Lectures on the Langlands Program and Conformal Field Theory
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Robert P. Langlands
The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2018 to Robert P. Langlands of the Institute for Advanced Study, Princeton, USA, “for his visionary program connecting representation theory to number theory.” The Langlands program predicts the existence of a tight The group GL(2) is the simplest example of a non- web of connections between automorphic forms and abelian reductive group. To proceed to the general Galois groups. case, Langlands saw the need for a stable trace formula, now established by Arthur. Together with Ngô’s proof The great achievement of algebraic number theory in of the so-called Fundamental Lemma, conjectured by the first third of the 20th century was class field theory. Langlands, this has led to the endoscopic classification This theory is a vast generalisation of Gauss’s law of of automorphic representations of classical groups, in quadratic reciprocity. It provides an array of powerful terms of those of general linear groups. tools for studying problems governed by abelian Galois groups. The non-abelian case turns out to be Functoriality dramatically unifies a number of important substantially deeper. Langlands, in a famous letter to results, including the modularity of elliptic curves and André Weil in 1967, outlined a far-reaching program that the proof of the Sato-Tate conjecture. It also lends revolutionised the understanding of this problem. weight to many outstanding conjectures, such as the Ramanujan-Peterson and Selberg conjectures, and the Langlands’s recognition that one should relate Hasse-Weil conjecture for zeta functions. representations of Galois groups to automorphic forms involves an unexpected and fundamental insight, now Functoriality for reductive groups over number fields called Langlands functoriality. -
Perverse Sheaves
Perverse Sheaves Bhargav Bhatt Fall 2015 1 September 8, 2015 The goal of this class is to introduce perverse sheaves, and how to work with it; plus some applications. Background For more background, see Kleiman's paper entitled \The development/history of intersection homology theory". On manifolds, the idea is that you can intersect cycles via Poincar´eduality|we want to be able to do this on singular spces, not just manifolds. Deligne figured out how to compute intersection homology via sheaf cohomology, and does not use anything about cycles|only pullbacks and truncations of complexes of sheaves. In any derived category you can do this|even in characteristic p. The basic summary is that we define an abelian subcategory that lives inside the derived category of constructible sheaves, which we call the category of perverse sheaves. We want to get to what is called the decomposition theorem. Outline of Course 1. Derived categories, t-structures 2. Six Functors 3. Perverse sheaves—definition, some properties 4. Statement of decomposition theorem|\yoga of weights" 5. Application 1: Beilinson, et al., \there are enough perverse sheaves", they generate the derived category of constructible sheaves 6. Application 2: Radon transforms. Use to understand monodromy of hyperplane sections. 7. Some geometric ideas to prove the decomposition theorem. If you want to understand everything in the course you need a lot of background. We will assume Hartshorne- level algebraic geometry. We also need constructible sheaves|look at Sheaves in Topology. Problem sets will be given, but not collected; will be on the webpage. There are more references than BBD; they will be online. -
MORSE THEORY and TILTING SHEAVES 1. Introduction This
MORSE THEORY AND TILTING SHEAVES DAVID BEN-ZVI AND DAVID NADLER 1. Introduction This paper is a result of our trying to understand what a tilting perverse sheaf is. (See [1] for definitions and background material.) Our basic example is that of the flag variety B of a complex reductive group G and perverse sheaves on the Schubert stratification S. In this setting, there are many characterizations of tilting sheaves involving both their geometry and categorical structure. In this paper, we propose a new geometric construction of such tilting sheaves. Namely, we show that they naturally arise via the Morse theory of sheaves on the opposite Schubert stratification Sopp. In the context of a flag variety B, our construction takes the following form. Given a sheaf F on B, a point p ∈ B, and a germ F of a function at p, we have the ∗ corresponding local Morse group (or vanishing cycles) Mp,F (F) which measures how F changes at p as we move in the family defined by F . If we restrict to sheaves F constructible with respect to Sopp, then it is possible to find a sheaf ∗ Mp,F on a neighborhood of p which represents the functor Mp,F (though Mp,F is not constructible with respect to Sopp.) To be precise, for any such F, we have a natural isomorphism ∗ ∗ Mp,F (F) ' RHomD(B)(Mp,F , F). Since this may be thought of as an integral identity, we call the sheaf Mp,F a Morse kernel. The local Morse groups play a central role in the theory of perverse sheaves. -
