Projective superspace Ariunzul Davgadorj Masaryk University, Czech - PowerPoint PPT Presentation

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Projective superspace Ariunzul Davgadorj Masaryk University, Czech Republic New Frontiers in String Theory 2018 August 2, 2018 based on works with Rikard von Unge Ariunzul Davgadorj Projective superspace 1

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary References If you are interested in Projective superspace check for example: 1 U.L, M.R; Commun.Math.Phys 128 (1990) 191 2 F.G-R, M.R, S.W, U.L, R.von U; arxiv: hep-th 9710250 3 F.G-R; arxiv: hep-th 9712128 4 A.D, R.von U; arxiv: hep-th 1706.07000 5 others... Ariunzul Davgadorj Projective superspace 2

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Outline Motivation for using Projective superspace 1 Review of Projective superspace 2 Super Yang-Mills theory in Projective superspace framework 3 Gauge connection and transformations Yang-Mills action in chiral representation Classic action from 1-loop divergent diagram Abelian limit Summary 4 Ariunzul Davgadorj Projective superspace 3

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Motivation Standard R 4 | 8 superspace is not sufficient in realizing off-shell N = 2 supersymmetric action. - A supersymmetric theory needs a given set of auxiliary fields to be described off-shell. - For N = 2 supersymmetry, it’s complicated to construct a multiplet with a finite number of auxiliaries. - Constraints to eliminate those auxiliary fields also put the physical fields on mass-shell. - Need some other relaxed constraints or an unconstrained description in some other space. - Perhaps an infinite number of auxilaries? Ariunzul Davgadorj Projective superspace 4

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Extend the conventional N = 2 susy group by its automorphism group SU (2) over U (1) subgroup and introduce: - Isotwistors on full C 2 \{ 0 } space - Harmonic space on sphere S 2 in spinor harmonic basis - Projective space over Riemann sphere C P 1 in terms of a holomorphic variable Ariunzul Davgadorj Projective superspace 5

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Review of Projective superspace N = 2 susy algebra: α , D j α , D j { D i β } = { D i β } = 0 , { D i β } = i δ i α , D j ˙ j ∂ α ˙ ˙ ˙ β ∂ ∂ ıθ ˙ α D i α = ∂θ i α + ˙ α , D 1 ≡ D D 2 ≡ Q i ∂ x α ˙ Introducing a bosonic coordinate on manifold CP 1 : ζ to parameterize the N = 2 Grassmann coordinates. Θ α = θ 2 α − ζθ 1 α , Θ ˙ α = θ 1 ˙ α + ζθ 2 ˙ α Ariunzul Davgadorj Projective superspace 6

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Define projective supercovariant derivatives that annihilate projective superfields Ω(Θ , Θ). ∇ α ≡ ( D α + ζ Q α )Ω = 0 ∇ ˙ α ≡ ( Q ˙ α − ζ D ˙ α )Ω = 0 Further define the second set of orthogonal derivatives. ∆ α ≡ ( Q α − 1 ∂ ζ D α ) ∼ ∂ Θ α α + 1 ∂ ∆ ˙ α ≡ ( D ˙ ζ Q ˙ α ) ∼ ∂ Θ ˙ α Ariunzul Davgadorj Projective superspace 7

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Projective supercovariant algebra follows: {∇ , ∇} = {∇ , ∇} = { ∆ , ∆ } = { ∆ , ∆ } = {∇ , ∆ } = 0 {∇ α , ∆ ˙ α } = −{ ∆ α , ∇ ˙ α } = 2˙ ı∂ α ˙ α {∇ α ( ζ 1 ) , ∇ ˙ α ( ζ 2 ) } = ˙ ı ( ζ 1 − ζ 2 ) ∂ α ˙ α A conjugation of an object is defined by applying Hermitian conjugation and an antipodal map onto the Riemann sphere. f ( ζ ) = ( − 1) p ζ p f ∗ ( − 1 ζ ) Ariunzul Davgadorj Projective superspace 8

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Grassmann integration is the same as differentiation. Using the ζ parameterized Grassmann coordinates the measure acting on a function of projective superfields Ω(Θ , Θ) will be constructed: 2 2 ∼ ∆ 2 ∆ 2 ∼ (1 ζ ∇ − 2 ζ D ) 2 (1 ∂ ∂ ζ ∇ + 2 D ) 2 f (Ω) ∼ D 2 D 2 f (Ω) ∂ Θ α ∂ Θ ˙ α � � � � � d ζ � � d 4 x d 4 θ d 4 xD 2 D 2 P = d ζ ζ The chiral measure will be: � � � � � d ζ � � d 4 x d 4 θ d 4 xD 2 Q 2 C = d ζ ζ Ariunzul Davgadorj Projective superspace 9

