School of Physics, University of Western Australia Second ANZAMP Annual Meeting Mooloolaba, November 27 29, 2013 Based on: SMK, arxiv:1212.6179
Background and motivation Exact results (partition functions, Wilson loops etc.) in rigid supersymmetric field theories on curved backgrounds (e.g., S 3, S 4, S 3 S 1 etc.) using localization techniques V. Pestun (2007, 2009) A. Kapustin, B. Willett & I. Yaakov (2010) D. Jafferis (2010) Necessary technical ingredients: Curved space M should admit some unbroken rigid supersymmetry (supersymmetric background); Rigid supersymmetric field theory on M should be off-shell. These developments have inspired much interest in the construction and classification of supersymmetric backgrounds that correspond to off-shell supergravity formulations.
Classification of supersymmetric backgrounds in off-shell supergravity Component approaches G. Festuccia and N. Seiberg (2011) B. Jia and E. Sharpe (2011) H. Samtleben and D. Tsimpis (2012) C. Klare, A. Tomasiello and A. Zaffaroni (2012) T. Dumitrescu, G. Festuccia and N. Seiberg (2012) D. Cassani, C. Klare, D. Martelli, A. Tomasiello and A. Zaffaroni (2012) T. Dumitrescu and G. Festuccia (2012) A. Kehagias and J. Russo (2012) Formalism to determine (conformal) isometries of curved superspaces I. Buchbinder & SMK, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace, IOP, Bristol, 1995 (1998)
Superspace formalism is universal, for it is geometric and may be generalized to supersymmetric backgrounds associated with any supergravity theory formulated in superspace. Applications of the formalism: Rigid supersymmetric field theories in 5D N = 1 AdS superspace SMK & G. Tartaglino-Mazzucchelli (2007) Rigid supersymmetric field theories in 4D N = 2 AdS superspace SMK & G. Tartaglino-Mazzucchelli (2008) D. Butter & SMK (2011) D. Butter, SMK, U. Lindström & G. Tartaglino-Mazzucchelli (2012) Rigid supersymmetric field theories in 3D (p, q) AdS superspaces SMK, & G. Tartaglino-Mazzucchelli (2012) SMK, U. Lindström & G. Tartaglino-Mazzucchelli (2012) D. Butter, SMK & G. Tartaglino-Mazzucchelli (2012)
Three formulations for gravity Metric formulation Gauge field: metric g mn (x) Gauge transformation: δg mn = m ξ n + n ξ m with ξ = ξ m (x) m a vector field generating an infinitesimal diffeomorphism. Vielbein formulation Gauge field: vielbein e a m (x), e := det(e a m ) 0 The metric is a composite field g mn = e a m e b n η ab Gauge transformation: δ a = [ξ b b + 1 2 K bc M bc, a ] Gauge parameters: ξ a = ξ m e a m (x) and K ab (x) = K ba (x) Covariant derivatives (M bc the Lorentz generators) a = e a m m + 1 2 ω a bc M bc, [ a, b ] = 1 2 R ab cd M cd e a m the inverse vielbein, e a m e m b = δ a b ; ω a bc [e] the Lorentz connection
Three formulations for gravity Weyl transformations The torsion-free constraint T ab c = 0 [ a, b ] T ab c c + 1 2 R ab cd M cd = 1 2 R ab cd M cd is invariant under Weyl (local scale) transformations a a = e σ( a + ( b σ)m ba ), with the parameter σ(x) being completely arbitrary. e a m e σ e a m, e m a e σ e m a, g mn e 2σ g mn
