This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich’s homological mirror symmetry conjecture, and . This is the second of two books that provide the scientific record of the school. The first book, Strings and Geometry, edited by Michael R. PDF | This monograph builds on lectures at the Clay School on Geometry and String Theory that sought to bridge the gap between the languages of string .
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Langlands dualitygeometric Langlands dualityquantum geometric Langlands duality.
Seiberg dualityAGT conjecture. Green-Schwarz mechanismdifferential string structure. T-dualitymirror symmetry.
S-dualityelectric-magnetic duality. D0-braneD2-braneD4-brane.
D1-braneD3-braneD5-brane. RR-fielddifferential K-theory. This is called mirror symmetry.
Dirichlet Branes and Mirror Symmetry
At least in some cases this can be understood as a special case of T-duality Strominger-Yau-Zaslow In this form mirror symmetry remains a conjecture, not the least because for the moment there is no complete construction of these SCFTs. The topological A-model can be expressed in terms of symplectic geometry of a variety and the topological B-model can be expressed in terms of the algebraic geometry of a variety.
These topological theories are easier to understand and do retain a little bit of the information encoded in the full SCFTs. In terms of these the statement of mirror symmetry says that passing to mirror CYs exchanges the A-model with the B B -model and conversely:.
Mathematics > Algebraic Geometry
This categorical formulation was introduced by Maxim Kontsevich in under the name homological mirror symmetry. The mirror symmetry conjecture roughly claims that every Calabi-Yau 3-fold has a mirror. In fact one considers mirror symmetry for degenerating families for Calabi-Yau 3-folds in large volume limit which may be expressed precisely via the Gromov-Hausdorff metric. The appropriate definition of an appropriate version of the Fukaya category of a symplectic manifold is difficult to achieve in desired generality.
Dirichlet Branes and Mirror Symmetry : Bennett Chow :
Mirror symmetry is related to the T-duality on each fiber of an associated Lagrangian fibration Strominger-Yau-Zaslow Although the non-Calabi-Yau case may be of lesser interest to physics, one can still formulate some mirror symmetry statements for, for instance, Fano manifolds.
Then the statements are: A few of the relevant names: Ballard, Meet homological mirror symmetry arxiv: The relation to T-duality was established in. Cumrun MirroorShing-Tung Yau eds.
Zaslow, Mirror symmetry is T T -duality as pages —. Nauk 59no.
Surveys 59no. Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrationsmath. Discussion in the context of derived Morita equivalence includes. Here is a list with references that give complete proofs of homological mirror symmetry on certain types of spaces. Smith, Homological mirror symmetry for the four-torusDuke Math.
Dirichlet branes and mirror symmetry – INSPIRE-HEP
Zaslow, Categorical mirror symmetry: Orlov, Mirror symmetry for abelian varietiesJ. Seidel, Homological mirror symmetry for the quartic surfacearXiv: Efimov, Homological mirror symmetry for curves of higher genusInventiones Math. OrlovMirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves; Mirror symmetry for weighted projective planes and their noncommutative deformationsAnn. Paul SeidelHomological mirror symmetry for the genus two curveJ.
Algebraic Geometry, to appear, arXiv: Computation via topological recursion in matrix models and all- genus? Deriving the matrix model arXiv: The spectral curve and mirror geometry arXiv: Last revised on April 11, at See the history of this page for a list of all contributions to it. Context Duality duality abstract duality: This site is running on Instiki 0.