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Topological Mirrors and Quantum Rings

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TLDR
In this paper, the importance of loop spaces for a deeper understanding of the geometrical origin of duality in string theory is discussed, and mirror symmetry can be reformulated in very simple terms as the statement of equivalence of two classes of topological theories: topological sigma models and topological Landau-Ginzburg models.
Abstract
Aspects of duality and mirror symmetry in string theory are discussed. We emphasize, through examples, the importance of loop spaces for a deeper understanding of the geometrical origin of dualities in string theory. Moreover we show that mirror symmetry can be reformulated in very simple terms as the statement of equivalence of two classes of topological theories: Topological sigma models and topological Landau-Ginzburg models. Some suggestions are made for generalization of the notion of mirror symmetry.

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Citations
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Book ChapterDOI

Geometry of 2D topological field theories

TL;DR: In this paper, the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories is studied, where WDVV equations and Frobenius manifolds are discussed.
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Geometric Engineering, Mirror Symmetry and 6d (1,0) -> 4d, N=2

TL;DR: In this article, the authors study compactification of 6-dimensional (1-0) theories on T^2 and obtain arbitrary genus curves with punctures from toroidal compactification, where the curve of the class S theory emerges through mirror symmetry.
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Geometric engineering, mirror symmetry and $ 6{\mathrm{d}}_{\left(1,0\right)}\to 4{\mathrm{d}}_{\left(\mathcal{N}=2\right)} $

TL;DR: In this article, the authors study compactification of 6-dimensional (1-0) theories via F-theory and employ mirror symmetry technology to solve for the effective 4d = 2-approximation for a large number of the theories including those associated with conformal matter.
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Bethe/gauge correspondence on curved spaces

TL;DR: In this article, the handle gluing operator ℋ is computed for supersymmetric vacua of massive gauge theories invariant under the two-dimensional Bethe/gauge correspondence and the Gaudin conjecture on the norm of Bethe states.
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Gravitational Quantum Cohomology

TL;DR: In this article, the theory of quantum cohomology may be generalized to ''gravitational quantum cohology'' by studying topological sigma models coupled to two-dimensional gravity, and the subspace of the phase space where only a marginal perturbation (with a parameter $t$) is turned on is constructed.
References
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Journal ArticleDOI

Fusion rings and geometry

TL;DR: The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper, where a formalism which closely parallels classical tools in the study of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories.
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