Narrow Results

Let $X$ be a smooth projective toric variety with $k$ ample line bundles. Let $Z$ be the zero locus of $k$ generic sections. It is well known that the ambient quantum ${𝒟}$-module of $Z$ is cyclic i.e. is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of $X$ and the line bundles. This description can be seen as a “left cancellation procedure”. We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum ${𝒟}$-module.

In this paper, we present explicit toric construction of moduli space of quasi maps from ${\mathbb{P}}^1$ with two marked points to ${\mathbb{P}}^1\times {\mathbb{P}}^1$, which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.

In this paper, we propose a geometrical proof of the generalized mirror transformation of genus $0$ Gromov–Witten invariants of degree $k$ hypersurface in $C{P}^{N-1}$.

Narasimhan–Ramanan branes, introduced by the authors in a previous paper, consist of a family of (BBB)-branes inside the moduli space of Higgs bundles, and a family of complex Lagrangian subvarieties. It was conjectured that these complex Lagrangian subvarieties support the (BAA)-branes that are mirror dual to the Narasimhan–Ramanan (BBB)-branes. In this paper, we show that the support of these branes intersects nontrivially the locus of wobbly Higgs bundles.

We discuss an quantum mechanics with or without a central charge. A representation is constructed with the number of bosonic degrees of freedom less than one-half of the fermionic degrees of freedom. We suggest a systematic method of reducing the bosonic degrees of freedom called "dynamical reduction". Our consideration opens a problem of a general classification of nonstandard representations of superalgebra.

According to the open-closed mirror symmetry and generalized Gel'fand–Kapranov–Zelerinski (GKZ) system, we calculate off-shell D-brane/F-theory effective superpotentials of four compact Calabi–Yau (CY) manifolds by integrating special periods of subsystem, and extracting the open Ooguri–Vafa invariants from the expansion of off-shell D-brane superpotentials.

This is an exposition of recent progress in the categorical approach to D-brane physics. I discuss the physical underpinnings of the appearance of homotopy categories and triangulated categories of D-branes from a string field theoretic perspective, and with a focus on applications to homological mirror symmetry.

In this paper, we explicitly derive the generalized mirror transformation of quantum cohomology of general type projective hypersurfaces, proposed in our previous article, as an effect of coordinate change of the virtual Gauss–Manin system.

We consider topological sigma models with generalized Kähler target spaces. The mirror map is constructed explicitly for a special class of target spaces and the topological A and B model are shown to be mirror pairs in the sense that the observables, the instantons and the anomalies are mapped to each other. We also apply the construction to open topological models and show that A branes are mapped to B branes. Furthermore, we demonstrate a relation between the field strength on the brane and a two-vector on the mirror manifold.

We argue that presymmetry, a hidden predynamical electroweak quark–lepton symmetry that explains the fractional charges and triplication of families, must be extended beyond the Standard Model as to have a residual presymmetry that embraces partner particles and includes the strong sector, so accounting for the twin or mirror partners proposed to alleviate the naturalness problem of the weak scale. It leads to the full duplication of fermions and gauge bosons of the Standard Model independently of the ultraviolet completion of the theory, even if the Higgs particle is discarded by experiment, which adds robustness to twin and mirror symmetries. The established connection is so strongly motivated that the search for twin or mirror matter becomes the possible test of presymmetry. If the physics beyond the Standard Model repairs its left–right asymmetry, mirror symmetry should be the one realized in nature.

We explicitly apply localization results to study the interpolation between three- and two-dimensional mirror symmetries for Abelian gauge theories with four supercharges. We first use the ellipsoid partition functions to verify the mirror symmetry between a pair of general three-dimensional 𝒩 = 2 Abelian Chern–Simons quiver gauge theories. These expressions readily factorize into holomorphic blocks and their antiholomorphic copies, so we can also obtain the partition functions on S¹×S² via fusion procedure. We then demonstrate S¹×S² partition functions for the three-dimensional Abelian gauge theories can be dimensionally reduced to the S² partition functions of 𝒩 = (2, 2) GLSM and Landau–Ginzburg model for the corresponding two-dimensional mirror pair, as anticipated previously in M. Aganagic et al., J. High Energy Phys.0107, 022 (2001). We also comment on the analogous interpolation for the non-Abelian gauge theories and compute the K-theory vortex partition function for a simple limit to verify the prediction from holomorphic block.

