1/59

JJIIJI

Back

Close

Non-Commutative Geometry and QuantumPhysicsS.N. Bose Institute, Kolkata, Jan 4-11, 2006

Quantization of Contact Manifoldsand Liquid Crystal Flow

S. G. Rajeev

Department of Physics and AstronomyUniversity of Rochester, Rochester, NY14627email: rajeev@pas.rochester.edu

Prepared using pdfslide developed by C. V. Radhakrishnan of River Valley Technologies,Trivandrum, India

2/59

JJIIJI

Back

Close

What is a Contact Manifold?

A contact manifold is the odd dimensional analogue of a symplectic

manifold. This idea emerges in classical mechanics in two different ways:

1. A system whose hamiltonian that depends explicitly on time; time

must be included as a co-ordinate in the phase space, hence it is

odd dimensional.

2. For a conservative system, the orbits lie in the surface of constant

energy, which is again odd dimensional.

Contact manifolds also arise in geometry: the co-sphere bundle of a

Riemannian manifold is a contact manifold. (It is the phase space of

the geodesic equations.)

3/59

JJIIJI

Back

Close

Contact Structure

Defn.

A contact form on a manifold of dimension n = 2k + 1 is a one-form

that satisfies

α ∧ (dα)k 6= 0

at each point. Two such forms are considered equivalent if they only

differ by multiplication by a positive function (a ‘gauge transformation’):

α ∼ fα.

A contact manifold has a contact form in each co-ordinate patch, there

being such positive functions that relate the contact forms on overlap-

ping patches.

4/59

JJIIJI

Back

Close

Examples

The analogue of Darboux’s theorem says that there is a local co-

ordinate system ( and choice of gauge) in which a contact form is

α = dz +

k∑i=1

pidqi.

Thus R2k+1 with this choice is the basic example of a contact manifold.

Given a symplectic manifold and a function w on it, the surface w =

constant is (up to minor technicalities) a contact manifold. Locally we

can choose the symplectic form to be of the form

ω = dθ, θ = w(dz + pidqi)

5/59

JJIIJI

Back

Close

The sub-manifold w = constant carries the induced one-form

α = dz + pidqi;

the ambiguity in the choice of co-ordinate is refected in the gauge trans-

formation α → fα.

A special case is a co-sphere bundle of a Riemannian manifold: the

bundle of 1-forms of unit length. The co-tangent bundle of any manifold

is a symplectic manifold. The length of the co-tangent vector set to one

is the constraint.

6/59

JJIIJI

Back

Close

Contrast with Symplectic Reduction

A single constraint W = constant is always first class. Dirac’s philos-

ophy of symplectic reduction is to regard as the true or ‘reduced’ phase

space the quotient of this constrained space by the orbits of the canoni-

cal transformation generated by W . Alternately, we allow as observables

only functions of the total space that have zero Poisson bracket with

W .

This procedure is not physically correct in all cases: we will show an

example from the theory of liquid crystals where we need to consider all

functions on the constrained phase space,not just those that commute

with W .

7/59

JJIIJI

Back

Close

A Fatal Flow in Symplectic Reduction

The quotient of the constained phase space by the orbits of w simply

may not exist as a manifold: the orbit can be dense in w. In fact

generically such orbits do not exist when w is the hamiltonian of a

conservative mechanical system due to chaos.

For example, the co-sphere bundle of a Riemann surface of constant

negative curvature does not admit a quotient by the orbits, as the orbits

fill all of the manifold.

We need a way to think of the odd dimensional space itself as a phase

space.

8/59

JJIIJI

Back

Close

Lightning Review of Mechanics

A symplectic form is a closed two-form that is non-degenerate:

dω = 0, ivω = 0 ⇒ v = 0.

This of course requires the manifold carrying ω to be even dimensional.

A symplectomorphism (‘canonical transformation’) is a diffeomor-

phism that preserves the sympletic form, φ∗ω = ω. Infinitesimally,

a symplectic vector field satisfies Lvω = 0. Since Lvω = d(ivω), this

implies that locally there is a function (‘generating function’) such that

ivω = gv.

