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The noncommutative Chern-Simons action and the Seiberg-Witten map Conference on higher dimensional QHE, Chern-Simons theory and noncomutative geometry in condensed matter and field theory ICTP, Trieste 2005
´ E. Grandi Nicolas Universidad Nacional de La Plata
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 1/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D Is there some map relating them? Aˆµ
←→
Aµ
θ µν 6= 0
←→
θ µν = 0
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D Yes!, and it is even more general Aˆµ
←→
Aµ
θ µν 6= 0
←→
θ ′µν 6= 0
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D Yes!, and it is even more general Aˆµ
←→
Aµ
θ µν 6= 0
←→
θ ′µν 6= 0
Seiberg-Witten map relates two noncommutative gauge theories θ µν ←→ θ ′µν
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4 For a general action, the dynamics is not invariant S θ [Aˆµ ] ←→ S θ+δθ [Aµ ] + δS[Aµ ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4 For a general action, the dynamics is not invariant S θ [Aˆµ ] ←→ S θ+δθ [Aµ ] + δS[Aµ ] For example, for the Yang-Mills action δSY M
Z 1 1 = δθ ρσ Fµν F µρ F νσ − Fµν {Aρ , Dσ F µν + ∂σ F µν } 6= 0 2 4 M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4 For a general action, the dynamics is not invariant S θ [Aˆµ ] ←→ S θ+δθ [Aµ ] + δS[Aµ ] But, for the Chern-Simons action we have Z δSCS = ∂µ f µ M
Then, in a manifold without boundary, the CS action is invariant
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0 l θ ˆ SCS [A]
←→
Fµν = 0 l θ′ SCS [A]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
l
l
θ ˆ SCS [A]
θ SW g] ZW [ˆ
θ′ SCS [A]
′
θ SW ZW [g]
←→ ′
S θ [ψ] ≡ S θ [ψ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
l
l
θ ˆ SCS [A]
θ′ SCS [A]
l θ SW g] ZW [ˆ
l ′
θ SW ZW [g]
←→ ′
S θ [ψ] ≡ S θ [ψ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
l θ ˆ SCS [A]
l θ′ SCS [A]
⇐⇒
l θ SW g] ZW [ˆ
l ′
θ SW ZW [g]
←→ ′
S θ [ψ] ≡ S θ [ψ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) ˙ Tr ǫij Aˆ0 ∗ Fˆij − Aˆi ∗ Aˆj + Bij 4π M
where the boundary term is (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 23 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ Z Z κ ~ti (ˆ g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)+ ǫij (ˆ − g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)∗(ˆ g −1∗∂j gˆ) 4π ∂M M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ Z Z κ ~ti (ˆ g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)+ ǫij (ˆ − g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)∗(ˆ g −1∗∂j gˆ) 4π ∂M M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ Z Z κ ~ti (ˆ g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)+ ǫij (ˆ − g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)∗(ˆ g −1∗∂j gˆ) 4π ∂M M Aˆ0 also enforces on the boundary an infinite set of nonlinear constraints involving Aˆi and all its derivatives
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ ˙ (1) (2) θ ij δSCS = Tr ǫ δ Aˆ0 Fˆij + Aˆ0 δ Fˆij − 2δ Aˆi Aˆj + δBij +δBij 4π M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ θ Tr ∂µ f µ δSCS = 4π M Then in a boudaryless manifold, the CS action is invariant under the Seiberg-Witten map
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ (1) (2) ij kl θ Tr ǫ δθ ∂i 2Aj ∂k Al A0 + Aj Ak A˙ l + δBij + δBij δSCS = 4π M Then in a boudaryless manifold, the CS action is invariant under the Seiberg-Witten map
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ (1) (2) ij kl θ Tr ǫ δθ ∂i 2Aj ∂k Al A0 + Aj Ak A˙ l + δBij + δBij δSCS = 4π M Then in a boudaryless manifold, the CS action is invariant under the Seiberg-Witten map
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064).
