We recall the notion of a singular foliation (SF) on a manifold M , viewed as an appropriate submodule of \mathfrak{X}(M) , and adapt it to the presence of a Riemannian metric g , yielding a module version of a singular Riemannian foliation (SRF). Following Garmendia–Zambon on Hausdorff Morita equivalence of SFs, we define the Morita equivalence of SRFs (both in the module sense as well as in the more traditional geometric one of Molino) and show that the leaf spaces of Morita equivalent SRFs are isomorphic as pseudo-metric spaces.In a second part, we introduce the category of \mathcal{I} -Poisson manifolds. Its objects and morphisms generalize Poisson manifolds and morphisms in the presence of appropriate ideals \mathcal{I} of the smooth functions on the manifold such that two conditions are satisfied: (i) The category of Poisson manifolds becomes a full subcategory when choosing \mathcal{I}=0 and (ii) there is a reduction functor from this new category to the category of Poisson algebras, which generalizes coistropic reduction to the singular setting.Every SF on M gives rise to an \mathcal{I} -Poisson manifold on T^{*}M and g enhances this to an SRF if and only if the induced Hamiltonian lies in the normalizer of \mathcal{I} . This perspective provides, on the one hand, a simple proof of the fact that every module SRF is a geometric SRF and, on the other hand, a construction of an algebraic invariant of singular foliations: Hausdorff Morita equivalent SFs have isomorphic reduced Poisson algebras.

Given a commutative algebra 𝒪, a proper ideal ℐ, and a resolution of 𝒪/ ℐ by projective 𝒪-modules, we construct an explicit Koszul-Tate resolution. We call it the arborescent Koszul-Tate resolution since it is indexed by decorated trees. When the 𝒪-module resolution has finite length, only finitely many operations are needed in our constructions – this is to be compared with the classical Tate algorithm, which requires infinitely many such computations if ℐ is not a complete intersection. As a by-product of our construction, the initial projective 𝒪-module resolution becomes equipped with an explicit A_∞-algebra.

The famous singular leaf decomposition ℒ_OH of ℝ^16≅𝕆^2 induced by the Hopf construction for octonions 𝕆 has no known Lie group action generating it. In this article we construct a G_2-equivariant Lie groupoid 𝒢⇒𝕆^2 whose orbits coincide with ℒ_OH. Its Lie algebroid E=Lie(𝒢) is of the form 𝕆^4 →𝕆^2 with polynomial structure functions. Its sheaf of sections induces a singular foliation ℱ_OH := ρ(Γ(E)) on 𝕆^2, which we call the singular octonionic Hopf foliation (SOHF). ℱ_OH is shown to be maximal among all singular foliations ℱ generating ℒ_OH – in the polynomial, the real analytic, as well as in the smooth setting. We extend E to a Lie 3-algebroid, which is a minimal length representative of the universal Lie ∞-algebroid of the SOHF. This permits to prove that E is the minimal rank Lie algebroid and that 𝒢 the lowest dimensional Lie groupoid which generate the SOHF. The leaf decomposition ℒ_OH is one of the few known examples of a singular Riemannian foliation in the sense of Molino which cannot be generated by local isometries (local non-homogeneity). We improve this result by showing that any smooth singular foliation ℱ inducing ℒ_OH cannot be even Hausdorff Morita equivalent to a singular foliation ℱ_M on a Riemannian manifold (M,g) generated by local isometries. Furthermore, we show that there is no real analytic singular foliation ℱ generating ℒ_OH which turns (ℝ^16, g_st, ℱ) into a module singular Riemannian foliation as defined in .

