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

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

  • 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 …