In this paper, we extend the non-Abelian mirror proposal of two of the authors from two-dimensional gauge theories with connected gauge groups to the case of $O(k)$ gauge groups with discrete theta angles. We check our proposed extension by counting and comparing vacua in mirrors to the known dual two-dimensional $(S)O(k)$ gauge theories. The mirrors in question are Landau–Ginzburg orbifolds, and for mirrors to $O(k)$ gauge theories, the critical loci of the mirror superpotential often intersect fixed-point loci, so that to count vacua, one must take into account the twisted sector contributions. This is a technical novelty relative to the mirrors of gauge theories with connected gauge groups, for which critical loci do not intersect fixed-point loci and so no orbifold twisted sector contributions are pertinent. The vacuum computations turn out to be a rather intricate test of the proposed mirrors, in particular as untwisted sector states in the mirror to one theory are often exchanged with twisted sector states in the mirror to the dual. In cases with nontrivial IR limits, we also check that the central charges computed from the Landau–Ginzburg mirrors match those expected for the IR SCFTs.

This survey covers recent developments on the geometry and physics of Looijenga pairs, namely pairs $(X,D)$ with $X$ a complex algebraic surface and $D$ a singular anticanonical divisor in it. I will describe a surprising web of correspondences linking together several a priori distant classes of enumerative invariants associated to $(X,D)$, including the log Gromov–Witten invariants of the pair, the Gromov–Witten invariants of an associated higher dimensional Calabi–Yau variety, the open Gromov–Witten invariants of certain special Lagrangians in toric Calabi–Yau threefolds, the Donaldson–Thomas theory of a class of symmetric quivers, and certain open and closed BPS-type invariants. I will also discuss how these correspondences can be effectively used to provide a complete closed-form solution to the calculation of all these invariants.

In this paper, we briefly survey some developments in gauged linear sigma models (GLSMs). Specifically, we give an overview of progress on constructions of GLSMs for various geometries, GLSM-based computations of quantum cohomology, quantum sheaf cohomology, and quantum K theory rings, as well as two-dimensional abelian and nonabelian mirror constructions.

In this paper, we describe non-Abelian T-dualities for a two-dimensional (2D) gauged linear sigma model (GLSM) with $\mathcal{𝒩}=2$ supersymmetry (SUSY). We start with the case of left and right SUSY $(2,2)$, $U(1)$ gauge group, and global non-Abelian symmetries. Our analysis applies to the specialization of the GLSM with the global group $SU(2)\times SU(2)$, whose original model is the resolved conifold. We analyze the dual model and compute the periods on the dual geometry, matching the effective potential for the $U(1)$ gauge field, and discussing the coincident singularity at the conifold point. We further present the non-Abelian T-dualization of models with $(0,2)$ SUSY, analyzing the case of global symmetry $SU(2)$. In this case, the full non-Abelian duality can be solved. The dual geometries with and without nonperturbative corrections to the superpotential coincide. The structure of the dual nonperturbative corrections is determined based on symmetry arguments. The non-Abelian T-dualities reviewed here lead to potential new symmetries between physical theories.

We study the problem of computing Gopakumar–Vafa (GV) invariants for multiparameter families of symmetric Calabi–Yau threefolds admitting flops to diffeomorphic manifolds. There are infinite Coxeter groups, generated by permutations and flops, that act as symmetries on the GV-invariants of these manifolds. We describe how these groups are related to symmetries in GLSMs, and the existence of multiple mirrors. Some representation theory of these Coxeter groups is also discussed. The symmetries provide an infinite number of relations between the GV-invariants for each fixed genus. This remarkable fact is of assistance in obtaining higher-genus invariants via the BCOV recursion. This paper is based on joint work with Philip Candelas and Xenia de la Ossa.

We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.

In this paper, we generalize Walcher's computation of the open Gromov–Witten invariants of the quintic hypersurface to Fano and Calabi–Yau projective hypersurfaces. Our main tool is the open virtual structure constants. We also propose the generalized mirror transformation for the open Gromov–Witten invariants, some parts of which are proven explicitly. We also discuss possible modification of the multiple covering formula for the case of higher-dimensional Calabi–Yau manifolds. The generalized disk invariants for some Calabi–Yau and Fano manifolds are shown and they are certainly integers after resummation by the modified multiple covering formula. This paper also contains the direct integration method of the period integrals for higher-dimensional Calabi–Yau hypersurfaces in the Appendix.

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface. Regardless of monodromy, the middle degree homology of intersection spaces is always a subspace of the homology of the deformation, yet itself contains the middle intersection homology group, the ordinary homology of the singular space, and the ordinary homology of the regular part.

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory.