New multiplets in four dimensional N=2 conformal supergravity Subramanya Hegde PRD 97, 066026 (2018) (SH, Ivano Lodato, Bindusar Sahoo), JHEP 03 (2018) 154 (Daniel Butter, SH, Ivano Lodato, Bindusar Sahoo) 1/42 Introduction Introduction N extended conformal supergravity in four dimensions is a theory of gravity which is a representation of the N -fold extended superconformal algebra su(2; 2jN ). Due to the conformal symmetry, it contains off-shell representations with smaller number of components than the Poincare supergravity theory. On gauge fixing some of the conformal symmetries using compensator fields, one can obtain the corresponding Poincare supergravity theory. 2/42 Introduction Conformal gravity and gauge equivalence Conformal gravity as a gauge theory SU(2,2) conformal algebra contains the generators Pa;Mab; D; Ka. [D; Pa] = Pa; [D; Ka] = −Ka, [Pa;Kb] = ηabD − 2Mab, [Pa;Mbc] = 2P[bηc]a, [Ka;Mbc] = 2K[bηc]a, [Mab;Mcd] = 2η[a[cMb]d] a ab Introduce a gauge field corresponding to each generator: eµ, !µ , bµ, a fµ . 3/42 Introduction Conformal gravity and gauge equivalence Transformation rule for the above fields is obtained by using the structure constants of the conformal algebra. A A C B A δhµ = @µ" + " hµ fBC a a a 1 bc d a For eg, fd[bc] = 2ηa[bδc]. Therefore δM eν = 2 λ eνfd[bc] . Thus, a ab δM eν = −λ eνb. Conformal curvatures: A A C B A Rµν = 2@[µhν] + hν hµ fBC a a ab For eg: Rµν = 2@[µeν] + 2b[µeν] + 2![µ eν]b 4/42 Introduction Conformal gravity and gauge equivalence Demand translations to act as general coordinate transformations. a a ν a δP eµ = δcoveµ + ξ Rµν(P ) a ab ν Conventional constraints: Rµν(P ) = 0, R(M)µν eb = 0. ab a !µ and fµ : Dependent gauge fields. ab ab [a b]ν !(e; b)µ = !(e)µ − 2eµ e bν 1 1 f a = R(e; b)a − R(e; b)ea µ 2 µ 12 µ a Number of independent field components : eµ(16); bµ(4) Number of gauge transformation parameters for su(2; 2) : 15. Therefore off-shell d.o.f = 20 − 15 = 5. But for Poincare gravity, we need 6 off-shell d.o.f! 5/42 Introduction Conformal gravity and gauge equivalence Consider µ L = −eφD Dµφ φ has Weyl weight +1. Dµφ = @µφ − bµφ, a a ab a DµD φ = (@µ − 2bµ)D φ − !µ Dbφ + fµ φ p p Gauge fixing: bµ = 0, φ = 6=( 2κ) 1 L = −e R 2κ2 6/42 Introduction Conformal gravity and gauge equivalence Outline 1 Introduction Conformal gravity and gauge equivalence N extended conformal supergravity Matter multiplets and multiplet calculus 2 Real Scalar multiplet 24+24 matter multiplet Restricted real scalar multiplet = tensor multiplet 3 Construction of the action for the real scalar multiplet 4 Construction of the real scalar multiplet dilaton Weyl multiplet in five dimensions Supercurrent multiplet for the tensor multiplet and linearized tranformation rules 5 Dilaton Weyl multiplet in four dimensions 6 Conclusions and future work 7/42 Introduction Conformal gravity and gauge equivalence N extended superconformal algebra in four dimensions Contains Q and S supercharges. i i i 1 i [Ka;Q ] = γaS , [Pa;S ] = 2 γaQ , i a i fQ ; Q¯jg = −(I − γ5)γ Paδ j It contains an R symmetry algebra SU(N)R × U(1)R. i ¯ 1 ab i fQ ; Sjg = 2 (I − γ5)(2σ Mab + D − iA − 2V j) Fermionic charges are Majorana Fermions. Hence, the R symmetry algebra is chiral. i 1 i i 1 i [D; Q ] = 2 Q , [D; S ] = 2 S ,... 