Quantization via mirror symmetry

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and (B, B, B) branes play an important role in the B-model approach to quantization.     

Ribbon graphs and mirror symmetry

Given a ribbon graph \(\Gamma \) with some extra structure, we define, using constructible sheaves, a dg category \(\mathrm {CPM}(\Gamma )\) meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by \(\Gamma .\) When \(\Gamma \) is appropriately decorated and admits a combinatorial “torus fibration with section,” we construct from \(\Gamma \) a one-dimensional algebraic stack \(\widetilde{X}_\Gamma \) with toric components. We prove that our model is equivalent to \(\mathcal {P}\mathrm {erf}(\widetilde{X}_\Gamma )\) , the dg category of perfect complexes on \(\widetilde{X}_\Gamma \) .     

On mirror symmetry in Wess-Zumino-Novikov-Witten models

We examine the problem of constructing N=2 superconformal algebras out of N=1 non-semi-simple affine Lie algebras. These N=2 superconformal theories share the property that the super Virasoro central charge depends only on the dimension of the Lie algebra. We find, in particular, a construction having a central charge c=9. This provides a possible internal space for string compactification and where mirror symmetry might be explored.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 55–63, July, 1995.     

Homological mirror symmetry for singularities of type D

We prove homological mirror symmetry for Lefschetz fibrations obtained as Sebastiani–Thom sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under Sebastiani–Thom summation of a polynomial of type D.     

Weighted blowups and mirror symmetry for toric surfaces

This paper explores homological mirror symmetry for weighted blowups of toric varietes. It will be shown that both the A-model and B-model categories have natural semi-orthogonal decompositions. An explicit equivalence of the right orthogonal categories will be shown for the case of toric surfaces.     

String cohomology of Calabi-Yau hypersurfaces via mirror symmetry

We propose a construction of string cohomology spaces for Calabi-Yau hypersurfaces that arise in Batyrev's mirror symmetry construction. The spaces are defined explicitly in terms of the corresponding reflexive polyhedra in a mirror-symmetric manner. We draw connections with other approaches to the string cohomology, in particular with the work of Chen and Ruan.     

Elliptic genera of toric varieties and applications to mirror symmetry

The paper contains a proof that elliptic genus of a Calabi-Yau manifold is a Jacobi form, finds in which dimensions the elliptic genus is determined by the Hodge numbers and shows that elliptic genera of a Calabi-Yau hypersurface in a toric variety and its mirror coincide up to sign. The proof of the mirror property is based on the extension of elliptic genus to Calabi-Yau hypersurfaces in toric varieties with Gorenstein singularities. Oblatum 12-V-1999 & 4-XI-1999?Published online: 21 February 2000     

Applications of homological mirror symmetry to hypergeometric systems: Duality conjectures

Homological mirror symmetry for crepant resolutions of Gorenstein toric singularities leads to a pair of conjectures on certain hypergeometric systems of PDEs. We explain these conjectures and verify them in some cases.     

Landau-Ginzburg/Calabi-Yau correspondence,global mirror symmetry and Orlov equivalence

We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following a proposal of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of $\widehat {\Gamma }$ -integral structures applied to the derived category of the hypersurface {W=0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.     

An integral structure in quantum cohomology and mirror symmetry for toric orbifolds

We introduce an integral structure in orbifold quantum cohomology associated to the K-group and the -class. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the Landau-Ginzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan's crepant resolution conjecture [Yongbin Ruan, The cohomology ring of crepant resolutions of orbifolds, in: Contemp. Math., vol. 403, Amer. Math. Soc., Providence, RI, 2006, pp. 117-126].     

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Given a smooth projective toric variety X, we construct an A_∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.