9/59

JJIIJI

Back

Close

The commutator of two symplectic vector fields is also symplectic.

This defines a commutator (‘Poisson bracket’) on the generating func-

tions:

g1, g2 = r(dg1, dg2)

where r is the inverse tensor of ω. These brackets satisfy the axioms of

a Poisson Algebra

10/59

JJIIJI

Back

Close

Poisson Algebra

Defn.

A Poisson Algebra is a commutative algebra A with identity along

with a bilinear , : A⊗ A → A that satisfies

1. g1, g2 = −g2, g2

2. g1, g2, g3 + g2, g3, g1 + g3, g1, g2 = 0 Jacobi identity

3. g1, g2g3 = g1, g2g3 + g2g1, g3 Leibnitz Rule

The last property implies that the Poisson bracket of a constant with

any function is zero.

11/59

JJIIJI

Back

Close

Examples of Poisson Algebras

The basic example is the algebra of functions on the plane:

g1, g2 = ∂xg1∂yg2 − ∂yg1∂xg2.

More generally the set of function on a symplectic manifold form a

Poisson algebra, with bracket we gave earlier. Conversely, if the Pois-

son algebra is non-degenerate (the only elements that have zero Poisson

bracket with everything are constants) it arises from a symplectic man-

ifold this way.

12/59

JJIIJI

Back

Close

Also, the set of functions on the dual of a Lie algebra is a Poisson

algebra (Kirillov):

F, G(ξ) = iξ[dF, dG]

13/59

JJIIJI

Back

Close

A Generalization of Poisson Algebra

Defn

A commutative algebra A with identity and a bilinear , : A×A → Ais a Generalized Poisson Algebra if

1. g1, g2 = −g2, g2

2. g1, g2, g3 + g2, g3, g1 + g3, g1, g2 = 0

3. g1, g2g3 = g1, g2g3 + g2g1, g3 + 1, g1g2g3.

Generalized Leibnitz Rule

The main point is that the constant function is no longer in the center.

The commutant of the constant function is a Poisson algebra.

14/59

JJIIJI

Back

Close

Applications

1. The quantization of time dependent systems: phase space includes

time

2. Quantization of the phase space with constant energy in a conser-

vative system

3. Cholesteric Liquid Crystals- We will focus on this example in this

talk.

15/59

JJIIJI

Back

Close

Contact Manifolds Give Generalized Poisson Algebras

We will restrict to the case of R3 for simplicity. Let α be a contact

form. Then the condition on a vector field v to preserve the contact

structure is that there exist a function fv with

Lvα = fvα ⇒ α ∧ Lvα = 0.

Then, fv will satisfy the condition for a one-cocycle:

Lvfv − Lvfv = f[v,v]

It is useful to write the condition Lvα = fvα in the more elementary

notation of vector calculus:

∇(α · v) + β × v = fvα

16/59

JJIIJI

Back

Close

17/59

JJIIJI

Back

Close

where β = curl α. Taking cross product with α, we can solve for v

since α. curl α 6= 0.

v =α×∇gv + βgv

α · β, gv = α · v.

Thus the component of the velocity along the director determines the

others. Putting this back into Lvα,

fv =β · ∇gv

α · β

18/59

JJIIJI

Back

Close

Gauge Invariance

Recall the contact form is only given up to multiplication by a non-

zero function, α → λα. Thus all observables such as velocity must be

gauge invariant under this transformation. In fact

gv → λgv, α×∇gv → α×∇λ gv + λα×∇gv

β → ∇λ× α + λβ, α → λ2α · β,

imply that the above formula for v is gauge invariant. We can regard

α×∇g + βg as a kind of covariant derivative.

19/59

JJIIJI

Back

Close

The Commutator

The commutator of vector fields induces a bracket on the generating

functions:

g[v,v] = Lvgv − fvgv

Or more explicitly, we have an analogue of the Poisson bracket for func-

tions on a three dimensional manifold:

g, g =α · [∇g ×∇g] + β[g∇g −∇g g]

α · β

This is obviously anti-symmetric and the Jacobi identity can also be

checked.

