A Noncommutative Sigma Model

A Noncommutative σ-ModelM.Sc. Studies

M. van den Worm1, R. Duvenhage1

1Department of PhysicsUniversity of Pretoria

Joint Congress of AMS and SAMS at PE

Mauritz van den Worm (UP) A Noncommutative σ-Model 2011 1 / 28

Outline

1 Motivation

2 The Irrational Rotation Algebra (Quantum Torus)

3 Derivations and the Smooth *-albegra A∞θ

4 Noncommutative Polyakov Action

5 Existence Theorems


Motivation

String Theory - A (very) Brief Introduction

Σ

X

ϕ

x

y

t

τ

σ


Motivation

String Theory - A (very) Brief Introduction

Polyakov ActionNatural choice in modern string theory is the Polyakov action

S = −14πα′

∫ √−hhαβ∂αX µ∂βX νηµνdτdσ

Equivalent to Nambu-Goto action and gives the same equations ofmotionWe can rewrite the action as

S =

∫∂αXµ∂αXµdσ1dσ2.


Motivation

What do we want to do?

Goal:Replace Σ and X with noncommutative C∗-algebrasDetermine noncommutative generalization of the Polyakov actionDo these mappings exist?


The Irrational Rotation Algebra (Quantum Torus)


The Irrational Rotation Algebra AθIs the C∗-algebra generated by two unitary operators U and V that

satisfy the commutation relation

UV = e2πiθVUwhere θ is some irrational.

Classical limit, θ = 0UV = VUIn the classical limit, A0

∼= C(T2)



The Unique Trace On Aθ

Canonical trace on AθDefine the linear functional τ by: for any a ∈ Aθ

τ(a) := 〈Ω,aΩ〉 =

∫aΩ(x , y)dxdy

τ is a trace on Aθτ is unique (...the canonical trace)


Derivations and the Smooth *-albegra A∞θ

Derivations

Definition of a DerivationLet A be a ∗-algebra with δ a linear mapping from A to itself. δ is a∗-derivation if

δ(xy) = δ(x)y + xδ(y),

δ(x∗) = δ(x)∗

for every x , y ∈ A.

Why derivations?Derivations can be regarded as the noncommutative counterparts ofpartial derivatives of operator valued functions.



The Smooth *-Algebra A∞θ

Definition of A∞θLet (Aθ,R2, α) be a C∗-dinamical system. We say that x ∈ Aθ is ofclass C∞, if and only if the mapping g 7→ αg(x) from R2 to the normedspace Aθ, is C∞ (ie. partial derivatives of any order exist and is welldefined). The smooth irrational rotation algebra is defined as

A∞θ := a ∈ Aθ : a is of the class C∞ .

Properties of A∞θA∞θ is a *-algebraA∞θ is dense in AθWe define derivations on A∞θ



The derivations on A∞θ

Derivation on A∞θLet a ∈ A∞θ and define

δ1(a) :=ddrαr ,0(a)

∣∣∣∣r=0

, δ2(a) :=ddsα0,s(a)

∣∣∣∣s=0

δi ’s are derivation for i = 1,2In the classical limit where θ = 0, we observe that

δ1 =∂

∂x, δ2 =

∂

∂y


Noncommutative Polyakov Action

The Noncommutative Polyakov Action

Let Σ and Rn be the two dimensional parameter space and spacetime respectively

g : Σ→ Rn, g(σ1, σ2) =(

X 1(σ1, σ2), · · · ,X n(σ1, σ2))

We want to work in algebraic frameworkWe want to use *-homomorphism in the action

ϕ : C(Rn)→ C(Σ) : f 7→ f g.


Noncommutative Polyakov Action

The Noncommutative Polyakov Action

The Noncommutative Polyakov ActionThe natural generalization of the Polyakov action to thenoncommutative C∗-algebra Aθ is given by

S(ϕ) = τ [δ1(ϕ(U))∗δ1(ϕ(U)) + δ2(ϕ(U))∗δ2(ϕ(U))

+δ1(ϕ(V ))∗δ1(ϕ(V )) + δ2(ϕ(V ))∗δ2(ϕ(V ))]

where ϕ : AΘ → Aθ and δi are normal derivations.


Existence Theorems

Existence Theorems

What is next?Replace Σ and X with respectively AΘ and AθWhere in general Θ 6= θ

Do *-homomorphisms of the form ϕ : AΘ → Aθ exist?

Without such mappings we cannothave a noncommutative σ-model!