Refined Invariants in Geometry, Topology and String Theory Jun 2
Refined invariants in geometry, topology and string theory Jun 2 - Jun 7, 2013 MEALS Breakfast (Bu↵et): 7:00–9:30 am, Sally Borden Building, Monday–Friday Lunch (Bu↵et): 11:30 am–1:30 pm, Sally Borden Building, Monday–Friday Dinner (Bu↵et): 5:30–7:30 pm, Sally Borden Building, Sunday–Thursday Co↵ee Breaks: As per daily schedule, in the foyer of the TransCanada Pipeline Pavilion (TCPL) Please remember to scan your meal card at the host/hostess station in the dining room for each meal. MEETING ROOMS All lectures will be held in the lecture theater in the TransCanada Pipelines Pavilion (TCPL). An LCD projector, a laptop, a document camera, and blackboards are available for presentations. SCHEDULE Sunday 16:00 Check-in begins @ Front Desk, Professional Development Centre 17:30–19:30 Bu↵et Dinner, Sally Borden Building Monday 7:00–8:45 Breakfast 8:45–9:00 Introduction and Welcome by BIRS Station Manager 9:00–10:00 Andrei Okounkov: M-theory and DT-theory Co↵ee 10.30–11.30 Sheldon Katz: Equivariant stable pair invariants and refined BPS indices 11:30–13:00 Lunch 13:00–14:00 Guided Tour of The Ban↵Centre; meet in the 2nd floor lounge, Corbett Hall 14:00 Group Photo; meet in foyer of TCPL 14:10–15:10 Dominic Joyce: Categorification of Donaldson–Thomas theory using perverse sheaves Co↵ee 15:45–16.45 Zheng Hua: Spin structure on moduli space of sheaves on CY 3-folds 17:00-17.30 Sven Meinhardt: Motivic DT-invariants of (-2)-curves 17:30–19:30 Dinner Tuesday 7:00–9:00 Breakfast 9:00–10:00 Alexei Oblomkov: Topology of planar curves, knot homology and representation -
Intersection Spaces, Perverse Sheaves and Type Iib String Theory
INTERSECTION SPACES, PERVERSE SHEAVES AND TYPE IIB STRING THEORY MARKUS BANAGL, NERO BUDUR, AND LAURENT¸IU MAXIM Abstract. The method of intersection spaces associates rational Poincar´ecomplexes to singular stratified spaces. For a conifold transition, the resulting cohomology theory yields the correct count of all present massless 3-branes in type IIB string theory, while intersection cohomology yields the correct count of massless 2-branes in type IIA the- ory. For complex projective hypersurfaces with an isolated singularity, we show that the cohomology of intersection spaces is the hypercohomology of a perverse sheaf, the inter- section space complex, on the hypersurface. Moreover, the intersection space complex underlies a mixed Hodge module, so its hypercohomology groups carry canonical mixed Hodge structures. For a large class of singularities, e.g., weighted homogeneous ones, global Poincar´eduality is induced by a more refined Verdier self-duality isomorphism for this perverse sheaf. For such singularities, we prove furthermore that the pushforward of the constant sheaf of a nearby smooth deformation under the specialization map to the singular space splits off the intersection space complex as a direct summand. The complementary summand is the contribution of the singularity. Thus, we obtain for such hypersurfaces a mirror statement of the Beilinson-Bernstein-Deligne decomposition of the pushforward of the constant sheaf under an algebraic resolution map into the intersection sheaf plus contributions from the singularities. Contents 1. Introduction 2 2. Prerequisites 5 2.1. Isolated hypersurface singularities 5 2.2. Intersection space (co)homology of projective hypersurfaces 6 2.3. Perverse sheaves 7 2.4. Nearby and vanishing cycles 8 2.5. -
Shtukas for Reductive Groups and Langlands Correspondence for Functions Fields
