Mirror symmetry is a phenomenon arising in string theory in which two very different manifolds give rise to equivalent physics. Such a correspondence has significant mathematical consequences, the most familiar of which involves the enumeration of holomorphic curves inside complex manifolds by solving differential equations obtained from a ""mirror"" geometry. The inclusion of D-brane states in the equivalence has led to further conjectures involving calibrated submanifolds of the mirror pairs and new (conjectural) invariants of complex manifolds: the Gopakumar Vafa invariants. This book aims to give a single, cohesive treatment of mirror symmetry from both the mathematical and physical viewpoint. Parts 1 and 2 develop the necessary mathematical and physical background ``from scratch,'' and are intended for readers trying to learn across disciplines. The treatment is focussed, developing only the material most necessary for the task. In Parts 3 and 4 the physical and mathematical proofs of mirror symmetry are given. From the physics side, this means demonstrating that two different physical theories give isomorphic physics. Each physical theory can be described geometrically, and thus mirror symmetry gives rise to a ""pairing"" of geometries. The proof involves applying $R\leftrightarrow 1/R$ circle duality to the phases of the fields in the gauged linear sigma model. The mathematics proof develops Gromov-Witten theory in the algebraic setting, beginning with the moduli spaces of curves and maps, and uses localization techniques to show that certain hypergeometric functions encode the Gromov-Witten invariants in genus zero, as is predicted by mirror symmetry. Part 5 is devoted to advanced topics in mirror symmetry, including the role of D-branes in the context of mirror symmetry, and some of their applications in physics and mathematics: topological strings and large $N$ Chern-Simons theory; geometric engineering; mirror symmetry at higher genus; Gopakumar-Vafa invariants; and Kontsevich's formulation of the mirror phenomenon as an equivalence of categories. This book grew out of an intense, month-long course on mirror symmetry at Pine Manor College, sponsored by the Clay Mathematics Institute. The lecturers have tried to summarize this course in a coherent, unified text.

Table of Contents:
Part 1. Mathematical Preliminaries: Differential geometry Algebraic geometry Differential and algebraic topology Equivariant cohomology and fixed-point theorems Complex and Kahler geometry Calabi-Yau manifolds and their moduli Toric geometry for string theory Part 2. Physics Preliminaries: What is a QFT? QFT in $d=0$ QFT in dimension 1: Quantum mechanics Free quantum field theories 1 + 1 dimensions $\mathcal{N} = (2,2)$ supersymmetry Non-linear sigma models and Landau-Ginzburg models Renormalization group flow Linear sigma models Chiral rings and topological field theory Chiral rings and the geometry of the vacuum bundle BPS solitons in $\mathcal{N}=2$ Landau-Ginzburg theories D-branes Part 3. Mirror Symmetry: Physics Proof: Proof of mirror symmetry Part 4. Mirror Symmetry: Mathematics Proof: Introduction and overview Complex curves (non-singular and nodal) Moduli spaces of curves Moduli spaces $\bar{\mathcal M}_{g,n}(X,\beta)$ of stable maps Cohomology classes on $\bar{\mathcal M}_{g,n}$ and ($\bar{\mathcal M})_{g,n}(X,\beta)$ The virtual fundamental class, Gromov-Witten invariants, and descendant invariants Localization on the moduli space of maps The fundamental solution of the quantum differential equation The mirror conjecture for hypersurfaces I: The Fano case The mirror conjecture for hypersurfaces II: The Calabi-Yau case Part 5. Advanced Topics: Topological strings Topological strings and target space physics Mathematical formulation of Gopakumar-Vafa invariants Multiple covers, integrality, and Gopakumar-Vafa invariants Mirror symmetry at higher genus Some applications of mirror symmetry Aspects of mirror symmetry and D-branes More on the mathematics of D-branes: Bundles, derived categories and Lagrangians Boundary $\mathcal{N}=2$ theories References Bibliography Index