N-flat connections

Deformed graded Poisson structures, Generalized Geometry and Supergravity

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the $2$-graded symplectic manifold $T^{*}[2]T[1]M$ that are based on a metric $g$, a closed Neveu-Schwarz $3$-form $H$ (locally expressed in terms of a Kalb-Ramond 2-for…

A compactness theorem for stable flat $SL(2,\mathbb{C})$ connections on $3$-folds

Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are $non$-$degenerate$. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a previous result in \cite{Huang}, we prove that the moduli space of the stable flat $SL(2,\mathbb{C})$ connections is disconnected under certain technical assumptions.

Cocalibrated G_2-manifolds with Ricci flat characteristic connection

Any 7-dimensional cocalibrated G_2-manifold admits a unique connection $\nabla$ with skew symmetric torsion. We study these manifolds under the additional condition that the $\nabla$-Ricci tensor vanishes. In particular, we describe their geometry in case of a maximal number of $\nabla$-parallel vector fields.

Flat connections and Wigner-Yanase-Dyson metrics

On the manifold of positive definite matrices, we investigate the existence of pairs of flat affine connections, dual with respect to a given monotone metric. The connections are defined either using the $α$-embeddings and finding the duals with respect to the metric, or by means of contrast functionals. We show that in both cases, the existence of such a pair of connections is possible if and only if the metric is given by the Wigner-Yanase-Dyson skew information.

A note on flat metric connections with antisymmetric torsion

In this short note we study flat metric connections with antisymmetric torsion $T \neq 0$. The result has been originally discovered by Cartan/Schouten in 1926 and we provide a new proof not depending on the classification of symmetric spaces. Any space of that type splits and the irreducible factors are compact simple Lie group or a special connection on $S^7$. The latter case is interesting from the viewpoint of $G_2$-structures and we discuss its type in the sense of the Fernandez-Gray classification. Moreover, we investigate flat metric connections of vectorial type.

Graded Lie algebra of Hermitian tangent valued forms

In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special functions" of the phase space. Indeed, this is the source of the covariant quantisation of the above special functions. In the original version of the theory, this result was formulated and proved in a rather involved way; now, we have achieved a more direct and simple approach to the…

Flat linear connections in terms of flat linear transports in tensor bundles

The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.

$\mathbb{Z}$-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures

This thesis studies the representation theory and linear structures of $\mathcal{Q}$-manifolds and higher Lie algebroids. We introduce differential graded modules (or for short DG-modules) of $\mathcal{Q}$-manifolds and the equivalent notion of representations up to homotopy in the case of Lie $n$-algebroids ($n\in\mathbb{N}$), as generalisations of the homonymous structures that exist already in the case of ordinary Lie algebroids. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy of Lie $n$-algebr…

作为微分双复形与Gauss-Manin联络的双平坦F-结构

Bi-flat F-structures as differential bicomplexes and Gauss-Manin connections

本研究证明双平坦F-结构可诱导切层值微分形式上的微分双复形，其递推向量场序列对应关联平坦联络族的平坦截面展开系数，且在Dubrovin-Frobenius流形中该联络可与平坦度量束的Levi-Civita联络等同。

更高复结构与平坦连接

Higher Complex Structures and Flat Connections

本文结合更高复结构与抛物平坦连接，研究其几何起源及与Toda可积系统的关系。

A lossless reduction of geodesics on supermanifolds to non-graded differential geometry

Let $\mathcal M= (M,\mathcal O_\mathcal M)$ be a smooth supermanifold with connection $\nabla$ and Batchelor model $\mathcal O_\mathcal M\congΓ_{ΛE^\ast}$. From $(\mathcal M,\nabla)$ we construct a connection on the total space of the vector bundle $E\to{M}$. This reduction of $\nabla$ is well-defined independently of the isomorphism $\mathcal O_\mathcal M \cong Γ_{ΛE^\ast}$. It erases information, but however it turns out that the natural identification of supercurves in $\mathcal M$ (as maps from $ \mathbb R^{1|1}$ to $\mathcal M$) with curves in $E$ restricts to a 1 to 1 correspondence on g…

