Gravity from BRST squared copy BRST Double-copy in a non-flat - PowerPoint PPT Presentation

Gravity from BRST squared Silvia Nagy Set-up Problems with the double Gravity from BRST squared copy BRST Double-copy in a non-flat Silvia Nagy background University of Nottingham based on work in collaboration with L. Borsten, I. Jubb, and V. Makwana December 13, 2019

Gravity from BRST squared Table of Contents Silvia Nagy Set-up Problems with the double copy 1 Set-up BRST Double-copy in a non-flat background 2 Problems with the double copy 3 BRST 4 Double-copy in a non-flat background

Gravity from BRST squared How general is the double copy? Silvia Nagy Set-up Problems with the double copy BRST • Can we construct the full space of the gravity theory Double-copy without resorting to special gauge choices and coordinate in a non-flat background choices ? • We are seeking a dictionary that continues to hold when we perfom gauge/coordinate transformations. • Can we construct a double copy dictionary in set-ups where we don’t have guidance from amplitudes ?

Gravity from BRST squared Minimal YM 2 Silvia Nagy Set-up Problems with • Schematically: the double copy BRST A µ ∗ ˜ A ν ≡ h µν + B µν + η µν φ Double-copy in a non-flat background • Want to construct the full theory with arbitrary boundary conditions of graviton, dilaton and two-form from the double copy. • If this is achievable, it should be possible to extract (pure) gravitational solutions by requiring B µν = φ = 0 • Start by constructing Lorenz-covariant dictionaries for all the fields compatible with symmetries and equations of motion.

Gravity from BRST squared Local symmetries Silvia Nagy Set-up Problems with the double • We tensor left and right off-shell linearised gauge fields copy with arbitrary non-Abelian gauge groups G L and G R . BRST • At linear level, want to reproduce Double-copy in a non-flat background graviton: δ h µν = ∂ µ ξ ν + ∂ ν ξ µ two-form: δ B µν = ∂ µ Λ ν − ∂ ν Λ µ dilaton: δϕ = 0 from the (linearised) YM gauge field µ = ∂ µ α i + f i δ A i jk A j µ θ k

Gravity from BRST squared Local symmetries Silvia Nagy • We define [Anastasiou, Borsten, Duff, Hughes, SN 2014] : Set-up Problems with Z µν ( x ) = [ A µ i ⋆ Φ − 1 ii ′ ⋆ ˜ A ν i ′ ]( x ) the double copy BRST where Φ ii ′ is the “spectator” bi-adjoint scalar field Double-copy introduced by [Hodges 2013] and [Cachazo 2014] in a non-flat background • The convolution is defined as � d 4 yf ( y ) g ( x − y ) . [ f ⋆ g ]( x ) = and is a consequence of the momentum-space origin of squaring: product in momentum space is convolution in position space! • Importantly, is doesn’t obey the Leibnitz rule: ∂ µ ( f ⋆ g ) = ( ∂ µ f ) ⋆ g = f ⋆ ( ∂ µ g )

Gravity from BRST squared Field dictionary [Cardoso,Inverso,SN,Nampuri’18] Silvia Nagy Set-up Problems with the double copy Write down most general dictionary for the product of two YM BRST fields: Double-copy in a non-flat background A ρ − 1 � � h µν = A µ ◦ ˜ A ν + A ν ◦ ˜ A ρ ◦ ˜ � ( ∂ · A ) ◦ ( ∂ · ˜ A µ + q η µν A ) B µν = A µ ◦ ˜ A ν − A ν ◦ ˜ A µ A ρ − 1 φ = A ρ ◦ ˜ � ( ∂ · A ) ◦ ( ∂ · ˜ A )

Gravity from BRST squared Field dictionary [Cardoso,Inverso,SN,Nampuri’18] Silvia Nagy Set-up Problems with the double copy • Reproduces the correct local transformations at linear BRST level: Double-copy δ h µν = ∂ µ ξ ν + ∂ ν ξ µ in a non-flat background δ B µν = ∂ µ Λ ν − ∂ ν Λ µ δφ = 0 • Allows us to read off parameter dictionaries: ξ µ = α ◦ ˜ A µ + A µ ◦ ˜ α, Λ µ = α ◦ ˜ A µ − A µ ◦ ˜ α,

Gravity from BRST squared Table of Contents Silvia Nagy Set-up Problems with the double copy 1 Set-up BRST Double-copy in a non-flat background 2 Problems with the double copy 3 BRST 4 Double-copy in a non-flat background

Gravity from BRST squared Problem 1: graviton - dilaton Silvia Nagy Couple arbitrary external sources to all equations: Set-up • Yang-Mills Problems with the double copy ∂ µ F i µν = j i ∂ µ ( ∗ F i ∂ µ j i ν , µν ) = 0 , µ = 0 BRST Double-copy in a non-flat • gravity background R µν − 1 2 η µν R = j ( h ) µν ∂ ρ H ρµν = j ( B ) µν � φ = j ( φ ) • spectator � Φ ii ′ = j (Φ) ii ′ Sources have a dual role : they ensure proper fall-off for the fields to be convoluted and they will illuminate a fundamental issue of the double copy.

