K. Ulker - Non-commutative Gauge Theories and Seiberg Witten Map

Non-commutative Gauge Theories
and
Seiberg–Witten Map to All Orders1

¨
Kayhan ULKER
Feza G¨rsey Institute *
u
Istanbul, Turkey
(savefezagursey.wordpress.com)

The SEENET-MTP Workshop JW2011

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

NC Field Theory

Outline

Moyal ∗–product.
ˆ ˆ
Non-commutative Yang–Mills (A , Λ)
ˆ ˆ ˆ ˆ
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
ˆ ˆ
SW map of other ﬁelds (Ψ → Ψ(A, ψ, θ))
Conclusion

¨

NC Field Theory Moyal *-product

Moyal ∗-product

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

i µν ∂ ∂
f (x) ∗ g (x) ≡ exp θ f (x)g (y )|y →x
2 ∂x µ ∂y ν
i
= f (x) · g (x) + θµν ∂µ f (x)∂ν g (x) + · · ·
2
for real CONSTANT antisymmetric parameter θ !
*–commutator of the ordinary coordinates :

[x µ , x ν ]∗ ≡ x µ ∗ x ν − x ν ∗ x µ = iθµν .

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

NC Field Theory NC-YM theories

NC Yang–Mills Theory

The action of NC YM theory is written as :

ˆ 1 ˆ ˆ 1 ˆ ˆ
S = − Tr d 4 x F µν ∗ Fµν = − Tr d 4 x F µν Fµν
4 4
where
ˆ ˆ ˆ ˆ ˆ
Fµν = ∂µ Aν − ∂ν Aµ − i[Aµ , Aν ]∗
ˆ
is the NC ﬁeld strength of the NC gauge ﬁeld A.

¨

NC Field Theory NC-YM theories

NC Yang–Mills Theory

The action of NC YM theory is written as :

ˆ 1 ˆ ˆ 1 ˆ ˆ
S = − Tr d 4 x F µν ∗ Fµν = − Tr d 4 x F µν Fµν
4 4
where
ˆ ˆ ˆ ˆ ˆ
Fµν = ∂µ Aν − ∂ν Aµ − i[Aµ , Aν ]∗
ˆ
is the NC ﬁeld strength of the NC gauge ﬁeld A.
The action is invariant under the NC gauge transformations:
ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ
δΛ Aµ = ∂µ Λ − i[Aµ , Λ]∗ ≡ Dµ Λ
ˆ ˆ ˆ ˆ
δ ˆ Fµν = i[Λ, Fµν ]∗ .
Λ

ˆ
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 string
theory by using diﬀerent regularization procedures (SW’99).
ˆ ˆ
Let Aµ and α to be the ordinary counterparts of Aµ and Λ respectively.

⇒There must be aˆmap from a commutative gauge ﬁeld A to a
noncommutative one A, that arises from the requirement that gauge
invariance should be preserved !

ˆ ˆˆ ˆ ˆ
A(A) + δΛ A(A) = A(A + δα A)
where δα is the ordinary gauge transformation :

δα Aµ = ∂µ α − i[Aµ , α] ≡ Dµ α.

¨


This map can be rewritten as
ˆˆ ˆ ˆ ˆ ˆ
δΛ Aµ (A; θ) = Aµ (A + δα A; θ) − Aµ (A; θ) = δα Aµ (A; θ)

Since SW map imposes the following functional dependence :
ˆ ˆ
Aµ = Aµ (A; θ) , ˆ ˆ
Λ = Λα (α, A; θ).

⇒ one has to solve
ˆˆ ˆ ˆ
δΛ Aµ (A; θ) = δα Aµ (A; θ)

ˆ ˆ
simultaneously for Aµ and Λα and it is diﬃcult !

¨


Now, remember the ordinary gauge consistency condition

δα δβ − δβ δα = δ−i[α,β] .