The Graded Differential Geometry of Mixed Symmetry Tensors

We show how the theory of $\mathbb{Z}_2^n$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

Flat $\mathfrak{so}(p,q)$-Connections for Manifolds of Non-Euclidean Signature

The well-known fact that $S^1$, $S^3$ and $S^7$ are parallelizable manifolds admitting flat connections is revisited. The role of torsion in the construction of those flat connections is made explicit, and the possibilities allowed by different metric signatures are examined. A necessary condition for parallelizability in an open region is that the torsion tensor must be covariantly constant. This property can be used to obtain a relation between a torsion-free and flat connections. Our treatment covers Riemannian and pseudo-Riemannian (non-Euclidean signature) hyperbolic manifolds of dimensio…

The volume of the moduli space of flat connections on a nonorientable 2-manifold

We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat connections on an orientable surface using Haar measure.

规范理论中的基本曲率与Atiyah上圈

Basic curvature and the Atiyah cocycle in gauge theory

本文在分级几何框架下研究Lie与Courant代数胚上的联络，定义基本曲率张量，并揭示高规范理论中规范变换由Kapranov L$_{\infty}[1]$代数的Atiyah上圈所支配。

Projectively flat connections and flat connections on homogeneous spaces

We show a correspondence between the set of all G-invariant projectively flat connections on a homogeneous apace $M=G/K$, and the one of all {G}^~-invariant flat connections on a homogeneous space {M}^~={G}^~/K, where {G}^~ is a central extension of G.

厄米特杨-米尔斯联络的半稳定性与局部穿墙

Semi-stability and local wall-crossing for hermitian Yang-Mills connections

本文在紧致凯勒流形上半稳定全纯向量丛的分次对象自同构群为阿贝尔群的条件下，给出极化邻域的半代数腔室分解，并证明稳定腔室中HYM联络收敛到分次对象上的HYM联络。

The prequantum line bundle on the moduli space of flat $SU(N)$ connections on a Riemann surface and the homotopy of the large $N$ limit

We show that the prequantum line bundle on the moduli space of flat $SU(2)$ connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat $SU(N)$ connections, in the limit as $N$ tends to infinity, and $\mathbb{C}P^\infty$. Applications to the stable moduli space of flat unitary connections are also discussed.

On Flat Connections Induced Over Covering Maps

Flat connections induced over covering maps are studied and the trivial ones among them are described. In the sequel, we deal with the resulting holonomy bundles.

有限半单纯集、微分分次代数与胞层

Finite semisimplicial sets, differential graded algebras, and cellular sheaves

本研究构建有限非奇异半单纯集及其胞层的微分分次模型，建立范畴等价并推广图与一阶微分演算的对应，刻画胞层的微分分次模描述，还研究了联络与曲率的性质及粘合准则。

爆破埃尔米特杨-米尔斯联络

Blowing-up Hermitian Yang--Mills connections

本文研究Kähler流形爆破后拉回向量丛的埃尔米特杨-米尔斯联络，在温和假设下给出了该丛存在相应联络序列的充要数值判据，并证明联络收敛于分次对象的拉回联络。

Courant代数胚上的平坦广义联络

Flat Generalized Connections on Courant Algebroids

本文研究传递Courant代数胚上含典范Levi-Civita联络的度量广义联络的平坦性条件，指出其构造单元为紧单李群，刻画了对应李群上的左不变平坦Levi-Civita广义联络并证明非平坦联络存在。

Tertiary classes for a one-parameter variation of flat connections on a smooth manifold

In this note, we extend the theory of Chern-Cheeger-Simons to construct canonical invariants for a one-parameter family of flat connections on a smooth manifold. These invariants lie in degrees $(2p-2)$-cohomology with $\C/\Z$-cohomology, for $p\geq 2$. Furthermore, they are shown to be rigid in a variation of paths (parametrising flat connections), in degrees at least three.