Gravity from BRST squared Problem 1: graviton - dilaton Silvia Nagy Set-up Problems with the double copy BRST Making use of the field dictionaries and the eom, we can read Double-copy off the surce dictionaries: in a non-flat background � η µν + ∂ µ ∂ ν � j ( h ) µν = − j ( µ ◦ ˜ j ρ ◦ ˜ j ρ j ν ) + ( q + 1) � j ( B ) µν = 2 j [ µ ◦ ˜ j ν ] j ( φ ) = j ρ ◦ ˜ j ρ

Gravity from BRST squared Problem 1: graviton - dilaton Silvia Nagy Set-up Problems with the double copy � η µν + ∂ µ ∂ ν � j ( h ) µν = − j ( µ ◦ ˜ j ρ ◦ ˜ j ρ j ν ) + ( q + 1) BRST � Double-copy j ( B ) µν = 2 j [ µ ◦ ˜ in a non-flat j ν ] background j ( φ ) = j ρ ◦ ˜ j ρ Note that out theory is constrained: j φ ∝ − T ( h ) ρ ρ Obstruction to obtaining pure gravity - even for the most general dictionary !!

Gravity from BRST squared Related issue: d.o.f. counting Silvia Nagy Set-up • On-shell, the counting is 2 ∗ 2 = 4: Problems with the double A + ⊗ ˜ A + = g ++ A − ⊗ ˜ A − = g −− copy BRST A + ⊗ ˜ A − = φ A − ⊗ ˜ A + = B Double-copy in a non-flat background • Looking at the off-shell counting, 3 ∗ 3 � = 10: [ A µ ] = [˜ A µ ] = 4 − 1 = 3 ( A µ → A µ + ∂ µ α ) [ h µν ] = 10 − 4 = 6 ( h µν → h µν + ∂ ( µ ξ ν ) ) [ B µν ] = 6 − (4 − 1) = 3 ( B µν → B µν + ∂ [ µ Λ ν ] , Λ µ → Λ µ + ∂ µ Λ) [ φ ] = 1 • Similar issues for double copy of SUSY multiplets.

Gravity from BRST squared Problem 2: gauge mapping Silvia Nagy Set-up Problems with the double copy BRST • The double-copy is usually formulated with some specific Double-copy in a non-flat gauge fixing on both the YM and the gravity side. background • There is no general procedure determining a mapping between these corresponding gauge choices. • This can lead to issues, particularly when studying off-shell or gauge-dependent objects [Plefka,Shi,Steinhoff,Wang’19] .

Gravity from BRST squared Table of Contents Silvia Nagy Set-up Problems with the double copy 1 Set-up BRST Double-copy in a non-flat background 2 Problems with the double copy 3 BRST 4 Double-copy in a non-flat background

Gravity from BRST squared BRST Silvia Nagy Set-up • Introduced to avoid issues caused by gauge symmetry in Problems with the double path integrals. copy BRST • Schematically Double-copy in a non-flat � background � � � d D x G [ f ] − ξ S BRST = L 0 [ f ] + b − ¯ cQ ( G [ f ]) 2 b − fj ( f ) +¯ � jc + ¯ cj , where L 0 [ f ] is the classical action for the field f , G [ f ] is the gauge-fixing functional and b is the Lautrup-Nakanishi Lagrange multiplier field. • Note that, unlike in the standard treatment, we have coupled sources to the ghost and anti-ghost .

Gravity from BRST squared BRST-symmetries Silvia Nagy Set-up As before, we require that the BRST symmetries of YM system: Problems with the double c = 1 copy QA µ = ∂ µ c , Qc = 0 , Q ¯ ξ G ( A ) BRST Double-copy induce the correct symmetries for the gravitational fields: in a non-flat background 1 Qh µν = 2 ∂ ( µ c ν ) , Qc µ = 0 , Q ¯ c µ = ξ ( h ) G µ [ h , ϕ ] , Q ¯ 1 QB µν = 2 ∂ [ µ d ν ] , Qd µ = ∂ µ d , d µ = ξ ( B ) G µ [ B , η ] , Q ϕ = 0 . Let us now make a choice of gauge fixing functional on the YM side, and set G [˜ A ] ≡ ∂ µ ˜ G [ A ] ≡ ∂ µ A µ , A µ .

Gravity from BRST squared BRST dictionary [Anastasiou,Borsten,Duff,SN,Zoccali’18] Silvia Nagy Set-up Problems with the double The most general dictionary in the absence of � − 1 terms and copy compatible with symmetries is: BRST Double-copy in a non-flat A ρ ◦ ˜ A ρ + ξ c α ◦ ˜ h µν = A µ ◦ ˜ A ν + A ν ◦ ˜ � � A µ + a η µν c α , background B µν = A µ ◦ ˜ A ν − A ν ◦ ˜ A µ , ϕ = A ρ ◦ ˜ A ρ + ξ c α ◦ ˜ c α . where we have introduced the OSp(2) ghost singlet c α ◦ ˜ c α = c ◦ ˜ c ◦ ˜ ¯ c − ¯ ¯ c .

Gravity from BRST squared Source issue - problem 1 Silvia Nagy Set-up Problems with the double copy • On the YM side, the e.o.m. are BRST � A µ − ξ +1 Double-copy ξ ∂ µ ( ∂ A ) = j µ in a non-flat background � c = j ( c ) , c = j (¯ c ) � ¯ • On the gravity side, we have ξ ( h ) +2 � � 2 ∂ ρ ∂ ( µ h ν ) ρ − ∂ µ ∂ ν h = j µν � h µν − ξ ( h ) � ϕ = j ( ϕ )