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Über dieses Buch

This book consists of a series of introductory lectures on mirror symmetry and its surrounding topics. These lectures were provided by participants in the PIMS Superschool for Derived Categories and D-branes in July 2016. Together, they form a comprehensive introduction to the field that integrates perspectives from mathematicians and physicists alike.

These proceedings provide a pleasant and broad introduction into modern research topics surrounding string theory and mirror symmetry that is approachable to readers new to the subjects. These topics include constructions of various mirror pairs, approaches to mirror symmetry, connections to homological algebra, and physical motivations. Of particular interest is the connection between GLSMs, D-branes, birational geometry, and derived categories, which is explained both from a physical and mathematical perspective. The introductory lectures provided herein highlight many features of this emerging field and give concrete connections between the physics and the math.

Mathematical readers will come away with a broader perspective on this field and a bit of physical intuition, while physicists will gain an introductory overview of the developing mathematical realization of physical predictions.



Derived Categories and Related Topics in Algebraic Geometry


Abelian and Triangulated Categories

The purpose of these notes is to provide enough background information to define triangulated categories. We provide background information on monomorphisms, epimorphisms, kernels and cokernels. We use this information to define additive and abelian categories, and provide several examples. We give a full definition of triangulated categories and the axioms TR1TR4. We conclude by describing some basic results on triangulated categories, leading to a long exact sequence of morphism groups.
Chantelle Hanratty

Derived Categories and Derived Functors

This is an introduction to the language of derived categories and understanding the triangulated structure on the derived category. We also introduce the analogue of functors on a derived category, i.e. derived functors.
Nitin Kumar Chidambaram

Introduction to Quivers

Quivers are directed graphs which are commonly used in fields such as representation theory and noncommutative geometry. This paper is meant to provide a short introduction for quivers and algebras produced from those quivers, called path algebras. We first look at basic definitions of quivers Q and path algebras kQ. We also cover some algebraic properties of path algebras in order to have a better understanding of the category of finite representations of a quiver Q. In fact, such category is equivalent the category of finitely generated left kQ-module corresponding to the quiver Q. As an example, we briefly describe how to obtain a representation of Q from a left kQ-module. At the end, we take a look at a bounded quiver Q (a.k.a. a quiver Q with a set of relations R) and its path algebra kQ / I where I is a two sided ideal generated by R. We use the Beilinson quiver for \(\mathscr {P}^2\) with relations as an example to illustrate the bounded quiver and its corresponding path algebra.
Minako Chinen

Semi-orthogonal Decompositions of Derived Categories

In this survey, we are mostly interested in:
\({\text {D}}^{{\text {b}}}(X):={\text {D}}^{{\text {b}}}(\mathop {\text{ Coh }}(X))\), where X is a smooth projective variety over a filed k and
\({\text {D}}^{{\text {b}}}(Q,I):={\text {D}}^{{\text {b}}}(\mathop {\text {mod-}}kQ/I) \cong {\text {D}}^{{\text {b}}}(\mathop {\text{ rep }}(Q,I)^{op})\), where Q is a quiver and k is a field.
The second one it rather easy to understand while the first one is relatively hard to understand in general. We are going to see in some nice cases, they are equivalent. The theory of semi-orthogonal decompositions and exceptional collections will give you an idea how one might study the structure of derived categories: we decompose the category into smaller pieces. We hope these pieces are as simple as possible - as the simplest derived category \({\text {D}}^{{\text {b}}}(pt) \cong {\text {D}}^{{\text {b}}}(\text {Vec}_{k}^{\text{ f }.d.})\), which is given exactly by exceptional objects.
Yijia Liu

Introduction to Stability Conditions

Let X be a smooth projective Calabi–Yau variety over \(\mathbb {C}\). Then \(\mathcal {D}^b(X)\), the derived category of coherent sheaves on X, is equivalent to the category of D-branes on X [9]. In [10], Douglas defined a notion of stability for D-branes on X called \(\Pi \)-stability. This notion of stability was meant to pick out BPS-branes on X. In [7], Bridgeland aimed to define a notion of stability directly for objects in \(\mathcal {D}^b(X)\) which would correspond to \(\Pi \)-stability for D-branes. Bridgeland’s stability can be defined on any triangulated category, and hence has been studied in other cases, such as for varieties which are not Calabi–Yau.
Rebecca Tramel

A Brief Introduction to Geometric Invariant Theory

We provide a brief introduction to Geometric Invariant Theory. Specifically, we discuss some foundational concepts and results and illustrate the general theory by way of examples.
Nathan Grieve

Birational Geometry and Derived Categories

The aim of these notes is to describe some relations between the birational geometry of algebraic varieties and their associated derived categories. This is a large subject with several diverging paths, so we’ll restrict our focus to the realm of topics discussed at the Alberta superschool. Being lecture notes, the discussion here is somewhat informal, and we only attempt a general overview, and referring the reader to the relevant original research articles for details. Given the background of the participants of the Alberta superschool, these notes take the somewhat unorthodox approach of assuming that the reader has modest familiarity with derived and triangulated categories, but is perhaps not as familiar with the more cabalistic aspects of birational geometry.
Colin Diemer