20/59

JJIIJI

Back

Close

This is not a Poisson Algebra

However, the Leibnitz rule is not satisfied!

g1, g2g3 − g1, g2g3 − g2g1, g3 =β · ∇g1

α · βg2g3 6= 0.

In particular,

1, g =β · ∇g

α · β.

So we have instead of the Leibnitz rule, a new identity

g1, g2g3 = g1, g2g3 + g2g1, g3 + 1, g1g2g3.

Note that if g2 = 1 both sides reduce to g1, g3.

21/59

JJIIJI

Back

Close

Commutation RelationsWith the choice α = dz+ 1

2[xdy−ydx] the analogue of the canonical

commutation relations can be worked out:

x, y = w, w, x = 0, w, y = 0

z, x =1

2x, z, y =

1

2y, z, w = w

The element w (which correspond to the constant function 1) is not

central anymore. w, x, y span a Heisenberg sub-algebra. z generates

the automorphism of the Heisenberg algebra which scales w, x, y. Since

the Leibnitz rule is replaced by the new identity, we have to be careful

about using these commutation relations to derive brackets for more

general functions. e.g., z, xy = 0.

22/59

JJIIJI

Back

Close

Representation of Commutation Rela-tions

A unitary representation of this Lie algebra can be constructed using

the usual ideas of induced representation theory. On functions of two

variables, w, y (which are represented by multiplication):

x = −iw∂

∂y, z = − i

2x

∂

∂x− i

2y

∂

∂y− iw

∂

∂w.

23/59

JJIIJI

Back

Close

Fluid Mechanics

Fluids display many phenomena that remain puzzling to physicists.The

motion of individual molecules in the fluid follows simple well-understood

laws of classical mechanics. The collective motion of a large number of

these molecules causes pheneomens such as

1. Turbulence. Onset of turbulence could be a kind of critical phenomenon-

still poorly understood.Large scale disorder.

2. Chaos. Extreme sensitivity to initial conditions. “ A butterfly flaping

its wings in Australia can cause rain in New Jersey”

3. Stable Vortices How can hurricanes and the ‘Red Eye of Jupiter’-

stable over long distances and times-arise out of chaos? .

24/59

JJIIJI

Back

Close

Collective Variables

We need to give up on describing a fluid in terms of the position and

momentum of each molecule, and instead pass to macroscopic variables

such as pressure, density and velocity of fluid elements that contain

averges over many molecules. This was done already in the eighteenth

century by Leonhard Euler, “the Master of us all” mathematical physi-

cists.

He applied this philosophy also to a rigid body, a collection of molecules

that move together,keeping the distance between two of them fixed.

There is a surprising mathematical similarity between these two systems,

a connection that could not have escaped Euler: In modern language,

they both describe geodesic motion on subgroups of the diffeomorphism

group. V. I. Arnold and B. Khesin Topological Methods in Hydrodynamics

25/59

JJIIJI

Back

Close

Mechanical Systems out of Lie Algebras

Given a Lie algebra G and a positive (not invariant) bilinear A on it,

we can define a mechanical system. The dual of the Lie algebra defines

a Poisson bracket on the functions of its dual (Kirillov):

ωa, ωb = ω[a,b], F, G(ω) =< ω, [dF, dG] >

The positive bilinear defines a function H = 12 < ω, A−1ω > which serves

as the hamiltonian (kinetic energy). Together they yield the equations

of motiondω

dt= ad∗A−1ωω.

Geometrically, this describes geodesic motion on the group G of the Lie

algebra, with respect to the left-invariant (but not isotropic) metric A.

26/59

JJIIJI

Back

Close

Examples: Various Collective Motions

Each collective motion is determined by the physical quantity it leaves

invariant:

1. Rigid Body: Isometries, Lvgij = 0 ⇐⇒ ∂ivj + ∂jvi = 0.

2. One dimensional compressible flow: Diff(R)

3. Two dimensional incompressible flow: Canonical Transformations,

Lvω = 0 ⇐⇒ vi = ωij∂jf .

4. Liquid Crystals: transformations preserving a direction Lvα = fα.

Nematic: dα ∧ α = 0 chiral nematic:dα ∧ α 6= 0.

