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 !
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 3 / 1
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 4 / 1
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 4 / 1
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 5 / 1
NC Field Theory SW–map
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 !
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 6 / 1
NC Field Theory SW–map
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 !
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 7 / 1
NC Field Theory SW–map
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
NC Field Theory SW–map
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 9 / 1
NC Field Theory SW–map
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).
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 10 / 1
NC Field Theory SW–map
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 11 / 1
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 14 / 1
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
NC Field Theory SW Differential Equation
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!θµ1ν1θµ2ν2 · · · θµkνk
(∂k−1
∂θµ2ν2 · · · ∂θµkνk{A(k)
µ1, ∂ν1 A
(k)γ + F (k)
ν1γ}∗)
θ=0
.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 16 / 1
NC Field Theory SW Differential Equation
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 17 / 1
NC Field Theory SW Differential Equation
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 .
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 18 / 1
NC Field Theory SW Differential Equation
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 .
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 19 / 1
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 + · · ·
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 20 / 1
NC Field Theory SW Map for matter fields
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
NC Field Theory SW Map for matter fields
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 .
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 22 / 1
NC Field Theory SW Map for matter fields
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!θµ1ν1θµ2ν2 · · · θµkνk
(∂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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 23 / 1
NC Field Theory SW Map for matter fields
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 24 / 1
NC Field Theory SW Map for matter fields
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 .
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 25 / 1
NC Field Theory SW Map for matter fields
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)
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 26 / 1
NC Field Theory SW Map for matter fields
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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 27 / 1
NC Field Theory SW Map for matter 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.
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 28 / 1
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).
Kayhan ULKER (Feza Gursey Institute*) All order solution of SW–map BW’11 29 / 1