The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential function, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of doubly stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors.

In this paper we describe some of the constructions of FJRW theory. We also briefly describe its relation to Saito-Givental theory via Landau-Ginzburg mirror symmetry and its relation to Gromov-Witten theory via the Landau-Ginzburg/Calabi-Yau correspondence. We conclude with a discussion of some of the recent results in the field, including the gauged linear sigma model, which is expected to provide a geometric framework for unifying many of these ideas.

The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold $F$ defines a characteristic class $A_F$ of $F$, called the principal asymptotic class. Gamma conjecture [29] of Vasily Golyshev and the present authors claims that the principal asymptotic class $A_F$ equals the Gamma class $\widehat{\Gamma}_F$ associated to Euler's $\Gamma$-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.

In this survey, the relationship between local and relative BPS state counts is explored in light of a conjecture on log mirror symmetry by N. Takahashi. This is based on the paper [14] by the author and the integrality result of [14] is reported.

In these notes, we provide an introduction to the hemisphere partition function of 2d $(2,2)$ supersymmetric gauge theories, and discuss its relation to the "D-brane central charge" which were studied in superstring theory, in 2d supersymmetric quantum field theory, and in topological string theory. We also discuss relation to "macroscopic loop" in matrix models. They are mostly reviews of the work by the authors, but contains some new results such as the partition function for a rotated supersymmetry as well as the differential equations.

Hurwitz spaces are spaces of meromorphic functions on algebraic curves. B. Dubrovin introduced Frobenius structures on Hurwitz spaces, which serve as one of the most spectacular examples of such structures. On the other hand, Hurwitz spaces admit natural compactification by stable maps. The main goal of the present paper is to formulate a question concerning the behavior of Dubrovin's Frobenius structures on Hurwitz spaces on the boundary of the compactification. Sample computations justifying the question are given.

This note comes out of the author's lecture presented at the workshop Primitive forms and related subjects, Feb 10-14 2014.

The text is devoted to explanation of the concept of Topological Quantum Field Theory (TQFT), its application to homological algebra and to the relation with the theory of good section from K. Saito's theory of Primitive forms. TQFT is explained in Dirac-Segal framework, 1 dimensional examples are explained in detail. As a first application we show how it can be used in explicit construction of reduction of $\infty$-structure after contraction of a subcomplex. Then we explain Associativity and Commutativity equations using this language. We use these results to construct solutions to Commutativity equations and find a new proof of for the fact that tree level BCOV theory solved Oriented Associativity equations.

The paper [2] proposed a construction of a twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularity. The main difficulty in extending the above construction to an arbitrary isolated singularity is in the so called phase factors – the scalar functions produced by composing two vertex operators. They are certain family of multivalued analytic functions on the space of miniversal deformations. The first result in this paper is an explicit formula for the unperturbed phase factors in terms of the classical monodromy operator and the polylogorithm functions. Our second result is that with respect to the deformation parameters the phase factors are analytic functions on the monodromy covering space.

This note discusses certain deformation theoretic aspects of the symplectic topology of pairs $(Y,D)$, where $Y$ is a smooth projective variety and $D$ is an ample simple normal crossings divisor. Motivated by homological mirror symmetry, we formulate a precise prediction concerning the symplectic cohomology of the affine variety $Y \setminus D$ for a wide class of examples. We show how our answer unites various mirror symmetry predictions in the literature.

We briefly describe a deep relationship known as the Landau–Ginzburg/Calabi–Yau correspondence for the famous mirror quintic via global mirror symmetry.

This is a report on the calculation of some quantum invariants, including Gromov-Witten invariants and FJRW invariants, via tautological relations on the moduli space of stable pointed curves.

It is one of the most important problems in mirror symmetry to obtain functorially Frobenius manifolds from smooth compact Calabi–Yau $A_\infty$-categories. This paper gives an approach to this problem based on the theory of primitive forms and, in particular, reduces it to a formality conjecture of certain homotopy algebra. Under this formality conjecture, a formal primitive form for a non-negatively graded connected smooth compact Calabi–Yau dg algebra can be constructed, which enable us to have a formal Frobenius manifold.