5. Three dimensional incompressible flow: Lvρ = 0 ⇐⇒ div v = 0

27/59

JJIIJI

Back

Close

Poisson Algebras in Fluid Mechanics

Poisson algebras and their generalizations appear in fluid mechanics

at two levels: as a hamiltonian system, the fluid has a phase space with

a Poisson bracket and hamiltonian.

But also the velocity fields of the fluid is also sometimes required to

preserve a symplectic or contact form. Incompressible vector fields in

two dimensions are the same as symplectic vector fields: the area form

is the symplectic form. We will see that three dimensional analogue

is a contact vector field, which decsribes the flow of cholesteric liquid

crystals.

28/59

JJIIJI

Back

Close

The Euler Equations

The Euler equations of an incompressible inviscid fluid are

∂

∂tv + (v · ∇)v = −∇p, div v = 0.

We can eliminate pressure p by taking a curl. Defining the vorticity

ω = curl v and using (v · ∇)v = ω × v + 12∇v2, we get,

∂

∂tω + curl [ω × v] = 0.

We can regard ω as the basic dynamic variable of the system, since v

determined by it 1.

1 We assume that the circulation∫

Cv around con-contactible loops (if any) C are zero and

that the boundary values of the velocity are given.

29/59

JJIIJI

Back

Close

Incompressible Flows

If the velocity of the fluid is small compared to the speed of sound in

it, the density of the fluid cannot change. This imposes a constraint of

incompressibility, weaker than that of a rigid body:

det∂φ

∂x= 1.

The velocity has to satisfy the infinitesimal version of this condition,

div v = 0. There are still an infinite number of such vectors.

See V. I. Arnold and B. Khesin Topological Methods in Hydrodynam-

ics for more on the geometrical picture of fluid flow.

30/59

JJIIJI

Back

Close

Geodesic Motion

There is a natural way to measure the ‘size’ of an incompressible

vector field:

||v||2 =

∫v2(x)dx.

It has a simple physical meaning as well: it is proportional to the total

kinetic energy of the fluid. This defines a metric on the tangent space

of the group of volume preserving diffeomorphisms. By left translation

this becomes a homogenous but not isotropic metric on the group itself.

Euler’s equations simply state that the fluid always moves along the

geodesics of this metric!

31/59

JJIIJI

Back

Close

Poisson Brackets

The Poisson bracket of two functions of vorticity is defined to be

F, G =

∫ω ·

[curl

δF

δω× curl

δG

δω

]d3x

This is just the natural Poisson bracket on the dual of the Lie algebra

of incompressible vector fields: recall that the dual of any Lie algebra

has a natural Poisson structure (Kirillov).

Using the identity div [a× b] = b · curl a− a · curl b this may also

be written as

F, G =

∫curl

[ω × curl

δF

δω

]· δGδω

d3x

32/59

JJIIJI

Back

Close

The Hamiltonian Formalism

The hamiltonian is

H =1

2

∫v · vd3x.

Using the identity

curlδF

δω=

δF

δvwe get H, ω = curl [ω × v] as needed to get the Euler equations.

33/59

JJIIJI

Back

Close

Analogy to Rigid Body Mechanics

The Euler equations of a rigid body are

dL

dt= L× Ω, L = AΩ

Its analogy with his equations of a fluid could not have escaped Euler.

Vorticity ω is analogous to angular momentum L; fluid velocity v to

angular velocity Ω; the curl operator is analogous to moment of inertia

tensor A.

34/59

JJIIJI

Back

Close

Hamiltonian Formulation of the RigidBody

The Poisson brackets are those of the dual of the Lie algebra of rota-

tions

Li, Lj = εijkLk

The hamiltonian that gives the Euler equations is

H =1

2< L,A−1L >

It is well known that these are the geodesics of a homogenous but not

always isotropic metric on the rotation group, determined by A.

35/59

JJIIJI

Back

Close

Sensitivity to Initial Conditions

In flat space, geodesics are straight lines: a small change in the initial

point or direction only grows as a power of time. But if the space

has negative curvature, geodesics that start out close depart from each

other exponentially fast. ( They focus towards each other for positive

curvature, as in the case of great circles on a sphere).

