# Mirror Symmetry

Kentaro Hori
American Mathematical Society, 2003 - Mathematics - 929 pages
This thorough and detailed exposition is the result of an intensive month-long course on mirror symmetry sponsored by the Clay Mathematics Institute. It develops mirror symmetry from both mathematical and physical perspectives with the aim of furthering interaction between the two fields. The material will be particularly useful for mathematicians and physicists who wish to advance their understanding across both disciplines. 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 gives a single, cohesive treatment of mirror symmetry. Parts 1 and 2 develop the necessary mathematical and physical background from ''scratch''. The treatment is focused, 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 topi This one-of-a-kind book is suitable for graduate students and research mathematicians interested in mathematics and mathematical and theoretical physics.

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#### Review: Mirror Symmetry (Clay Mathematics Monographs, V. 1)

User Review  - Jon Paprocki - Goodreads

This enormous book would require several years of study to get through, so I'm evaluating based on the handful of sections that I have read. It is certainly a fully self-contained account of the idea ... Read full review

### References from web pages

Mirror symmetry - Wikipedia, the free encyclopedia
In physics and mathematics, mirror symmetry is a surprising relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such ...
en.wikipedia.org/ wiki/ Mirror_symmetry

SYMPLECTIC GEOMETRY AND MIRROR SYMMETRY
Mirror symmetry had been discovered by physicists in string theory as a duality ... The theory of mirror symmetry was further enhanced by physicists in the ...
www.wspc.com.sg/ books/ mathematics/ 4778.html

Workshop on Mirror Symmetry
Main topics will be Strominger-Yau-Zaslow conjecture, homological mirror symmetry, as well as number theoretic aspects. New directions such as (0,2) mirror ...
www.fields.utoronto.ca/ programs/ scientific/ 04-05/ string-theory/ mirror/ index.html

Opening Mirror Symmetry on the Quintic
Abstract: Aided by mirror symmetry, we determine the number of holomorphic disks ... This is the first exact result on open string mirror symmetry for a ...

Mirror Symmetry
This book is a product of a month-long school on mirror symmetry that CMI held at Pine Manor College in Brookline, Massachusetts in the Spring of 2000. ...
www.claymath.org/ publications/ Mirror_Symmetry/

Atlas: Mirror symmetry formula for elliptic genus of some Fano ...
Hori and Vafa recently suggested a mirror symmetry construction for some manifolds with non-negative first Chern class. The mirror partner of such a ...
atlas-conferences.com/ c/ a/ p/ e/ 09.htm

ESI Workshop on Homological Mirror Symmetry | The String Coffee Table
Read the post Kontsevich Lectures on Mirror Symmetry, I Weblog: The String Coffee Table ... Read the post Kapustin on SYM, Mirror Symmetry and Langlands, I ...
golem.ph.utexas.edu/ string/ archives/ 000832.html

arxiv1.org - High Energy Physics - Theory - Mirror Symmetry - E-prints
We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds ... This provides not only a proof of mirror symmetry in the case of (local and ...
www.arxiv1.org/ modules/ news/

Monopole Operators and Mirror Symmetry in Three-Dimensional Gauge ...
Predictions of three-dimensional mirror symmetry for the ... 1 are free fields. This amounts. to a proof of 3-D mirror symmetry in these special cases. ...
etd.caltech.edu/ etd/ available/ etd-04062004-015855/ unrestricted/ thesis.pdf

arxiv:hep-th/0209138 v1 17 Sep 2002
In [1] a proof of mirror symmetry was provided in terms of equivalence of linear ... a perturbative derivation of mirror symmetry, which can be viewed as a ...
arxiv.org/ pdf/ hep-th/ 0209138