(check for instance for δα ψ = iαψ)
Let, Λ be a Lie algebra valued gauge parameter Λ = Λa T a . For the
non-commutative case, we get

ˆ 1 ˆ 1 ˆ
(δΛα δΛβ −δΛβ δΛα )Ψ = [T a , T b ]{Λα,a , Λβ,b }∗ ∗Ψ+ {T a , T b }[Λα,a , Λβ,b ]∗ ∗Ψ
2 2

Only a U(N) gauge theory allows to express {T a , T b } again in terms of
T 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 :
ˆ
Λ = αa T a + Λ1 : T a T b : + · · · Λa1 ···an : T a1 · · · T an : + · · ·
n−1
ab

let all NC ﬁelds and parameters depend only on Lie algebra valued
ﬁelds A, ψ, · · · and parameter α i.e.
ˆ ˆ
Aµ ≡ Aµ (A) , ˆ ˆ
Ψµ ≡ Ψµ (A, ψ) , ˆ ˆ
Λ = Λ(A, α)

impose NC gauge consistency condition:
ˆ ˆ ˆ ˆ ˆ
iδα Λβ − iδβ Λα − [Λα , Λβ ]∗ = i Λ−i[α,β] .

Note that, above construction of Wess et.al. is obtained entirely
independent of string theory !
¨


In contrast to SW we now have an equation only for the gauge parameter
ˆ ˆ ˆ ˆ ˆ
iδα Λβ − iδβ Λα − [Λα , Λβ ]∗ = i Λ−i[α,β] .

and once we solve it we can then solve
ˆˆ ˆ ˆ

only for Aµ .
ˆ ˆ
To ﬁnd the solutions we expand Λα and Aµ as formal power series,
ˆ
Λα = α + Λ 1 + · · · + Λ n + · · · ,
α α
ˆ µ = Aµ + A1 + · · · + An + · · · ,
A µ µ

The superscript n denotes the order of θ.

¨


NC gauge consistency condition and gauge equivalence condition
ˆ ˆ ˆ ˆ ˆ
iδα Λβ − iδβ Λα − [Λα , Λβ ]∗ = i Λ−i[α,β] , ˆˆ ˆ ˆ

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

iδα Λn − iδβ Λn −
β α [Λp , Λq ]∗r = i Λn
α β
ˆ
−i[α,β]
p+q+r =n

δα An = ∂µ Λn − i
µ α [Ap , Λq ]∗r
µ α
p+q+r =n

respectively. Here, ∗r denotes :
r
1 i
f (x) ∗r g (x) ≡ θµ1 ν1 · · · θµr νr ∂µ1 · · · ∂µr f (x)∂ν1 · · · ∂νr g (x).
r! 2

¨
u All order solution of SW–map BW’11 10 / 1


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

ˆ ˆα ˆ ˆα ˆ
∆Λn ≡ iδα Λn − iδβ Λn − [α, Λn ] − [Λn , β] − i Λn
β β −i[α,β] = [Λp , Λq ]∗r
α β
p+q+r =n,
p,q=n

∆α An ≡ δα An − i[α, An ] = ∂µ Λn + i
µ µ µ α [Λp , Aq ]∗r
α µ
p+q+r =n,
q=n

so that the l.h.s contains only the n-th order components.
However, note that one can extract the homogeneous parts such that
˜α
∆Λn = 0 , ˜µ
∆α An = 0

˜α ˜ µ
It is clear that one can add any homogeneous solution Λn , An to the
inhomogeneous solutions Λα µn , An with arbitrary coeﬃcients.

¨

NC Field Theory 1-st order solution

First order solution given in the original paper (SW, JHEP’99 ) :
1
Λ1 = − θκλ {Aκ , ∂λ α}
α
4
1
A1 = − θκλ {Aκ , ∂λ Aγ + Fλγ }.
γ
4
One can find the field strength form the definition :
1
Fγρ = − θκλ {Aκ , ∂λ Fγρ + Dλ Fγρ } − 2{Fγκ , Fρλ } .
1
4

These solutions are not unique since one can add homogeneous
solutions with arbitrary coefficients. i.e.
˜ ˜ρ
Λ1 = ic1 θµν [Aµ , ∂ν α] , A1 = c2 θµν Dρ Fµν