The reason why hydrodynamics is so unstable is that the curvature

of the metric of the configuration space is negative all except a finite

number of directions. This means that even small fluctuations in initial

data cannot ever be ignored. Rigid body mechanics by contrast has only

one unstable dierection ( rotation around the eigenvector of the middle

eigenvalue of A).

36/59

JJIIJI

Back

Close

Entropy

We deal with such systems all the time in statistical mechanics. The

basic idea is to look for ‘macroscopic’ variables (density, pressure etd.)

that are averages over large numbers of particles so that fluctuations

average out. But then we have to live with partial information: with

entropy. The most likely state of the system will be one that maximizes

this entropy for a given value of the consderved quantities.

Is there a similar notion of entropy for turbulent fluctuations?

37/59

JJIIJI

Back

Close

Matrix Models

It has emerged recently that many of the above examples of collective

motion of molecules also arise as collective motion of matrix models.

Perhaps the most well-known is the case of a hermitean matrix model

(Sakita,Jevicki, Das, Karabali..). The motion of eigenvalues is mapped

first to a system of free fermions and then to a compressible one dimen-

sional fluid with a non-local ( integro-differential) equation of motion of

the Benjamin-Ono type.

Two dimensional incompressible flow is also connected to a hermitean

matrix model, with a different hamiltonian which is not invariant un-

der unitary transformations(Fairley, Zachos,Dowker,Rajeev). Indeed the

eigenvalues flow trivially (are conserved quantities). In a sense this is

the opposite of the previous case.

38/59

JJIIJI

Back

Close

Regularization by Matrix Models

It is not unusual that very different microscopic dynamics can lead

to the same collective motion. Both molecular dynamics and Matrix

models lead to the same kind of fluid flow as collective motion: they

can both be thought of as regularizations of fluid flow with a finite

number of degrees of freedom.

The regularization using matrix models is more elegant as it retains a

geometric flavor: in essence the underlying manifold is replaced by a non-

commutative manifold. This preserves all the conservation laws, which

is difficult to do in other regularization schemes for fluids. Practically

useful for numerical solutions.

39/59

JJIIJI

Back

Close

Two-Dimensional Euler Equations

It is useful to look first at the much simpler example of two dimensional

incompressible flow.

Area preserving transformations are the same as canonical transfor-

mations. The corresponding Lie algebra is the algebra of symplectic

transformations on the plane:

[f1, f2] = εab∂af1∂bf2.

The above Lie bracket of functions gives the Poisson bracket for vorticity:

ω(x), ω(y) = εab∂bω(x)∂aδ(x− y).

40/59

JJIIJI

Back

Close

Hamiltonian Formulation

Euler equations follow from these if we postulate the hamiltonian to

be the total energy of the fluid,

H =1

2

∫u2d2x =

1

2

∫G(x, y)ω(x)ω(y)d2xd2y.

The quantities Qk =∫

ωk(x)d2x are conserved for any k = 1, 2 · · ·:these are the Casimir invariants. In spite of the infinite number of

conservation laws, two dimensional fluid flow is chaotic.

41/59

JJIIJI

Back

Close

Lie Algebra of Canonical Transforma-tions

The Lie algebra of symplectic transformations can be thought of as

the limit of the unitary Lie algebra as the rank goes to infinity. To see

this, impose periodic boundary conditions and write in terms of Fourier

coefficients as

H = (L1L2)2

∑m 6=(0,0)

1

m2|ωm|2, ωm, ωn = − 2π

L1L2εabmanbωm+n.

42/59

JJIIJI

Back

Close

Non-Commutative Regularization

Using an idea of Fairlie and Zachos, we now truncate this system

by imposing a discrete periodicity mod N in the Fourier index m; the

structure constants must be modified to preserve peridocity and the

Jacobi identity:

ωm, ωn =1

θsin[θ(m1n2 −m2n1)]ωm+n mod N , θ =

2π

N.