Symplectically Flat Connections and Their Functionals

We continue our study of symplectically flat bundles. We broaden the notion of symplectically flat connections on symplectic manifolds to $ζ$-flat connections on smooth manifolds. These connections on principal bundles can be represented by maps from an extension of the base's fundamental group to the structure group. We introduce functionals with zeroes being symplectically flat connections and study their critical points. Such functionals lead to novel geometric flows. We also describe some characteristic classes of the $ζ$-flat bundles.

Projectively flat and affinely flat parabolic subgroups of special linear groups

A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is projectively equivalent to a flat affine connection.

Prequantization of the moduli space of flat connections over a four-manifold

We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli space of flat connections we shall construct a hermitian line bundle with connection whose curvature is given by the symplectic form. This is the Chern-Simons prequantum line bundle. The group of gauge transformations on the boundary of the base manifold acts on the moduli space of…

On CR Submanifolds of Maximal CR Dimension with Flat Normal Connection of a Complex Projective Space

In this paper, we study the CR submanifolds of maximal CR dimension with flat normal connection of a complex projective space. We first investigate the position of the umbilical normal vector in the normal bundle, especially for the submanifolds of dimension 3. Then as the application, we prove the non-existence of a class of CR submanifolds of maximal CR dimension with flat normal connection.

Kaehler manifolds admitting a flat complex conformal connection

We prove that any Kaehler manifold admitting a flat complex conformal connection is a Bochner-Kaehler manifold with special scalar distribution and zero geometric constants. Applying the local structural theorem for such manifolds we obtain a complete description of the Kaehler manifolds under consideration.

Hermitian Manifolds with Flat Associated Connection

A local classification of the Hermitian manifolds with flat associated connection is given. Hermitian manifolds admitting locally a conformal metric with flat associated connection are characterized by a curvature identity. Locally conformal Kaehler manifolds as well as Hermitian surfaces with vanishing associated conformal curvature tensor are characterized as locally conformal to a Kaehler manifold of constant holomorphic sectional curvatures.

四维空间形式中具有平坦法向连接的曲面

Surfaces with flat normal connection in 4-dimensional space forms

本文研究了四维空间形式中具有平坦法向连接的曲面的特征化问题，探讨了其法向向量场的性质及相关几何结构。

Moduli spaces of flat connections on 2-manifolds, cobordism, and Witten's volume formulas

We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

Finite Nocommutative Coverings and Flat Connections

Any flat connection on a principal fibre bundle comes from a linear representation of the fundamental group. The noncommutative analog of this fact is discussed here.

Cohomological invariants of a variation of flat connection

In this paper, we apply the theory of Chern-Cheeger-Simons to construct canonical invariants associated to a $r$-simplex whose points parametrize flat connections on a smooth manifold $X$. These invariants lie in degrees $(2p-r-1)$-cohomology with $C/Z$-coefficients, for $p> r\geq 1$. In turn, this corresponds to a homomorphism on the higher homology groups of the moduli space of flat connections, and taking values in $C/Z$-cohomology of the underlying smooth manifold $X$.

里奇平坦度量、平坦联络与$G_{2}$流形

Ricci Flat Metrics, Flat Connections and $G_{2}$-Manifolds

受M理论启发，证明了以紧致平坦三维流形T³/K为底的超凯勒ALE四维流形族构成的七维流形上，复化无挠G₂结构模空间与T³/K上平坦G^ℂ_ADE联络的特定模空间等价。

Seiberg-Witten monopoles and flat PSL(2;R)-connections

I show that flat PSL(2;R)-connections on three-manifolds satisfying certain 'stability condition' can be interpreted as solutions of the Seiberg-Witten equations with two spinors. This is used to construct explicit examples of the Seiberg-Witten moduli spaces. Also, I show that in this setting blow up sets satisfy certain non-trivial topological restrictions.

Moduli spaces of flat Lie algebroid connections

We shall prove that a moduli space of flat irreducible Lie algebroid connections over a compact manifold has locally a natural structure of a smooth differentiable space. This is a generalization of some well known results for the moduli space of holomorphic structures on a complex vector bundle over a compact complex manifold.

