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, 2|N ).

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, ωµ , 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 = ∂µε + ε 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] + 2ω[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 ωµ and fµ : Dependent gauge fields.

ab ab [a b]ν ω(e, b)µ = ω(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 − ωµ Dbφ + fµ φ √ √ 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 {Q , Q¯j} = −(I − γ5)γ Paδ j

It contains an R symmetry algebra SU(N)R × U(1)R. i ¯ 1 ab i {Q , Sj} = 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 , ωµ , bµ, fµ , i i i a ab a i i i Vµj, Aµ, ψµ , 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 = ∂µε + ε 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(Rˆ (Q)i + γ χi) = 0, µν 2 µν ˜ 1 3 eνRˆ (M) b − iRˆ (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: ψµ(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 (ε) + 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 ε = ¯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 = εikεjlB . 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 – ψi 2 -3/2 +1/2 +1 ξijk 4 3/2 -1/2 +1

18/42 Real Scalar multiplet 24+24 matter multiplet

k δV = ¯kψ + h.c. , i i i j ij i δψ = D/V − A/ j − 2K j − 2V η , 2 1 1 δKij = 2V ¯(iχj) − ¯(iD/ψj) + ¯kξ εilεjm + ¯(iγ · T −ψ εj)l 3 3 lmk 12 l − 2¯η(iψj) , 2 1 δA i = V ¯ γ χi + ¯ γ D/ψi − 2¯ D ψi − εliεnk¯ γ ξ a j j a 3 j a j a 3 n a ljk 1 + ¯ γ γ · T −ψ εik − η¯ γ ψi − (h.c; traceless) , 24 j a k j a

19/42 Real Scalar multiplet 24+24 matter multiplet

3 δξ = D Aal ε ε m − 3D A l ε ε γabm ijk 2 a (i j|m| k)l a b (i j|m| k)l 3 − V γ · R(V )l ε ε m + 6D/K lm ε ε − C l 2 (i j|m| k)l (i j|l| k)m ijkl 3 3 − γ · T −nK ε − R¯(Q)l ψ γabmε ε 4 (ij k)n 2 ab (i j|l| k)m 3 3 − χ¯lψ mε ε + χ¯lγ ψ γabmε ε 2 (i j|l| k)m 2 ab (i j|l| k)m 3 + χ¯ ψlmε ε + 3¯χmγ ψlγa ε ε 2 (i j|l| k)m a (i j|l| k)m nm m n − 6K η(iεj|n|εk)m − 6A/ (iη εj|m|εk)n , ˜ (m ˜npq) δCijkl = ¯(iΓjkl) + εimεjnεkpεlq¯ Γ − 4¯η(iξjkl) , where, ˜ lm Γijk = −2D/ξijk + 12χ(iK εj|l|εk)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. Can we restrict this multiplet to obtain an 8+8 multiplet?

In the 24+24 real scalar multiplet, Eij, has chiral weight −1. Therefore, we can not impose the ’reality’ condition on Eij. −iσ/2 However, we can impose the constraint Eij = e Lij where Lij are ’real’. −iσ This can be rephrased in the form Rij = E¯ij − e Eij = 0 where kl E¯ij = εikεjlE .

21/42 Real Scalar multiplet Restricted real scalar multiplet = tensor multiplet

Supersymmetric variation of the above constraint gives us the full set of 16+16 constraints. We are left with a restricted real scalar multiplet with 8+8 degrees of i freedom (Lij, Habc, φ, σ, Λ ) This multiplet is equivalent to the 8+8 tensor multiplet. i.e. it contains the gauge invariant objects of the tensor multiplet.