Three formulations for gravity Conformal formulation Gauge fields: vielbein e a m (x), e := det(e a m ) 0 & conformal compensator ϕ(x), ϕ 0 Gauge transformations (K := ξ b b + 1 2 K bc M bc ) δ a = [ξ b b + 1 2 K bc M bc, a ] + σ a + ( b σ)m ba (δ K + δ σ ) a, δϕ = ξ b b ϕ + σϕ (δ K + δ σ )ϕ Gauge-invariant gravity action (in four spacetime dimensions) S = 1 2 d 4 x e ( a ϕ a ϕ + 1 6 Rϕ2 + λϕ 4) Weyl gauge condition: ϕ = 6/κ = const. Action turns into the Einstein-Hilbert action with a cosmological term. S = 1 2κ 2 d 4 x e R Λ κ 2 d 4 x e
Conformal isometries Conformal Killing vector fields A vector field ξ = ξ m m = ξ a e a, with e a := e a m m, is conformal Killing if there exist local Lorentz, K bc [ξ], and Weyl, σ[ξ], parameters such that [ξ b b + 1 2 K bc [ξ]m bc, a ] + σ[ξ] a + ( b σ[ξ])m ba = 0 A short calculation gives K bc [ξ] = 1 2 Conformal Killing equation ( b ξ c c ξ b), σ[ξ] = 1 4 bξ b a ξ b + b ξ a = 2η ab σ[ξ] Equivalent spinor form ( a α α and ξ a ξ α α ) (α ( α ξ β) β) = 0
Conformal isometries Lie algebra of conformal Killing vector fields Conformally related spacetimes ( a, ϕ) and ( a, ϕ) a = e ρ( a + ( b ρ)m ba ), ϕ = e ρ ϕ have the same conformal Killing vector fields ξ = ξ a e a = ξ a ẽ a. The parameters K cd [ ξ] and σ[ ξ] are related to K cd [ξ] and σ[ξ] as follows: Conformal field theories K[ ξ] := ξ b b + 1 2 K cd [ ξ]m cd = K[ξ], σ[ ξ] = σ[ξ] ξρ
Isometries Killing vector fields A vector field ξ = ξ m m = ξ a e a, with e a := e a m m, is Killing if there exist local Lorentz K bc [ξ] and Weyl σ[ξ] parameters such that [ξ b b + 1 2 K bc [ξ]m bc, a ] + σ[ξ] a + ( b σ[ξ])m ba = 0, ξϕ + σ[ξ]ϕ = 0 These Killing equations are Weyl invariant in the following sense: Given a conformally related spacetime ( a, ϕ) a = e ρ( a + ( b ρ)m ba ), ϕ = e ρ ϕ, the above Killing equations have the same functional form when rewritten in terms of ( a, ϕ), in particular ξ ϕ + σ[ ξ] ϕ = 0
Isometries Because of Weyl invariance, we can work with a conformally related spacetime such that ϕ = 1 Then the Killing equations turn into [ξ b b + 1 ] 2 K bc [ξ]m bc, a = 0, σ[ξ] = 0 Standard Killing equation: a ξ b + b ξ a = 0 Lie algebra of Killing vector fields Rigid symmetric field theories in curved space
The Wess-Zumino superspace geometry 4D N = 1 curved superspace M 4 4 parametrized by z M = (x m, θ µ, θ µ ). The superspace geometry is described by covariant derivatives of the form D A = (D a, D α, D α ) = E A + Ω A. E A is the inverse vielbein, E A = E A M M Ω A is the Lorentz connection, Ω A = 1 2 Ω A bc M bc = Ω A βγ M βγ + Ω A β γ M β γ, with M bc (M βγ, M β γ ) the Lorentz generators. Supergravity gauge transformation (U matter tensor superfield) δ K D A = [K, D A ], δ K U = KU The gauge parameter K is K = ξ B D B + 1 2 K bc M bc
The Wess-Zumino superspace geometry To describe supergravity, the superspace torsion tensor must obey nontrivial constraints that generalize the torsion-free condition, T ab c = 0, in ordinary gravity. The covariant derivatives obey the following anti-commutation relations: {D α, D α } = 2iD α α, {D α, D β } = 4 RM αβ, { D α, D β} = 4R M α β, [ D γ α, D β β] = iε α β (R D β + G β D γ ( D γ G δ β ) M ) γ δ + 2W γδ β M γδ i(d β R) M α β, Torsion tensors R, G a = Ḡa and W αβγ = W (αβγ) satisfy the Bianchi identities D α R = 0, D α W αβγ = 0, D γ G α γ = D α R, D γ W αβγ = i D (α γ G β) γ.