8/42 Introduction N extended conformal supergravity Superconformal gauge theory Begin with a superconformal gauge theory: Associate a gauge field to a ab a each charge from the superconformal algebra: eµ , !µ , bµ, fµ , i i i a ab a i i i Vµj, Aµ, µ , φµ with P , M , D, K , V j, A, Q , S respectively. Transformation rule for the above fields is obtained by using the structure constants of the superconformal algebra. A A C B A δhµ = @µ" + " hµ fBC A A C B A Conformal curvatures: Rµν = 2@[µhν] + hν hµ fBC In this theory, the superconformal transformations act as internal symmetries. 9/42 Introduction N extended conformal supergravity To realize this gauge theory as a theory of supergravity, impose i constraints on conformal curvatures. Add matter fields (D; Tab; χ ) such that the bosonic and fermionic degrees of freedom match. Conventional constraints: a Rµν(P ) = 0; 1 γµ(R^ (Q)i + γ χi) = 0; µν 2 µν ~ 1 3 eνR^ (M) b − iR^ (A) + T +T −b − De = 0 b µν a µa 4 ab µ 2 µa These constraints make some of the gauge fields to be dependent fields. 10/42 Introduction N extended conformal supergravity Weyl Multiplet Multiplet of fields obtained in this manner is known as the Weyl multiplet. This is the minimal multiplet containing the gauge fields of the superconformal algebra. Independent Bosonic fields: a i ij eµ(16); bµ(4);Aµ(4); Vµj(12);Tab(6);D(1). Number of bosonic gauge a a parameters: Mab(6);P (4);K (4);D(1);V (3);A(1). Off-shell bosonic d.o.f=24. i i Independent Fermionic fields: µ(32); χ (8) Number of fermionic gauge parameters: Qi(8);Si(8) Off-shell fermionic degrees of freedom = 24. 11/42 Introduction N extended conformal supergravity Algebra on the fields: (cov) [δQ(1); δQ(2)] = δ (ξ) + δM (") + δK (ΛK ) + δS(η) + δgauge i ab i [δS(η); δQ()] = δM (2¯η σ i + h.c) + δD(¯ηi + h.c) i i + δA(iη¯i + h.c) + δV (−2¯η j − (h.c;traceless)) a i [δS(η1); δS(η2)] = δK (¯η2iγ η1 + h.c) (cov) P µ where δ (ξ) = δgct(ξ) + T δT (−ξ hµ(T )). The field dependent transformation parameters are given by µ i µ ξ = 2 ¯2 γ 1i + h.c ab i j ab " = ¯12Tij + h.c 3 Λa = ¯i j D T ba − ¯i γa D + h.c K 1 2 b ij 2 2 1i i i j η = 6¯[12]χj 12/42 Introduction Matter multiplets and multiplet calculus Matter multiplets There is also an 8+8 tensor multiplet, on which the above algebra is realized, with field content G(A complex scalar), φi(SU(2) doublet of chiral fermions), Eµν(A two form gauge field) and Lij(SU(2) triplet kl ij ∗ of scalars with 'reality' condition Lij = ikjlL , (L ) = Lij). There are other 8+8 multiplets in N = 2 conformal supergravity, such as the vector multiplet, non-linear multiplet etc. The above matter multiplets can be used as compensator multiplets to obtain the physical Poincare supergravity. 13/42 Introduction Matter multiplets and multiplet calculus Story of the chiral multiplet: multiplet calculus There is a 16+16 components chiral multiplet which reduces to the 8+8 restricted chiral multiplet when a consistent set of 8+8 constraints are imposed. There is a chiral weight 0 complex triplet of scalars Bij, which is kl constrained to satisfy a 'reality' condition. Bij = "ik"jlB . Other contraints can be obtained by supersymmetric variation of this constraint. The restricted chiral multiplet is equivalent to tensor (vector) multiplet. i.e. gauge invariant quantities of these multiplets can be embedded in the restricted chiral multiplet. 