k. ulker - non-commutative gauge theories and seiberg witten map

Non-commutative Gauge Theoriesand

Seiberg–Witten Map to All Orders1

Kayhan ULKER

Feza Gursey Institute *Istanbul, Turkey

(savefezagursey.wordpress.com)

The SEENET-MTP Workshop JW2011

1K. Ulker, B Yapiskan Phys. Rev. D 77, 065006 (2008) [arXiv:0712.0506 ]Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 1 / 1

NC Field Theory

Outline

Moyal ∗–product.

Non-commutative Yang–Mills (A , Λ)

Seiberg – Witten Map (A→ A(A, θ) , Λ→ Λ(A, α, θ))

Construction of the maps order by order leads to all order solutions

All order solutions can be obtained from SW differential equation

SW map of other fields (Ψ→ Ψ(A, ψ, θ))

Conclusion

Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 2 / 1

NC Field Theory Moyal *-product

Moyal ∗-product

The simplest way to introduce non-commutativity is to use (Moyal)*–product

f (x) ∗ g(x) ≡ exp

(i

2θµν

∂

∂xµ∂

∂yν

)f (x)g(y)|y→x

= f (x) · g(x) +i

2θµν∂µf (x)∂νg(x) + · · ·

for real CONSTANT antisymmetric parameter θ !

*–commutator of the ordinary coordinates :

[xµ , xν ]∗ ≡ xµ ∗ xν − xν ∗ xµ = iθµν .

NC QFT models can then be obtained by replacing the ordinary productwith the *–product !


NC Field Theory NC-YM theories

NC Yang–Mills Theory

The action of NC YM theory is written as :

S = −1

4Tr

∫d4xFµν ∗ Fµν = −1

4Tr

∫d4xFµν Fµν

whereFµν = ∂µAν − ∂νAµ − i [Aµ, Aν ]∗

is the NC field strength of the NC gauge field A.

The action is invariant under the NC gauge transformations:

δΛAµ = ∂µΛ− i [Aµ, Λ]∗ ≡ DµΛ

δΛFµν = i [Λ, Fµν ]∗.

Here, Λ is the NC gauge parameter.


NC Field Theory SW–map

Seiberg–Witten Map

One can derive both conventional and NC gauge theories from stringtheory by using different regularization procedures (SW’99).

Let Aµ and α to be the ordinary counterparts of Aµ and Λ respectively.

⇒There must be a map from a commutative gauge field A to anoncommutative one A, that arises from the requirement that gaugeinvariance should be preserved !

A(A) + δΛA(A) = A(A + δαA)

where δα is the ordinary gauge transformation :

δαAµ = ∂µα− i [Aµ, α] ≡ Dµα.



This map can be rewritten as

δΛAµ(A; θ) = Aµ(A + δαA; θ)− Aµ(A; θ) = δαAµ(A; θ)

Since SW map imposes the following functional dependence :

Aµ = Aµ(A; θ) , Λ = Λα(α,A; θ).

⇒ one has to solve

δΛAµ(A; θ) = δαAµ(A; θ)

simultaneously for Aµ and Λα and it is difficult !



Now, remember the ordinary gauge consistency condition

δαδβ − δβδα = δ−i [α,β].

(check for instance for δαψ = iαψ)Let, Λ be a Lie algebra valued gauge parameter Λ = ΛaT

a. For thenon-commutative case, we get

(δΛαδΛβ−δΛβδΛα)Ψ =1

2[T a,T b]{Λα,a,Λβ,b}∗∗Ψ+

1

2{T a,T b}[Λα,a,Λβ,b]∗∗Ψ

Only a U(N) gauge theory allows to express {T a,T b} again in terms ofT a. Therefore, two gauge transformations do not close in general !



To able to generalize to any gauge group (J. Wess et.al. EPJ’01)

let the parameters to be in the enveloping algebra of the Lie algebra :

Λ = αaTa + Λ1

ab : T aT b : + · · ·Λn−1a1···an : T a1 · · ·T an : + · · ·

let all NC fields and parameters depend only on Lie algebra valuedfields A, ψ, · · · and parameter α i.e.

Aµ ≡ Aµ(A) , Ψµ ≡ Ψµ(A, ψ) , Λ = Λ(A, α)

impose NC gauge consistency condition:

iδαΛβ − iδβΛα − [Λα, Λβ]∗ = i Λ−i [α,β].

Note that, above construction of Wess et.al. is obtained entirely

independent of string theory !Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 8 / 1


In contrast to SW we now have an equation only for the gauge parameter

iδαΛβ − iδβΛα − [Λα, Λβ]∗ = i Λ−i [α,β].

and once we solve it we can then solve

δΛAµ(A; θ) = δαAµ(A; θ)

only for Aµ.

