arXiv subjects

Daniel Álvarez

Publications and source records attributed to Daniel Álvarez.

8 recordsLinked to original sources

arXiv2026

Morita equivalences, moduli spaces and flag varieties

Double Bruhat cells in a connected complex semisimple Lie group $G$ emerged as a crucial concept in the work of S. Fomin and A. Zelevinsky on total positivity and cluster algebras. These cells are special instances of a broader class of cluster varieties known as generalized double Bruhat cells, which can be studied collectively as Poisson subvarieties of $\widetilde{F}_{2n} = \mathcal{B}^{2n-1} \times G$, where $\mathcal{B}$ is the flag variety of $G$. The spaces $\widetilde{F}_{2n}$ are Poisson groupoids over $\mathcal{B}^n$ and were introduced by J.-H. Lu, V. Mouquin, and S. Yu in the study of configuration Poisson groupoids of flags. In this work, we describe the spaces $\widetilde{F}_{2n}$ as decorated moduli spaces of flat $G$-bundles over a disc. This perspective yields the following results: (1) We explicitly integrate the Poisson groupoids $\widetilde{F}_{2n}$ to symplectic double groupoids, which are complex algebraic varieties. Furthermore, we show that these integrations are symplectically Morita equivalent for all $n$. (2) Using this construction, we integrate the Poisson subgroupoids of $\widetilde{F}_{2n}$ formed by unions of generalized double Bruhat cells to explicit symplectic double groupoids. As a corollary, we obtain integrations for the top-dimensional generalized double Bruhat cells contained therein. (3) Finally, we relate our integration to the work of P. Boalch on meromorphic connections. We lift the torus actions on $\widetilde{F}_{2n}$ to the double groupoid level and show that they correspond to the quasi-Hamiltonian actions on the fission spaces of irregular singularities.

math.SG↗

arXiv2026

Transitive Courant algebroids and double symplectic groupoids

In this work we extend the Lu-Weinstein construction of double symplectic groupoids to any Lie bialgebroid such that its associated Courant algebroid is transitive and its Atiyah algebroid integrable. We illustrate this result by showing how it generalises many of the examples of double symplectic groupoids that have appeared in the literature. As preliminary steps for this construction, we give a classification of exact twisted Courant algebroids over Lie groupoids (CA-groupoids for short) and we show the existence of a foliation by twisted Courant algebroids on the base of a twisted CA-groupoid.

math.DG↗

arXiv2026

Shifted lagrangian structures in Poisson geometry

This paper develops new aspects of the interplay between shifted symplectic geometry and classical Poisson geometry, focusing on lagrangian morphisms into 2-shifted symplectic groups. We establish a Lie-type correspondence between such morphisms and Dirac structures in transitive Courant algebroids given by the product of an exact Courant algebroid and a quadratic Lie algebra. As a key application, we identify the global objects integrating quasi-Poisson manifolds, which we call multiplicative D-valued moment maps; this extends the integration of Poisson manifolds to symplectic groupoids and the lifting of Poisson actions to multiplicative hamiltonian actions. We devise systematic constructions of quasi-symplectic groupoids via fibred products of 2-shifted lagrangians, extending classical reduction procedures. This places known constructions, such as the integrations of Poisson homogeneous spaces and Poisson quotients, into a broader, conceptual framework, while yielding new examples.

math.SG↗

arXiv2025

Symplectic double groupoids and the generalized Kähler potential

A description of the fundamental degrees of freedom underlying generalized Kähler geometry, which separates its holomorphic moduli from its compatible Riemannian metric in a similar way to the Kähler case, has been sought since its discovery in 1984. In this paper, we describe a full solution to this problem for arbitrary generalized Kähler manifolds. We discover that the holomorphic structure underlying a generalized Kähler manifold is a holomorphic symplectic Morita double bimodule between double symplectic groupoids, and that each compatible Riemannian metric is given by a Lagrangian submanifold forming a bisection of the real symplectic core of this double bimodule. In other words, a generalized Kähler manifold has an associated holomorphic symplectic manifold of quadruple dimension and equipped with an anti-holomorphic involution; the generalized Kähler metric is then determined by the choice of a Lagrangian submanifold of the fixed point locus of this involution. This resolves affirmatively a long-standing conjecture by physicists concerning the existence of a generalized Kähler potential. We demonstrate the theory by constructing explicitly the above Morita double bimodule and Lagrangian bisection for the well-known generalized Kähler structures on compact even-dimensional semisimple Lie groups, which have until now escaped such analysis. We construct the required holomorphic symplectic manifolds by expressing them as moduli spaces of flat connections on surfaces with Lagrangian boundary conditions, through a quasi-Hamiltonian reduction.

math.DG↗

arXiv2024

Homological vector fields over differentiable stacks

In this work we solve the problem of providing a Morita invariant definition of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry, we view these structures as instances of vector fields on graded groupoids which are homological up to homotopy. We describe such vector fields in general from two complementary viewpoints: firstly, as Maurer-Cartan elements in a differential graded Lie algebra of multivector fields and, secondly, we also view them from a categorical approach, in terms of functors and natural transformations. Thereby, we obtain a unifying conceptual framework for studying LA-groupoids, $L_2$-algebroids (including semistrict Lie 2-algebras and 2-term representations up to homotopy), infinitesimal gerbe prequantizations, higher gauge theory (specifically, 2-connections on 2-bundles), quasi-Poisson groupoids and (twisted) multiplicative Courant algebroids.

math.DG↗

arXiv2023

Poisson groupoids and moduli spaces of flat bundles over surfaces

Let $Σ$ be a compact connected and oriented surface with nonempty boundary and let $G$ be a Lie group equipped with a bi-invariant pseudo-Riemannian metric. The moduli space of flat principal $G$-bundles over $Σ$ which are trivialized at a finite subset of $\partialΣ$ carries a natural quasi-Hamiltonian structure which was introduced by Li-Bland and Severa. By a suitable restriction of the holonomy over $\partial Σ$ and of the gauge action, which is called a decoration of $\partial Σ$, it is possible to obtain a number of interesting Poisson structures as subquotients of this family of quasi-Hamiltonian structures. In this work we use this quasi-Hamiltonian structure to construct Poisson and symplectic groupoids in a systematic fashion by means of two observations: (1) gluing two copies of the same decorated surface along suitable subspaces of their boundaries determines a groupoid structure on the moduli space associated to the new surface, this procedure can be iterated by gluing four copies of the same surface, thereby inducing a double Poisson groupoid structure; (2) on the other hand, we can suppose that $G$ is a Lie 2-group, then the groupoid structure on $G$ descends to a groupoid structure on the moduli space of flat $G$-bundles over $Σ$. These two observations can be combined to produce up to three distinct and compatible groupoid structures on the associated moduli spaces. We illustrate these methods by considering symplectic groupoids over Bruhat cells, twisted moduli spaces and Poisson 2-groups besides the classical examples.

math.DG↗

arXiv2020

Integrability of quotients in Poisson and Dirac geometry

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We also give explicit constructions of Lie groupoids integrating two interesting families of geometric structures: (i) a special class of Poisson homogeneous spaces of symplectic groupoids integrating Poisson groups and (ii) Dirac homogeneous spaces.

math.DG↗

arXiv2020

Leaves of stacky Lie algebroids

We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining an interesting class of symplectic groupoids coming from their symplectic foliations. In particular, we show that for a (strict) Lie 2-group the coadjoint orbits of the units in the dual of its Lie 2-algebra are symplectic groupoids, meaning that the classical Kostant-Kirillov-Souriau symplectic forms on these special coadjoint orbits are multiplicative.

math.SG↗