Therefore,
A1 + ic1 θµν ([∂ρ Aµ , Aν ] − i[[Aρ , Aµ ], Aν ]) + c2 θµν Dρ Fµν
ρ
is also a solution
¨

NC Field Theory 1-st order solution

2nd order solutions (Moller’04 ) α A2
and Λ2 can be written in terms of lower
µ
¨
order solutions : (K.U ,B. Yapiskan PRD’08.)
1 i
Λ2 = − θκλ {A1 , ∂λ α} + {Aκ , ∂λ Λ1 } − θκλ θµν [∂µ Aκ , ∂ν ∂λ α]
α κ α
8 16

1
A2
γ = − θκλ {A1 , ∂λ Aγ + Fλγ } + {Aκ , ∂λ A1 + Fλγ }
κ γ
1
8
i
− θκλ θµν [∂µ Aκ , ∂ν (∂λ Aγ + Fλγ )].
16

The ﬁeld strength at the second order can also be written in terms of ﬁrst
order solutions :
2 1
Fγρ = − θκλ {Aκ , ∂λ Fγρ + (Dλ Fγρ )1 }
1
8
+{A1 , ∂λ Fγρ + Dλ Fγρ } − 2{Fγκ , Fρλ } − 2{Fγκ , Fρλ }
κ
1 1

i κλ µν
− θ θ [∂µ Aκ , ∂ν ∂λ Fγρ + Dλ Fγρ ] − 2[∂µ Fγκ , ∂ν Fρλ ] .
16
Where, (Dλ Fγρ )1 = Dλ Fγρ − i[A1 , Fγρ ] + 1 θµν {∂µ Aλ , ∂ν Fγρ }.
1
λ 2
¨

NC Field Theory n-th order solutions

By analyzing ﬁrst two order solutions one can conjecture the general
¨
structure (K.U ,B. Yapiskan PRD’08.) :
1
Λn+1 = −
α θκλ {Ap , ∂λ Λq }∗r
κ α
4(n + 1) p+q+r =n

1 q
An+1 = −
γ θκλ {Ap , ∂λ Aq + Fλγ }∗r .
κ γ
4(n + 1) p+q+r =n

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

¨

NC Field Theory SW Differential Equation

Solution of Seiberg-Witten Differential Equation
Let us vary the deformation parameter infinitesimally

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

ˆ ˆ ˆ
δ Aγ (θ) = Aγ (θ + δθ) − Aγ (θ)
ˆ
∂ Aγ 1 ˆ ˆ ˆ
= δθµν µν = − δθκλ {Aκ , ∂λ Aγ + Fλγ }∗
∂θ 4

ˆ ˆ ˆ
δ Λ(θ) = Λ(θ + δθ) − Λ(θ)
∂Λˆ 1 ˆ ˆ
= δθµν µν = − θκλ {Aκ , ∂λ Λ}∗ (1)
∂θ 4
These differential equations are commonly called SW differential equations.
¨


To find solutions of the differential equation
expand NC gauge parameter and NC gauge field into a Taylor series:
ˆα
Λ(n) = α + Λ1 + · · · + Λn ,
α α
ˆµ
A(n) = Aµ + A1 + · · · + An .
µ µ

ˆ (n) ˆ (n)
here Λα and Aµ denotes the sum up to order n !
Then it is possible to write (Wulkenhaar et.al’01) :
n+1
ˆα 1 1 µ1 ν1 µ2 ν2 ∂ k−1 ˆ ˆα
Λ(n+1) = α − θ θ · · · θ µ k νk {A(k) , ∂ν1 Λ(k) }∗
4 k! ∂θµ2 ν2 · · · ∂θµk νk µ1 θ=0
k=1

n+1
ˆγ 1 1 µ1 ν1 µ2 ν2 ∂ k−1 ˆ ˆγ ˆ (k)
A(n+1) = Aγ − θ θ · · · θµk νk {A(k) , ∂ν1 A(k) + Fν1 γ }∗
4 k! ∂θµ2 ν2 · · · ∂θµk νk µ1 θ
k=1

¨


Contrary to the solutions presented in the previous section, these
expressions 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 above
expressions.

