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Mirror symmetry
About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.
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TL;DR: The moduli space of N = 4,4 string theories with a K3 target space is determined in this article, and it is shown that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20).
Abstract: The moduli space of N=(4,4) string theories with a K3 target space is determined, establishing in particular that the discrete symmetry group is the full integral orthogonal group of an even unimodular lattice of signature (4,20). The method combines an analysis of the classical theory of K3 moduli spaces with mirror symmetry. A description of the moduli space is also presented from the viewpoint of quantum geometry, and consequences are drawn concerning mirror symmetry for algebraic K3 surfaces.
16 citations
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TL;DR: In this article, the authors analyzed the genus zero picture and showed that wall-crossing in this case significantly generalizes toric mirror symmetry (the toric cases correspond to abelian groups G) and gave a geometric interpretation of the mirror map as a generating series of quasimap invariants.
Abstract: For each positive rational number epsilon, the theory of epsilon-stable quasimaps to certain GIT quotients W//G developed in arXiv:1106.3724[math.AG] gives rise to a Cohomological Field Theory. Furthermore, there is an asymptotic theory corresponding to epsilon --> 0. For epsilon >1 one obtains the usual Gromov-Witten theory of W//G, while the other theories are new. However, they are all expected to contain the same information and in particular the numerical invariants should be related by wall-crossing formulas. In this paper we analyze the genus zero picture and find that the wall-crossing in this case significantly generalizes toric mirror symmetry (the toric cases correspond to abelian groups G). In particular, we give a geometric interpretation of the mirror map as a generating series of quasimap invariants. We prove our wall-crossing formulas for all targets W//G which admit a torus action with isolated fixed points, as well as for zero loci of sections of homogeneous vector bundles on such W//G.
16 citations
TL;DR: In this article, the authors discuss various aspects of this approach from the point of view of non-commutative algebraic geometry in the tensor category of graded vector spaces.
Abstract: Homological mirror symmetry aims to explain the phenomenon of mirror symmetry in the language of A∞-categories and their deformation theory. In these notes I discuss various aspects of this approach from the point of view of noncommutative algebraic geometry in the tensor category of graded vector spaces.
16 citations
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TL;DR: The connections amongst quivers whose representation varieties are Calabi-Yau, the combinatorics of bipartite graphs on Riemann surfaces, and the geometry of mirror symmetry have engendered a rich subject at whose heart is the physics of gauge/string theories as mentioned in this paper.
Abstract: The connections amongst (1) quivers whose representation varieties are Calabi-Yau, (2) the combinatorics of bipartite graphs on Riemann surfaces, and (3) the geometry of mirror symmetry have engendered a rich subject at whose heart is the physics of gauge/string theories.
We review the various parts of this intricate story in some depth, for a mathematical audience without assumption of any knowledge of physics, emphasizing a plethora of results residing at the intersection between algebraic geometry, representation theory and number theory.
16 citations
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TL;DR: In this article, the authors established a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for k3 surfaces, and computed various examples of mirror moduli families for the moduli space of degree 2n polarized K3 surface.
Abstract: We establish a relationship between mirror symmetry for K3 surfaces and Arnold's strange duality for K3 surfaces. We compute various examples of mirror families. Among them the mirror moduli family for the moduli space of degree 2n polarized K3 surfaces. It turns out to be related to the moduli space of elliptic curves with level n.
16 citations