Let R be a non-commutative field. We prove that generic triples of flags in an m-dimensional R-vector space are described by flat R-line bundles on the honeycomb graph with (m-1)(m-2)/2 holes. Generalising this, we prove that the non-commutative mo
duli space X(m,S) of (twisted) framed flat R-vector bundles of rank m on a decorated surface S is birationally identified with the moduli spaces of (twisted) flat line bundles on a spectral surface $Sigma_Gamma$ assigned to certain bipartite graphs $Gamma$ on S. We introduce non-commutative cluster Poisson varieties related to bipartite ribbon graphs. They carry a canonical non-commutative Poisson structure. The result above just means that the space X(m, S) has a structure of a non-commutative cluster Poisson variety, equivariant under the action of the mapping class group of S. For bipartite graphs on a torus, we get the non-commutative dimer cluster integrable system. We develop a parallel dual story of non-commutative cluster A-varieties related to bipartite ribbon graphs. They carry a canonical non-commutative 2-form. The dual non-commutative moduli space A(m,S) of twisted decorated local systems on S carries a cluster A-variety structure, equivariant under the action of the mapping class group of S. The non-commutative cluster A-coordinates on the space A(m,S) are expressed as ratios of Gelfand-Retakh quasideterminants. In the case m=2 this recovers the Berenstein-Retakh non-commutative cluster algebras related to surfaces. We introduce stacks of admissible dg-sheaves on surfaces, and use them to give an alternative microlocal proof of the above results.
We propose a new axiom system for unitary quantum field theories on curved space-time backgrounds, by postulating that the partition function and the correlators extend analytically to a certain domain of complex-valued metrics. Ordinary Riemannian m
etrics are contained in the allowable domain, while Lorentzian metrics lie on its boundary.
We introduce the notion of multiplication kernels of birational and $D$-module type and give various examples. We also introduce the notion of a semi-classical multiplication kernel associated with an integrable system and discuss its quantization. F
inally, we discuss geometric and algebraic aspects of method of separation of variables, and describe hypothetically a cyclic $D$-module for the generalized multiplication kernels for Hitchin systems for groups $GL_r$.
We prove $L_{infty}$-formality for the higher cyclic Hochschild complex $chH$ over free associative algebra or path algebra of a quiver. The $chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau structure. We show th
at cohomologies of this complex are pure in case of free algebras (path algebras), concentrated in degree zero. It serves as a main ingredient for the formality proof. For any smooth algebra we choose a small qiso subcomplex in the higher cyclic Hochschild complex, which gives rise to a calculus of highly noncommutative monomials, we call them $xidelta$-monomials. The Lie structure on this subcomplex is combinatorially described in terms of $xidelta$-monomials. This subcomplex and a basis of $xidelta$-monomials in combination with arguments from Groebner bases theory serves for the cohomology calculations of the higher cyclic Hochschild complex. The language of $xidelta$-monomials in particular allows an interpretation of pre-Calabi-Yau structure as a noncommutative Poisson structure.
We prove that $p$-determinants of a certain class of differential operators can be lifted to power series over $mathbb{Q}$. We compute these power series in terms of monodromy of the corresponding differential operators.
We introduce the notion of analytic stability data on the Lie algebra of vector fields on a torus. We prove that the subspace of analytic stability data is open and closed in the topological space of all stability data. We formulate a general conject
ure which explains how analytic stability data give rise to resurgent series. This conjecture is checked in several examples.
We introduce a framework in noncommutative geometry consisting of a $*$-algebra $mathcal A$, a bimodule $Omega^1$ endowed with a derivation $mathcal Ato Omega^1$ and with a Hermitian structure $Omega^1otimes bar{Omega}^1to mathcal A$ (a noncommutativ
e Kahler form), and a cyclic 1-cochain $mathcal Ato mathbb C$ whose coboundary is determined by the previous structures. These data give moment map equations on the space of connections on an arbitrary finitely-generated projective $mathcal A$-module. As particular cases, we obtain a large class of equations in algebra (Kings equations for representations of quivers, including ADHM equations), in classical gauge theory (Hermitian Yang-Mills equations, Hitchin equations, Bogomolny and Nahm equations, etc.), as well as in noncommutative gauge theory by Connes, Douglas and Schwarz. We also discuss Nekrasovs beautiful proposal for re-interpreting noncommutative instantons on $mathbb{C}^nsimeq mathbb{R}^{2n}$ as infinite-dimensional solutions of Kings equation $$sum_{i=1}^n [T_i^dagger, T_i]=hbarcdot ncdotmathrm{Id}_{mathcal H}$$ where $mathcal H$ is a Hilbert space completion of a finitely-generated $mathbb C[T_1,dots,T_n]$-module (e.g. an ideal of finite codimension).
We give an explicit formula showing how the double Poisson algebra introduced in cite{VdB} appears as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan eq
uation on $Aoplus A^*$. Specific part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structures. The result holds for any associative algebra $A$ and emphasizes the special role of the fourth component of a pre-Calabi-Yau structure in this respect. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson brackets on representation spaces $({rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.
We discuss quantum analogues of minimal surfaces in Euclidean spaces and tori.
We introduce new invariants in equivariant birational geometry and study their relation to modular symbols and cohomology of arithmetic groups.
