Instantons in gauge theories with N=1/2 supersymmetry Oleg Lunin - PDF document

Instantons in gauge theories with N=1/2 supersymmetry Oleg Lunin Institute for Advanced Study R. Britto, B. Feng, O. L., S–J. Rey, hep-th/0311275

Outline • Noncommutative superspace. • Gauge theory on NS – classical aspects – perturbative regime – instanton solutions and supersymmetry • One instanton solution in U(N) gauge theory. – procedure for deforming the instanton – geometry of the deformed moduli space • Chiral ring and gluino condensate • Summary

String in graviphoton field • String in flat space � 1 α � L = 1 ˙ ∂θ α + ¯ α + ˜ θ α + ˜ α ∂ ˜ ∂x µ ∂x µ + p α ˜ ˜ α ˜ ∂ ¯ p α ∂ ˜ ¯ θ ˙ ¯ p ˙ p ˙ θ 2 α ′ Berkovits ’96 • Euclidean target space: independent θ, ¯ θ, p, ¯ p : � � � � � � α ) → ( − ∂ ∂θ α , − ∂ ∂ α → − ∂ ¯ � � � ( p α , ¯ p ˙ α ) , q α → , d ˙ � � � ∂ ¯ α ˙ θ ˙ ∂θ α ∂ ¯ θ x y y α ˙ y µ = x µ + iθ α σ µ α + i ˜ α ˜ α ¯ θ α σ µ ¯ θ ˙ θ α ˙ α ˙ p → ¯ • Change of variables: p → q, ¯ d : α ∂x µ + 1 2 θθ∂θ α − 3 q α = − p α − iσ µ 2 ∂ ( θ α θθ ) α ˙ • New Lagrangian: � 1 α � L = 1 ˙ ∂θ α + ¯ α − ˜ θ α + ˜ α ∂ ˜ ∂y µ ∂y µ − q α ˜ ˜ α ˜ ∂ ¯ q α ∂ ˜ ¯ ¯ θ ˙ d ˙ d ˙ θ 2 α ′ • D brane: θ = ˜ θ, q = ˜ q at z = ˜ z • Preserved SUSY: � � qdz + ˜ ( qdz + ˜ qd ˜ z ) , (¯ qd ˜ ¯ z )

Graviphoton and Noncommutative Superspace • Adding graviphoton field to the Lagrangian: � � L 1 = 1 ∂θ α − ˜ θ α + α ′ F αβ q α ˜ − q α ˜ q α ∂ ˜ q β α ′ • To avoid gravitational backreaction: F ˙ β = 0 α ˙ • Effective Lagrangian: � 1 � θ α ˜ ∂ ˜ ∂θ β L eff = α ′ F αβ • Boundary conditions at z = ˜ z : � 1 � θ α = ˜ θ α , θ α δθ β + ˜ ( ∂ ˜ ∂θ α δ ˜ θ β ) = 0 : θ α = − ˜ ∂ ˜ α ′ F ∂θ α αβ • Propagators: w ) � = α ′ 2 πiF αβ log ˜ z − w � θ α ( z, ˜ z ) θ β ( w, ˜ z − ˜ w � θ α ( τ ) θ β ( τ ′ ) � = α ′ 2 F αβ sign ( τ − τ ′ ) { θ α , θ β } = α ′ 2 F αβ = C αβ , [ y µ , y ν ] = 0 Seiberg ’03

Gauge theory on Noncommutative Superspace • Noncommutative superspace: y m = x m + iθ α σ m α ¯ { θ α , θ β } = C αβ , θ ˙ α α ˙ • Star product: finite number of terms � � ← − − − − → − C αβ ∂ ∂ f ( θ ) ⋆ g ( θ ) = f ( θ ) exp g ( θ ) ∂θ α ∂θ β 2 • Modification of SUSY algebra: ∂ 2 { ¯ α , ¯ β } ⋆ = − 4 C αβ σ m α σ n Q ˙ Q ˙ α ˙ β ˙ β ∂y m ∂y n • Gauge field: � � W α = − 1 e − V ⋆ D α e V → e − i Λ ⋆ W α ⋆ e i Λ 4 DD ⋆ ⋆ ⋆ ⋆ • WZ gauge: C –dependent corrections to V . • Action for “ N = 1 2 ” SYM: � iτ � � iτ � � � 8 π W α ⋆ W α d 4 x Tr d 4 x Tr S = − θ 2 + 8 π W ˙ α ⋆ W ˙ α ¯ θ 2