¨


ˆ (n+1) :
Let us write the n + 1–st component of Λα

1 ∂n ˆ ˆα
Λn+1 = −
α θµν θµ1 ν1 · · · θµn νn {A(n) , ∂ν1 Λ(n) }∗ .
4(n + 1)! ∂θµ1 ν1 · · · ∂θµn νn µ1 θ=0

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

1 ∂n
Λn+1 = −
α θµν θµ1 ν1 · · · θµn νn {Ap , ∂ν Λq }∗r
µ α .
4(n + 1)! ∂θµ1 ν1 · · · ∂θµn νn p+q+r =n

It is then an easy exercise to show that the above equation reduces to the
recursive formula :
1
Λn+1 = −
α θµν {Ap , ∂ν Λq }∗r .
µ α
4(n + 1) p+q+r =n

¨


With the same algebraic manipulation one can also derive the same
recursive formula for the gauge ﬁeld
1
An+1 = −
γ θµν θµ1 ν1 · · · θµn νn ×
4(n + 1)!
∂n ˆ ˆγ ˆ (n)
× {A(n) , ∂ν1 A(n) + Fν1 γ }∗
∂θµ1 ν1 · · · ∂θµn νn µ1 θ=0
1
= − θµν {Ap , ∂ν Aq + Fνγ }∗r .
µ γ
q
4(n + 1) p+q+r =n

¨

NC Field Theory SW Map for matter fields

Seiberg–Witten Map for Matter Fields

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

ˆˆ ˆ ˆ
δΛ Ψ(ψ, A; θ) = δα Ψ(ψ, A; θ).

The solution of this gauge equivalence relation can be found order by order
ˆ
by after expanding the NC field Ψ as formal power series in θ
ˆ ˆ ˆ
Ψ = ψ + Ψ1 + · · · + Ψn + · · ·

¨


Fundamental Representation :
Ordinary gauge transformation of ψ is written as

δα ψ = iαψ.

NC gauge transformation is deﬁned with the help of ∗–product :
ˆˆ ˆ ˆ ˆ
δΛ Ψ = i Λα ∗ Ψ.

Following the general strategy the gauge equivalence relation reads

∆α Ψn ≡ δα Ψn − iαΨn = i Λp ∗r Ψq ,
α
p+q+r =n,
q=n

for all orders.
˜
As discussed before, one is free to add any homogeneous solution Ψn of
the equation
˜
∆α Ψn = 0
to the solutions Ψn .
¨


A solution for the first order is given by Wess et.al :
1
Ψ1 = − θκλ Aκ (∂λ + Dλ )ψ
4
where Dµ ψ = ∂µ ψ − iAµ ψ .

The SW differential equation from the first order solution reads :

ˆ
∂Ψ 1
δθµν ˆ ˆ ˆ ˆ
= − δθκλ Aκ ∗ (∂λ Ψ + Dλ Ψ)
∂θ µν 4

which can also be written as
∂Ψˆ 1ˆ 1ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
= − Aκ ∗ (∂λ Ψ + Dλ Ψ) + Aλ ∗ (∂κ Ψ + Dκ Ψ)
∂θκλ 8 8

The NC covariant derivative here is defined as :
ˆ ˆ ˆ ˆ ˆ
Dµ Ψ = ∂µ Ψ − i Aµ ∗ Ψ .

¨


ˆ
By using a similar method, we expand Ψ in Taylor series :
ˆ
Ψ(n+1) = ψ + Ψ1 + Ψ2 + · · · + Ψn+1
n+1
1 µ 1 ν1 µ 2 ν2 ∂k ˆ
= ψ+ θ θ · · · θµk νk Ψ(n+1)
k! ∂θµ1 ν1 · · · ∂θµk νk θ=0
k=1

From the diﬀerential equation we get
n+1
ˆ 1 1 µ 1 ν1 µ 2 ν2
Ψ(n+1) = ψ − θ θ · · · θµk νk ×
4 k!
k=1
∂ k−1 ˆ ˆ ˆ ˆ
× A(k) ∗ (∂ν1 Ψ(k) + (Dν1 Ψ)(k) )
∂θµ2 ν2 · · · ∂θµk νk µ1 θ=0

where
ˆ ˆ ˆ ˆµ ˆ
(Dµ Ψ)(n) = ∂µ Ψ(n) − i A(n) ∗ Ψ(n) .

