2013/07/01 by Bertrand Toën, B. Toën, Toën, B.
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #math.CT
paper · pdf · doi:10.48550/arxiv.1307.0405
Minor corrections, 55 Pages
openalex publication_date 2013/07/01 · arxiv created 2013/10/23 · arxiv updated 2013/10/24 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
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 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).