Existence Theorems

Existence Theorems

The proofs are technical and make use ofC∗-algebrasK-theory, enMorita equivalence

We ommit the proofs here

Theorem (Simplest Case)

Fix Θ and θ in (0,1), both irrational. There is a unital *-homomorphismϕ : AΘ → Aθ if and only if Θ = cθ + d for some c,d ∈ Z, c 6= 0. Such a*-homomorphism ϕ can be chosen to be an isomorphism onto itsimage if and only if c = ±1.


Existence Theorems

Existence Theorems

Theorem (More General Case - Still Unital)

Fix Θ and θ in (0,1) both irrational and n ∈ N,n ≥ 1. There is a unital*-homomorphism

ϕ : AΘ → Mn(Aθ)

if and only if nΘ = cθ + d for some c,d ∈ Z and c 6= 0. Such a*-homomorphism can be chosen to be an isomorphism onto its imageif and only if n = 1 and c = 1.


Existence Theorems

Conclusion

To summarize:QFT and string theory needs solid mathematical foundationsMost natural choice is operator algebrasWhy noncommutative?

At high energies (Large Hardron Collider) energy lanscape isnoncommutativeNatural choice is operator algebras

classical physics ⊂ quantum physics ⊂ quantum string theory


Existence Theorems

Acknowledgments

Thanks to...Dr. Rocco Duvenhage and the Operator Algebra groupDepartement of Physics at UPDepartement of Mathematics and Applied Mathematics at UPNITheP for funding


Existence Theorems


Finite Dimensional Representation

M2(C) case

The M2(C) case

Consider the special case of 2× 2 matrices with complex entriesConsider the matrices

u =

(1 00 −1

), v =

(0 11 0

)that obeys the relations

uv = qvu, q = exp(

ikr2

)= exp (iπ) = −1



M2(C) case

Consider a *-homomorphism ϕ : M2(C)→ M2(C)

All such *-homomorphisms necassarily has to be*-automorphisms

Small technical aspect

Finite dimensions imply B(H) = K (H)

All *-automorphisms of K (H) have the form

AdU : A 7→ UAU∗

where U is a unitary operator in K (H)

We only have to consider mappings A 7→ UAU∗




Parametrize SU(2)

We want to calculate (noncommutative) path integralsFor normal path integral we integrate over all the fieldsHere we want to integrate over all AdU with U ∈ SU(2)

Parametrize all the matrices in SU(2)



Finite Dimensional Reprentation

Parametrize SU(2)

Any unitary g ∈ SU(2) can be expressed as a function of Eulerangles as:

g(φ, θ, ψ) =

(cos θ

2ei φ+ψ2 i sin θ

2ei φ−ψ2

i sin θ2e−i φ−ψ2 cos θ

2e−i φ+ψ2

).

0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, 0 ≤ ψ ≤ 4πWe can determine the measure

dµ(g) =1

16π2 sin θdθdφdψ met∫

SU(2)dµ(g) = 1




Path Integrals on SU(2)

The parametrization gives all the elements of SU(2)

The Polyakov action is rewritten (with Mathematica) as

S(ϕ) =2∑

k=1

2∑l=1

Tr [δk (gulg∗)∗ δk (gulg∗)]

=1

16r2 e−2i(φ+ψ)[−4(−1 + e4iφ

)(−1 + e4iψ

)cos(θ)−(

1 + e2iφ)2 (

1 + e2iψ)2

cos(2θ)+

4e2i(φ+ψ)(21 + cos(2φ)(1− 3 cos(2ψ)) + cos(2ψ))].




Path Integral on SU(2)

Path integral is just a partition functionUse action to construct partition function

Z (ϕ) =

∫SU(2)

e−S(ϕ)dµ(g)

=1

16π2

∫ 4π

0

∫ 2π

0

∫ π

0e−S(ϕ) sin θdθdφdψ.

Graphical representation?




2 4 6 8 10

r

0.2

0.4

0.6

0.8

1.0

ZHAdUL

Figure: Partition function as a function of r




2 4 6 8 10

r

0.5

1.0

1.5

2.0

2.5

3.0

3.5

SminHAdUL

Figure: Minimum of thepartition function as a functionof r . We can determine anexact relationship: Smin = 4r−2

2 4 6 8 10

r

1

2

3

4

5

SmaxHAdUL

Figure: Maximum of thepartition function as a functionof r . We can determine anexact relationship:Smax = 6r−2




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6 6 6 666

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Figure: Contour plot of theaction for ψ = 0 en r = 1.

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0 1 2 3 4 5 6

0

2

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12

Figure: Contour plot of theaction for θ = π

2 en r = 1.




What does all this mean?Classical σ-models have finite number of critical pointsThe Critical points are those that satisfy the Euler-LagrangeequationsClassically these are the physical trajectories of particlesM2(C)-case shows we have infinitely many critical pointsGauge theory?Details elude us...


A Noncommutative Sigma Model

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