On canonical Cartan connections associated to filtered G-structures

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making it into a nilpotent graded Lie algebra. Assuming that these symbol algebras are the same for all points, one obtains a natural frame bundle for the associated graded to the tangent bundle, and filtered G--structures are defined as reductions of structure group of this bundle. Ge…

平坦偏联络的射流

Jets of flat partial connections

该研究在光滑叶状结构与局部自由层平坦偏联络的背景下定义并研究平坦偏联络的射流，分别刻画了余维1叶状结构的横截仿射、射影结构，还利用法层Bott联络的射流定义任意余维叶状结构横截射影结构的延拓并构造奇异结构。

On compact Hermitian manifolds with flat Gauduchon connections

Given a Hermitian manifold $(M^n,g)$, the Gauduchon connections are the one parameter family of Hermitian connections joining the Chern connection and the Bismut connection. We will call $\nabla^s = (1-\frac{s}{2})\nabla^c + \frac{s}{2}\nabla^b$ the $s$-Gauduchon connection of $M$, where $\nabla^c$ and $\nabla^b$ are respectively the Chern and Bismut connections. It is natural to ask when a compact Hermitian manifold could admit a flat $s$-Gauduchon connection. This is related to a question asked by Yau \cite{Yau}. The cases with $s=0$ (a flat Chern connection) or $s=2$ (a flat Bismut connecti…

Tanaka structures (non holonomic G-structures) and Cartan connections

Let \gh = \gh_{-k}\oplus \cdots \oplus \gh_{l} (k >0, l \geq 0) be a finite dimensional real graded Lie algebra, with a Euclidian metric \langle \cdot , \cdot \rangle adapted to the gradation. The metric \langle\cdot , \cdot \rangle is called admissible if the codifferentials \partial^{*} : C^{k+1}(\gh_{-}, \gh ) \ra C^{k} (\gh_{-}, \gh) (k\geq 0) are Ad_{Q}-invariant (Lie(Q) = \gh_{0}\oplus \gh_{+}). We find necessary and sufficient conditions for a Euclidian metric, adapted to the gradation, to be admissible, and we develop a theory of normal Cartan connections, when these conditions are sat…

Classification Theorem for a Class of Flat Connections and Representations of Kahler Groups

The paper presents a classification theorem for the class of flat connections with triangular (0,1)-components on a topologically trivial complex vector bundle over a compact Kahler manifold. As a consequence we obtain several results on the structure of Kähler groups, i.e., the fundamental groups of compact Kahler manifolds.

Hermitian manifolds with flat Gauduchon connections

We complete the classification of compact Hermitian manifolds admitting a flat Gauduchon connection. In particular, we establish a conjecture of Yang and Zheng, showing that apart from the cases of a flat Chern or Bismut connection, such manifolds are Kähler. More generally, we prove the same result holds when the flatness assumption is replaced by the so-called Kähler-like condition, proving a conjecture of Angella, Otal, Ugarte and Villacampa. We also treat the non-compact case.

Stable pairs, flat connections and Gopakumar-Vafa invariants

Using the interpretation of certain generalised Donaldson-Thomas invariants, including stable pairs curve counts, as the monodromy of a flat connection on a formal principal bundle, we show that the conjectural Gopakumar-Vafa contributions of all genera to the Gromov-Witten partition function appear in the asymptotics of the corresponding flat sections. The Fourier-Laplace integrals used to produce flat sections lead naturally to the GW/DT change of variable -q = e^{i u}.

Generalized classical dynamical Yang-Baxter equations and moduli spaces of flat connections on surfaces

In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new finite dimensional description of the Atiyah-Bott symplectic structure on the moduli space of flat connections on a surface.

Perturbed geodesics on the moduli space of flat connections and Yang-Mills theory

If we consider the moduli space of flat connections of a non trivial principal SO(3)-bundle over a surface, then we can define a map from the set of perturbed closed geodesics, below a given energy level, into families of perturbed Yang-Mills connections depending on a small parameter. In this paper we show that this map is a bijection and maps perturbed geodesics into perturbed Yang-Mills connections with the same Morse index.