The famous singular leaf decomposition $\mathcal{L}_{OH}$ of $\mathbb{R}^{16}\cong \mathbb{O}^2$ induced by the Hopf construction for octonions $\mathbb{O}$ has no known Lie group action generating it. In this article we construct a $\mathrm{G}_2$-equivariant Lie groupoid $\mathcal{G} \Rightarrow \mathbb{O}^{2}$ whose orbits coincide with $\mathcal{L}_{OH}$. Its Lie algebroid $E=\mathrm{Lie}(\mathcal{G})$ is of the form $\mathbb{O}^4 \to \mathbb{O}^2$ with polynomial structure functions. Its sheaf of sections induces a singular foliation $\mathcal{F}_{OH} := \rho(\Gamma(E))$ on $\mathbb{O}^{2}$, which we call the singular octonionic Hopf foliation (SOHF). $\mathcal{F}_{OH}$ is shown to be maximal among all singular foliations $\mathcal{F}$ generating $\mathcal{L}_{OH}$ -- in the polynomial, the real analytic, as well as in the smooth setting. We extend $E$ to a Lie $3$-algebroid, which is a minimal length representative of the universal Lie $\infty-$algebroid of the SOHF. This permits to prove that $E$ is the minimal rank Lie algebroid and that $\mathcal{G}$ the lowest dimensional Lie groupoid which generate the SOHF. The leaf decomposition $\mathcal{L}_{OH}$ is one of the few known examples of a singular Riemannian foliation in the sense of Molino which cannot be generated by local isometries (local non-homogeneity). We improve this result by showing that any smooth singular foliation $\mathcal{F}$ inducing $\mathcal{L}_{OH}$ cannot be even Hausdorff Morita equivalent to a singular foliation $\mathcal{F}_M$ on a Riemannian manifold $(M,g)$ generated by local isometries. Furthermore, we show that there is no real analytic singular foliation $\mathcal{F}$ generating $\mathcal{L}_{OH}$ which turns $(\mathbb{R}^{16}, g_{st}, \mathcal{F})$ into a module singular Riemannian foliation as defined in \cite{NS24}.

We consider mechanical systems on T ∗ M with possibly irregular and reducible first class contraints linear in the momenta, which thus correspond to singular foliations on M . According to a recent result, the latter ones have a Lie-infinity algebroid ( M , Q ) covering them, where we restrict to the case of Lie-2 algebroids. We propose to consider T ∗ M as a potential BFV extended phase space of the constrained system, such that the canonical lift of the nilpotent vector field Q yields automatically a solution to the BFV master equation. We show that in this case, the BFV extension of the Hamiltonian, providing a second corner stone of the BFV formalism, may be obstructed. We identify the corresponding complex governing this second extension problem explicitly (the first extension problem was circumvented by means of the lift of the Lie-2 algebroid structure). We repeatedly come back to the example of angular momenta on T ∗ R 3 : in this procedure, the standard free Hamiltonian does not have a BFV extension—while it does so on T ∗ ( R 3 \{ 0 } ), with a relatively involved ghost contribution singular at the origin.

We present the construction of the classical Batalin–Vilkovisky (BV) action for topological Dirac sigma models. The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess–Zumino–Novikov–Witten model and the Poisson sigma model. Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. Special attention is paid to target space covariance, which requires the introduction of two connections with torsion on the Dirac structure.

One of the main assumptions in textbooks about constrained systems—see, e.g., [1]—is that the constraint functions Ga are regular on the unconstrained symplectic manifold. Important existence theorems about BRSTBV [2, 3] or BFV [4, 5] extensions have been proven in this setting only. But not a single physically realistic system satisfies this condition. (In a system where it is satisfied, one hardly needs a gauge theoretic description, albeit the non-linearity of the quotient space, if smooth, may, admittedly, also pose problems). In realistic situations, group actions often have fixed points and there the corresponding constraints are not regular. In YangMills gauge theories, for example, the constraints become non-regular at reducible connections. On the other hand, without irreducibility and, in particular, regularity assumptions, often the BV and BFV extensions become hard mathematical problems. Let us illustrate this statement in the finite dimensional setting: Suppose you are given a set va of vector fields on a manifold M satisfying

Equation (5.20), should read as follows.

