3,815 research outputs found

    A Physical Origin for Singular Support Conditions in Geometric Langlands Theory

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    We explain how the nilpotent singular support condition introduced into the geometric Langlands conjecture by Arinkin and Gaitsgory arises naturally from the point of view of N = 4 supersymmetric gauge theory. We define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum, both in functorial field theory and in the language of factorization algebras. For B-twisted N = 4 gauge theory with gauge group G, the moduli space of vacua is equivalent to h*/W , and the nilpotent singular support condition arises by restricting to the vacuum 0 in h*/W. We then investigate the categories obtained by restricting to points in larger strata, and conjecture that these categories are equivalent to the geometric Langlands categories with gauge symmetry broken to a Levi subgroup, and furthermore that by assembling such for the groups GL_n for all positive integers n one finds a hidden factorization structure for the geometric Langlands theory.Comment: 55 pages, 5 figures, more improvements to the expositio

    On 2-Holonomy

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    We construct a cycle in higher Hochschild homology associated to the 2-dimensional torus which represents 2-holonomy of a non-abelian gerbe in the same way the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez-Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module mu: g -> h of the principal 2-bundle, the Lie algebra h is abelian, up to equivalence of crossed modules

    Lie Groupoids and Lie algebroids in physics and noncommutative geometry

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    The aim of this review paper is to explain the relevance of Lie groupoids and Lie algebroids to both physicists and noncommutative geometers. Groupoids generalize groups, spaces, group actions, and equivalence relations. This last aspect dominates in noncommutative geometry, where groupoids provide the basic tool to desingularize pathological quotient spaces. In physics, however, the main role of groupoids is to provide a unified description of internal and external symmetries. What is shared by noncommutative geometry and physics is the importance of Connes's idea of associating a C*-algebra C*(G) to a Lie groupoid G: in noncommutative geometry C*(G) replaces a given singular quotient space by an appropriate noncommutative space, whereas in physics it gives the algebra of observables of a quantum system whose symmetries are encoded by G. Moreover, Connes's map G -> C*(G) has a classical analogue G -> A*(G) in symplectic geometry due to Weinstein, which defines the Poisson manifold of the corresponding classical system as the dual of the so-called Lie algebroid A(G) of the Lie groupoid G, an object generalizing both Lie algebras and tangent bundles. This will also lead into symplectic groupoids and the conjectural functoriality of quantization.Comment: 39 pages; to appear in special issue of J. Geom. Phy

    The enveloping algebra of a Lie algebra of differential operators

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    The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a D\operatorname{D}-Lie algebra. A D\operatorname{D}-Lie algebra L~\tilde{L} is a Lie-Rinehart algebra over A/kA/k equipped with an AkAA\otimes_k A-module structure that is compatible with the Lie-structure. It may be viewed as a simultaneous generalization of a Lie-Rinehart algebra and an Atiyah algebra with additional structure. Given a D\operatorname{D}-Lie algebra L~\tilde{L} and an arbitrary connection (ρ,E)(\rho, E) we construct the universal ring U~(L~,ρ)\tilde{U}^{\otimes}(\tilde{L},\rho) of the connection (ρ,E)(\rho, E). The associative unital ring U~(L~,ρ)\tilde{U}^{\otimes}(\tilde{L},\rho) is in the case when AA is Noetherian and L~\tilde{L} and EE finitely generated AA-modules, an almost commutative Noetherian sub ring of Diff(E)\operatorname{Diff}(E) - the ring of differential operators on EE. It is constructed using non-abelian extensions of D\operatorname{D}-Lie algebras. The non-flat connection (ρ,E)(\rho, E) is a finitely generated U~(L~,ρ) \tilde{U}^{\otimes}(\tilde{L},\rho)-module, hence we may speak of the characteristic variety Char(ρ,E)\operatorname{Char}(\rho,E) of (ρ,E)(\rho, E) in the sense of DD-modules. We may define the notion of holonomicity for non-flat connections using the universal ring U~(L~,ρ) \tilde{U}^{\otimes}(\tilde{L},\rho). This was previously done for flat connections. We also define cohomology and homology of arbitrary non-flat connections. The cohomology and homology of a non-flat connection (ρ,E)(\rho,E) is defined using Ext\operatorname{Ext} and Tor\operatorname{Tor}-groups of a non-Noetherian ring U\operatorname{U}. In the case when the AA-module EE is finitely generated we may always calculate cohomology and homology using a Noetherian quotient of U\operatorname{U}. This was previously done for flat connections.Comment: 24.3.2019: Corrections made on the definition of the universal ring and some new proofs added. 24.09.2019: Extended introduction and minor changes. 03.11.2019: A significant extension - 15 pages added. 21.07.2020: An example on finite dimensionality of cohomology and homology groups added (Ex. 3.21

    Towards differentiation and integration between Hopf algebroids and Lie algebroids

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    In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie-Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie-Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors form the category of Lie-Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented along the exposition.Comment: Minor changes. Comments are very welcome
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