# What is Quantization?What is Quantization? 2020-01-01¢ Quantization of Algebraic Curves...

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What is Quantization?What is Quantization?What is Quantization?

“Quantum Symplectic Geometry”

symplectic manifold (Hilbert space)H

algebra of functions on

(algebra of operators)A

Lagrangian submanifold vector P 2222 H

A = 0 ^

P = 0 i

A i

(M, ω)

M

L 3333 M

The simplest example

symplectic manifold:

Lagrangian submanifold vector P 2222 H

A(x,y) = 0 P = 0

parametrized by

A (x,y;q)^ ^ ^

What is expected …

PA(x,y)=0

A

The First Surprise

A(x,y) polynomial in x,y

and

Quantization of Algebraic Curves

• For any closed cycle:

[S.G., P.Sulkowski]

• Has an elegant interpretation in terms of algebraic K-theory and the Bloch

group of �

Quantization of Algebraic Curves

• For any closed cycle:

Example:

A(x,y) = 1 - (x - x - 2 - x + x )y + y-4 -2 42 2

B(x,y) = 1 - (x - x - 2 - x + x )y + y-6 -2 62 2

Has all the symmetries, but is NOT A-polynomial of any knot

Geometric Representation Theory

e.g. SL(2,�)

e.g. SL(2,�)

e.g. SU(2)

G = real form of G� �

G = complexification of G�

G = (simple) compact Lie group

• Let O (x) = G , x be a coadjoint orbit

through an element x , where � �

an example of a symplectic manifold!

More Surprises

• there exist unitary representations that

don’t appear to correspond to orbits,

e.g. complementary series

• conversely, there are real orbits that don’t seem to correspond to unitary representations.

M = O �

real coadjoint orbit of G�

representation of G�

: [Borel-Weil-Bott] [Harish-Chandra]

: [B.Kostant]

[A.A.Kirillov] [D.Vogan]

[R.Brylinski] [N.Berline, M.Vergne]

:

Puzzles (lessons for quantization):

More Surprises

M = O �

real coadjoint orbit of G�

representation of G�

Puzzles (lessons for quantization):

= minimal orbit of SO(p,q) type B

(p+q = odd)

exists if p #### 3 or q #### 3 does not exist if p,q PPPP 4

[D.Vogan, …]

More Surprises

M = O �

real coadjoint orbit of G�

representation of G�

Puzzles (lessons for quantization):

• Both of these issues can be resolved at the cost of replacing classical geometric objects (namely, coadjoint orbits) with their quantum or “stringy” analogs (branes).

[S.G., E.Witten]

Brane Quantization

Puzzle 1: complementary series?

M = O �

real coadjoint orbit of G�

representation of G�

Puzzle 2: = minimal orbit of SO(p,q) type B

There is a corresponding A-brane!

Lagrangian brane supported on M is a good

object in the Fukaya category of Y = M only if

is a mod 2 reduction of

a torsion class in the integral cohomology of M.

�

Quantization: Uniqueness?Quantization: Uniqueness?Quantization: Uniqueness?

Integrable Systems

Spectral curve:

Baxter equation

A(x,y) = 0

Q = 0

in

A (x,y;q)^ ^ ^

XXZ magnet and sinh-Gordon

Spectral curve:

Baxter equation: Q = 0A (x,y;q)^ ^ ^

Trigonometric Ruijsenaars-Schneider

Baxter equation: Q = 0A (x,y;q)^ ^ ^

Spectral curve:

has a simple solution:

Two Different Quantizations

[A.Gadde, S.G., P.Putrov]

P = 0A (x,y;q)^ ^ ^

Q = 0A ^

Baxter

Complex Chern-SimonsComplex ChernComplex Chern--SimonsSimons

Quantization of

LLLL = line bundle

over with

E.Verlinde

Example: G=SU(2) g=2

Quantization of

Complexification

-valued connectionG�

moduli space of Higgs bundleshyper-Kahler

ω

I

ω

J

ω

K

I J K conj.

Complexification

-valued connectionG�

circle action

Equivariant Verlinde Formula

Example: G=SU(2)

[S.G., D.Pei]

New 2d/ 3d TQFT

• “equivariant Higgs vertex”

• 3d-3d correspondence (equivariant integration over Hitchin moduli space)

• Equivariant G/G model on S

• Abelian 2d theory on the Coulomb branch

• Topological twist of 3d N=2 adjoint SQCD (equivariant quantum K-theory of vortex moduli space)

Classifying Phases of MatterClassifying Phases of MatterClassifying Phases of Matter

Quiver Chern-Simons theory

cf. [D.Belov, G.Moore] [A.Kapustin, N.Saulina]

[J.Fuchs, C.Schweigert, A.Valentino] :

a vertex U(1) Chern-Simons at level a

a

edge

a i

a j

Quiver Chern-Simons theory

integrate out A

3d Kirby moves

3d Kirby moves

A is Lagrange multiplier

Integrating out A makes B pure gauge and removes all its Chern-Simons couplings

Plumbing graphs

Intersection form on :

Kaluza-Klein compactification

6d fivebrane theory

on � x M 6-n

n

“effective” theory T[M ]

in 6-n dimensions n

depends on topology and

geometry of M n

4-manifolds VOA’s44--manifolds manifolds VOAVOA’’ss

6 = 2 + 4

4-manifold M4 2d N = (0,2) theory

T[M ] 4

6d fivebrane theory

on � x M 2

4 depends on topology and

geometry of M 4

6 = 2 + 4

4-manifold M4 2d N = (0,2) theory

T[M ] 4

• representations of affine Kac-Moody algebras

• moonshine module / monster symmetry

• affine W-algebras

• chiral de Rham complex

Vertex Operator Algebras:

6 = 2 + 4

4-manifold M4 Vertex Operator Algebra

Heisenberg algebra0-handle

affine Kac-Moodyadding 2-handles: ADE

more general plumbing graphs, Kirby moves, …

VOAs associated with even positive lattices …

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