Geometric Langlands duality and representations of algebraic groups over commutative rings
TL;DR: In this article, the basic relationship between G and G is discussed, and a canonical construction of G, starting from G, is presented, which leads to a rather explicit construction of a Hopf algebra by Tannakian formalism.
Abstract: As such, it can be viewed as a first step in the geometric Langlands program. The connected complex reductive groups have a combinatorial classification by their root data. In the root datum the roots and the co-roots appear in a symmetric manner and so the connected reductive algebraic groups come in pairs. If G is a reductive group, we write G for its companion and call it the dual group G. The notion of the dual group itself does not appear in Satake's paper, but was introduced by Langlands, together with its various elaborations, in [LI], [L2] and is a cornerstone of the Langlands program. It also appeared later in physics [MO], [GNO]. In this paper we discuss the basic relationship between G and G. We begin with a reductive G and consider the affine Grassmannian Qx, the Grassmannian for the loop group of G. For technical reasons we work with formal algebraic loops. The affine Grassmannian is an infinite dimen sional complex space. We consider a certain category of sheaves, the spherical perverse sheaves, on ?r. These sheaves can be multiplied using a convolution product and this leads to a rather explicit construction of a Hopf algebra, by what has come to be known as Tannakian formalism. The resulting Hopf algebra turns out to be the ring of functions on G. In this interpretation, the spherical perverse sheaves on the affine Grassman nian correspond to finite dimensional complex representations of G. Thus, instead of defining G in terms of the classification of reductive groups, we pro vide a canonical construction of G, starting from G. We can carry out our construction over the integers. The spherical perverse sheaves are then those with integral coefficients, but the Grassmannian remains a complex algebraic object.
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Book•
21 Sep 2017TL;DR: The first comprehensive introduction to the theory of algebraic group schemes over fields was given in this paper, which includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry.
Abstract: Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.
233 citations
TL;DR: In this article, an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory is given. But the connections between the Langlands Program and two-dimensional conformal field theory are not discussed.
Abstract: These lecture notes give an overview of recent results in geometric Langlands correspondence which may yield applications to quantum field theory. We start with a motivated introduction to the Langlands Program, including its geometric reformulation, addressed primarily to physicists. I tried to make it as self-contained as possible, requiring very little mathematical background. Next, we describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality.
231 citations
TL;DR: In this paper, a survey of the theory of perverse sheaves is presented, concluding with the decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber.
Abstract: We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples.
222 citations
Posted Content•
TL;DR: In this paper, the Coulomb branch is defined as an affine algebraic variety with a singularity and a Coulomb action in the form Ω( √ √ N 2 ).
Abstract: Consider the $3$-dimensional $\mathcal N=4$ supersymmetric gauge theory associated with a compact Lie group $G_c$ and its quaternionic representation $\mathbf M$. Physicists study its Coulomb branch, which is a noncompact hyper-Kahler manifold with an $\mathrm{SU}(2)$-action, possibly with singularities. We give a mathematical definition of the Coulomb branch as an affine algebraic variety with $\mathbb C^\times$-action when $\mathbf M$ is of a form $\mathbf N\oplus\mathbf N^*$, as the second step of the proposal given in arXiv:1503.03676.
191 citations
Posted Content•
01 Jan 2015
TL;DR: In this article, a construction for the quantum-corrected Coulomb branch of a general 3D gauge theory with N = 4 supersymmetry is proposed in terms of local coordinates associated with an abelianized theory.
Abstract: We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory withN = 4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a xed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkahler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.
167 citations
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CERN1
TL;DR: When quantized, the magnetic monopole soliton solutions constructed by 't Hooft and Polyakov, as modified by Prasad, Sommerfield and Bogomolny, form a gauge triplet with the photon, corresponding to a Lagrangian similar to the original Georgi-Glashow one, with magnetic replacing electric charge as discussed by the authors.
1,067 citations
TL;DR: In this article, a generalization to singular spaces of the Poincare-Lefschetz theory of intersections of homology cycles on manifolds is presented, which can be summarized in three fundamental propositions: 0.
771 citations
TL;DR: In this article, the magnetic field for an exact gauge group H (assumed compact and connected) exhibits an inverse square law behaviour at large distances, and the generalized magnetic charge, appearing as the coefficient, completely determines the topological quantum number of the solution.
752 citations
Book•
28 Jun 1994
TL;DR: In this paper, the DG-modules and equivariant cohomology of toric varieties have been studied, and the derived category D G (X) and functors have been defined.
Abstract: Derived category D G (X) and functors.- DG-modules and equivariant cohomology.- Equivariant cohomology of toric varieties.
604 citations