Non-commutative Gauge Theories and Seiberg Witten Map to All Orders 1

Transcript

1 Non-commutative Gauge Theories and Seiberg Witten Map to All Orders 1 Kayhan ÜLKER Feza Gürsey Institute * Istanbul, Turkey (savefezagursey.wordpress.com) The SEENET-MTP Workshop JW K. Ulker, B Yapiskan Phys. Rev. D 77, (2008) [arxiv: ] Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 1 / 1

2 Outline Moyal product. Non-commutative Yang Mills (Â, ˆΛ) Seiberg Witten Map (Â Â(A, θ), ˆΛ ˆΛ(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 ( ˆΨ ˆΨ(A, ψ, θ)) Conclusion Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 2 / 1

3 Moyal *-product Moyal -product The simplest way to introduce non-commutativity is to use (Moyal) * product ( i f (x) g(x) exp ) 2 θµν x µ y ν f (x)g(y) y x = f (x) g(x) + i 2 θµν µ f (x) ν g(x) + for real CONSTANT antisymmetric parameter θ! * commutator of the ordinary coordinates : [x µ, x ν ] x µ x ν x ν x µ = iθ µν. NC QFT models can then be obtained by replacing the ordinary product with the * product! Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 3 / 1

4 NC-YM theories NC Yang Mills Theory The action of NC YM theory is written as : Ŝ = 1 4 Tr d 4 x ˆF µν ˆF µν = 1 4 Tr d 4 x ˆF µν ˆF µν where ˆF µν = µ Â ν ν Â µ i[â µ, Â ν ] is the NC field strength of the NC gauge field Â. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 4 / 1

5 NC-YM theories NC Yang Mills Theory The action of NC YM theory is written as : Ŝ = 1 4 Tr d 4 x ˆF µν ˆF µν = 1 4 Tr d 4 x ˆF µν ˆF µν where ˆF µν = µ  ν ν  µ i[â µ,  ν ] is the NC field strength of the NC gauge field Â. The action is invariant under the NC gauge transformations: ˆδˆΛ µ = µˆλ i[â µ, ˆΛ] ˆD µˆλ ˆδˆΛ ˆF µν = i[ˆλ, ˆF µν ]. Here, ˆΛ is the NC gauge parameter. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 4 / 1

6 SW map Seiberg Witten Map One can derive both conventional and NC gauge theories from string theory by using different regularization procedures (SW 99). Let A µ and α to be the ordinary counterparts of  µ and ˆΛ respectively. There must be a map from a commutative gauge field A to a noncommutative one Â, that arises from the requirement that gauge invariance should be preserved! Â(A) + ˆδˆΛÂ(A) = Â(A + δ α A) where δ α is the ordinary gauge transformation : δ α A µ = µ α i[a µ, α] D µ α. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 5 / 1

7 SW map This map can be rewritten as ˆδˆΛ µ (A; θ) =  µ (A + δ α A; θ)  µ (A; θ) = δ α  µ (A; θ) Since SW map imposes the following functional dependence :  µ =  µ (A; θ), ˆΛ = ˆΛ α (α, A; θ). one has to solve ˆδˆΛµ(A; θ) = δ α  µ (A; θ) simultaneously for  µ and ˆΛ α and it is difficult! Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 6 / 1

8 SW map Now, remember the ordinary gauge consistency condition δ α δ β δ β δ α = δ i[α,β]. (check for instance for δ α ψ = iαψ) Let, Λ be a Lie algebra valued gauge parameter Λ = Λ a T a. For the non-commutative case, we get (δ Λα δ Λβ δ Λβ δ Λα ) ˆΨ = 1 2 [T a, T b ]{Λ α,a, Λ β,b } ˆΨ+ 1 2 {T a, T b }[Λ α,a, Λ β,b ] ˆΨ 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! Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 7 / 1

9 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 : ˆΛ = α a T a + Λ 1 ab : T a T b : + Λ n 1 a 1 a n : T a1 T an : + let all NC fields and parameters depend only on Lie algebra valued fields A, ψ, and parameter α i.e.  µ µ(A), ˆΨµ ˆΨ µ (A, ψ), ˆΛ = ˆΛ(A, α) impose NC gauge consistency condition: iδ αˆλ β iδ β ˆΛ α [ˆΛ α, ˆΛ β ] = i ˆΛ i[α,β]. Note that, above construction of Wess et.al. is obtained entirely independent of string theory! Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 8 / 1

