Here are my nominees for the 2006 paper of the year award in mathematics & mathematical physics : in math.RA : math.RA/0606241 : Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I by Maxim Kontsevich and Yan Soibelman. Here is the abstract :

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne’s conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product (this action is more or less equivalent to the structure of 2-dimensional Topological Field Theory associated with an “abstract” Calabi-Yau manifold).

why ? : Because this paper probably gives the correct geometric object associated to a non-commutative algebra (a huge coalgebra) and consequently the right definition of a map between noncommutative affine schemes. In a previous post (and its predecessors) I’ve tried to explain how this links up with my own interpretation and since then I’ve thought more about this, but that will have to wait for another time. in hep-th : hep-th/0611082 : Children’s Drawings From Seiberg-Witten Curves by Sujay K. Ashok, Freddy Cachazo, Eleonora Dell’Aquila. Here is the abstract :

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called “dessins d’enfants” or “children’s drawings” on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of N=1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck’s programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories.

why ? : Because this paper gives the best introduction I’ve seen to Grothendieck’s dessins d’enfants (slightly overdoing it by giving a crash course on elementary Galois theory in appendix A) and kept me thinking about dessins and their Galois invariants ever since (again, I’ll come back to this later).

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arxiv, Calabi-Yau, coalgebras, Galois, geometry, Grothendieck, Kontsevich, moduli, non-commutative, noncommutative, Riemann, superpotential

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Posted in geometry
Written on Fri, 29 December 2006 at 9:42 am
Tags: arxiv, Calabi-Yau, coalgebras, Galois, geometry, Grothendieck, Kontsevich, moduli, non-commutative, noncommutative, Riemann, superpotential
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