arXiv subjects
Publications and source records attributed to Chenchang Zhu.
We introduce the concept of $m$-shifted symplectic Lie $n$-groupoids and symplectic Morita equivalences between them. We then build various models for the 2-shifted symplectic structure on the classifying stack in this setting and construct explicit symplectic Morita equivalences between them.
The main goal of this work is to introduce derived Lie n-groupoids and their shifted symplectic structures. We further define shifted lagrangian structures and prove that their composition is well defined under suitable conditions. As an application, we show that our framework incorporates several reduction procedures at critical values, including: classical Hamiltonian reduction, group valued moment maps, Poisson Lie group valued moment maps and Mikami-Weinstein for proper symplectic groupoids.
Differentiating an Lie $n$-groupoid via the differential-geometric fat point a priori only yielads a presheaf of graded manifolds. In this article we prove that this presheaf is representable by the tangent complex of the Lie $n$-groupoid. As an immediate consequence we obtain that the tangent complex carries the structure of a Lie $n$-algebroid.
We introduce a notion of ``$n$-dual'' to a simplicial vector space for $n\ge 0$. Coming with it, there is a canonical pairing, which we show to be non-degenerate up to homotopy for homotopy $n$-types. As a result this notion of duality is reflexive up to homotopy for $n$-types. In particular the same properties hold for $n$-groupoid objects in vector spaces, whose $n$-duals are again such $n$-groupoid objects. We study this construction in the context of the Dold-Kan correspondence and we reformulate the Eilenberg-Zilber theorem, which classically controls monoidality of the Dold-Kan functors, in terms of internal homs. We compute explicitly the 1-dual of a groupoid object and the 2-dual of a 2-groupoid object in the category of vector spaces. As the 1-dual of a groupoid object, we recover its dual as a $\mathsf{VB}$ groupoid over a point.
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid, and every Lie algebroid can be integrated into a Weinstein groupoid. Furthermore, we apply this general result to Jacobi manifolds and construct contact groupoids for Jacobi manifolds. There are further applications in prequantization and integrability of Poisson bivectors.
We study the nonlinear $σ$-model in ${(d+1)}$-dimensional spacetime with connected target space $K$ and show that, at energy scales below singular field configurations (such as vortices), it has an emergent non-invertible higher symmetry. The symmetry defects of the emergent symmetry are described by the $d$-representations of a discrete $d$-group $\mathbb{G}^{(d)}$ (i.e. the emergent symmetry is the dual of the invertible $d$-group $\mathbb{G}^{(d)}$ symmetry). The $d$-group $\mathbb{G}^{(d)}$ is determined such that its classifying space $B\mathbb{G}^{(d)}$ is given by the $d$-th Postnikov stage of $K$. In $(2+1)$D and for finite $\mathbb{G}^{(2)}$, this symmetry is always holo-equivalent to an invertible ${0}$-form (ordinary) symmetry with potential 't Hooft anomaly. The singularity-free disordered phase of the nonlinear $σ$-model spontaneously breaks this symmetry, and when $\mathbb{G}^{(d)}$ is finite, it is described by the deconfined phase of $\mathbb{G}^{(d)}$ higher gauge theory. We consider examples of such disordered phases. We focus on a singularity-free $S^2$ nonlinear $σ$-model in ${(3+1)}$D and show that it has an emergent non-invertible higher symmetry. As a result, its disordered phase is described by axion electrodynamics and has two gapless modes corresponding to a photon and a massless axion. Notably, this non-perturbative result is different from the results obtained using the $S^N$ and $\mathbb{C}P^{N-1}$ nonlinear $σ$-models in the large-$N$ limit.
In this paper, we give the necessary and sufficient conditions of the integrability of relative Rota-Baxter Lie algebras via double Lie groups, matched pairs of Lie groups and factorization of diffeomorphisms respectively. We use the integrability of Rota-Baxter operators to characterize whether the Poisson-Lie group integrating a factorizable Lie bialgebra is again factorizable. We thoroughly study the integrability of Rota-Baxter operators on the unique nontrivial 2-dimensional Lie algebra. As a byproduct, we construct a matched pair of Lie algebras that can not be integrated to a matched pair of Lie groups.
The notion of post-groups was introduced by Bai, Guo and the first two authors recently, which are the global objects corresponding to post-Lie algebras, equivalent to skew-left braces, and can be used to construct set-theoretical solutions of the Yang-Baxter equation. In this paper, first we introduce the notion of post-groupoids, which consists of a group bundle and some other structures satisfying some compatibility conditions. Post-groupoids reduce to post-groups if the underlying base is one point. An action of a group on a set gives rise to the natural example of post-groupoids. We show that a post-groupoid gives rise to a groupoid (called the Grossman-Larson groupoid), and an action on the original group bundle. Then we introduce the notion of relative Rota-Baxter operators on a groupoid with respect to an action on a group bundle. A relative Rota-Baxter operator naturally gives rise to a post-groupoid and a matched pair of groupoids. One important application of post-groupoids is that they give rise to quiver-theoretical solutions of the Yang-Baxter equation on the quiver underlying the Grossman-Larson groupoid. We also introduce the notion of a skew-left bracoid, which consists of a group bundle and a groupoid satisfying some compatibility conditions. A skew-left bracoid reduces to a skew-left brace if the underlying base is one point. We give the one-to-one correspondence between post-groupoids and skew-left bracoids. Finally, we show that post-Lie groupoids give rise to post-Lie algebroids via differentiation.