The Wess-Zumino superspace geometry Super-Weyl transformations δ σ D α = ( 1 2 σ σ)d α (D β σ) M αβ, δ σ D α = ( 1 2 σ σ) D α ( D β σ) M α β, δ σ D α α = 1 2 (σ + σ)d α α i 2 ( D α σ)d α i 2 (D ασ) D α (D β β ασ)m αβ (D α σ) M α β, with the scalar parameter σ being covariantly chiral, D α σ = 0
Conformal isometries A supervector field ξ = ξ B E B on M 4 4 is called conformal Killing if there exists a bivector K bc = K cb and a covariantly chiral scalar σ such that Implications (I): (δ K + δ σ )D A = 0. δ α β ξ β = i 4 D αξ β β ξ α = i 8 D αξ α α, K αβ = D (α ξ β) i 2 ξ β (α G β) β, σ = 1 3 (Dα ξ α + 2 D α ξ α iξ a G a ), Implications (II): D α ξ α = i 2 ξ α α R, D α K βγ = iξ α α W αβγ, D α K βγ = δ α (β D γ) σ 4δ α (β ξ γ) R + i 2 δ α (β ξ γ) γ D γ R + i 2 ξ α αd (β G γ) α.
The real vector ξ a is the only independent parameter. It obeys the equation D (α ξ β) β = 0 ( ) which in fact contains all information about the conformal Killing supervector field. In particular, it implies the ordinary conformal Killing equation D a ξ b + D b ξ a = 1 2 η abd c ξ c Alternative definition of the conformal Killing supervector field: It is a real supervector field ξ = ξ A E A, ξ A = (ξ a, i 8 D β ξ α β, i ) 8 Dβ ξ β α obeying the equation ( ).
Isometries Each off-shell formulation for N = 1 supergravity can be realized as a super-weyl invariant coupling of the old minimal supergravity, discussed above, to a scalar compensator Ψ and its conjugate Ψ (if the compensator is complex) with a super-weyl transformation of the form δ σ Ψ = (pσ + q σ)ψ, where p and q are fixed parameters which are determined by the off-shell structure of Ψ. The compensator is assumed to be nowhere vanishing. Killing supervector field A supervector field ξ = ξ B E B on M 4 4 is called Killing if there exists a bivector K bc = K cb and a covariantly chiral scalar σ such that (δ K + δ σ )D A = 0, (δ K + δ σ )Ψ = 0
Supersymmetric backgrounds Look for those curved backgrounds which admit some unbroken (conformal) supersymmetries. By definition, such a superspace possesses a (conformal) Killing supervector field ξ A with the property ξ a = 0, ɛ α := ξ α 0, where U denotes the θ, θ independent part of a tensor superfield U(z) = U(x m, θ µ, θ µ ), U := U θ µ = θ µ =0 By definition, supersymmetric backgrounds are defined to have no covariant fermionic fields D α R = 0, D α G β β = 0, W αβγ = 0. As a result, the gravitino can be gauged away. The remaining supergravity fields are e a m and R = 1 3 M, G a = 2 3 b a.
Conformal supersymmetry 2 a ɛ β i } {(σ a ζ)β + (σ a ɛ) β M 2(σ ac ɛ) β b c + 2b a ɛ β = 0 3 ɛ β & ɛ β Q-supersymmetry parameters ζ β & ζ β S-supersymmetry parameters (ζ β := D β σ ). Introducing a gauge-covariant derivative D a ɛ β := ( a i 2 b a)ɛ β and expressing ζ β & ζ β in terms of ɛ β & ɛ β gives D a ɛ β + 1 4 (σ a σ c D c ɛ) β = 0 D α γ ɛ β + D β γ ɛ α = 0 ɛ β is a charged conformal Killing spinor Given a commuting conformal Killing spinor ɛ β, the null vector V β β := ɛ β ɛ β is a conformal Killing vector field (α ( α V β) β) = 0
Rigid supersymmetry In the non-conformal case, ζ α = 0 and the equation for unbroken rigid supersymmetry 2 a ɛ β i } {(σ a ɛ) β M 2(σ ac ɛ) β b c + 2b a ɛ β = 0 3 Nontrivial consistency conditions on the backgrounds fields M & b c
Rigid supersymmetry Spacetimes admitting four supercharges a M = 0, M b c = 0, a b c = 0, C abcd = 0, with C abcd the Weyl tensor. The Riemann tensor proves to be R abcd = 1 { } b c (b a η bd b b η ad ) b d (b a η bc b b η ac ) b 2 (η ac η bd η ad η bc ) 9 This gives the Ricci tensor 1 9 M M(η ac η bd η ad η bc ) R ab = 2 9 (b ab b b 2 η ab ) 1 3 M Mη ab G. Festuccia and N. Seiberg (2011)