10 SW map In contrast to SW we now have an equation only for the gauge parameter iδ αˆλ β iδ β ˆΛ α [ˆΛ α, ˆΛ β ] = i ˆΛ i[α,β]. and once we solve it we can then solve ˆδˆΛµ(A; θ) = δ α  µ (A; θ) only for A µ. To find the solutions we expand ˆΛ α and  µ as formal power series, ˆΛ α = α + Λ 1 α + + Λ n α +,  µ = A µ + A 1 µ + + A n µ +, The superscript n denotes the order of θ. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW 11 9 / 1

11 SW map NC gauge consistency condition and gauge equivalence condition iδ αˆλβ iδ β ˆΛα [ˆΛ α, ˆΛ β ] = i ˆΛ i[α,β], ˆδˆΛµ(A; θ) = δ α  µ (A; θ) can be written for the n th order components of ˆΛ α and  µ as iδ α Λ n β iδ βλ n α p+q+r=n δ α A n µ = µ Λ n α i [Λ p α, Λ q β ] r = i ˆΛ n i[α,β] p+q+r=n [A p µ, Λ q α] r respectively. Here, r denotes : f (x) r g(x) 1 ( ) i r θ µ 1ν1 θ µr νr µ1 r! 2 µr f (x) ν1 νr g(x). Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

12 SW map We can rearrange the above Eq.s for any order n Λ n iδ αˆλ n β iδ β ˆΛ n α [α, ˆΛ n β ] [ˆΛ n α, β] i ˆΛ n i[α,β] = α A n µ δ α A n µ i[α, A n µ] = µ Λ n α + i p+q+r=n, q n p+q+r=n, p,q n [Λ p α, A q µ] r [Λ p α, Λ q β ] 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 Λ n α = 0, α Ã n µ = 0 It is clear that one can add any homogeneous solution Λ n α, Ã n µ to the inhomogeneous solutions Λ n α, A n µ with arbitrary coefficients. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

13 1-st order solution First order solution given in the original paper (SW, JHEP 99) : Λ 1 α = 1 4 θκλ {A κ, λ α} A 1 γ = 1 4 θκλ {A κ, λ A γ + F λγ }. One can find the field strength form the definition : Fγρ 1 = 1 ) 4 θκλ( {A κ, λ F γρ + D λ F γρ } 2{F γκ, F ρλ }. These solutions are not unique since one can add homogeneous solutions with arbitrary coefficients. i.e. Λ 1 = ic 1 θ µν [A µ, ν α], Ã 1 ρ = c 2 θ µν D ρ F µν Therefore, A 1 ρ + ic 1 θ µν ([ ρ A µ, A ν ] i[[a ρ, A µ ], A ν ]) + c 2 θ µν D ρ F µν is also a solution Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

14 1-st order solution 2nd order solutions (Moller 04) A 2 µ and Λ 2 α can be written in terms of lower order solutions : (K.Ü,B. Yapiskan PRD 08.) Λ 2 α = 1 8 θκλ ( {A 1 κ, λ α} + {A κ, λ Λ 1 α} ) i 16 θκλ θ µν [ µ A κ, ν λ α] A 2 γ = 1 8 θκλ ( {A 1 κ, λ A γ + F λγ } + {A κ, λ A 1 γ + F 1 λγ} ) i 16 θκλ θ µν [ µ A κ, ν ( λ A γ + F λγ )]. The field strength at the second order can also be written in terms of first order solutions : Fγρ 2 = 1 8 θκλ( {A κ, λ Fγρ 1 + (D λ F γρ ) 1 } ) +{A 1 κ, λ F γρ + D λ F γρ } 2{F γκ, Fρλ} 1 2{Fγκ, 1 F ρλ } i 16 θκλ θ µν( ( ) ) [ µ A κ, ν λ F γρ + D λ F γρ ] 2[ µ F γκ, ν F ρλ ]. Where, (D λ F γρ ) 1 = D λ F 1 γρ i[a 1 λ, F γρ] θµν { µ A λ, ν F γρ }. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

15 n-th order solutions By analyzing first two order solutions one can conjecture the general structure (K.Ü,B. Yapiskan PRD 08.) : A n+1 γ Λ n+1 α 1 = 4(n + 1) θκλ 1 = 4(n + 1) θκλ p+q+r=n p+q+r=n {A p κ, λ Λ q α} r {A p κ, λ A q γ + F q λγ } r. The overall constant 1/4(n + 1) is fixed uniquely with third order solutions. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