SHTUKAS FOR REDUCTIVE GROUPS AND LANGLANDS CORRESPONDENCE FOR FUNCTIONS FIELDS VINCENT LAFFORGUE This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [49] is a conjecture of utmost impor- tance, concerning global fields, i.e. number fields and function fields. Many excellent surveys are available, for example [39, 14, 13, 79, 31, 5]. The Langlands correspondence belongs to a huge system of conjectures (Langlands functoriality, Grothendieck’s vision of motives, special val- ues of L-functions, Ramanujan-Petersson conjecture, generalized Rie- mann hypothesis). This system has a remarkable deepness and logical coherence and many cases of these conjectures have already been es- tablished. Moreover the Langlands correspondence over function fields admits a geometrization, the “geometric Langlands program”, which is related to conformal field theory in Theoretical Physics. Let G be a connected reductive group over a global field F . For the sake of simplicity we assume G is split. The Langlands correspondence relates two fundamental objects, of very different nature, whose definition will be recalled later, • the automorphic forms for G, • the global Langlands parameters , i.e. the conjugacy classes of morphisms from the Galois group Gal(F =F ) to the Langlands b dual group G(Q`). b For G = GL1 we have G = GL1 and this is class field theory, which describes the abelianization of Gal(F =F ) (one particular case of it for Q is the law of quadratic reciprocity, which dates back to Euler, Legendre and Gauss). -
RELATIVE LANGLANDS This Is Based on Joint Work with Yiannis
RELATIVE LANGLANDS DAVID BEN-ZVI This is based on joint work with Yiannis Sakellaridis and Akshay Venkatesh. The general plan is to explain a connection between physics and number theory which goes through the intermediary: extended topological field theory (TFT). The moral is that boundary conditions for N = 4 super Yang-Mills (SYM) lead to something about periods of automorphic forms. Slogan: the relative Langlands program can be explained via relative TFT. 1. Periods of automorphic forms on H First we provide some background from number theory. Recall we can picture the upper-half-space H as in fig. 1. We are thinking of a modular form ' as a holomorphic function on H which transforms under the modular group SL2 (Z), or in general some congruent sub- group Γ ⊂ SL2 (Z), like a k=2-form (differential form) and is holomorphic at 1. We will consider some natural measurements of '. In particular, we can \measure it" on the red and blue lines in fig. 1. Note that we can also think of H as in fig. 2, where the red and blue lines are drawn as well. Since ' is invariant under SL2 (Z), it is really a periodic function on the circle, so it has a Fourier series. The niceness at 1 condition tells us that it starts at 0, so we get: X n (1) ' = anq n≥0 Date: Tuesday March 24, 2020; Thursday March 26, 2020. Notes by: Jackson Van Dyke, all errors introduced are my own. Figure 1. Fundamental domain for the action of SL2 (Z) on H in gray. -
Arxiv:1804.05014V3 [Math.AG] 25 Nov 2020 N Oooyo Ope Leri Aite.Frisac,Tedeco the U Instance, for for Varieties
PERVERSE SHEAVES ON SEMI-ABELIAN VARIETIES YONGQIANG LIU, LAURENTIU MAXIM, AND BOTONG WANG Abstract. We give a complete (global) characterization of C-perverse sheaves on semi- abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell’s work on perverse sheaves on complex abelian varieties, as well as Gabber-Loeser’s results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties. 1. Introduction Perverse sheaves are fundamental objects at the crossroads of topology, algebraic geometry, analysis and differential equations, with important applications in number theory, algebra and representation theory. They provide an essential tool for understanding the geometry and topology of complex algebraic varieties. For instance, the decomposition theorem [3], a far-reaching generalization of the Hard Lefschetz theorem of Hodge theory with a wealth of topological applications, requires the use of perverse sheaves. Furthermore, perverse sheaves are an integral part of Saito’s theory of mixed Hodge module [31, 32]. Perverse sheaves have also seen spectacular applications in representation theory, such as the proof of the Kazhdan-Lusztig conjecture, the proof of the geometrization of the Satake isomorphism, or the proof of the fundamental lemma in the Langlands program (e.g., see [9] for a beautiful survey). A proof of the Weil conjectures using perverse sheaves was given in [20]. However, despite their fundamental importance, perverse sheaves remain rather mysterious objects. In his 1983 ICM lecture, MacPherson [27] stated the following: The category of perverse sheaves is important because of its applications. -
Contents 1 Root Systems
Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations. -
Questions and Remarks to the Langlands Program