This can be thought of as a ‘quantum deformation’ of the Lie algebra of symplectic transformations. This is the Lie

algebra of U(N).

43/59

JJIIJI

Back

Close

Regularized hamiltonian

The hamiltonian also has a periodic truncation

H =1

2

∑λ(m)6=0

1

λ(m)|ωm|2,

λ(m) =

N

2πsin

[2π

Nm1

]2

+

N

2πsin

[2π

Nm2

]2

.

This hamiltonian with the above Poisson brackets describe the geodesics

on U(N) with respect to an anisotropic metric. We are writing it in a

basis in which the metric is diagonal.

44/59

JJIIJI

Back

Close

Entropy of Matrix Models

The constants Qm =∫

ωm(x)d2x define some infinite dimensional

surface in the space of all functions on the plane. The micro-canonical

entropy (a la Boltzmann) of this system will be the log of the volume of

this surface. How to define this volume of an infinite dimensional man-

ifold? We can compute it in the regularization and take the limit. It is

convenient to regard ω as a matrix by defining ω =∑

m ωmU(m) where

U(m) = Um11 Um2

2 with the defining relations U1U2 = e2πiN U2U1. In this

basis, the Lie bracket of vorticity is just the usual matrix commutator.

Then the regularized constants of motion are Qk = 1N trωk. The

set of herimtean matrices with a fixed value of these constants has a

finite volume,known from random matrix theory:∏

k<l(λk − λl)2, the

λk being the eigenvalues.

45/59

JJIIJI

Back

Close

Entropy of Turbulent Flow

The entropy is thus

S =1

N 2

∑k 6=l

log |λk − λl| = P∫

log |λ− λ′|ρ(λ)ρ(λ′)

where ρ(λ)dλdλ′ = 1N

∑k δ(λ− λk).

In the continuum limit this tends to a remarkably simple formula:

S = P∫

log |ω(x)− ω(y)|d2xd2y

Even without a complete theory based on the Fokker-Plank equa-

tion (assuming that its continuum limit does exist) we can make some

predictions about two dimensional turbulence.

46/59

JJIIJI

Back

Close

Entropy for Two Dimensional Flow

Savitri V. Iyer and S.G. Rajeev Mod.Phys.Lett.A17:1539-1550,2002;

physics/0206083

We found a remarkably simple formula that measures the turbulent

entropy for two dimensional incompressible flow:∫log |ω(x)− ω(y)|dxdy.

We don’t have one yet for three dimensional flow yet.

47/59

JJIIJI

Back

Close

The Profile of a Hurricane

What is the vorticity profile that maximizes entropy for fixed value of

mean vorticity Q1 and enstrophy 2? This should be the most probable

configuration for a vortex profile. Using a simple variational argument

we get the answer in parametric form

ω(r) = 2σ sin φ + Q1, r2 =1

2[a2

1 + a22]± [a2

2− a21]

1

π

[φ +

1

2sin (2φ)

]in the region a1 ≤ r ≤ a2. We should expect this to be the vorticity

distribution of the tornados and hurricanes.

2 Q2 = σ2 + Q21

48/59

JJIIJI

Back

Close

Vorticity vs Radial Distance

2 4 6 8 10 12 14radius

4

5

6

7

Vorticity

Notice the ‘eye’of the hurricane: the flow is fastest right inside the

eyewall and then decreases gradually to the outer wall.

49/59

JJIIJI

Back

Close

Reverse Cascade

In three dimensions, vortices tend to break up into smaller scales,

leading to very complicated behavior (the ‘cascade’ effect). In two di-

mensional flow, the opposite can be observed: vortices tend to combine

into a few big ones. We have a simple explanation for this phenomenon:

The entropy of vortices increases as they combine!

The hurricane is stable in a chaotic environment because it maximises

the entropy.

50/59

JJIIJI

Back

Close

What Lies in Between Two and ThreeDimensions?

Such a non-commutative geometric regularization is not completely

understood for three dimensional flow yet. I will describe a case in-

termediate between two and three dimensions, that of liquid crystals,

where we can find a ‘matrix regularization’ of fluid motion. There are

also interesting connections to ideas of differential geometry.

