Formal deformations, contractions and moduli spaces of Lie algebras

算子代数的模问题

Moduli problems for operadic algebras

本文推广了Pridham和Lurie关于特征零下形式模问题与李代数等价的定理，建立了Koszul算子代数参数化的形式模问题与其Koszul对偶算子代数的对应关系，并给出了pre-Lie结构对应置换形式模问题及算子形式模问题对应增广算子等实例。

Maurer-Cartan moduli and models for function spaces

We set up a formalism of Maurer-Cartan moduli sets for L-infinity algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of Chevalley-Eilenberg and Harrison cohomology. We apply the developed technology to construct rational homotopy models for function spaces.

Computations in formal symplectic geometry and characteristic classes of moduli spaces

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spher…

Formal weights in Kontsevich's formality construction and multiple zeta values

We construct a functor that associates to any dg cooperad of dg commutative algebras (satisfying some conditions) an augmented commutative algebra. When applied to the cohomology operad of Francis Brown's moduli spaces it produces an algebra that formally models the algebra of multiple zeta values. We prove that there is an injection from the graded dual of the Grothendieck-Teichmueller Lie algebra into the indecomposables of the algebra associated to the Gerstenhaber cooperad, and that there is a morphism from the algebra associated to Brown's moduli spaces to the algebra associated to the Ge…

The moduli space of N=1 superspheres with tubes and the sewing operation

Within the framework of complex supergeometry and motivated by two-dimensional genus-zero holomorphic N=1 superconformal field theory, we define the moduli space of N=1 genus-zero super-Riemann surfaces with oriented and ordered half-infinite tubes, modulo superconformal equivalence. We define a sewing operation on this moduli space which gives rise to the sewing equation and normalization and boundary conditions. To solve this equation, we develop a formal theory of infinitesimal N=1 superconformal transformations based on a representation of the N=1 Neveu-Schwarz algebra in terms of superder…

Quantization of Poisson structures on $\R^2$

An 'isomorphism' between the 'moduli space' of star products on $\R^2$ and the 'moduli space' of all formal Poisson structures on $\R^2$ is established.

Moduli of quantum Riemannian geometries on <= 4 points

We classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously. The full moduli space is found for $\le 3$ points, and a restricted moduli space for 4 points. The topological part of the moduli space is found for $\le 9$ points based on the known atlas of regular graphs. We also discuss aspects of the quantum theory defined by functional integration.

Quantum Moduli Spaces of Flat Connections

Using the formalism of discrete quantum group gauge theory, one can construct the quantum algebras of observables for the Hamiltonian Chern-Simons model. The resulting moduli algebras provide quantizations of the algebra of functions on the moduli spaces of flat connections on a punctured 2-dimensional surface. In this note we describe some features of these moduli algebras with special emphasis on the natural action of mapping class groups. This leads, in particular, to a closed formula for representations of the mapping class groups on conformal blocks.

$\mathbb{E}_n$-单向量范畴的变形理论

Deformation Theory of $\mathbb{E}_n$-Monoidal Categories

本文研究了$\mathbb{E}_n$-单向量范畴的变形理论，证明了其变形问题与非单位$\mathbb{E}_{n+2}$-代数的关系，并探讨了其在特定变形问题中的应用。

Moduli stacks of algebraic structures and deformation theory

We connect the homotopy type of simplicial moduli spaces of algebraic structures to the cohomology of their deformation complexes. Then we prove that under several assumptions, mapping spaces of algebras over a monad in an appropriate diagram category form affine stacks in the sense of Toen-Vezzosi's homotopical algebraic geometry. This includes simplicial moduli spaces of algebraic structures over a given object (for instance a cochain complex). When these algebraic structures are parametrised by properads, the tangent complexes give the known cohomology theory for such structures and there i…

The equivariant Verlinde formula on the moduli of Higgs bundles

We prove an analog of the Verlinde formula on the moduli space of semistable meromorphic G-Higgs bundles over a smooth curve for a reductive group G whose fundamental group is free. The formula expresses the graded dimension of the space of sections of a positive line bundle as a finite sum whose terms are indexed by formal solutions of a generalized Bethe ansatz equation on the maximal torus of G. In an appendix, Constantin Teleman proves a vanishing theorem for the higher cohomology of positive line bundles on the stack of Higgs bundles.