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2017 | Buch

Algebra, Geometry, and Physics in the 21st Century

Kontsevich Festschrift

herausgegeben von: Prof. Denis Auroux, Prof. Ludmil Katzarkov, Prof. Tony Pantev, Prof. Yan Soibelman, Prof. Yuri Tschinkel

Verlag: Springer International Publishing

Buchreihe : Progress in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim’s vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim’s heart.

Contributors:

S. Donaldson

A. Goncharov

D. Kaledin

M. Kapranov

A. Kapustin

L. Katzarkov

A. Noll

P. Pandit

S. Pimenov

J. Ren

P. Seidel

C. Simpson

Y. Soibelman

R. Thorngren

Inhaltsverzeichnis

Frontmatter

Adiabatic Limits of Co-associative Kovalev–Lefschetz Fibrations

Abstract

We study co-associative fibrations of G ₂-manifolds. We propose that the adiabatic limit of this structure should be given locally by a maximal submanifold in a space of indefinite signature and set up global versions of the constructions.

Simon Donaldson

Ideal Webs, Moduli Spaces of Local Systems, and 3d Calabi–Yau Categories

Abstract

A decorated surface S is an oriented surface with punctures, and a finite set of marked points on the boundary, considered modulo isotopy. We assume that each boundary component has a marked point. We introduce ideal bipartite graphs on S. Each of them is related to a group G of type A_m or GL _m, and gives rise to cluster coordinate systems on certain moduli spaces of G-local systems on S. These coordinate systems generalize the ones assigned in [FG1] to ideal triangulations of S.

A bipartite graph W on S gives rise to a quiver with a canonical potential. The latter determines a triangulated 3d Calabi–Yau A _∞-category C _W with a cluster collection S _W – a generating collection of spherical objects of special kind [KS1].

Let W be an ideal bipartite graph on S of type G.We define an extension Г_G,S of the mapping class group of S, and prove that it acts by symmetries of the category CW.

There is a family of open CY threefolds over the universal Hitchin base BG,S, whose intermediate Jacobians describe Hitchin’s integrable system [DDDHP], [DDP], [G], [KS3], [Sm]. We conjecture that the 3d CY category with cluster collection (C _W, S _W) is equivalent to a full subcategory of the Fukaya category of a generic threefold of the family, equipped with a cluster collection of special Lagrangian spheres. For G = SL ₂ a substantial part of the story is already known thanks to Bridgeland, Keller, Labardini-Fragoso, Nagao, Smith, and others, see [BrS], [Sm].

We hope that ideal bipartite graphs provide special examples of the Gaiotto–Moore–Neitzke spectral networks [GMN4].

A. B. Goncharov

Spectral Sequences for Cyclic Homology

Abstract

We prove the non-commutative Hodge-to-de Rham Degeneration Conjecture of Kontsevich and Soibelman.

D. Kaledin

Derived Varieties of Complexes and Kostant’s Theorem for

Abstract

Given a graded vector space V, the variety of complexes Com(V) consists of all differentials making V into a cochain complex. This variety was first introduced by Buchsbaum and Eisenbud and later studied by Kempf, De Concini, Strickland and many other people. It is highly singular and can be seen as a proto-typical singular moduli space in algebraic geometry. We introduce a natural derived analog of Com(V) which is a smooth derived scheme RCom(V). It can be seen as the derived scheme classifying twisted complexes. We study the cohomology of the dg-algebra of regular functions on RCom(V). It turns out that the natural action of the group GL(V) (automorphisms of V as a graded space) on the cohomology has simple spectrum. This generalizes the known properties of Com(V) and the classical theorem of Kostant on the Lie algebra cohomology of upper triangular matrices.

M. Kapranov, S. Pimenov

Higher Symmetry and Gapped Phases of Gauge Theories

Abstract

We study topological field theory describing gapped phases of gauge theories where the gauge symmetry is partially Higgsed and partially confined. The TQFT can be formulated both in the continuum and on the lattice and generalizes Dijkgraaf–Witten theory by replacing a finite group by a finite 2-group. The basic field in this TQFT is a 2-connection on a principal 2-bundle. We classify topological actions for such theories as well as loop and surface observables. When the topological action is trivial, the TQFT is related to a Dijkgraaf–Witten theory by electric-magnetic duality, but in general it is distinct.We propose the existence of new phases of matter protected by higher symmetry.

Anton Kapustin, Ryan Thorngren

Constructing Buildings and Harmonic Maps

Abstract

In a continuation of our previous work [21], we outline a theory which should lead to the construction of a universal pre-building and versal building with a φ-harmonic map from a Riemann surface, in the case of twodimensional buildings for the group SL₃. This will provide a generalization of the space of leaves of the foliation defined by a quadratic differential in the classical theory for SL₂. Our conjectural construction would determine the exponents for SL₃ WKB problems, and it can be put into practice on examples.

Ludmil Katzarkov, Alexander Noll, Pranav Pandit, Carlos Simpson

Cohomological Hall Algebras, Semicanonical Bases and Donaldson–Thomas Invariants for 2-dimensional Calabi–Yau Categories (with an Appendix by Ben Davison)

Abstract

We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the “dimensional reduction” from 3-dimensional Calabi–Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson–Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi–Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi–Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).

Jie Ren, Yan Soibelman

Fukaya A ∞-structures Associated to Lefschetz Fibrations. II

Abstract

Consider the Fukaya category associated to a Lefschetz fibration. It turns out that the Floer cohomology of the monodromy around ∞ gives rise to natural transformations from the Serre functor to the identity functor, in that category. We pay particular attention to the implications of that idea for Lefschetz pencils.

Paul Seidel

Titel: Algebra, Geometry, and Physics in the 21st Century
herausgegeben von: Prof. Denis Auroux
Prof. Ludmil Katzarkov
Prof. Tony Pantev
Prof. Yan Soibelman
Prof. Yuri Tschinkel
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-59939-7
Print ISBN: 978-3-319-59938-0
DOI: https://doi.org/10.1007/978-3-319-59939-7