Related papers: Fukaya Algebra over $\mathbb{Z}$

When $A$ in the Kauffman bracket skein relation is a primitive $2N$th root of unity, where $N\geq 3$ is odd, the Kauffman bracket skein algebra $K_N(F)$ of a finite type surface $F$ is a ring extension of the $SL_2\mathbb{C}$-characters…

In this paper we show how the rich algebraic formalism of Eliashberg-Givental-Hofer's symplectic field theory (SFT) can be used to define higher algebraic structures in Hamiltonian Floer theory. Using the SFT of Hamiltonian mapping tori we…

We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an…

In these notes we construct a quantization functor, associating an Hilbert space H(V) to a finite dimensional symplectic vector space V over a finite field F_q. As a result, we obtain a canonical model for the Weil representation of the…

The goal of these notes is to give a short introduction to Fukaya categories and some of their applications. The first half of the text is devoted to a brief review of Lagrangian Floer (co)homology and product structures. Then we introduce…

The 2016 papers of J. Solomon and S. Tukachinsky use bounding chains in Fukaya's $A_{\infty}$-algebras to define numerical disk counts relative to a Lagrangian under certain regularity assumptions on the moduli spaces of disks. We present a…

Let L be an exact Lagrangian submanifold inside the cotangent bundle of a closed manifold N. We prove that if N satisfies a mild homotopy assumption then the image of \pi_2(L) in \pi_2(N) has finite index. We make no assumption on the…

This article introduces and provides the mathematical foundation of the open string Floer theory of Landau-Ginzburg model viaWitten equation. We introduce the concept of regular tame exact Landau-Ginzburg system on a noncompact Kaehler…

We study Dehn twists along Lagrangian submanifolds that are finite quotients of spheres. We decribe the induced auto-equivalences to the derived Fukaya category and explain its relation to twists along spherical functors.

We give an affirmative answer to the question whether there exist Lie algebras for suitable closed subgroups of the unitary group $U(\mathcal{H})$ in a Hilbert space $\mathcal{H}$ with $U(\mathcal{H})$ equipped with the strong operator…

We construct the cyclic open--closed map for the big (i.e., bulk-deformed) relative Fukaya category, in the semipositive case, and show that it is a morphism of `polarized variations of semi-infinite Hodge structures'. We also give a…

This article is an expanded version of talks given by the authors in Oberwolfach, Bochum, and at the Fano Conference in Torino. Some new results (e. g. the material concerning flag varieties, Quot spaces over $\P^1$, and the generalized…

Recently Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a…

Given an algebraic hypersurface $H=f^{-1}(0)$ in $(\mathbb{C}^*)^n$, homological mirror symmetry relates the wrapped Fukaya category of $H$ to the derived category of singularities of the mirror Landau-Ginzburg model. We propose an enriched…

We generalize a result of M. Kapranov, O. Schiffmann, and E. Vasserot by showing that, for a number field $K$ with class number one, the spherical Hall algebra of $\overline{\operatorname{Spec}(\mathcal{O}_K)}$, where $\mathcal{O}_K$ is the…

In this article we introduce a generalization of the Khovanov--Lauda Rouquier algebras, the electric KLR algebras. These are superalgebras which connect to super Brauer algebras in the same way as ordinary KLR-algebras of type $A$ connect…

Starting from a recently-introduced algebraic structure on spin foam models, we define a Hopf algebra by dividing with an appropriate quotient. The structure, thus defined, naturally allows for a mirror analysis of spin foam models with…

We introduce the shifted quantum affine algebras. They map homomorphically into the quantized $K$-theoretic Coulomb branches of $3d\ {\mathcal N}=4$ SUSY quiver gauge theories. In type $A$, they are endowed with a coproduct, and they act on…

We define a diagram associated to any algebraic connection on a vector bundle on a Zariski open subset of the Riemann sphere, extending the definition of Boalch-Yamakawa to the general case featuring several irregular singularities,…

A universal symmetry algebra organizing the gravitational phase space has been recently found. It corresponds to the subset of diffeomorphisms that become physical at corners -- codimension-$2$ surfaces supporting Noether charges. It…