action on derived categories of brane moduli spaces, advancing the understanding of their algebraic structures and their role in deformation theory.

Findings

Constructed operad actions on derived categories of moduli spaces of maps.

Proposed a sketch for the proof of the higher formality conjecture.

Provided a positive answer to Kapustin's conjecture relating polyvector fields and deformations.

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 $\D(X)$.