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The stated equivalence holds at the classical level. The quantum result would require the study of the of the behavior of the measure DAµ under the map (Kaminsky, Okawa, Ooguri hep-th/0101133, Kaminsky hep-th/0310011). The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The stated equivalence holds at the classical level. The quantum result would require the study of the of the behavior of the measure DAµ under the map (Kaminsky, Okawa, Ooguri hep-th/0101133, Kaminsky hep-th/0310011). The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The stated equivalence holds at the classical level. The quantum result would require the study of the of the behavior of the measure DAµ under the map (Kaminsky, Okawa, Ooguri hep-th/0101133, Kaminsky hep-th/0310011). The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
´ E. Grandi Nicolas Universidad Nacional de La Plata
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 1/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D Is there some map relating them? Aˆµ
←→
Aµ
θ µν 6= 0
←→
θ µν = 0
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D Yes!, and it is even more general Aˆµ
←→
Aµ
θ µν 6= 0
←→
θ ′µν 6= 0
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Open strings in a constant Neveu-Schwartz background Pauli-Villars
−→ Commutative gauge theory
Aµ → Aµ + Dµ Λ = Aµ + ∂µ Λ + Aµ Λ − Λ Aµ Point-Splitting −→ Noncommutative gauge theory ˆ µΛ ˆ = Aˆµ + ∂µ Λ ˆ + Aˆµ ∗ Λ ˆ −Λ ˆ ∗ Aˆµ Aˆµ → Aˆµ + D Yes!, and it is even more general Aˆµ
←→
Aµ
θ µν 6= 0
←→
θ ′µν 6= 0
Seiberg-Witten map relates two noncommutative gauge theories θ µν ←→ θ ′µν
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 2/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4 For a general action, the dynamics is not invariant S θ [Aˆµ ] ←→ S θ+δθ [Aµ ] + δS[Aµ ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4 For a general action, the dynamics is not invariant S θ [Aˆµ ] ←→ S θ+δθ [Aµ ] + δS[Aµ ] For example, for the Yang-Mills action δSY M
Z 1 1 = δθ ρσ Fµν F µρ F νσ − Fµν {Aρ , Dσ F µν + ∂σ F µν } 6= 0 2 4 M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Seiberg-Witten map Explicit form of the map
θ ρσ ←→ θ ρσ + δθ ρσ
1 ρσ ˆ Aµ ←→ Aµ + δθ {Aρ , ∂σ Aµ + Fσµ } 4 1 ρσ ˆ Fµν ←→ Fµν + δθ (−2 {Fµρ , Fνσ } + {Aρ , Dσ Fµν + ∂σ Fµν }) 4 For a general action, the dynamics is not invariant S θ [Aˆµ ] ←→ S θ+δθ [Aµ ] + δS[Aµ ] But, for the Chern-Simons action we have Z δSCS = ∂µ f µ M
Then, in a manifold without boundary, the CS action is invariant
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 3/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0 l θ ˆ SCS [A]
←→
Fµν = 0 l θ′ SCS [A]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
l
l
θ ˆ SCS [A]
θ SW g] ZW [ˆ
θ′ SCS [A]
′
θ SW ZW [g]
←→ ′
S θ [ψ] ≡ S θ [ψ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
l
l
θ ˆ SCS [A]
θ′ SCS [A]
l θ SW g] ZW [ˆ
l ′
θ SW ZW [g]
←→ ′
S θ [ψ] ≡ S θ [ψ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
The Noncommutative CS action The noncommutative Chern-Simons action is Z κ 2 θ SCS [Aˆµ ] = Tr ǫµνρ Aˆµ ∗ ∂ν Aˆρ + Aˆµ ∗ Aˆν ∗ Aˆρ 4π 3 M What should we expect? Fˆµν = 0
←→
Fµν = 0
l θ ˆ SCS [A]
l θ′ SCS [A]
⇐⇒
l θ SW g] ZW [ˆ
l ′
θ SW ZW [g]
←→ ′
S θ [ψ] ≡ S θ [ψ]
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 4/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) ˙ Tr ǫij Aˆ0 ∗ Fˆij − Aˆi ∗ Aˆj + Bij 4π M