Perturbative N = 1 2 SYM • Lagrangian for the component fields � � L = 1 − 1 σ m ∇ m λ + 1 4 F mn F mn − i ¯ 2 D 2 λ ¯ g 2 Tr � � − iC mn F mn λλ + C 2 + 1 8 ( λλ ) 2 g 2 Tr 2 Seiberg ’03 • Operators with ∆ = 5 : no renormalizability � � (∆ i − 4) − 1 Ω div = 4 + ( r l + d l + 4) 2 i ∈ L ext � � = 4 + (∆ i − 4) − s l ext i ∈ L • Assumption: new vertices are connected without changing external lines

Renormalization of N = 1 2 SYM • Features of the theory: – no hermiticity – R symmetry: λ → e iα λ • “Non–renormalizable” vertices: lines cannot terminate inside the diagram � � � � ′ ′ Ω div = 4 + (∆ i − 4) + (∆ i − 4) − s l − s l i ∈ L i ∈ L ext ext � � � ′ ( ˜ = 4 + (∆ i − 4) + ∆ i − 4) − s l i ∈ L i ∈ L ext • ˜ ∆ < 4 accounts for R charge flow • SYM is renormalizable: no new vertices. O L, Rey

Instantons in N = 1 2 SYM • Instantons in N = 1 SYM – minimal action in a given topological sector – solutions preserving N = 1 2 SUSY • SUSY transformations in N = 1 2 theory � � F αβ + i δλ α = iε α D + 2 2 C αβ λλ ε β ˙ ˙ β β δD = − ε α ∇ α ˙ δF αβ = − iε ( α ∇ β ) ˙ β λ β λ • Instantons preserving SUSY F αβ + i β = 0 , ˙ 2 C αβ λλ = 0 , ∇ α ˙ λ α = D = 0 β λ • Alternative derivation: rewrite the action as � � � � 2 � S = 1 mn + i − iλ σ m ∇ m ¯ d 4 x Tr F (+) λ + D 2 − 2 C mn λλ g 2 � − iτ Tr F ∧ F. 4 π • “Instanton number” is negative • “Holomorphic instanton” – no deformation: β = 0 , αβ λ β = 0 , α ˙ F ˙ ∇ ˙ α = D = 0 λ ˙

Constructing Deformed Instantons • Equations to be solved F αβ + i β = 0 ˙ 2 C αβ λλ = 0 , ∇ α ˙ β λ • Perturbation theory in C αβ : truncated series • Example: one instanton for U (2) α = ¯ α + x ˙ ˙ ¯ β ¯ ¯ β , ξ ˙ ζ ˙ α α η α α = F ˙ λ ˙ ξ α ˙ – Fermi statistics: λλ ∈ U (1) – prepotential for the U (1) part: A m = C mn ∇ n Φ – solution of the Laplace equation for Φ : � � ρ 2 α + 1 α + ρ 2 η α η α ) ˙ ˙ Φ = − 8 i ( r 2 + ρ 2 ) 2 ξ ˙ α ξ r 2 + ρ 2 ( ζ ˙ α ζ Imaanpur

One Instanton for U(N) • k instantons for U ( N ) : 2 kN zero modes • Generically series terminates at | C | kN • One instanton solution: series up to | C | 3 • Zero modes for one instanton: � α η α � β + x ˙ ˙ λ (0) = F (0) ¯ ¯ β ζ α ˙ α ˙ ˙ β α χ i χ i i = a = ε a ˙ ¯ λ (0) λ (0) ¯ ¯ ( x 2 + ρ 2 ) 3 / 2 δ a ( x 2 + ρ 2 ) 3 / 2 αa ˙ αi ˙ α ˙ • Global U ( N − 2) rotation: χ 4 = . . . = χ N = 0 • Exact solution found by perturbation theory � Φ (1) +Φ (2) +Φ (3) � � Φ (1) , ∇ n Φ (1) � + i A m = A (0) m + C mn ∇ n 16 C kl C kl � � � � α �� − C kl C kl α = λ ˙ (0) ˙ α +¯ (0) ˙ Ψ (1) + Ψ (2) σ m ˙ αα C α β ∇ m Φ (1) , Φ (1) , λ λ β β 32 • Poisson equations for prepotentials ∇ 2 Φ ( m ) = J ( m ) , ∇ 2 Ψ ( m ) = K ( m ) α α