Heat flow on the moduli space of flat connections and Yang-Mills theory

It is known that there is a bijection between the perturbed closed geodesics, below a given energy level, on the moduli space of flat connections M and families of perturbed Yang-Mills connections depending on a small parameter. In this paper we study the heat flow on the loop space on M and the Yang-Mills L^2-flows for a 3-manifold N with partial rescaled metrics. Our main result is that the bounded Morse homology of the loop space on M is isomorphic to the bounded Morse homologies of the connections space of N.

Hermitian-Einstein connections on principal bundles over flat affine manifolds

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal $G$-bundle $E_G$ over $M$ admits a Hermitian-Einstein structure if and only if $E_G$ is polystable. A polystable flat principal $G$--bundle over $M$ admits a unique Hermitian-Einstein connection. We also prove the existence and uniqueness of a Harder-Narasimhan filtration for …

Normal forms and moduli stacks for logarithmic flat connections

We establish normal form theorems for a large class of singular flat connections on complex manifolds, including connections with logarithmic poles along weighted homogeneous Saito free divisors. As a result, we show that the moduli spaces of such connections admit the structure of algebraic quotient stacks. In order to prove these results, we introduce homogeneous Lie groupoids and study their representation theory. In this direction, we prove two main results: a Jordan-Chevalley decomposition theorem, and a linearization theorem. We give explicit normal forms for several examples of free div…

超平面配置的Hirzebruch二次型与平坦对数联络

The Hirzebruch quadratic form of a hyperplane arrangement and flat logarithmic connections

本文证明复超平面配置的Hirzebruch二次型在稳定权集上非正，并识别出零集为特殊类型的平坦对数联络，证明使用了Kempf-Ness和框架势不等式。

平坦球面上鞍连接的界

Bounds on saddle connections for flat spheres

针对带锥形奇点的平坦球面，在角缺损部分和不等于 $2π$ 的条件下，利用浸入圆盘几何给出至多含 $k$ 个自交点的鞍连接数量及归一化面积下长度的显式界。

1-minimal models for $C_{\infty}$-algebras and flat connections

Given a smooth manifold $M$ equipped with a properly and discontinuous smooth action of a discrete group $G$, the nerve $M_{\bullet}G$ is a simplicial manifold and its vector space of differential forms $\operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right)$ carry a $C_{\infty}$-algebra structure $m_{\bullet}$. We show that each $C_{\infty}$-algebra $1$-minimal model $g_{\bullet}\: : \: \left(W, {m'}_{\bullet} \right) \to \left( \operatorname{Tot}_{N}\left(A_{DR}(M_{\bullet}G)\right),m_{\bullet}\right) $ gives a flat connection $\nabla$ on a smooth trivial bundle $E$ on $M$ where the fiber i…

幂零希格斯丛与平坦联络族

Nilpotent Higgs bundles and families of flat connections

本文研究首项为幂零希格斯场的平坦联络族，证明其单值性与首项为正则希格斯丛的族相同，并由此得到和乐迹关于参数有理幂的渐近指数性质。

Irreducible flat SL(2,R)-connections on the trivial holomorphic bundle

We construct an irreducible holomorphic connection with SL(2,R)-monodromy on the trivial holomorphic vector bundle of rank two over a compact Riemann surface. This answers a question of Calsamiglia, Deroin, Heu and Loray in \cite{CDHL}.

The derived moduli stack of logarithmic flat connections

We give an explicit finite-dimensional model for the derived moduli stack of flat connections on $\mathbb{C}^k$ with logarithmic singularities along a weighted homogeneous Saito free divisor. We investigate in detail the case of plane curves of the form $x^p = y^q$ and relate the moduli spaces to the Grothendieck-Springer resolution. We also discuss the shifted Poisson geometry of these moduli spaces. Namely, we conjecture that the map restricting a logarithmic connection to the complement of the divisor admits a shifted coisotropic structure and we construct a shifted Poisson structure on the…