To find the solutions we expand Λα and Aµ as formal power series,

Λα = α + Λ1α + · · ·+ Λn

α + · · · ,Aµ = Aµ + A1

µ + · · ·+ Anµ + · · · ,

The superscript n denotes the order of θ.



NC gauge consistency condition and gauge equivalence condition

iδαΛβ − iδβΛα − [Λα, Λβ]∗ = i Λ−i [α,β] , δΛAµ(A; θ) = δαAµ(A; θ)

can be written for the n–th order components of Λα and Aµ as

iδαΛnβ − iδβΛn

α −∑

p+q+r=n

[Λpα,Λ

qβ]∗r = i Λn

−i [α,β]

δαAnµ = ∂µΛn

α − i∑

p+q+r=n

[Apµ , Λq

α]∗r

respectively. Here, ∗r denotes :

f (x) ∗r g(x) ≡ 1

r !

(i

2

)r

θµ1ν1 · · · θµrνr∂µ1 · · · ∂µr f (x)∂ν1 · · · ∂νr g(x).



We can rearrange the above Eq.s for any order n

∆Λn≡ iδαΛnβ − iδβΛn

α − [α, Λnβ]− [Λn

α, β]− i Λn−i [α,β] =

∑p+q+r=n,

p,q 6=n

[Λpα,Λ

qβ]∗r

∆αAnµ≡ δαAn

µ − i [α,Anµ] = ∂µΛn

α + i∑

p+q+r=n,q 6=n

[Λpα,A

qµ]∗r

so that the l.h.s contains only the n-th order components.However, note that one can extract the homogeneous parts such that

∆Λnα = 0 , ∆αA

nµ = 0

It is clear that one can add any homogeneous solution Λnα, A

nµ to the

inhomogeneous solutions Λnα,A

nµ with arbitrary coefficients.


NC Field Theory 1-st order solution

First order solution given in the original paper (SW, JHEP’99) :

Λ1α = −1

4θκλ{Aκ, ∂λα}

A1γ = −1

4θκλ{Aκ, ∂λAγ + Fλγ}.

One can find the field strength form the definition :

F 1γρ = −1

4θκλ({Aκ, ∂λFγρ + DλFγρ} − 2{Fγκ,Fρλ}

).

These solutions are not unique since one can add homogeneoussolutions with arbitrary coefficients. i.e.

Λ1 = ic1θµν [Aµ, ∂να] , A1

ρ = c2θµνDρFµν

Therefore,

A1ρ + ic1θ

µν([∂ρAµ,Aν ]− i [[Aρ,Aµ],Aν ]) + c2θµνDρFµν

is also a solutionKayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 12 / 1

NC Field Theory 1-st order solution

2nd order solutions (Moller’04) A2µ and Λ2

α can be written in terms of lower

order solutions : (K.U ,B. Yapiskan PRD’08.)

Λ2α = −1

8θκλ

({A1

κ, ∂λα}+ {Aκ, ∂λΛ1α})− i

16θκλθµν [∂µAκ, ∂ν∂λα]

A2γ = −1

8θκλ

({A1

κ, ∂λAγ + Fλγ}+ {Aκ, ∂λA1γ + F 1

λγ})

− i

16θκλθµν [∂µAκ, ∂ν(∂λAγ + Fλγ)].

The field strength at the second order can also be written in terms of firstorder solutions :

F 2γρ = −1

8θκλ({Aκ, ∂λF

1γρ + (DλFγρ)1}

+{A1κ, ∂λFγρ + DλFγρ} − 2{Fγκ,F

1ρλ} − 2{F 1

γκ,Fρλ})

− i

16θκλθµν

([∂µAκ, ∂ν

(∂λFγρ + DλFγρ

)]− 2[∂µFγκ, ∂νFρλ]

).

Where, (DλFγρ)1 = DλF1γρ − i [A1

λ,Fγρ] + 12θ

µν{∂µAλ, ∂νFγρ}.Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 13 / 1

NC Field Theory n-th order solutions

By analyzing first two order solutions one can conjecture the generalstructure (K.U ,B. Yapiskan PRD’08.) :

Λn+1α = − 1

4(n + 1)θκλ

∑p+q+r=n

{Apκ, ∂λΛq

α}∗r

An+1γ = − 1

4(n + 1)θκλ

∑p+q+r=n

{Apκ, ∂λA

qγ + F q

λγ}∗r .

The overall constant −1/4(n + 1) is fixed uniquely with third ordersolutions.


