Mirror symmetry

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

No Mirror Symmetry in Landau-Ginzburg Spectra!

Abstract: We use a recent classification of non-degenerate quasihomogeneous polynomials to construct all Landau-Ginzburg (LG) potentials for N=2 superconformal field theories with c=9 and calculate the corresponding Hodge numbers. Surprisingly, the resulting spectra are less symmetric than the existing incomplete results. It turns out that models belonging to the large class for which an explicit construction of a mirror model as an orbifold is known show remarkable mirror symmetry. On the other hand, half of the remaining 15\% of all models have no mirror partners. This lack of mirror symmetry may point beyond the class of LG-orbifolds.

On N=1 4d Effective Couplings for F-theory and Heterotic Vacua

Abstract: We show that certain superpotential and Kahler potential couplings of N=1 supersymmetric compactifications with branes or bundles can be computed from Hodge theory and mirror symmetry. This applies to F-theory on a Calabi-Yau four-fold and three-fold compactifications of type II and heterotic strings with branes. The heterotic case includes a class of bundles on elliptic manifolds constructed by Friedmann, Morgan and Witten. Mirror symmetry of the four-fold computes non-perturbative corrections to mirror symmetry on the three-folds, including D-instanton corrections. We also propose a physical interpretation for the observation by Warner that relates the deformation spaces of certain matrix factorizations and the periods of non-compact 4-folds that are ALE fibrations.

Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds

Abstract: Let X be a Calabi-Yau 3-fold, T = D b (coh(X)) the derived category of coherent sheaves on X, and Stab(T) the complex manifold of Bridgeland stability conditions on T. It is conjectured that one can define invariants J α (Z, P) e Q for (Z, P) ∈ Stab(T) and a ∈ K(T) generalizing Donaldson-Thomas invariants, which "count" (Z, P)-semistable (complexes of) coherent sheaves on X, and whose transformation law under change of (Z, P) is known. This paper explains how to combine such invariants J α (Z, P), if they exist, into a family of holomorphic generating functions F α : Stab(T) → C for a ∈ K(T). Surprisingly, requiring the F α to be continuous and holomorphic determines them essentially uniquely, and implies they satisfy a p.d.e., which can be interpreted as the flatness of a connection over Stab(T) with values in an infinite-dimensional Lie algebra L. The author believes that underlying this mathematics there should be some new physics, in String Theory and Mirror Symmetry. String Theorists are invited to work out and explain this new physics.

Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory

Abstract: We introduce several families of $\mathcal{N}=(2,2)$ UV boundary conditions in 3d $\mathcal N=4$ gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d $\mathcal{N}=4$ gauge theories and their boundary conditions, we propose a physical origin for symplectic duality - an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces

Abstract: We investigate quantum corrections to the hypermultiplet moduli space \( \mathcal{M} \) in Calabi-Yau compactifications of type II string theories, with particular emphasis on instanton effects from Euclidean NS5-branes. Based on the consistency of D- and NS5-instanton corrections, we determine the topology of the hypermultiplet moduli space at fixed string coupling, as previewed in [1]. On the type IIB side, we compute corrections from (p, k)-fivebrane instantons to the metric on \( \mathcal{M} \) (specifically, the correction to the complex contact structure on its twistor space \( \mathcal{Z} \)) by applying S-duality to the D-instanton sum. For fixed fivebrane charge k, the corrections can be written as a non-Gaussian theta series, whose summand for k = 1 reduces to the topological A-model amplitude. By mirror symmetry, instanton corrections induced from the chiral type IIA NS5-brane are similarly governed by the wave function of the topological B-model. In the course of this investigation we clarify charge quantization for coherent sheaves and find hitherto unnoticed corrections to the Heisenberg, monodromy and S-duality actions on \( \mathcal{M} \), as well as to the mirror map for Ramond-Ramond fields and D-brane charges.