We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated, O-submodule of vector fields on the underlying manifold closed under Lie bracket. Here O can be the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie ∞-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie ∞-algebroid inducing the same foliation or any of its subfoliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie ∞-algebroid of the singular foliation. It can be chosen, locally, to be a Lie n-algebroid for real analytic or holomorphic singular foliations. We show that this universal structure encodes several aspects of the geometry of the leaves of a singular foliation. In particular, it contains the holonomy algebroid and groupoid of a leaf in the sense of Androulidakis and Skandalis. But even more, each leaf carries an isotropy Lie ∞-algebra structure that is unique up to isomorphism and that extends a minimal isotropy Lie algebra that can be associated to each leaf by higher brackets containing additional invariants of the foliation. As a byproduct, we construct an example of a foliation generated by r vector fields for which we show by these techniques that it cannot be generated by the image through the anchor map of a Lie algebroid of the minimal rank r.

An enhanced Leibniz algebra is an algebraic struture that arises in the context of particular higher gauge theories describing self-interacting gerbes. It consists of a Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, a bilinear form on $\mathbb{V}$ with values in another vector space $\mathbb{W}$, and a map $t \colon \mathbb{W} \to \mathbb{V}$, satisfying altogether four compatibility relations. Our structure theorem asserts that an enhanced Leibniz algebra is uniquely determined by the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ])$, an appropriate abelian ideal ${\mathfrak i}$ inside it, as well as a cohomology 2-class $[\Delta]$ which only effects the $\mathbb{W}$-valued product. Positive quadratic enhanced Leibniz algebras, as needed for the definition of a Yang-Mills type action functional, turn out to be rather restrictive on the underlying Leibniz algebra $(\mathbb{V},[ \cdot, \dot ])$: $\mathbb{V}$ has to be the hemisemidirect product of a positive quadratic Lie algebra ${\mathfrak g}$ with a ${\mathfrak g}$-module ${\mathfrak i}$, $\mathbb{V} \cong {\mathfrak g}\ltimes{\mathfrak i}$, with ${\mathfrak i}$ the above-mentioned ideal in this case. The second main result of this article is the construction of a functor from the category of such enhanced Leibniz algebras to the category of (semi-strict) Lie 2-algebras or, equivalentely, of two-term $L_\infty$-algebras.

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional "representation constraint"). Every Leibniz algebra gives rise to a Lie n-algebra in a canonical way (for every $$n\in \mathbb {N}\cup \{ \infty \}$$). It is the gauging of this $$L_\infty $$-algebra that explains the tensor hierarchy of the bosonic sector of gauged supergravity theories. The tower of p-from gauge fields corresponds to Lyndon words of the universal enveloping algebra of the free Lie algebra of an odd vector space in this construction. Truncation to some n yields the reduced field content needed in a concrete spacetime dimension.

A bstract The BFV formulation of a given gauge theory is usually significantly easier to obtain than its BV formulation. Based on foundational work by Fisch and Henneaux, Grigoriev and Damgaard introduced simple formulas for obtaining the latter from the former. Since BFV relies on the Hamiltonian version of the gauge theory, however, it does not come as a surprise that in general the resulting BV theory does not exhibit spacetime covariance. We provide an explicit example of this phenomenon in two spacetime dimensions and show how to restore covariance of the BV data by improving the Fisch-Henneaux-Grigoriev-Damgaard procedure with appropriate adaptations of their formulas.

We reformulate the compatibility condition between a generalized metric and a small (non-maximal rank) Dirac structure in an exact Courant algebroid found in the context of the gauging of strings and formulated by means of two connections in purely Dirac-geometric terms. The resulting notion, a transverse generalized metric, is also what is needed for the dynamics on the reduced phase space of a string theory.