NC Field Theory SW Differential Equation

Solution of Seiberg-Witten Differential Equation

Let us vary the deformation parameter infinitesimally

θ → θ + δθ

To get equivalent physics, A(θ) and Λ(θ) should change when θ is varied,(SW’99):

δAγ(θ) = Aγ(θ + δθ)− Aγ(θ)

= δθµν∂Aγ∂θµν

= −1

4δθκλ{Aκ, ∂λAγ + Fλγ}∗

δΛ(θ) = Λ(θ + δθ)− Λ(θ)

= δθµν∂Λ

∂θµν= −1

4θκλ{Aκ, ∂λΛ}∗ (1)

These differential equations are commonly called SW differential equations.Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 15 / 1


To find solutions of the differential equation

expand NC gauge parameter and NC gauge field into a Taylor series:

Λ(n)α = α + Λ1

α + · · ·+ Λnα,

A(n)µ = Aµ + A1

µ + · · ·+ Anµ.

here Λ(n)α and A

(n)µ denotes the sum up to order n !

Then it is possible to write (Wulkenhaar et.al’01) :

Λ(n+1)α = α− 1

4

n+1∑k=1

1

k!θµ1ν1θµ2ν2 · · · θµkνk

(∂k−1

∂θµ2ν2 · · · ∂θµkνk{A(k)

µ1, ∂ν1 Λ(k)

α }∗)

θ=0

A(n+1)γ = Aγ−

1

4

n+1∑k=1

1


(∂k−1

∂θµ2ν2 · · · ∂θµkνk{A(k)

µ1, ∂ν1 A

(k)γ + F (k)

ν1γ}∗)

θ=0

.



Contrary to the solutions presented in the previous section, theseexpressions explicitly contain

derivatives w.r.t. θ

∗–product itself

and given as a sum of all (n+1) orders.

However, it is possible to extract our recursive solutions from the aboveexpressions.



Let us write the n + 1–st component of Λ(n+1)α :

Λn+1α = − 1

4(n + 1)!θµνθµ1ν1 · · · θµnνn

(∂n

∂θµ1ν1 · · · ∂θµnνn{A(n)

µ1, ∂ν1Λ(n)

α }∗)θ=0

.

Since, θ is set to zero after taking the derivatives, the expression in theparanthesis can be written as a sum up to n-th order:

Λn+1α = − 1

4(n + 1)!θµνθµ1ν1 · · · θµnνn

(∂n

∂θµ1ν1 · · · ∂θµnνn

∑p+q+r=n

{Apµ, ∂νΛq

α}∗r

).

It is then an easy exercise to show that the above equation reduces to therecursive formula :

Λn+1α = − 1

4(n + 1)θµν

∑p+q+r=n

{Apµ, ∂νΛq

α}∗r .



With the same algebraic manipulation one can also derive the samerecursive formula for the gauge field

An+1γ = − 1

4(n + 1)!θµνθµ1ν1 · · · θµnνn ×

×(

∂n

∂θµ1ν1 · · · ∂θµnνn{A(n)

µ1, ∂ν1A

(n)γ + F (n)

ν1γ}∗)θ=0

= − 1

4(n + 1)θµν

∑p+q+r=n

{Apµ, ∂νA

qγ + F q

νγ}∗r .


NC Field Theory SW Map for matter fields

Seiberg–Witten Map for Matter Fields

SW–map of a NC field Ψ in a gauge invariant theory can be derived from(J. Wess et.al. EPJ’01):

δΛΨ(ψ,A; θ) = δαΨ(ψ,A; θ).

The solution of this gauge equivalence relation can be found order by orderby after expanding the NC field Ψ as formal power series in θ

Ψ = ψ + Ψ1 + · · ·+ Ψn + · · ·



Fundamental Representation :Ordinary gauge transformation of ψ is written as

δαψ = iαψ.

NC gauge transformation is defined with the help of ∗–product :

δΛΨ = i Λα ∗ Ψ.

Following the general strategy the gauge equivalence relation reads

∆αΨn ≡ δαΨn − iαΨn = i∑

p+q+r=n,q 6=n

Λpα∗rΨq,

for all orders.As discussed before, one is free to add any homogeneous solution Ψn ofthe equation

∆αΨn = 0

to the solutions Ψn.Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 21 / 1


A solution for the first order is given by Wess et.al :

Ψ1 = −1

4θκλAκ(∂λ + Dλ)ψ

where Dµψ = ∂µψ − iAµψ .

The SW differential equation from the first order solution reads :

δθµν∂Ψ

∂θµν= −1

4δθκλAκ ∗ (∂λΨ + DλΨ)

which can also be written as

∂Ψ

∂θκλ= −1

8Aκ ∗ (∂λΨ + DλΨ) +

1

8Aλ ∗ (∂κΨ + DκΨ)

The NC covariant derivative here is defined as :

DµΨ = ∂µΨ− i Aµ ∗ Ψ .



