Takuma N.C.T.

Graduate School of Mathematics, Nagoya University, 23 Jul. 2009

Non-compact Hopf Maps,

Quantum Hall Effect, and Twistor Theory

Takuma N.C.T.

Kazuki Hasebe

arXiv: 0902.2523, 0905.2792

Introduction

There are remarkable close relations between these two independently developed fields !

2. Quatum Hall Effect

Novel Quantum State of Matter

(Condensed matter:Non-relativistic Quantum Mechanics)

Quatum Spin Hall Effect, Quantum Hall Effect in Graphene etc.

1. Twistor Theory

Quantization of Space-Time

(Mathematical Physics: Relativistic Quantum Mechanics)

ADHM Construction, Integrable Models. Twistor String etc.

Light has special importance.

Monopole plays an important role. R. Laughlin (1983)

R. Penrose (1967)

Landau Quantization

2D - plane

Magnetic Field

Landau levels

LLL

1st LL

2nd LL

LLL projection ``massless limit’’

Cyclotron frequency Lev Landau (1930)

Quantum Hall Effect and Monopole

Stereographic projection

F.D.M. Haldane (1983)

Many-body state on a sphere in a monopole b.g.d.

SO(3) global symmetry

``Edge’’ breaks translational sym.

Heuristic Observation: Why Light & MonopoleMassless particle with helicity

To satisfy the SU(2) algebra

Massless particle ``sees’’ a charge monopole in p-space !

spin

momentum

The position of a massless particle with definite helicityis uncertain !

Bacry (1981)

Atre, Balachandran,Govindarajan (1986)

If

Brief Introduction to Twistor

Twistor ProgramRoger Penrose (1967)

Quantization of Space-Time

What is the fundamental variables ?

Light (massless-paticle) will play the role !

Space-Time Twistor Space

``moduli space of light’’

Massless Free Particle

Massless particle

Free particle

:

Gauge symmetry

Twistor Description

Suggests a fuzzy space-time.

Massless limit

Fundamental variable

Helicity:

: Incidence Relation

Hopf Maps and QHE

Dirac Monopole and 1st Hopf Map

The 1st Hopf map

P.A.M. Dirac (1931)

Dirac Monopole

Connection of bundle

Explicit Realization of 1st Hopf Map

Hopf spinor

One-particle Mechanics

LLL Lagrangian

Constraint

Constraint

Lagrangian

LLL

Fundamental variable

LLL PhysicsEmergence of Fuzzy Geometry

Holomorphic wavefunctions

Many-body Groudstate

Laughlin-Haldane wavefunction

On the QH groundstate, particles are distributed uniformly on the basemanifold.

The groundstate is invariant under SU(2) isometry of , and does not include complex conjugations.

: SU(2) singlet combination of Hopf spinors

Higher D. Hopf Maps

Topological maps from sphere to sphere with different dimensions.

Heinz Hopf (1931,1935)

1st

2nd

3rd

(Complex number)

(Quaternion)

(Octonion)

Quaternion and 2nd Hopf Map

Willian R. Hamilton (1843)

Quaternion

2nd Hopf map

Unit

1st Hopf map

Unit

C

C

The 2nd Hop Map & SU(2) Monopole

C.N. Yang (1978)

Yang MonopoleThe 2nd Hopf map

SO(5) global symmetry

4D QHE and Twistor

D. Mihai, G. Sparling, P. Tillman (2004)

S.C. Zhang, J.P. Hu (20

01)

Many-body problem on a four-sphere in a SU(2) monopole b.g.d.

In the LLL

Point out relations to Twistor theory

In particular, Sparling and his coworkers suggested the use of the ultra-hyperboloid

G. Sparling (2002)

D. Karabali, V.P. Nair (2002,2003) S.C. Zhang (2002)

Short Summary

QHE Hopf Map Division algebra

2D

4D

8D

1st

2nd

3rd

complex numbers

quaternions

octonions

LLLTwistor ??

QHE with SU(2,2) symmetry

Noncompact Version of the Hopf Map

Hopf maps

Non-compact groups

Non-compact Hopf maps !

Split-Complex number

Split-Quaternions

Split-Octonions

Complex number

Quaternions

Octonions

James Cockle (1848,49)

Split-Algebras Split-Complex number

Split-Quaternions

Non-compact Hopf Maps

1st

2nd

3rd

Ultra-Hyperboloid with signature (p,q)

:

p q+1

Non-compact 2nd Hopf Map

SO(3,2) Hopf spinor SO(3,2) Hopf spinor

Incidence Relation

generators

Stereographic coordinates

SO(3,2) symmetry

SU(1,1) monopole

One-particle action

One-particle Mechanics on Hyperboloid

(c.f.)

constraint

LLL projection

SU(2,2) symmetry

Symmetry is Enhanced from SO(3,2) to SU(2,2)!

LLL-limit

Fundamental variable

constraint

Realization of the fuzzy geometry

The space(-time) non-commutativity comes out from that of the more fundamental space.

First, the Hopf spinor space becomes fuzzy.

This demonstrates the philosophy of Twistor !

Then, the hyperboloid also becomes fuzzy.

Analogies

Complex conjugation = Derivative

Twistor QHE

More Fundamental Quantity than Space-Time

Massless Condition

Noncommutative Geometry,

SU(2,2) Enhanced Symmetry

Holomorphic functions

Quantize and rather than !

Summary Table Non-compact 4D QHE Twistor Theory

Fundamental Quantity

Quantized value Monopole charge Helicity

Base manifold Hyperboloid Minkowski space

Original symmetry

Hopf spinor Twistor

Fuzzy Hyperboloid Fuzzy Twistor space Noncommutative Geometry

Emergent manifold

Enhanced symmetry

Special limit

Poincare

LLL zero-mass

Physics of the non-compact 4D QHE

One-particle Problem

Landau problem on a ultra-hyperboloid

: fixedThermodynamic limit

Many-body Groudstate

Laughlin-Haldane wavefunction

On the QH groundstate, particles are distributed uniformly on the basemanifold.

The groundstate is invariant under SO(3,2) isometry of , and does not include complex conjugations.

Topological Excitations Topological excitations are generated by flux penetrations.

Membrane-like excitations !

The flux has SU(1,1) internal structures.

Perspectives

Uniqueness

Everything is uniquely determined by the geometry of the Hopf map !

Base manifold

Gauge Symmetry

Global Symmetry in LLL

(For instance)n-c. 2nd Hopf map

Extra-Time Physics ?

Sp(2,R) gauge symmetry is required to eliminate the negative norms.

Base manifold

Gauge Symmetry

2T

The present model fulfills this requirement from the very beginning ! There may be some kind of ``duality’’ ??

This set-up exactly corresponds to 2T physics developed by I. Bars !

C.M. Hull (99)

Magic Dimensions of Space-Time ? Compact Hopf maps Non-compact maps

1st

2nd

3rd

After ALL

Split-algebras

Higher D. quantum liquid

Membrane-like excitation

Non-compact Hopf Maps

Non-commutative Geometry Twistor Theory

Uniqueness

Extra-time physics Magic Dimensions

END

Deep mathematical structure exists behind the model !

Entire picture is still Mystery !