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?

Transcript of What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves •...

Page 1: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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

Page 2: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

“Quantum Symplectic Geometry”

symplectic manifold(Hilbert space)H

algebra of functionson

(algebra of operators)A

Lagrangian submanifoldvector P 2222 H

A = 0^

P = 0i

A i

(M, ω)

M

L 3333 M

Page 3: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

The simplest example

symplectic manifold:

Lagrangian submanifoldvector P 2222 H

A(x,y) = 0 P = 0

parametrized by

A (x,y;q)^ ^^

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What is expected …

PA(x,y)=0

A

Page 5: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

The First Surprise

A(x,y) polynomial in x,y

and

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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 �

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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

Page 8: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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!

Page 9: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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):

Page 10: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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

[D.Vogan, …]

Page 11: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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]

Page 12: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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.

Page 13: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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

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Integrable Systems

Spectral curve:

Baxter equation

A(x,y) = 0

Q = 0

in

A (x,y;q)^ ^^

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XXZ magnet and sinh-Gordon

Spectral curve:

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

Page 16: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

Trigonometric Ruijsenaars-Schneider

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

Spectral curve:

has a simple solution:

Page 17: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

Two Different Quantizations

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

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

^Baxter

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Complex Chern-SimonsComplex ChernComplex Chern--SimonsSimons

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Quantization of

LLLL = line bundle

overwith

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E.Verlinde

Example: G=SU(2) g=2

Quantization of

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Complexification

-valued connectionG�

moduli space of Higgs bundleshyper-Kahler

ω

I

ω

J

ω

K

I J K conj.

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Complexification

-valued connectionG�

circle action

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Equivariant Verlinde Formula

Example: G=SU(2)

[S.G., D.Pei]

Page 24: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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)

Page 25: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

Classifying Phases of MatterClassifying Phases of MatterClassifying Phases of Matter

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Quiver Chern-Simons theory

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

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

avertex U(1) Chern-Simons at level a

a

edge

ai

aj

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Quiver Chern-Simons theory

integrate out A

Page 28: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

3d Kirby moves

Page 29: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

3d Kirby moves

A is Lagrange multiplier

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

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Plumbing graphs

Intersection form on :

Page 31: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

Kaluza-Klein compactification

6d fivebrane theory

on � x M6-n

n

“effective” theory T[M ]

in 6-n dimensionsn

depends on topology and

geometry of Mn

Page 32: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

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

Page 33: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

6 = 2 + 4

4-manifold M42d N = (0,2) theory

T[M ]4

6d fivebrane theory

on � x M2

4depends on topology and

geometry of M4

Page 34: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

6 = 2 + 4

4-manifold M42d 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:

Page 35: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y

6 = 2 + 4

4-manifold M4Vertex Operator

Algebra

Heisenberg algebra0-handle

affine Kac-Moodyadding 2-handles: ADE

more general plumbinggraphs, Kirby moves, …

VOAs associated witheven positive lattices …

Page 36: What is Quantization?What is Quantization? · 2020-01-01 · Quantization of Algebraic Curves • For any closed cycle: Example: A(x,y ) = 1 - (x - x - 2 - x + x -4 -2 42 2)y + y