By using a similar method, we expand Ψ in Taylor series :

Ψ(n+1) = ψ + Ψ1 + Ψ2 + · · ·+ Ψn+1

= ψ +n+1∑k=1

1


(∂k

∂θµ1ν1 · · · ∂θµkνk(

Ψ(n+1)))

θ=0

.

From the differential equation we get

Ψ(n+1) = ψ − 1

4

n+1∑k=1

1

k!θµ1ν1θµ2ν2 · · · θµkνk ×

×( ∂k−1

∂θµ2ν2 · · · ∂θµkνkA(k)µ1∗ (∂ν1Ψ(k) + (Dν1Ψ)(k))

)θ=0

where(DµΨ)(n) = ∂µΨ(n) − i A(n)

µ ∗ Ψ(n).

For the abelian case this solution differs from the one given in Wulkenhaaret.al. by a homogeneous solution.



To find the all order recursive solution we write the n + 1–st component :

Ψn+1 = − 1

4(n + 1)!θµνθµ1ν1 · · · θµnνn ×

×(

∂n

∂θµ1ν1 · · · ∂θµnνnA(n)µ ∗ (∂νΨ(n) + (DνΨ)(n))

)θ=0

= − 1

4(n + 1)!θµνθµ1ν1 · · · θµnνn ×

×

(∂n

∂θµ1ν1 · · · ∂θµnνn

∑p+q+r=n

Apµ∗r (∂νΨ(q) + (DνΨ)q)

)

where(DµΨ)n = ∂Ψn − i

∑p+q+r=n

Apµ ∗r Ψq.



After taking derivatives w.r.t. θ’s we obtain the all order recursive solutionof the gauge equivalence relation :

ψn+1 = − 1

4(n + 1)θκλ

∑p+q+r=n

Apκ∗r (∂λΨ(q) + (DλΨ)q).

The first order solution of Wess et.al. is obtained by setting n = 0.

By setting n = 1 we find the second order solution of Moller .



Adjoint representation :The gauge transformation reads as

δαψ = i [α,ψ]

Non–commutative generalization can be written as

δΛΨ = i [Λα, Ψ]∗ .

Following the general strategy, the gauge equivalence relation can bewritten as

∆αΨn := δαΨn − i [α,Ψ] = i∑

p+q+r=n,q 6=n

[Λpα,Ψ

q]∗r

for all orders. The solutions can be found

either by directly solving this equation order by order

by solving the respective differential equation

However, possibly the easiest way to obtain the solution is to usedimensional reduction. (K.U, Saka PRD’07)



By setting the components of the deformation parameter θ on thecompactified dimensions zero, i.e.

ΘMN =

(θµν 00 0

),

the trivial dimensional reduction (i.e. from six to four dimensions) leads tothe general n–th order solution for a complex scalar field :

ψn+1 = − 1

4(n + 1)θκλ

∑p+q+r=n

{Apκ, (∂λΨ(q) + (DλΨ)q)}∗r .

Since the structure of the solutions are the same for both the scalar andfermionic fields, this solution can also be used for the fermionic fields.



Note that, after introducing the anticommutators / commutators properly,the form of the solution is similar with the one given for the fundamentalcase except that now

Dµψ = ∂µψ − i [Aµ, ψ]

and hence(DµΨ)n = ∂µΨn − i

∑p+q+r=n

[Apµ,Ψ

q]∗r .

The same result can also be obtained by solving the differential equation :

∂Ψ

∂θκλ= −1

8{Aκ, (∂λΨ + DλΨ)}∗ +

1

8{Aλ, (∂κΨ + DκΨ)}∗.

Therefore, the result obtained above via dimensional reduction can also bethought as an independent check of the aforementioned results.


NC Field Theory Conclusion

CONCLUSION

Equations,Aµ(A; θ) + δΛAµ(A; θ) = Aµ(A + δαA; θ).

Ψ(A, ψ; θ) + δΛΨ(A, ψ; θ) = Ψ(A + δαA, ψ + δαψ; θ).

can be solved as

Λn+1α = − 1

4(n + 1)θκλ

∑p+q+r=n

{Apκ, ∂λΛq

α}∗r

An+1γ = − 1

4(n + 1)θκλ

∑p+q+r=n

{Apκ, ∂λA

qγ + F q

λγ}∗r .

ψn+1 = − 1

4(n + 1)θκλ

∑p+q+r=n

Apκ∗r (∂λΨ(q) + (DλΨ)q).


k. ulker - non-commutative gauge theories and seiberg witten map

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