{"product_id":"mirror-symmetry-paperback-softback-9780821834879","title":"Mirror Symmetry (Paperback \/ softback) 9780821834879","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eMirror Symmetry\u003c\/font\u003e\u003cbr\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003c\/p\u003e\n\u003cp\u003e\u003cfont size=\"4\"\u003eKentaro Hori (Author), Sheldon Katz (Author), Albrecht Klemm (Author), Rahul Pandharipande (Author), Richard Thomas (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9780821834879\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003ePaperback \/ softback, published 31 January 2003\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e929 pages\u003cbr\u003e25.9 x 18.8 x 5 cm, 1.688 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cem\u003e\u003cfont size=\"3\"\u003e“This book, a product of the collective efforts of the lecturers at the School organized ... by the Clay Mathematics Institute, is a valuable contribution to the continuing intensive collaboration of physicists and mathematicians. It will be of great value to young and mature researchers in both communities interested in this fascinating modern grand unification project.” - Yuri Manin, Max Planck Institute for Mathematics, Bonn, Germany\u003c\/font\u003e\u003c\/em\u003e\u003c\/p\u003e\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003eMirror 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.\u003cbr\u003e\u003cbr\u003eThis 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.\u003cbr\u003e\u003cbr\u003eThis 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.\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e\u003cul\u003e\n\u003cli\u003ePart 1. Mathematical Preliminaries: Differential geometry\u003c\/li\u003e\n\u003cli\u003eAlgebraic geometry\u003c\/li\u003e\n\u003cli\u003eDifferential and algebraic topology\u003c\/li\u003e\n\u003cli\u003eEquivariant cohomology and fixed-point theorems\u003c\/li\u003e\n\u003cli\u003eComplex and Kahler geometry\u003c\/li\u003e\n\u003cli\u003eCalabi-Yau manifolds and their moduli\u003c\/li\u003e\n\u003cli\u003eToric geometry for string theory\u003c\/li\u003e\n\u003cli\u003ePart 2. Physics Preliminaries: What is a QFT?\u003c\/li\u003e\n\u003cli\u003eQFT in $d=0$\u003c\/li\u003e\n\u003cli\u003eQFT in dimension 1: Quantum mechanics\u003c\/li\u003e\n\u003cli\u003eFree quantum field theories 1 + 1 dimensions\u003c\/li\u003e\n\u003cli\u003e$\\mathcal{N} = (2,2)$ supersymmetry\u003c\/li\u003e\n\u003cli\u003eNon-linear sigma models and Landau-Ginzburg models\u003c\/li\u003e\n\u003cli\u003eRenormalization group flow\u003c\/li\u003e\n\u003cli\u003eLinear sigma models\u003c\/li\u003e\n\u003cli\u003eChiral rings and topological field theory\u003c\/li\u003e\n\u003cli\u003eChiral rings and the geometry of the vacuum bundle\u003c\/li\u003e\n\u003cli\u003eBPS solitons in $\\mathcal{N}=2$ Landau-Ginzburg theories\u003c\/li\u003e\n\u003cli\u003eD-branes\u003c\/li\u003e\n\u003cli\u003ePart 3. Mirror Symmetry: Physics Proof: Proof of mirror symmetry\u003c\/li\u003e\n\u003cli\u003ePart 4. Mirror Symmetry: Mathematics Proof: Introduction and overview\u003c\/li\u003e\n\u003cli\u003eComplex curves (non-singular and nodal)\u003c\/li\u003e\n\u003cli\u003eModuli spaces of curves\u003c\/li\u003e\n\u003cli\u003eModuli spaces $\\bar{\\mathcal M}_{g,n}(X,\\beta)$ of stable maps\u003c\/li\u003e\n\u003cli\u003eCohomology classes on $\\bar{\\mathcal M}_{g,n}$ and ($\\bar{\\mathcal M})_{g,n}(X,\\beta)$\u003c\/li\u003e\n\u003cli\u003eThe virtual fundamental class, Gromov-Witten invariants, and descendant invariants\u003c\/li\u003e\n\u003cli\u003eLocalization on the moduli space of maps\u003c\/li\u003e\n\u003cli\u003eThe fundamental solution of the quantum differential equation\u003c\/li\u003e\n\u003cli\u003eThe mirror conjecture for hypersurfaces I: The Fano case\u003c\/li\u003e\n\u003cli\u003eThe mirror conjecture for hypersurfaces II: The Calabi-Yau case\u003c\/li\u003e\n\u003cli\u003ePart 5. Advanced Topics: Topological strings\u003c\/li\u003e\n\u003cli\u003eTopological strings and target space physics\u003c\/li\u003e\n\u003cli\u003eMathematical formulation of Gopakumar-Vafa invariants\u003c\/li\u003e\n\u003cli\u003eMultiple covers, integrality, and Gopakumar-Vafa invariants\u003c\/li\u003e\n\u003cli\u003eMirror symmetry at higher genus\u003c\/li\u003e\n\u003cli\u003eSome applications of mirror symmetry\u003c\/li\u003e\n\u003cli\u003eAspects of mirror symmetry and D-branes\u003c\/li\u003e\n\u003cli\u003eMore on the mathematics of D-branes: Bundles, derived categories and Lagrangians\u003c\/li\u003e\n\u003cli\u003eBoundary $\\mathcal{N}=2$ theories\u003c\/li\u003e\n\u003cli\u003eReferences\u003c\/li\u003e\n\u003cli\u003eBibliography\u003c\/li\u003e\n\u003cli\u003eIndex\u003c\/li\u003e\n\u003c\/ul\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\u003c\/font\u003e","brand":"American Mathematical Society","offers":[{"title":"Brand New","offer_id":52523092771096,"sku":"9780821834879","price":110.38,"currency_code":"GBP","in_stock":true}],"url":"https:\/\/freshlyprintedbooks.co.uk\/products\/mirror-symmetry-paperback-softback-9780821834879","provider":"Freshly Printed Books","version":"1.0","type":"link"}