Liquid crystals are very convenient media on which to perform exper-

iments. Phenomena are simpler than in three dimensions yet complex

enough to demonstrate turbulence and chaos.

51/59

JJIIJI

Back

Close

Flow of Liquid Crystals

Liquid crystals are molecules that are so long that they cannot freely

move past each other. In the simplest phase (‘nematic’) the molecules

are all aligned in the same direction. Only motion that preserves this

direction can happen at small cost of energy. Just as we can deform

even a rigid body at some cost of elastic energy, the direction can also

be changed: by exerting external forces by an electric field for example-

how an LCD works.

Thus liquid crystal flow is an intermediate case in between two and

three dimensional flow. Geometrically, the direction of the molecules is

best described as a one-form α (the director). However if we multiply

α by an everywhere non-zero function the orientation described by it

doesn’t change.

52/59

JJIIJI

Back

Close

Liquid Crystal Groups

The flow of a liquid crystals is described by diffeomorphisms that

preserve the director up to such a multiplication:

φ∗α = fα.

We need to distinguish two very different cases:

α ∧ dα = 0

the nematic phases and the cholesteric or chiral nematic phase

α ∧ dα 6= 0.

53/59

JJIIJI

Back

Close

Frobenius Integrabiity

One is tempted to call the nematic phase where α∧dα = 0 the Frobe-

nius phase. Its geometric meaning is that the tangent planes transversal

to α at each point can be integrated (locally) into a two dimensional

submanifold. Frobenius theorem tells us that locally there are func-

tions τ and σ such that α = τdσ Reminds one of thermodynamics a

la Caratheodory! Unlike in thermodynamics, we have the freedom to

multiply α by a non-zero function.

We can even choose ‘the gauge’ α = dσ The vector fields transversal

to α are tangential to the surfaces

σ = constant.

54/59

JJIIJI

Back

Close

Contact Geometry

The cholesteric phase is the exact opposite: the integrability condi-

tion for the transversal planes to fit into a local submanifold is violated

everywhere. In this case, the director defines a contact structure on

R3. The analogue of Darboux theorem for contact manifolds says that

locally there are functions z, x, y such that

α = dz +1

2[xdy − ydx].

55/59

JJIIJI

Back

Close

Flow Vector Fields of the Nematic phase

Suppose α ∧ dα = 0. The allowed flows are such that

Lvα = fvα

where fv is a function that depends linearly on v. Suppose we choose

the z-axis along the director. Then α = dz an the condition above

becomes dvz = fvdz. This means that

∂vz

∂x= 0 =

∂vz

∂y.

There is no condition on the transversal components vx, vy. Such vector

fields form a Lie algebra.

56/59

JJIIJI

Back

Close

Cholesteric Flows preserve Contact Structure

A contact structure on a three dimensional manifold is a one form

(modulo multiplication by a non-zero function) satisfying α ∧ dα 6= 0.

( see V. I. Arnold Classical Mechanics). Thus the group of flows of

the cholesteric phase (satisfying φ∗α = fφα) are precisely the ‘contact

transformations’ familiar from classical mechanics.

57/59

JJIIJI

Back

Close

Cholesteric Equations of MotionIn the absence of external forces, the liquid crystal will flow subject

to internal forces necessary to maintain the constraint α ∧ Lvα = 0. If

we impose the condition with a Lagrange multiplier p ( a vector field

playing the same role that pressure plays for incompressible fluids), these

forces are given by

δ

δv

∫p · [α×∇(α · v) + βα · v − vα · β]d3x

Thus the analogues of Euler’s equations are

∂v

∂t+ v · ∇v = α[ div (α× p) + β · p]− α · β p

The constraint of preserving the director can be used to determine the

additional unknowns p.

58/59

JJIIJI

Back

Close

The Non-Commutative T 3

This leads us to a non-commutative analogue of the three torus:

XY = WY X, XN = Y N = WN = 1, (1)

ZXZ−1 = Xr ZY Z−1 = Y r, ZWZ−1 = W r2

where N is a prime number and 2 ≤ r ≤ N − 1.

What is the non-commutative S3?