We observe that a system of irreducible, fiber-linear, first-class constraints on \(T^*M\) is equivalent to the definition of a foliation Lie algebroid over M. The BFV formulation of the constrained system is given by the Hamiltonian lift of the Vaintrob description (E[1], Q) of the Lie algebroid to its cotangent bundle \(T^*E[1]\). Affine deformations of the constraints are parametrized by the first Lie algebroid cohomology \(H^1_Q\) and lead to irreducible constraints also for much more general Lie algebroids such as Dirac structures; the modified BFV function follows by the addition of a representative of the deformation charge. Adding a Hamiltonian to the system corresponds to a metric g on M. Evolution invariance of the constraint surface introduces a connection \(\nabla \) on E and one reobtains the compatibility of g with \((E,\rho ,\nabla )\) found previously in the literature. The covariantization of the Hamiltonian to a function on \(T^*E[1]\) serves as a BFV-Hamiltonian, iff, in addition, this connection is compatible with the Lie algebroid structure, turning \((E,\rho ,[ \cdot , \cdot ],\nabla )\) into a Cartan–Lie algebroid. The BV formulation of the system is obtained from BFV by a (time-dependent) AKSZ procedure.

We reformulate the compatibility condition between a generalized metric and a small (non-maximal rank) Dirac structure in an exact Courant algebroid found in the context of the gauging of strings and formulated by means of two connections in purely Dirac-geometric terms. The resulting notion, a transverse generalized metric, is also what is needed for the dynamics on the reduced phase space of a string theory. 1. Let E be an exact Courant algebroid over M , characterized by the class [H ] ∈ H dR(M) [11, 12] and let V ⊂ E be a generalized metric, i.e. a positive definite, rank n = dimM subbundle of E. These data are equivalent to the choice of a Riemannian metric g and a representative closed 3-form H on M (since there is a unique splitting of ρ : E → TM such that V can be written as the graph of a symmetric 2-tensor). They are also the data needed on the target space for the definition of a standard sigma model with Wess-Zumino term. Choose a small Dirac structure, i.e. an involutive, isotropic C(M)-submodule D of Γ(E). In this note we only consider regular D’s, i.e. those of the form D = Γ(D) for some sub-vector bundle D ⊂ E. We call a rank n subbundle W ⊂ E a pre-D-transverse generalized metric if D ⊂ W ⊂ D and 〈w,w〉 > 0 for every w ∈ W with w 6∈ D. This becomes a D-transverse generalized metric, or simply a transverse generalized metric, if in addition the invariance property [Γ(D),Γ(W )] ⊂ Γ(W ) (1) holds true. 2. If D is such that ρ|D : D → TM is injective (in which case we call D projectable), then a D-transverse generalized metric is equivalent to a Riemannian metric and a closed 3-form, both on the space of leaves of the resulting foliation F := ρ(D) ⊂ TM . In more detail, we have: Proposition 1. Suppose that the leaves of the foliation F = ρ(D) generated by a projectable small Dirac structure D are the fibers of a surjective submersion π : M → Q. If W ⊂ E is a D-transverse generalized metric, then there is a unique splitting E ∼= (T ⊕T )M such that the resulting 3-form is of the form πHQ and W is the graph of π gQ, where, respectively, HQ and gQ are a closed 3-form and a Riemannian metric on Q. Proof. There is a unique splitting identifying E with (T ⊕ T )M such that W is the graph of a (degenerate) symmetric bilinear form h on TM . Using this splitting, one has D = F = ker h. The condition [Γ(D),Γ(W )] ⊂ Γ(W ) means

