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@inbook{Thomas:2004:B0-12-512666-2/00037-7,
author = {Thomas, RP},
booktitle = {Encyclopedia of Mathematical Physics: Five-Volume Set},
doi = {B0-12-512666-2/00037-7},
pages = {439--448},
title = {Mirror Symmetry: A Geometric Survey},
url = {http://dx.doi.org/10.1016/B0-12-512666-2/00037-7},
year = {2004}
}

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TY  - CHAP
AB  - Mirror symmetry was discovered in the late 1980s by physicists studying superconformal field theories (SCFTs). One way to produce SCFTs is from closed string theory; in the Riemannian (rather than Lorentzian) theory the string's world line gives a map of a Riemannian 2-manifold into the target with an action which is conformally invariant, so the 2-manifold can be thought of as a Riemann surface with a complex structure. Making sense of the infinities in the quantum theory (supersymmetry and anomaly cancelation) forces the target to be 10-dimensional - Minkowski space times by a 6-manifold X - and X to be (to first order) Ricci flat and so to have holonomy in SU(3). That is X is a Calabi-Yau 3-fold (Formula;). So SCFTs come from -models (mapping Riemann surfaces into Calabi-Yau 3-folds) but, it turns out, in two different ways - the A-model and the B-model. Deformations of the SCFT and either -model are isomorphic, so over an open set the two coincide. Thus, it was natural to conjecture that almost all of the relevant SCFTs came from geometry - from an A or B -model.
AU  - Thomas,RP
DO  - B0-12-512666-2/00037-7
EP  - 448
PY  - 2004///
SN  - 9780125126663
SP  - 439
TI  - Mirror Symmetry: A Geometric Survey
T1  - Encyclopedia of Mathematical Physics: Five-Volume Set
UR  - http://dx.doi.org/10.1016/B0-12-512666-2/00037-7
ER  -

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