3,815 research outputs found
A Physical Origin for Singular Support Conditions in Geometric Langlands Theory
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
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
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
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 -Lie algebra. A -Lie
algebra is a Lie-Rinehart algebra over equipped with an
-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 -Lie
algebra and an arbitrary connection we construct the
universal ring of the connection . The associative unital ring is in
the case when is Noetherian and and finitely generated
-modules, an almost commutative Noetherian sub ring of
- the ring of differential operators on . It is
constructed using non-abelian extensions of -Lie algebras.
The non-flat connection is a finitely generated -module, hence we may speak of the
characteristic variety of in the
sense of -modules. We may define the notion of holonomicity for non-flat
connections using the universal ring .
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 is defined using and
-groups of a non-Noetherian ring . In the
case when the -module is finitely generated we may always calculate
cohomology and homology using a Noetherian quotient of . 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
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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