Questions and remarks to the Langlands program1 A. N. Parshin (Uspekhi Matem. Nauk, 67(2012), n 3, 115-146; Russian Mathematical Surveys, 67(2012), n 3, 509-539) Introduction ....................................... ......................1 Basic fields from the viewpoint of the scheme theory. .............7 Two-dimensional generalization of the Langlands correspondence . 10 Functorial properties of the Langlands correspondence. ................12 Relation with the geometric Drinfeld-Langlands correspondence . 16 Direct image conjecture . ...................20 A link with the Hasse-Weil conjecture . ................27 Appendix: zero-dimensional generalization of the Langlands correspondence . .................30 References......................................... .....................33 Introduction The goal of the Langlands program is a correspondence between representations of the Galois groups (and their generalizations or versions) and representations of reductive algebraic groups. The starting point for the construction is a field. Six types of the fields are considered: three types of local fields and three types of global fields [L3, F2]. The former ones are the following: 1) finite extensions of the field Qp of p-adic numbers, the field R of real numbers and the field C of complex numbers, arXiv:1307.1878v1 [math.NT] 7 Jul 2013 2) the fields Fq((t)) of Laurent power series, where Fq is the finite field of q elements, 3) the field of Laurent power series C((t)). The global fields are: 4) fields of algebraic numbers (= finite extensions of the field Q of rational numbers), 1I am grateful to R. P. Langlands for very useful conversations during his visit to the Steklov Mathematical institute of the Russian Academy of Sciences (Moscow, October 2011), to Michael Harris and Ulrich Stuhler, who answered my sometimes too naive questions, and to Ilhan˙ Ikeda˙ who has read a first version of the text and has made several remarks. -
LANGLANDS' CONJECTURES for PHYSICISTS 1. Introduction This Is
LANGLANDS’ CONJECTURES FOR PHYSICISTS MARK GORESKY 1. Introduction This is an expanded version of several lectures given to a group of physicists at the I.A.S. on March 8, 2004. It is a work in progress: check back in a few months to see if the empty sections at the end have been completed. This article is written on two levels. Many technical details that were not included in the original lectures, and which may be ignored on a first reading, are contained in the end-notes. The first few paragraphs of each section are designed to be accessible to a wide audience. The present article is, at best, an introduction to the many excellent survey articles ([Ar, F, G1, G2, Gr, Kn, K, R1, T] on automorphic forms and Langlands’ program. Slightly more advanced surveys include ([BR]), the books [Ba, Be] and the review articles [M, R2, R3]. 2. The conjecture for GL(n, Q) Very roughly, the conjecture is that there should exist a correspondence nice irreducible n dimensional nice automorphic representations (2.0.1) −→ representations of Gal(Q/Q) of GL(n, AQ) such that (2.0.2) {eigenvalues of Frobenius}−→{eigenvalues of Hecke operators} The purpose of the next few sections is to explain the meaning of the words in this statement. Then we will briefly examine the many generalizations of this statement to other fields besides Q and to other algebraic groups besides GL(n). 3. Fields 3.1. Points in an algebraic variety. If E ⊂ F are fields, the Galois group Gal(F/E)is the set of field automorphisms φ : E → E which fix every element of F. -
Langlands Program, Trace Formulas, and Their Geometrization
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, Number 1, January 2013, Pages 1–55 S 0273-0979(2012)01387-3 Article electronically published on October 12, 2012 LANGLANDS PROGRAM, TRACE FORMULAS, AND THEIR GEOMETRIZATION EDWARD FRENKEL Notes for the AMS Colloquium Lectures at the Joint Mathematics Meetings in Boston, January 4–6, 2012 Abstract. The Langlands Program relates Galois representations and auto- morphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functorial- ity Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey of my recent joint work with Robert Langlands and NgˆoBaoChˆau on a new approach to proving the Functoriality Conjecture using the trace formulas, and on the geometrization of the trace formulas. In particular, I discuss the connection of the latter to the categorification of the Langlands correspondence. Contents 1. Introduction 2 2. The classical Langlands Program 6 2.1. The case of GLn 6 2.2. Examples 7 2.3. Function fields 8 2.4. The Langlands correspondence 9 2.5. Langlands dual group 10 3. The geometric Langlands correspondence 12 3.1. LG-bundles with flat connection 12 3.2. Sheaves on BunG 13 3.3. Hecke functors: examples 15 3.4. Hecke functors: general definition 16 3.5. Hecke eigensheaves 18 3.6. Geometric Langlands correspondence 18 3.7. Categorical version 19 4. Langlands functoriality and trace formula 20 4.1.