16 SW Differential Equation Solution of Seiberg-Witten Differential Equation Let us vary the deformation parameter infinitesimally θ θ + δθ To get equivalent physics, Â(θ) and ˆΛ(θ) should change when θ is varied, (SW 99): δâ γ (θ) = Â γ (θ + δθ) Â γ (θ) = δθ µν Â γ θ µν = 1 4 δθκλ {Â κ, λ Â γ + ˆF λγ } δˆλ(θ) = ˆΛ(θ + δθ) ˆΛ(θ) µν ˆΛ = δθ θ µν = 1 4 θκλ {Â κ, λˆλ} (1) These differential equations are commonly called SW differential equations. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

17 SW Differential Equation To find solutions of the differential equation expand NC gauge parameter and NC gauge field into a Taylor series: ˆΛ (n) α ˆΛ (n) α = α + Λ 1 α + + Λ n α, Â (n) µ = A µ + A 1 µ + + A n µ. here and Â(n) µ denotes the sum up to order n! Then it is possible to write (Wulkenhaar et.al 01) : ˆΛ (n+1) α = α 1 n+1 4 k=1 Â (n+1) γ = A γ 1 n+1 4 k=1 ( 1 k! θµ1ν1 θ µ2ν2 θ µ k ν k ( 1 k! θµ1ν1 θ µ2ν2 θ µ k ν k k 1 ) θ µ2ν2 θ µ k ν {Â(k) k µ 1, ˆΛ(k) ν1 α } θ=0 k 1 θ µ2ν2 θ µ k ν {Â(k) k µ 1, ν1 Â (k) γ + ) (k) ˆF ν 1γ} θ Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

18 SW Differential Equation 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. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

19 SW Differential Equation Let us write the n + 1 st component of ˆΛ (n+1) α : Λ n+1 α ( 1 = 4(n + 1)! θµν θ µ n ) 1ν1 θ µnνn θ µ 1ν 1 θ µ {Â(n) nν n µ 1, ˆΛ(n) ν1 α }. 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: Λ n+1 α ( 1 n = 4(n + 1)! θµν θ µ1ν1 θ µnνn θ µ1ν1 θ µnνn p+q+r=n {A p µ, ν Λ q α} r ). It is then an easy exercise to show that the above equation reduces to the recursive formula : Λ n+1 α 1 = 4(n + 1) θµν p+q+r=n {A p µ, ν Λ q α} r. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

20 SW Differential Equation With the same algebraic manipulation one can also derive the same recursive formula for the gauge field 1 γ = 4(n + 1)! θµν θ µ 1ν1 θ µnνn ( n ) θ µ 1ν 1 θ µ nν n {Â (n) µ 1, ν1 Â (n) γ + ˆF ν (n) 1 γ} θ=0 1 = 4(n + 1) θµν {A p µ, ν A q γ + Fνγ} q r. A n+1 p+q+r=n Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

21 SW Map for matter fields Seiberg Witten Map for Matter Fields SW map of a NC field ˆΨ in a gauge invariant theory can be derived from (J. Wess et.al. EPJ 01): ˆδˆΛ ˆΨ(ψ, A; θ) = δ α ˆΨ(ψ, A; θ). The solution of this gauge equivalence relation can be found order by order by after expanding the NC field ˆΨ as formal power series in θ ˆΨ = ψ + ˆΨ ˆΨ n + Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

22 SW Map for matter fields Fundamental Representation : Ordinary gauge transformation of ψ is written as δ α ψ = iαψ. NC gauge transformation is defined with the help of product : ˆδˆΛ ˆΨ = i ˆΛ α ˆΨ. Following the general strategy the gauge equivalence relation reads α Ψ n δ α Ψ n iαψ n = i Λ p α r Ψ q, p+q+r=n, q n for all orders. As discussed before, one is free to add any homogeneous solution Ψ n of the equation α Ψ n = 0 to the solutions Ψ n. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

