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This book collects various perspectives, contributed by both mathematicians and physicists, on the B-model and its role in mirror symmetry. Mirror symmetry is an active topic of research in both the mathematics and physics communities, but among mathematicians, the “A-model” half of the story remains much better-understood than the B-model. This book aims to address that imbalance. It begins with an overview of several methods by which mirrors have been constructed, and from there, gives a thorough account of the “BCOV” B-model theory from a physical perspective; this includes the appearance of such phenomena as the holomorphic anomaly equation and connections to number theory via modularity. Following a mathematical exposition of the subject of quantization, the remainder of the book is devoted to the B-model from a mathematician’s point-of-view, including such topics as polyvector fields and primitive forms, Givental’s ancestor potential, and integrable systems.
Symmetry (Mathematics) --- Invariance (Mathematics) --- Group theory --- Automorphisms --- Geometry, algebraic. --- Algebraic Geometry. --- Mathematical Physics. --- Algebraic geometry --- Geometry --- Algebraic geometry. --- Mathematical physics. --- Physical mathematics --- Physics --- Mathematics
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This book collects various perspectives, contributed by both mathematicians and physicists, on the B-model and its role in mirror symmetry. Mirror symmetry is an active topic of research in both the mathematics and physics communities, but among mathematicians, the “A-model” half of the story remains much better-understood than the B-model. This book aims to address that imbalance. It begins with an overview of several methods by which mirrors have been constructed, and from there, gives a thorough account of the “BCOV” B-model theory from a physical perspective; this includes the appearance of such phenomena as the holomorphic anomaly equation and connections to number theory via modularity. Following a mathematical exposition of the subject of quantization, the remainder of the book is devoted to the B-model from a mathematician’s point-of-view, including such topics as polyvector fields and primitive forms, Givental’s ancestor potential, and integrable systems.
Algebraic geometry --- Geometry --- Mathematical physics --- landmeetkunde --- wiskunde --- fysica --- geometrie
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An introduction to the theory of orbifolds from a modern perspective, combining techniques from geometry, algebraic topology and algebraic geometry. One of the main motivations, and a major source of examples, is string theory, where orbifolds play an important role. The subject is first developed following the classical description analogous to manifold theory, after which the book branches out to include the useful description of orbifolds provided by groupoids, as well as many examples in the context of algebraic geometry. Classical invariants such as de Rham cohomology and bundle theory are developed, a careful study of orbifold morphisms is provided, and the topic of orbifold K-theory is covered. The heart of this book, however, is a detailed description of the Chen-Ruan cohomology, which introduces a product for orbifolds and has had significant impact. The final chapter includes explicit computations for a number of interesting examples.
Orbifolds. --- Topology --- Manifolds (Mathematics) --- Orbifolds --- 512.7 --- 515.14 --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Geometry, Differential --- 515.14 Algebraic topology --- Algebraic topology --- 512.7 Algebraic geometry. Commutative rings and algebras --- Algebraic geometry. Commutative rings and algebras --- Topology. --- Homology theory. --- Quantum theory. --- String models. --- Models, String --- String theory --- Nuclear reactions --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Cohomology theory --- Contrahomology theory
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"We construct a global B-model for any quasi-homogeneous polynomial f that has properties similar to the properties of the physic's B-model on a Calabi-Yau manifold. The main ingredients in our construction are K. Saito's theory of primitive forms and Givental's higher genus reconstruction. More precisely, we consider the moduli space M[unfilled bullet]mar of the so-called marginal deformations of f. For each point [sigma] [is an element of] M[unfilled bullet]mar we introduce the notion of an opposite subspace in the twisted de Rham cohomology of the corresponding singularity f[sigma] and prove that opposite subspaces are in one-to-one correspondence with the splittings of the Hodge structure in the vanishing cohomology of f[sigma]. Therefore, according to M. Saito, an opposite subspace gives rise to a semi-simple Frobenius structure on the space of miniversal deformations of f[sigma]. Using Givental's higher genus reconstruction we define a total ancestor potential A[sigma](h,q) wh ose properties can be described quite elegantly in terms of the properties of the corresponding opposite subspace. For example, if the opposite subspace corresponds to the splitting of the Hodge structure given by complex conjugation, then the total ancestor potential is monodromy invariant and it satisfies the BCOV holomorphic anomaly equations. The coefficients of the total ancestor potential could be viewed as quasi-modular forms on M[unfilled bullet]mar in a certain generalized sense. As an application of our construction, we consider the case of a Fermat polynomial W that defines a Calabi-Yau hypersurface XW in a weighted-projective space. We have constructed two opposite subspaces and proved that the corresponding total ancestor potentials can be identified with respectively the total ancestor potential of the orbifold quotient XW/G̃W and the total ancestor potential of FJRW invariants corresponding to (W,GW). Here GW is the maximal group of diagonal symmetries of W and GW is a quotient of GW by the subgroup of those elements that act trivially on XW . In particular, our result establishes the so-called Landau-Ginzburg/Calabi-Yau correspondence for the pair (W,GW)"--
Gromov-Witten invariants. --- Calabi-Yau manifolds. --- Symplectic manifolds. --- Homology theory. --- Invariants de Gromov-Witten --- Calabi-Yau, Variétés de --- Variétés symplectiques --- Homologie
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