Approaches to Mirror Symmetry


Introduction to Mirror Symmetry

An introduction to mirror symmetry in Hodge diamonds, with a review of the prerequisite geometry.
Richard Derryberry

Batyrev Mirror Symmetry

We describe Batyrev’s construction of the mirror to a family of Calabi–Yau hypersurfaces in a Fano toric variety, based on polar duality for lattice polytopes. We revisit the example of the quintic threefold in this language, and briefly mention connections with later developments, such as the Batyrev–Borisov construction for complete intersections in Fano toric varieties, and the Gross–Siebert program.
Mattia Talpo

Introduction to Differential Graded Categories

Differential graded (dg) categories provide enhancements of triangulated categories that allow us to overcome some problems that come from working solely with the triangulated structure. In this talk, we present the definition of dg categories and describe some constructions that can be performed with them. We then consider how a dg category provides an enhancement of a triangulated category, and show how to compute some important invariants of the category using such a dg enhancement. Finally we’ll present some theorems about such invariants, and how to derive them using properties of the dg enhancement. This talk is purely expository and does not contain original material; it is mostly based on B. Keller’s excellent survey on dg categories [9], and whenever possible I have used notation compatible with that source. I also included material and examples from the other sources listed as references as well.
Alex A. Takeda

Introduction to Symplectic Geometry and Fukaya Category

We give a brief introduction to symplectic manifolds and Fukaya Category in this manuscript.
Alex Zhongyi Zhang

Introduction to Homological Mirror Symmetry

Mirror symmetry states that to every Calabi-Yau manifold \(X\) with complex structure and symplectic symplectic structure there is another dual manifold \(X^\vee \), so that the properties of \(X\) associated to the complex structure (e.g. periods, bounded derived category of coherent sheaves) reproduce properties of \(X^\vee \) associated to its symplectic structure (e.g. counts of pseudo holomorphic curves and discs).
Andrew Harder

The SYZ Conjecture via Homological Mirror Symmetry

These are expanded notes based on a talk given at the Superschool on Derived Categories and D-branes held at the University of Alberta in July of 2016. The goal of these notes is to give a motivated introduction to the Strominger-Yau-Zaslow (SYZ) conjecture from the point of view of homological mirror symmetry.
Dori Bejleri

Physical Motivations


The Derived Category of Coherent Sheaves and B-model Topological String Theory

This elementary survey article was prepared for a talk at the PIMS 2016 Alberta Superschool on Derived Categories and D-branes. The goal is to outline an identification of the (bounde d) derived category of coherent sheaves on a Calabi–Yau threefold X with the D-brane category in B-model topological string theory. This was originally conjectured by Kontsevich [1]. We begin by briefly introducing topological closed string theory to acquaint the reader with the basics of the non-linear sigma model. With the inclusion of open strings, we must specify boundary conditions for the endpoints; these are what we call D-branes. Most naïvely, a D-brane in the B-model is a holomorphic submanifold of X and a locally-free sheaf supported on it; such objects pushforward to the category of coherent sheaves on X. After briefly summarizing the necessary homological algebra and sheaf cohomology, we argue that one should think of a D-brane as a complex of coherent sheaves, and provide a physical motivation to identify complexes up to homotopy. Finally, we argue that renormalization group (RG) flow on the worldsheet provides a physical realization of quasi-isomorphism. This identifies an element in the derived category with a universality class of D-branes in physics. I aim for this article to be an approachable introduction to the subject for both mathematicians and physicists. As such, it is far from a complete account. The material is based largely on Eric Sharpe’s lecture notes [2] as well as Paul Aspinwall’s paper [3].
Stephen Pietromonaco

Introduction to Topological String Theories

These notes are for a talk in Superschool on derived categories and D-branes at University of Alberta in summer of 2016. The purpose of these notes is to give a main idea of topological string theories as one of examples of mirror symmetry without any technical details. This means that some definitions are somewhat mathematically less rigorous but we rather show intuitive analyses instead. Readers should be familiar with GR, QFT, SUSY, CFT and some basics of string theories.
Kento Osuga

An Overview of B-branes in Gauged Linear Sigma Models

We review the BPS D-branes in gauged linear sigma models corresponding to toric Calabi–Yau (CY) varieties preserving \({\mathscr {N}}=2_B\) supersymmetry, and their relation to stable low energy branes. The chiral sectors of these low energy branes are described mathematically by various derived categories in various parts of the CY Kähler moduli space \({\mathscr {M}}_K\). For a fixed \({\mathscr {M}}_K\), all these descriptions should in fact be equivalent in a categorical sense and we review some aspects of this equivalence from a physical perspective. This is a short summary of the results of the comprehensive work by Herbst, Hori and Page [3] with some elementary commentary.
Nafiz Ishtiaque
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