23 SW Map for matter fields A solution for the first order is given by Wess et.al : where D µ ψ = µ ψ ia µ ψ. Ψ 1 = 1 4 θκλ A κ ( λ + D λ )ψ The SW differential equation from the first order solution reads : which can also be written as δθ µν ˆΨ θ µν = 1 4 δθκλ  κ ( λ ˆΨ + ˆD λ ˆΨ) ˆΨ θ κλ = 1 8Âκ ( λ ˆΨ + ˆD λ ˆΨ) + 1 8Âλ ( κ ˆΨ + ˆD κ ˆΨ) The NC covariant derivative here is defined as : ˆD µ ˆΨ = µ ˆΨ i  µ ˆΨ. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

24 SW Map for matter fields By using a similar method, we expand ˆΨ in Taylor series : ˆΨ (n+1) = ψ + Ψ 1 + Ψ Ψ n+1 n+1 1 = ψ + k! θµ 1ν 1 θ µ 2ν2 θ µ kν k k=1 From the differential equation we get ˆΨ (n+1) = ψ 1 4 where n+1 k=1 ( 1 k! θµ 1ν 1 θ µ 2ν2 θ µ kν k k ( ˆΨ(n+1) θ µ 1ν 1 θ µ k ν k ))θ=0 ( k 1 θ µ 2ν 2 θ µ k ν k  (k) µ 1 ( ν1 ˆΨ (k) + ( ˆD ν1 ˆΨ) (k) ) ) θ=0 ( ˆD µ ˆΨ) (n) = µ ˆΨ(n) iâ(n) µ ˆΨ (n). For the abelian case this solution differs from the one given in Wulkenhaar et.al. by a homogeneous solution. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

25 SW Map for matter fields To find the all order recursive solution we write the n + 1 st component : Ψ n+1 1 = 4(n + 1)! θµν θ µ1ν1 θ µnνn ( n ) Â(n) θ µ1ν1 µnνn µ ( ν θ ˆΨ(n) + ( ˆD ν ˆΨ) (n) ) θ=0 1 = 4(n + 1)! θµν θ µ1ν1 θ µnνn ( ) n A p θ µ1ν1 θ µ r ( ν Ψ (q) + (D ν Ψ) q ) µnνn p+q+r=n where (D µ Ψ) n = Ψ n i A p µ r Ψ q. p+q+r=n Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

26 SW Map for matter fields After taking derivatives w.r.t. θ s we obtain the all order recursive solution of the gauge equivalence relation : ψ n+1 1 = 4(n + 1) θκλ p+q+r=n A p κ r ( λ Ψ (q) + (D λ Ψ) 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 Möller. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

27 Adjoint representation : The gauge transformation reads as SW Map for matter fields δ α ψ = i[α, ψ] Non commutative generalization can be written as ˆδˆΛ ˆΨ = i[ˆλ α, ˆΨ]. Following the general strategy, the gauge equivalence relation can be written as α Ψ n := δ α Ψ n i[α, Ψ] = i [Λ p α, Ψ q ] r 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) Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

28 SW Map for matter fields By setting the components of the deformation parameter θ on the compactified dimensions zero, i.e. ( ) Θ MN θ µν 0 =, 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 : ψ n+1 1 = 4(n + 1) θκλ p+q+r=n {A p κ, ( λ Ψ (q) + (D λ Ψ) q )} r. 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. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

29 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 fundamental case except that now D µ ψ = µ ψ i[a µ, ψ] and hence (D µ Ψ) n = µ Ψ n i [A p µ, Ψ q ] r. p+q+r=n The same result can also be obtained by solving the differential equation : ˆΨ θ κλ = 1 8 {Â κ, ( λ ˆΨ + ˆD λ ˆΨ)} {Â λ, ( κ ˆΨ + ˆD κ ˆΨ)}. Therefore, the result obtained above via dimensional reduction can also be thought as an independent check of the aforementioned results. Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1

30 Conclusion CONCLUSION Equations,  µ (A; θ) + ˆδˆΛµ(A; θ) =  µ (A + δ α A; θ). can be solved as ˆΨ(A, ψ; θ) + ˆδˆΛ ˆΨ(A, ψ; θ) = ˆΨ(A + δ α A, ψ + δ α ψ; θ). A n+1 γ Λ n+1 α 1 = 4(n + 1) θκλ 1 = 4(n + 1) θκλ ψ n+1 1 = 4(n + 1) θκλ p+q+r=n p+q+r=n p+q+r=n {A p κ, λ Λ q α} r {A p κ, λ A q γ + F q λγ } r. A p κ r ( λ Ψ (q) + (D λ Ψ) q ). Kayhan ÜLKER (Feza Gürsey Institute*) All order solution of SW map BW / 1