nn-th parafermion WN\mathcal{W}_N characters from U(N)U(N) instanton counting on C2/Zn{\mathbb {C}}^2/{\mathbb {Z}}_n

We propose, following the AGT correspondence, how the $\mathcal{W}^{\, para}_{N, n}$ ($n$-th parafermion $\mathcal{W}_N$) minimal model characters are obtained from the $U(N)$ instanton counting on ${\mathbb {C}}^2/{\mathbb {Z}}_n$ with $\Omega$-deformation by imposing specific conditions which remove the minimal model null states.Comment: 25 pages, 2 figures; v2: minor changes, references added; I would like to dedicate this paper to the memory of Professor Omar Fod

Macdonald topological vertices and brane condensates

We show, in a number of simple examples, that Macdonald-type $qt$-deformations of topological string partition functions are equivalent to topological string partition functions that are without $qt$-deformations but with brane condensates, and that these brane condensates lead to geometric transitions.Comment: 23 pages, 5 figures. v2: minor changes, published versio

Determinantal Calabi-Yau varieties in Grassmannians and the Givental II-functions

We examine a class of Calabi-Yau varieties of the determinantal type in Grassmannians and clarify what kind of examples can be constructed explicitly. We also demonstrate how to compute their genus-0 Gromov-Witten invariants from the analysis of the Givental $I$-functions. By constructing $I$-functions from the supersymmetric localization formula for the two dimensional gauged linear sigma models, we describe an algorithm to evaluate the genus-0 A-model correlation functions appropriately. We also check that our results for the Gromov-Witten invariants are consistent with previous results for known examples included in our construction.Comment: 50 page

Local B-model Yukawa couplings from A-twisted correlators

Using the exact formula for the A-twisted correlation functions of the two dimensional $\mathcal{N}=(2,2)$ gauged linear sigma model, we reconsider the computation of the B-model Yukawa couplings of the local toric Calabi-Yau varieties. Our analysis is based on an exact result that has been evaluated from the supersymmetric localization technique and careful treatment of its application. We provide a detailed description of a procedure to investigate the local B-model Yukawa couplings and also test our prescription by comparing the results with known expressions evaluated from the local mirror symmetry approach. In particular, we find that the ambiguities of classical intersection numbers of a certain class of local toric Calabi-Yau varieties discovered previously can be interpreted as degrees of freedom of the twisted mass deformations.Comment: 29 pages, 7 figures. v2: minor changes. v3: minor changes, published versio

Reconstructing GKZ via topological recursion

In this article, a novel description of the hypergeometric differential equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ equation) for Givental's $J$-function in the Gromov-Witten theory will be proposed. The GKZ equation involves a parameter $\hbar$, and we will reconstruct it as the WKB expansion from the classical limit $\hbar\to 0$ via the topological recursion. In this analysis, the spectral curve (referred to GKZ curve) plays a central role, and it can be defined as the critical point set of the mirror Landau-Ginzburg potential. Our novel description is derived via the duality relations of the string theories, and various physical interpretations suggest that the GKZ equation is identified with the quantum curve for the brane partition function in the cohomological limit. As an application of our novel picture for the GKZ equation, we will discuss the Stokes matrix for the equivariant $\mathbb{C}\textbf{P}^{1}$ model and the wall-crossing formula for the total Stokes matrix will be examined. And as a byproduct of this analysis we will study Dubrovin's conjecture for this equivariant model.Comment: 66 pages, 13 figures, 6 tables; v2: new subsections added, minor revisions, typos corrected; v3: minor revisions, typos correcte

Singular vector structure of quantum curves

We show that quantum curves arise in infinite families and have the structure of singular vectors of a relevant symmetry algebra. We analyze in detail the case of the hermitian one-matrix model with the underlying Virasoro algebra, and the super-eigenvalue model with the underlying super-Virasoro algebra. In the Virasoro case we relate singular vector structure of quantum curves to the topological recursion, and in the super-Virasoro case we introduce the notion of super-quantum curves. We also discuss the double quantum structure of the quantum curves and analyze specific examples of Gaussian and multi-Penner models.Comment: 33 pages; proceedings of the 2016 AMS von Neumann Symposiu