Mirror symmetry

About: Mirror symmetry is a research topic. Over the lifetime, 2422 publications have been published within this topic receiving 90786 citations.

Affine Manifolds, Log Structures, and Mirror Symmetry

Abstract: We outline work in progress suggesting an algebro-geometric version of the Strominger-Yau-Zaslow conjecture. We define the notion of a toric degeneration, a special case of a maximally unipotent degeneration of Calabi-Yau manifolds. We then show how in this case the dual intersection complex has a natural structure of an affine manifold with singularities. If the degeneration is polarized, we also obtain an intersection complex, also an affine manifold with singularities, related by a discrete Legendre transform to the dual intersection complex. Finally, we introduce log structures as a way of reversing this construction: given an affine manifold with singularities with a suitable polyhedral decomposition, we can produce a degenerate Calabi-Yau variety along with a log structure. Hopefully, in interesting cases, this object will have a well-behaved deformation theory, allowing us to use the discrete Legendre transform to construct mirror pairs of Calabi-Yau manifolds. We also connect this approach to the topological form of the Strominger-Yau-Zaslow conjecture.

Homogeneous coordinate rings and mirror symmetry for toric varieties

Abstract: In this paper we give some evidence for M Kontsevich’s homological mirror symmetry conjecture [13] in the context of toric varieties. Recall that a smooth complete toric variety is given by a simplicial rational polyhedral fan  such that jj D R and all maximal cones are non-singular (Fulton [10, Section 2.1]). The convex hull of the primitive vertices of the 1–cones of  is a convex polytope which we denote by P , containing the origin as an interior point, and may be thought of as the Newton polytope of a Laurent polynomial W W .C/ ! C. This Laurent polynomial is the Landau–Ginzburg mirror of X .

Holomorphic Blocks in Three Dimensions

Abstract: We decompose sphere partition functions and indices of three-dimensional N=2 gauge theories into a sum of products involving a universal set of "holomorphic blocks". The blocks count BPS states and are in one-to-one correspondence with the theory's massive vacua. We also propose a new, effective technique for calculating the holomorphic blocks, inspired by a reduction to supersymmetric quantum mechanics. The blocks turn out to possess a wealth of surprising properties, such as a Stokes phenomenon that integrates nicely with actions of three-dimensional mirror symmetry. The blocks also have interesting dual interpretations. For theories arising from the compactification of the six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a basis of wavefunctions in analytically continued Chern-Simons theory on M. For theories engineered on branes in Calabi-Yau geometries, the blocks offer a non-perturbative perspective on open topological string partition functions.

to complex geometry

Abstract: We construct from a real ane manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect that our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods.

Theory of interacting topological crystalline insulators

Abstract: We study the effect of electron interactions in topological crystalline insulators (TCIs) protected by mirror symmetry, which are realized in the SnTe material class and host multivalley Dirac fermion surface states. We find that interactions reduce the integer classification of noninteracting TCIs in three dimensions, indexed by the mirror Chern number, to a finite group ${Z}_{8}$. In particular, we explicitly construct a microscopic interaction Hamiltonian to gap eight flavors of Dirac fermions on the TCI surface, while preserving the mirror symmetry. Our construction builds on interacting edge states of $U(1)\ifmmode\times\else\texttimes\fi{}{Z}_{2}$ symmetry-protected topological phases of fermions in two dimensions, which we classify. Our work reveals a deep connection between three-dimensional topological phases protected by spatial symmetries and two-dimensional topological phases protected by internal symmetries.