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We define an action of the (double of) Cohomological Hall algebra of Kontsevich and Soibelman on the cohomology of the moduli space of spiked instantons of Nekrasov. We identify this action with the one of the affine Yangian of $\mathfrak{gl}(1)$. Based on that we derive the vertex algebra at the corner $\mathcal{W}_{r_1,r_2,r_3}$ of Gaiotto and Rapcak. We conjecture that our approach works for a big class of Calabi-Yau categories, including those associated with toric Calabi-Yau $3$-folds., Comment: 72 pages, 4 figures, v2: Added some clarifications and updated references 73 pages, 4 figures, v3: Corrected typos and clarified some minor points

A W-algebra action is constructed via Hecke transformations on the equivariant Borel–Moore homology of the Hilbert scheme of points on a nonreduced plane in three-dimensional affine space. The resulting W-module is then identified to the vacuum module. The construction is based on a generalization of the ADHM construction as well as the W-action on the equivariant Borel–Moore homology of the moduli space of instantons constructed by Schiffmann and Vasserot.

Let $X$ be a smooth projective curve over a field of characteristic zero and let $D$ be a non-empty set of rational points of $X$. We calculate the motivic classes of moduli stacks of semistable parabolic bundles with connections on $(X,D)$ and motivic classes of moduli stacks of semistable parabolic Higgs bundles on $(X,D)$. As a by-product we give a criteria for non-emptiness of these moduli stacks, which can be viewed as a version of the Deligne-Simpson problem.

We propose a new approach to the topological recursion of Eynard-Orantin based on the notion of Airy structure, which we introduce in the paper. We explain why Airy structure is a more fundamental object than the one of the spectral curve. We explain how the concept of quantization of Airy structure leads naturally to the formulas of topological recursion as well as their generalizations. The notion of spectral curve is also considered in a more general framework of Poisson surfaces endowed with foliation. We explain how the deformation theory of spectral curves is related to Airy structures. Few other topics (e.g. the Holomorphic Anomaly Equation) are also discussed from the general point of view of Airy structures.

Let X be a smooth projective curve over a field of characteristic zero. We calculate the motivic class of the moduli stack of semistable Higgs bundles on X. We also calculate the motivic class of the moduli stack of vector bundles with connections by showing that it is equal to the class of the stack of semistable Higgs bundles of the same rank and degree zero. We follow the strategy of Mozgovoy and Schiffmann for counting Higgs bundles over finite fields. The main new ingredient is a motivic version of a theorem of Harder about Eisenstein series claiming that all vector bundles have approximately the same motivic class of Borel reductions as the degree of Borel reduction tends to $-\infty$., Comment: Minor corrections and improvements; 48 pages

We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the “dimensional reduction” from 3-dimensional Calabi–Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson–Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi–Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi–Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).

We study algebraic structures ($L_\infty$ and $A_\infty$-algebras) introduced by Gaiotto, Moore and Witten in their recent work devoted to certain supersymmetric 2-dimensional massive field theories. We show that such structures can be systematically produced in any number of dimensions by using the geometry of secondary polytopes, esp. their factorization properties. In particular, in 2 dimensions, we produce, out of a polyhedral "coefficient system", a dg-category $R$ with a semi-orthogonal decomposition and an $L_\infty$-algebra $\mathfrak g$. We show that $\mathfrak g$ is quasi-isomorphic to the ordered Hochschild complex of $R$, governing deformations preserving the semi-orthogonal decomposition. This allows us to give a more precise mathematical formulation of the (conjectural) alternative description of the Fukaya-Seidel category of a Kahler manifold endowed with a holomorphic Morse function., 75 pages, 28 figures

The aim of this paper is to discuss a class of representations of Cohomological Hall algebras related to the notion of framed stable object of a category. The paper is an extended version of the talk the author gave at the workshop on Donaldson–Thomas invariants at the University Paris-7 in June 2013 and at the conference “Algebra, Geometry, Physics” dedicated to Maxim Kontsevich (June 2014, IHES). Because of the origin of the paper it contains more speculations than proofs.

Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is given a Calabi-Yau threefold interpretation. The motivic Donaldson-Thomas theory developed by M. Kontsevich and the third author then yields natural motivic invariants for algebraic knots. This construction is motivated by previous work of V. Shende, C. Vafa and the first author on the large $N$-duality derivation of the above conjecture.

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

We study refined and motivic wall-crossing formulas in N = 2 supersymmetric gauge theories with SU(2) gauge group and N _f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that “refined = motivic.”

Homological mirror symmetry aims to explain the phenomenon of mirror symmetry in the language of A∞-categories and their deformation theory. In these notes I discuss various aspects of this approach from the point of view of noncommutative algebraic geometry in the tensor category of graded vector spaces.

We discuss the possible relationship of homological mirror symmetry with deformation quantization. We speculate that after certain nonlinear “twist” the Fukaya category becomes equivalent to the category of holonomic modules over a quantized algebra of functions.

The paper is devoted to the notion of wall-crossing structure and its constructions and applications in various situations. It is motivated by our previous work on Mirror Symmetry and Donaldson–Thomas invariants (see [30, 31, 35, 36]) where examples of wall-crossing structures appeared for the first time.

We suggest to compactify the universal covering of the moduli space of complex structures by noncommutative spaces. The latter are described by certain categories of sheaves with connections which are flat along foliations. In the case of Abelian varieties, this approach gives quantum tori as a noncommutative boundary of the moduli space. Relations to mirror symmetry, modular forms and deformation theory are discussed.

BPS quivers for N=2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric Calabi-Yau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius D-brane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological Donaldson-Thomas invariants. We verify the conjectured absence of BPS states with "exotic" SU(2)_R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determine the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative Donaldson-Thomas invariants for framed quiver representations., Comment: 114 pages; v2:minor corrections

The universal quantumR-matrix is obtained in the case of the affine Kac-Moody Lie algebra sl(2).

Some recent results in the representation theory of the algebras of functions on compact quantum groups are presented. Applications to different subjects are given.

The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R,\(u \in \Lambda ^2 \mathfrak{h}_R\), and\(\mathfrak{h}_R\) is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a,u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universalR-matrix, are given.

We discuss examples of non-commutative spaces over non-archimedean fields. Those include non-commutative and quantum affinoid algebras, quantized K3 surfaces and quantized locally analytic p-adic groups.

We develop a geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. The geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from the geometric point of view. The chapter contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of the Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne’s conjecture which includes both Hochschild chains and cochains. We conclude the chapter with the description of an action of the PROP of singular chains of the topological PROP of two-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with scalar product (this action is more or less equivalent to the structure of two-dimensional Topological Field Theory associated with an “abstract” Calabi–Yau manifold).

In this paper we propose a way to construct an analytic space over a non-archimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of the Homological Mirror conjecture and the geometric Strominger-Yau-Zaslow conjecture. In particular, we glue from “flat pieces” an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group SO(1, 18) by piecewise-linear transformations on the two-dimensional sphere S 2 equipped with naturally defined singular affine structure.

In this paper we discuss two major conjectures in Mirror Symmetry: Strominger-Yau-Zaslow conjecture about torus fibrations, and the homological mirror conjecture (about an equivalence of the Fukaya category of a Calabi-Yau manifold and the derived category of coherent sheaves on the dual Calabi-Yau manifold). Our point of view on the origin of torus fibrations is based on the standard differential-geometric picture of collapsing Riemannian manifolds as well as analogous considerations for Conformal Field Theories. It seems to give a description of mirror manifolds much more transparent than the one in terms of D-branes. Also we make an attempt to prove the homological mirror conjecture using the torus fibrations. In the case of abelian varieties, and for a large class of Lagrangian submanifolds, we obtain an identification of Massey products on the symplectic and holomorphic sides. Tools used in the proof are of a mixed origin: not so classical Morse theory, homological perturbation theory and non-archimedean analysis., Comment: version accepted for publication