Geometric Langlands Conjecture: An Introduction

Coleman Dobson CMC3 South October 13, 2018

Grothendieck taught that to do geometry one does not need a space, it is enough to have a category of sheaves on that would be space; this idea has been transmitted to noncommutative algebra by Yuri Manin

IdeaWe will explore the magical contributions of Peter Scholze (University of Bonn) and Edward Frenkle (UCB) to the Geometric Langlands Conjecture, by way of p-adic number theory, perfectoid spaces, curves over finite fields, Riemann surfaces (curves over the field of complex numbers) and Galois representations.

Frenkle:“ It is tempting to think of it as a “grand unified theory” of mathematics, since it ties together so many different disciplines”

Andrew Wiles:“There exists a deep analogy between number theory and the

geometry of complex algebraic curves, with the theory of

algebraic curves over finite fields appearing as the go-between.

The Original TitansGeometric Langlands theory originated from the ideas of four people:

A. Beilinson, P. Deligne, V. Drinfeld and G. Laumon

Langlands Correspondence

Classical Langlands:

Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles

Geometric Langlands:

Equate representation of Galois Groups and p-adic modular forms. Method: Replace number fields with function fields and apply derived algebraic geometry

Categorical Langlands:

The categorical geometric Langlands conjecture is a categorical version of the geometric Langlands conjecture. It is formulated as an equivalence of stable (infinity,1)-categories of D-modules on the derived stack of G-bundles on a curve X, and ind-coherent sheaves on the derived stack of LG-equivariant local systems on X, where X is a smooth complete curve over a field of characteristic zero, and G is a reductive group and LG is its Langlands dual.

Quantum Q-Langlands: quantum K-theory of Nakajima quiver varieties

Frenkle: Grand Unified Theory of Mathematics

Andre WeilNumber theory side : galois groups / finite fields

Riemann surfaces side: curves over complex fields

(Geometry: with geometry we get cohomology, Functors, Varieties)

2 Sides connected with : etale cohomology

Algebraic analogues of cohomology groups of topological space (Weil conjectures)

Use to construct representations of finite groups of Lie Type

The 2nd TitansAndre Weil

D. Arinkin, D. Gaitsgory, Andrew Wiles

Peter Scholze: P-adic Fields, Prismatic Cohomology

Edward Frenkle

Andrei Okounkov

Mina Aganagic

Kyoto School

3 Columns of Langlands

Number Theory

Galois Groups Over Function Fields

Derived Algebraic Geometry - Principal Bundle (Fiber is a Lie Group :) )

Classical LC

Number Theory

Galois Groups over Finite Fields

Curves over Finite Fields

Categorical LC

Equate O-modules on LOC_LG withD-Modules on Bun_GX: Curve/CG: reductive algebra grp/CLG: Langlands dual group

Geometric LC

Curves over Function FieldsGalois Groups over Function FieldsCurves over Complex FieldsRiemann Surfaces

Classical Langlands CorrespondenceX = compact Riemann Surface; X/B = smooth projective curve char(k) = 0 (bc Working in D-Module)

Geometry Side Topology Side

Functional Analysis: Consider functions Bun(F_q) → Complex Fields } Automorphic Forms

Upgrade notion of Function: Geometric Langlands Correspondence

We want to construct a nice family of automorphic objects

(Function → D-Modules → Varieties )

Automorphic SideAdelesV. Bundles on Riemann SurfaceBun_GL(n) = {rank n v. bundles on X}

Galois SideRank n v bundle on X together with flat connectionRank n local systems of PDE’s on X

Geometric Langlands CorrespondenceGeometric Objects Topological Objects

Automorphic Object of Interest

Look at Space of Functions

Category of D-Modules/Algebraic systems of PDE’s/Quasicoherent Sheaves with Flat connections

Rough goal: find a family of special D-Modules on Bun_G that forms a basis.

Classically: 𝜋_1(x) → GL(n) {look at representations of 𝜋_1(x) and of GL(n)}

Fundamental group is incarnation of Galois Group

D-Modules on Bun_G

Bun_G is Algebraic Stack

Depends on Holomorphic Structure

LOC: Rank n Local Systems on X

Depends on Topological Structure

Monodromy Invariant

Compatible with FunctorsCategorical Equivalence is Compatible with Functors. 1 Side Hecke Functors

(parameterized by rep of dual group)

O_e: Skyscraper Sheaf: F_v,x(O_e) ~ V x(tensor) O_e

O-Modules on LOC_G

Frobenius Functors

O-e is Frobenius Eigensheaf

D-Modules on Bun_G

Hecke Functors (finite alg rep of LG) (multiply by stokes of perverse sheaf)

F_e is Hecke Eigensheaf

Fourier Mikai Transform2 copies of real line

e^itx

--------------------- x ---------------------- t

Delta Function FT Wave

D-Modules is a non-abelian FT of O-Modules (Supported at O_e)

Equivalent Geometric Problem: Find the non-abelian FT in geometry

All connected with Quantum PhysicsS Duality: String Theory- physics invariant under strong and weak coupling constant. Sheshmani: S Duality Modularity

Mirror Symmetry: Topologically distinct CY Manifolds identical under Mirror Flop

C.L.C

Same Equivalent MicroLocalization

HMS

**Commutative Triangle of Equivalent Categories: S-Duality connected to GLC

There is Geometric/Categorical LC, HMS from S Duality, Microlocalization connects D-modules on Bun_G and A-branes

on Hitchin Moduli Space M_h(G)

DO-modules on LOC_LG

F_v,x

A Brane on M_h(LG)

DD-Modules on Bun_G

H _v,x

B Brane on M_h(G)

ReferencesDennis Gaitsgory, ‘Recent Progress in Geometric Langlands Theory’ 2016

Frenkle, ‘Lectures on the Langlands Program and Conformal Field Theory’ 2005

Aganagic, ‘Quantum Q-Langlands Correspondence’ 2017

Thank you!!Uncountable thank you to the fellow Q’s : Justin, Stephen, LanLan, Brian, Natasha, Artems