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


To ﬁnd the all order recursive solution we write the n + 1–st component :
1
Ψn+1 = − θµν θµ1 ν1 · · · θµn νn ×
4(n + 1)!
∂n ˆ ˆ ˆ ˆ
× A(n) ∗ (∂ν Ψ(n) + (Dν Ψ)(n) )
∂θµ1 ν1 · · · ∂θµn νn µ θ=0
1
= − θµν θµ1 ν1 · · · θµn νn ×
4(n + 1)!
∂n
× Ap ∗r (∂ν Ψ(q) + (Dν Ψ)q )
µ
∂θµ1 ν1 · · · ∂θµn νn p+q+r =n

where
(Dµ Ψ)n = ∂Ψn − i Ap ∗r Ψq .
µ
p+q+r =n

¨


After taking derivatives w.r.t. θ’s we obtain the all order recursive solution
of the gauge equivalence relation :
1
ψ n+1 = − θκλ Ap ∗r (∂λ Ψ(q) + (Dλ Ψ)q ).
κ
4(n + 1) p+q+r =n

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 M¨ller .
o

¨


Adjoint representation :
The gauge transformation reads as
δα ψ = i[α, ψ]
Non–commutative generalization can be written as
ˆˆ ˆ ˆ ˆ
δΛ Ψ = i[Λα , Ψ]∗ .
Following the general strategy, the gauge equivalence relation can be
written as
∆α Ψn := δα Ψn − i[α, Ψ] = i [Λp , Ψq ]∗r
α
p+q+r =n,
q=n

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 use
dimensional reduction. (K.U, Saka PRD’07)
¨


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

θµν 0
ΘMN = ,
0 0

the trivial dimensional reduction (i.e. from six to four dimensions) leads to
the general n–th order solution for a complex scalar field :
1
ψ n+1 = − θκλ {Ap , (∂λ Ψ(q) + (Dλ Ψ)q )}∗r .
κ
4(n + 1) p+q+r =n

Since the structure of the solutions are the same for both the scalar and
fermionic 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 fundamental
case except that now
Dµ ψ = ∂µ ψ − i[Aµ , ψ]
and hence
(Dµ Ψ)n = ∂µ Ψn − i [Ap , Ψq ]∗r .
µ
p+q+r =n

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

∂Ψˆ 1 ˆ 1 ˆ
ˆ ˆ ˆ ˆ ˆ ˆ
= − {Aκ , (∂λ Ψ + Dλ Ψ)}∗ + {Aλ , (∂κ Ψ + Dκ Ψ)}∗ .
∂θ κλ 8 8

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

¨

NC Field Theory Conclusion

CONCLUSION
Equations,
ˆ ˆˆ ˆ ˆ
Aµ (A; θ) + δΛ Aµ (A; θ) = Aµ (A + δα A; θ).

ˆ ˆˆ ˆ ˆ
Ψ(A, ψ; θ) + δΛ Ψ(A, ψ; θ) = Ψ(A + δα A, ψ + δα ψ; θ).
can be solved as
1
Λn+1 = −
α θκλ {Ap , ∂λ Λq }∗r
κ α
4(n + 1) p+q+r =n

1 q
An+1 = −
γ θκλ {Ap , ∂λ Aq + Fλγ }∗r .
κ γ
4(n + 1) p+q+r =n

1
ψ n+1 = − θκλ Ap ∗r (∂λ Ψ(q) + (Dλ Ψ)q ).
κ
4(n + 1) p+q+r =n

¨

K. Ulker - Non-commutative Gauge Theories and Seiberg Witten Map

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