Motivation for using Projective superspace Review of Projective superspace Super Yang-Mills theory in Projective superspace framework Summary Projective superfields of particular interests for us are polar multiplets and the tropical multiplet ∞ � Υ n ζ n Υ = Arctic n =0 ∞ � Υ n ( − 1 ζ ) n Υ = Antarctic n =0 ∞ � υ n ζ n , υ − n = ( − 1) n υ n V ( ζ ) = Tropical −∞ A free action for the polar multiplets would be: � � � � d ζ � d 4 x d 4 θ ζ ΥΥ P Ariunzul Davgadorj Projective superspace 10

Motivation for using Projective superspace Gauge connection and transformations Review of Projective superspace Yang-Mills action in chiral representation Super Yang-Mills theory in Projective superspace framework Classic action from 1-loop divergent diagram Summary Abelian limit Gauge transformation and connection Under a gauge transformation polar multiplets transform: → e ˙ ı Λ Υ , Υ − → Υ e − ˙ ı Λ Υ − To make the action inv under the transformation introduce a real projective bridge V that converts Λ transformation to the Λ one. e V − ı Λ e V e − ˙ → e ˙ ı Λ Now using e V make all fields transform in arctic Λ or antarctic Λ representations. Υ A ≡ Υ e V − ı Λ � � � → � → e ˙ Υ e − ˙ ı Λ Υ A ≡ Υ − Υ , or Υ A ≡ e V Υ − ı Λ � � � → � → e ˙ Υ e − ˙ ı Λ Υ , Υ A ≡ Υ − Ariunzul Davgadorj Projective superspace 11

Motivation for using Projective superspace Gauge connection and transformations Review of Projective superspace Yang-Mills action in chiral representation Super Yang-Mills theory in Projective superspace framework Classic action from 1-loop divergent diagram Summary Abelian limit These newly defined polar multiplets are annihilated by ∇ and ∇ . Next, one can also define vector representation. Split: e V = e U e U e U − ı Λ , e U − → e ˙ ı K e U e − ˙ → e ˙ ı Λ e U e − ˙ ı K Now let the fields transform with the ζ -independent real field K . ı K � Υ vec ≡ Υ e U − � → e ˙ � → � Υ e − ˙ Υ vec ≡ e U Υ − ı K Υ , They are annihilated by: α ≡ e U ∇ α e − U = e − U ∇ α e U = ∇ α + Γ α ( ζ ) α e − U = e − U ∇ ˙ α e U = ∇ ˙ α ≡ e U ∇ ˙ α + Γ ˙ α ( ζ ) ˙ Ariunzul Davgadorj Projective superspace 12

Motivation for using Projective superspace Gauge connection and transformations Review of Projective superspace Yang-Mills action in chiral representation Super Yang-Mills theory in Projective superspace framework Classic action from 1-loop divergent diagram Summary Abelian limit One can identify Γ α ( ζ ) as a gauge connection. And notice that Γ α ( ζ ) = Γ 1 α + ζ Γ 2 α . Which means: D α = D α + Γ 1 α α = D α + ζ Q α with Q α = Q α + Γ 2 α D and Q are N = 2 gauge covariant derivatives. Their algebra is: { D α , Q β } = ˙ ı C αβ W , { D α , D ˙ α } = { Q α , Q ˙ α } = ˙ ı α ˙ α Projective gauge covariant derivatives satisfy: { α ( ζ 1 ) , β ( ζ 2 ) } = ˙ ı C αβ ( ζ 1 − ζ 2 ) W or equiv { α , [ ∂ ζ , β ] } = ˙ ı C αζ W W is a field strength in vector representation. Ariunzul Davgadorj Projective superspace 13

Motivation for using Projective superspace Gauge connection and transformations Review of Projective superspace Yang-Mills action in chiral representation Super Yang-Mills theory in Projective superspace framework Classic action from 1-loop divergent diagram Summary Abelian limit Find the field strength in arctic/antarctic representation. ı C αβ e − U W e U ≡ ˙ {∇ α , [ e − U ∂ ζ e U , ∇ β ] } = ˙ ı C αβ W ( ζ ) ı C αβ e U W e − U ≡ ˙ {∇ α , [ e U ∂ ζ e − U , ∇ β ] } = ˙ ı C αβ � W ( ζ ) From here we define the gauge covariant ζ -derivatives. D ζ = ∂ ζ + A ζ = e − U ∂ ζ e U , D ζ = ∂ ζ + � � A ζ = e U ∂ ζ e − U Field strengths are expressed by the ζ -connections: ı ∇ 2 � ı ∇ 2 A ζ , � W = − ˙ W = − ˙ A ζ , where e − V ( ∂ ζ e V ) = A ζ − e − V � A ζ e V Ariunzul Davgadorj Projective superspace 14