A quadratic Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ],\kappa)$ gives rise to a canonical Yang-Mills type functional $S$ over every space-time manifold. The gauge fields consist of 1-forms $A$ taking values in $\mathbb{V}$ and 2-forms $B$ with values in the subspace $\mathbb{W} \subset \mathbb{V}$ generated by the symmetric part of the bracket. If the Leibniz bracket is anti-symmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, $B\equiv 0$, and $S$ becomes identical to the usual Yang-Mills action functional. We describe this gauge theory for a general quadratic Leibniz algebra. We then prove its (classical and quantum) equivalence to a Yang-Mills theory for the Lie algebra ${\mathfrak{g}} = \mathbb{V}/\mathbb{W}$ to which one couples massive 2-form fields living in a ${\mathfrak{g}}$-representation. Since in the original formulation the B-fields have their own gauge symmetry, this equivalence can be used as an elegant mass-generating mechanism for 2-form gauge fields, thus providing a 'higher Higgs mechanism' for those fields.

A quadratic Leibniz algebra $(\mathbb{V},[ \cdot, \cdot ],\kappa)$ gives rise to a canonical Yang-Mills type functional $S$ over every space-time manifold. The gauge fields consist of 1-forms $A$ taking values in $\mathbb{V}$ and 2-forms $B$ with values in the subspace $\mathbb{W} \subset \mathbb{V}$ generated by the symmetric part of the bracket. If the Leibniz bracket is anti-symmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, $B\equiv 0$, and $S$ becomes identical to the usual Yang-Mills action functional. We describe this gauge theory for a general quadratic Leibniz algebra. We then prove its (classical and quantum) equivalence to a Yang-Mills theory for the Lie algebra ${\mathfrak{g}} = \mathbb{V}/\mathbb{W}$ to which one couples massive 2-form fields living in a ${\mathfrak{g}}$-representation. Since in the original formulation the B-fields have their own gauge symmetry, this equivalence can be used as an elegant mass-generating mechanism for 2-form gauge fields, thus providing a 'higher Higgs mechanism' for those fields.

Consider an anchored bundle (E, rho), i.e. a vector bundle E -> M equipped with a bundle map rho: E -> TM covering the identity. M. Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid FR(E) superset of E. We adapt his construction to the case of an anchored bundle equipped with an arbitrary connection, (E, del), and show that it gives rise to a unique connection, (del) over tilde on FR(E) which is compatible with its Lie algebroid structure, thus turning (FR(E), (del) over tilde) into a Cartan-Lie algebroid. Moreover, this construction is universal: any connection-preserving vector bundle morphism from (E, del) to a Cartan-Lie Algebroid (A, (del) over bar) factors through a unique Cartan-Lie algebroid morphism from (FR(E), (del) over tilde) to (A, (del) over bar). Suppose that, in addition, M is equipped with a geometrical structure defined by some tensor field t which is compatible with (E, rho, del) in the sense of being annihilated by a natural E-connection that one can associate to these data. For example, for a Riemannian base (M, g) of an involutive anchored bundle (E, rho), this condition implies that M carries a Riemannian foliation. It is shown that every E-compatible tensor field t becomes invariant with respect to the Lie algebroid representation associated canonically to the Cartan-Lie algebroid (FR(E), (del) over tilde). (C) 2018 Elsevier B.V. All rights reserved.

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the (possibly singular) foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized metric structure.

We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated O-submodule of vector fields on the underlying manifold closed under Lie bracket. Here O can be the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie ∞-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie ∞-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie ∞-algebroid of the singular foliation. For real analytic or holomorphic singular foliations, it can be chosen, locally, to be a Lie n-algebroid for some finite n. We show that this universal structure encodes several aspects of the geometry of the leaves of a singular foliation. In particular, it contains the holonomy algebroid and groupoid of a leaf in the sense of Androulidakis and Skandalis. But even more, each leaf carries an isotropy Lie ∞-algebra structure that is unique up to isomorphism. It extends a minimal isotropy Lie algebra, that can be associated to each leaf, by higher brackets, which give rise to additional invariants of the foliation. As a byproduct, we construct an example of a foliation generated by r vector fields for which we show by these techniques that it cannot be generated by the image through the anchor map of a Lie algebroid of the minimal rank r.