We discuss two generalizations of Lie groupoids. One consists of Lie $n$-groupoids defined as simplicial manifolds with trivial $π_{k\geq n+1}$. The other consists of stacky Lie groupoids $\cG\rra M$ with $\cG$ a differentiable stack. We build a 1-1 correspondence between Lie 2-groupoids and stacky Lie groupoids up to a certain Morita equivalence. We prove this in a general set-up so that the statement is valid in both differential and topological categories. \Equivalences of higher groupoids in various categories are also described.
In this paper, we establish a local Lie theory for relative Rota-Baxter operators of weight $1$. First we recall the category of relative Rota-Baxter operators of weight $1$ on Lie algebras and construct a cohomology theory for them. We use the second cohomology group to study infinitesimal deformations of relative Rota-Baxter operators and modified $r$-matrices. Then we introduce a cohomology theory of relative Rota-Baxter operators on a Lie group. We construct the differentiation functor from the category of relative Rota-Baxter operators on Lie groups to that on Lie algebras, and extend it to the cohomology level by proving the Van Est theorem between the two cohomology theories. We integrate a relative Rota-Baxter operator of weight 1 on a Lie algebra to a local relative Rota-Baxter operator on the corresponding Lie group, and show that the local integration and differentiation are adjoint to each other. Finally, we give two applications of our integration of Rota-Baxter operators: one is to give an explicit formula for the factorization problem, and the other is to provide an integration for matched pairs.
In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Kotov and Strobl's construction of an $L_\infty$-algebra from an embedding tensor, to a functor from the category of homotopy embedding tensors to that of Leibniz$_\infty$-algebras, and a functor further to that of $L_\infty$-algebras.
We construct a Chern-Simons type of theory using the $l_\infty$ algebra encoded by a Poisson structure on arbitrary Riemann surfaces with boundaries. A deformation quantization within the Batalin-Vilkovisky framework is performed by constructing propagators with Dirichlet boundary condition on Fulton-MacPherson compactified configuration space. Our results show that the BV quantization is independent of several gauge choices in propagators, which leads to global observables that are candidates for geometric invariants of Poisson structure and topological invariants for the worldsheet structure. At the level of local observables, a Swiss-Cheese algebra structure has been identified. If the Poisson structure is symplectic, the two-dimensional theory is homotopic to a boundary theory. This is known in the classical case, and we confirm that the quantum homotopy exists as well.
We introduce the notion of a Hamiltonian action of an étale Lie group stack on an étale symplectic stack and establish versions of the Kirwan convexity theorem, the Meyer-Marsden-Weinstein symplectic reduction theorem, and the Duistermaat-Heckman theorem in this context.
Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by integrating finite type Lie $n$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of Lie $n$-algebras introduced in the companion paper arXiv:1809.05999, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie $\infty$-groupoids form an `incomplete category of fibrant objects' in which the weak equivalences correspond to `stalkwise' weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler's results also hold in this more general context. As an application, we show that Henriques' integration functor is an exact functor with respect to a class of distinguished fibrations which we call `quasi-split fibrations'. Such fibrations include acyclic fibrations as well as fibrations that arise in string-like extensions. In particular, integration sends $L_\infty$ quasi-isomorphisms to weak equivalences, quasi-split fibrations to Kan fibrations, and preserves acyclic fibrations, as well as pullbacks of acyclic/quasi-split fibrations.
We use higher dimensional bosonization and fermion decoration to construct exactly soluble interacting fermion models to realize fermionic symmetry protected trivial (SPT) orders (which are also known as symmetry protected topological orders) in any dimensions and for generic fermion symmetries $G_f$, which can be a non-trivial $Z_2^f$ extension (where $Z_2^f$ is the fermion-number-parity symmetry). This generalizes the previous results from group superconhomology of Gu and Wen (arXiv:1201.2648), where $G_f$ is assumed to be a trivial $Z_2^f$ extension. We find that the SPT phases from fermion decoration construction can be described in a compact way using higher groups.
A discrete non-linear $σ$-model is obtained by triangulate both the space-time $M^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the complex homomorphisms $ϕ: M^{d+1} \to K$, with an partition function that is independent of space-time triangulation, then the corresponding non-linear $σ$-model will be called topological non-linear $σ$-model which is exactly soluble. Those exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects (i.e. fluctuations described by homomorphisms $ϕ$) usually produce a topologically ordered state and are topological phase transitions, while phase transitions induced by fluctuations with all the topological defects give rise to trivial product states and are not topological phase transitions. If $K$ is a space with only non-trivial first homotopy group $G$ which is finite, those topological non-linear $σ$-models can realize all 3+1D bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group $π_1(K)=G$. Here, we show that the 3+1D bosonic topological orders with emergent fermions can be realized by topological non-linear $σ$-models with $π_1(K)=$ finite groups, $π_2(K)=Z_2$, and $π_{n>2}(K)=0$. A subset of those topological non-linear $σ$-models corresponds to 2-gauge theories, which realize and classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections. The classification of 3+1D bosonic topological orders may correspond to a classification of unitary fully dualizable fully extended topological quantum field theories in 4-dimensions.
Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, i.e., actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples that naturally arise from non-integrable Lie algebroids.
Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative analogue of exact Courant algebroids. In this article, we explore what the "principal bundle" behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connection data. We prove a lifting theorem for the stack of string principal bundles with connections and show the multiplicity of the lifts once they exist. This is a differential geometrical refinement of what is known for string structures by Redden, Waldorf and Stolz-Teichner. We also extend the result of Bressler and Chen-Stiénon-Xu on extension obstruction involving transitive Courant algebroids to the case of transitive Courant algebroids with connections, as a lifting theorem with the description of multiplicity once liftings exist. At the end, we build a morphism between these two stacks. The morphism turns out to be neither injective nor surjective in general, which shows that the process of associating the "higher Atiyah algebroid" loses some information and at the same time, only some special transitive Courant algebroids come from string bundles.