Explicit Form of the Instanton • Undeformed solution b = − 2 iε ca ( A (0) x 2 + ρ 2 ( δ a β x b β + δ b β x a β ) c β ) ˙ ˙ β ˙ b = 8 iε ca ρ 2 ( F (0) ( x 2 + ρ 2 ) 2 ( δ a α δ b β + δ b α δ a β ) c β ) ˙ ˙ α ˙ ˙ ˙ ˙ • Prepotentials � � ρ 2 χ i χ i b = − 8 i 1 1 α + α + ρ 2 η α η α ) − ˙ ˙ (Φ (1) ) a δ b ( r 2 + ρ 2 ) 2 ξ ˙ α ξ r 2 + ρ 2 ( ζ ˙ α ζ r 2 + ρ 2 a 64 ρ 2 a ˙ a χ i χ i χ j 2 ξ ˙ 2 χ i ξ i ¯ i = − a = − j = (Φ (1) ) a (Φ (1) ) i (Φ (1) ) i ( r 2 + ρ 2 ) 3 / 2 ; ( r 2 + ρ 2 ) 3 / 2 ; r 2 + ρ 2 4 ρ 2 x m x n χ i χ i 1 b = − 2 iC mk (Φ (2) ) a ( ρ 2 + r 2 ) 3 ( σ kn ) b a ρ 2 � � α ˙ ζ ˙ α ζ ( r 2 + 2 ρ 2 ) − ρ 2 η α η α − η α x α ˙ α ˙ α ζ × ρ 2 � � χ i χ i 1 b ( xC ) a ζ a ( xC ) bα η α + ζ α η α + 2 i ( ρ 2 + r 2 ) 2 ρ 2 � � α ˙ χ i ( r 2 + 2 ρ 2 ) ζ ˙ α ζ a = − 8 α ˙ (Φ (2) ) i ( xC ) aα η α + η α η α ( xCx ) a α ζ ( r 2 + ρ 2 ) 5 / 2 ˙ ρ 2 � � α ˙ χ i i = − 8 ( r 2 + 2 ρ 2 ) ζ ˙ α ζ α ˙ (Φ (2) ) a α η α + η α η α ( xCx ) a ˙ ( xC ) a α ζ ( r 2 + ρ 2 ) 5 / 2 ρ 2 � � α ¯ 1 , 1 , 2 r 4 + 4 r 2 ρ 2 + ρ 4 η α η α ¯ α ¯ ζ ˙ χ i χ i Φ (3) = i C kl C kl ζ ˙ ( r 4 +6 r 2 ρ 2 +3 ρ 4 ) diag ρ 4 ( r 2 + ρ 2 ) 3 r 4 + 6 r 2 ρ 2 + 3 ρ 4 2

Metric on the Moduli Space • Motivation – measure on the moduli space – metric on MS and AdS/CFT – instanton MS in large N → bulk geometry – leading contribution: SU(2) instantons Dorey at ’96 • Problems with L 2 metric – no manifest gauge invariance – no conformal invariance • Information metric: � d 4 x ∂ A F ∂ B F G AB dZ A dZ B ≡ dZ A dZ B F Hitchin ’88 • Instanton density 1 F [ x, Z A ] = 16 π 2 Tr F ∧ F • AdS/CFT: bulk–to–boundary propagator: � ∆ Z F [ x, Z A ] = 0 , F [ x, Z A ] � ρ =0 = δ 4 ( x − X ) Balasubramanian et al ’98

Information Metric • Undeformed U (2) instanton – moduli space: ρ, X, ¯ ζ, η – instanton density and information metric 96 ρ 4 1 F = [( x − X ) 2 + ρ 2 ] 4 16 π 2 � dρ 2 � ρ 2 + dX 2 G AB dZ A dZ B = 128 ρ 2 5 Blau, Narain, Thompson ’01 • Instanton with C deformation – density is a function of ρ, X, ¯ ζ, η, χ, ¯ χ 1 1 0.4 0.5 0.5 0.2 0 0 0 -0.2 -0.5 -0.5 -0.4 -1 -1 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 – information metric � � � ρ 2 d � G AB dZ A dZ B = 128 6 ρ 6 C 2 S 1 − 96 ρ 2 C 2 S 2 1 + � ρ 2 5 7˜ 7˜ � � � + d � X 2 3 ρ 6 C 2 S 1 + 24 − C 2 13 ρ 2 C 2 S 2 ρ 7 T m d � 1 − ρ dX m � 14 � 7 � 14 � ρ 2 – determinant is C –independent

Chiral Ring & Gluino Condensate • Antichiral ring: [ Q, O ] = 0 . – ring property O ∼ O + [ Q, M } : � [ Q, M }O 1 . . . O n � = 0 – coordinate independence: ∂ O ∼ [ Q, [ Q, O}} – C –independence δ δL = { Q α , J α } δC αβ �O 1 . . . O n � = 0 – alternative ring [ D, O} = 0 is deformed Seiberg ’03 • No chiral ring since Q is not a symmetry • Gaugino condensate: perturbative and instanton corrections 4 4 4 λ λ λ Imaanpur ’03