where the boundary term is (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 23 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ Z Z κ ~ti (ˆ g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)+ ǫij (ˆ − g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)∗(ˆ g −1∗∂j gˆ) 4π ∂M M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ Z Z κ ~ti (ˆ g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)+ ǫij (ˆ − g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)∗(ˆ g −1∗∂j gˆ) 4π ∂M M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
Aˆ0 enforces the constraint Fˆij = 0 then Aˆi = gˆ−1 ∂i gˆ Z Z κ ~ti (ˆ g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)+ ǫij (ˆ − g −1∗∂i gˆ)∗(ˆ g −1∗∂t gˆ)∗(ˆ g −1∗∂j gˆ) 4π ∂M M Aˆ0 also enforces on the boundary an infinite set of nonlinear constraints involving Aˆi and all its derivatives
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
Z κ (1) (2) ˙ Tr ǫij Aˆ0 Fˆij − Aˆi ∗ Aˆj + Bij + Bij 4π M
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij )
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ ˙ (1) (2) θ ij δSCS = Tr ǫ δ Aˆ0 Fˆij + Aˆ0 δ Fˆij − 2δ Aˆi Aˆj + δBij +δBij 4π M
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ θ Tr ∂µ f µ δSCS = 4π M Then in a boudaryless manifold, the CS action is invariant under the Seiberg-Witten map
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ (1) (2) ij kl θ Tr ǫ δθ ∂i 2Aj ∂k Al A0 + Aj Ak A˙ l + δBij + δBij δSCS = 4π M Then in a boudaryless manifold, the CS action is invariant under the Seiberg-Witten map
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Explicit calculation Writing the CS action as θ SCS [Aˆ0 , Aˆi ] =
κ Tr 4π
Z
ǫij
M
(1) (2) ˙ Aˆ0 Fˆij − Aˆi Aˆj + Bij + Bij
where the boundary terms are (1) Bij
h i = ∂i (Aˆj ∗ Aˆ0 ) + [∂i Aˆj + 32 Aˆi ∗ Aˆj , Aˆ0 ] − 23 Aˆi ∗ Aˆ0 , Aˆj
(2) ˙ ˙ Bij = (Aˆ0 ∗ Fˆij − Aˆ0 Fˆij ) − (Aˆi ∗ Aˆj − Aˆj Aˆi )
Applying the Seiberg-Witten map we get Z κ (1) (2) ij kl θ Tr ǫ δθ ∂i 2Aj ∂k Al A0 + Aj Ak A˙ l + δBij + δBij δSCS = 4π M Then in a boudaryless manifold, the CS action is invariant under the Seiberg-Witten map
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 5/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064).
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The stated equivalence holds at the classical level. The quantum result would require the study of the of the behavior of the measure DAµ under the map (Kaminsky, Okawa, Ooguri hep-th/0101133, Kaminsky hep-th/0310011). The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The stated equivalence holds at the classical level. The quantum result would require the study of the of the behavior of the measure DAµ under the map (Kaminsky, Okawa, Ooguri hep-th/0101133, Kaminsky hep-th/0310011). The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
Closing remarks In a boundaryless manifold, the noncommutative Chern-Simons action is invariant under the Seiberg-Witten map, this meaning (N.E.G and G.A. Silva hep-th/0310113) θ SCS
←→
θ′ SCS
In a manifold with boundary, we need to impose a infinite set of nonlinear relations involving all the derivatives of the gauge field as boundary conditions (A.R. Lugo hep-th/0111064). The "boundary theory" is not a NC chiral Wess-Zumino-Witten model. In fact it is not a boundary theory at all because it involves infinite derivatives of the fields in the directions of the bulk.
The stated equivalence holds at the classical level. The quantum result would require the study of the of the behavior of the measure DAµ under the map (Kaminsky, Okawa, Ooguri hep-th/0101133, Kaminsky hep-th/0310011). The noncommutative Chern-Simons action and the Seiberg-Witten map – p. 6/