14/42 Introduction Matter multiplets and multiplet calculus This was used to write a superconformal action for the (improved) tensor multiplet, vector multiplet and the Weyl multiplet. Obtaining an action for one multiplet through an action for another multiplet is known as multiplet calculus. This allowed for constructions of minimal Poincare supergravity theories as well as construction of supersymmetric higher derivative actions. The study of all the off-shell representations of the superconformal algebra and the corresponding actions is interesting, in this context. 15/42 Real Scalar multiplet Table of Contents 1 Introduction Conformal gravity and gauge equivalence N extended conformal supergravity Matter multiplets and multiplet calculus 2 Real Scalar multiplet 24+24 matter multiplet Restricted real scalar multiplet = tensor multiplet 3 Construction of the action for the real scalar multiplet 4 Construction of the real scalar multiplet dilaton Weyl multiplet in five dimensions Supercurrent multiplet for the tensor multiplet and linearized tranformation rules 5 Dilaton Weyl multiplet in four dimensions 6 Conclusions and future work 16/42 Real Scalar multiplet 24+24 matter multiplet 24+24 matter multiplet: Real scalar multiplet Field content of the multiplet is given by Table: Field content of the 24+24 multiplet Field SU(2) Irreps Weyl Chiral Chirality weight weight (w) (c) φ 1 1 0 { i Sa j 3 1 0 { Eij 3 1 -1 { Cijkl 5 2 0 { Λi 2 1/2 +1/2 +1 Ξijk 4 3/2 -1/2 +1 It is the generalization of the flat space 24+24 multiplet constructed by Howe et al1 to include coupling to conformal supergravity. 1P S Howe, K S Stelle, P K Townsend, Nucl. Phys. B 1983 17/42 Real Scalar multiplet 24+24 matter multiplet Field redefinitions to simplify the transformation rule The Q-transformation of the above multiplet is highly non-linear in the fields, although the field components are S-invariant except Λi which transforms as δSΛi = −2ηi. Q transformations are simplified by redefinition of the fields. Field content of the redefined multiplet is as follows. Table: Field content of the redefined 24+24 multiplet Field SU(2) Irreps Weyl Chiral Chirality weight weight (w) (c) V 1 -2 0 { i Aa j 3 -1 0 { Kij 3 -1 -1 { Cijkl 5 0 0 { i 2 -3/2 +1/2 +1 ξijk 4 3/2 -1/2 +1 18/42 Real Scalar multiplet 24+24 matter multiplet k δV = ¯k + h:c: ; i i i j ij i δ = D=V − A= j − 2K j − 2V η ; 2 1 1 δKij = 2V ¯(iχj) − ¯(iD= j) + ¯kξ "il"jm + ¯(iγ · T − "j)l 3 3 lmk 12 l − 2¯η(i j) ; 2 1 δA i = V ¯ γ χi + ¯ γ D= i − 2¯ D i − "li"nk¯ γ ξ a j j a 3 j a j a 3 n a ljk 1 + ¯ γ γ · T − "ik − η¯ γ i − (h:c; traceless) ; 24 j a k j a 19/42 Real Scalar multiplet 24+24 matter multiplet 3 δξ = D Aal " " m − 3D A l " " γabm ijk 2 a (i jjmj k)l a b (i jjmj k)l 3 − V γ · R(V )l " " m + 6D=K lm " " − C l 2 (i jjmj k)l (i jjlj k)m ijkl 3 3 − γ · T −nK " − R¯(Q)l γabm" " 4 (ij k)n 2 ab (i jjlj k)m 3 3 − χ¯l m" " + χ¯lγ γabm" " 2 (i jjlj k)m 2 ab (i jjlj k)m 3 + χ¯ lm" " + 3¯χmγ lγa " " 2 (i jjlj k)m a (i jjlj k)m nm m n − 6K η(i"jjnj"k)m − 6A= (iη "jjmj"k)n ; ~ (m ~npq) δCijkl = ¯(iΓjkl) + "im"jn"kp"lq¯ Γ − 4¯η(iξjkl) ; where, ~ lm Γijk = −2D=ξijk + 12χ(iK "jjlj"k)m : 20/42 Real Scalar multiplet Restricted real scalar multiplet = tensor multiplet 8+8 restricted real scalar multiplet The multiplet is a 24+24 multiplet rather than the more common 8+8.