22/42 Real Scalar multiplet Restricted real scalar multiplet = tensor multiplet The 8+8 tensor multiplet

Contains a complex scalar G, a triplet of ’real’ scalars Lij (reality kl ∗ ij condition Lij = εikεjlL where (Lij) = L ), a two form gauge field i Eµν and a doublet of Majorana fermions φ . The transformation rules are

(k l) δLij = 2¯(iϕj) + 2εikεjl¯ ϕ , i ij ij i ij δϕ = D/L j + H/ε j − G + 2L ηj , 1 δG = −2¯ D/ϕi − 6¯ χ Lij + εij¯ γ · T +ϕ + 2¯η ϕi , i i j 4 i j i i j jk i δEµν = i¯ γµνϕ εij + 2iLijε ¯ γ[µψν]k + h.c .

23/42 Real Scalar multiplet Restricted real scalar multiplet = tensor multiplet Restricted real scalar multiplet=tensor multiplet

Consider the following combinations of tensor multiplet fields, φ4 = L2 i −2 ij Λ = −2L L ϕj −4 kl −2 Eij = L LijL ϕ¯kϕl − L GL¯ ij i −2 ik −4 ik m −2 i Sa j = 2L HaL εkj + 4L L Ljmϕ¯ γaϕk − L ϕ¯ γaϕj 1   − L−2δi ϕ¯mγ ϕ + L−2 LikD L − L D Lik 2 j a m a jk jk a −6 l mn −4 l Ξijk = −24L ϕ L ϕ¯mϕnL(ijLk)l + 6L ϕ ϕ¯lϕ(iLjk) −4 ln −4 nm − 6L L D/Ll(iLjk)ϕn + 6L L H/ϕnL(ijεk)m −4 ¯ l −2 −2 l + 12L GL(ijLk)lϕ + 6L D/ϕ(iLjk) + 18L L(ijLk)lχ 3 − L−2γ · T −ϕlL ε 4 (ij k)l −4 ¯ Cijkl = 6L GGL(ijLkl) + ....

24/42 Real Scalar multiplet Restricted real scalar multiplet = tensor multiplet

These fields transform exactly like the real scalar multiplet fields, but with 8+8 off-shell degrees of freedom. From the above identification, one can read off the relation between the tensor multiplet fields and the restricted real scalar multiplet fields.

φ4 = L2 , i −2 ij Λ = −2L L ϕj , Z 1/2 e−iσ/2 = , Z¯

Lij = |Z| Lij ,

Ha ∼ Ha , Up to fermion bilinears . −4 kl −2 Z = L L ϕ¯kϕl − L G.¯

25/42 Construction of the action for the 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

26/42 Construction of the action for the real scalar multiplet Action for the 24+24 multiplet and multiplet calculus: Work in progress

As the 24+24 multiplet admits a tensor multiplet embedding, this enables us to do multiplet calculus. Can we construct combinations of the real scalar multiplet fields to obtain a chiral multiplet? ik jl Aˆ = EijEklε ε is a chiral field. Further variation gives the other chiral multiplet components. This gives us a higher derivative action for the tensor multiplet. This is the same action one would obtain for the tensor multiplet from the chiral density formula. Can we develop a density formula for the real scalar multiplet, independent of the chiral multiplet. The answer appears to be yes. (Work in progress).

27/42 Construction of the 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

28/42 Construction of the real scalar multiplet dilaton Weyl multiplet in five dimensions Motivation: Dilaton Weyl multiplet in five dimensions

In d = 6 and 5, it was found that there is not one, but two possible Weyl multiplets. The new Weyl multiplet constructed had a dilaton field with Weyl weight +1, and hence was called dilaton Weyl multiplet. Two routes to construction of dilaton Weyl multiplet in five dimensions2: a) Multiplet of supercurrents of a non conformal rigid supersymmetry multiplet - vector multiplet in five dimensions. b) Coupling the improved vector multiplet to conformal supergravity. In four dimensions, the second route gives a 24+24 dilaton Weyl multiplet3.

2E Bergshoeff, S Cucu, M Derix, T de Wit, R Halbersma, A van Proyen, JHEP 2001 3D Butter, SH, I Lodato, B Sahoo, JHEP, 2018 29/42 Construction of the real scalar multiplet dilaton Weyl multiplet in five dimensions Current multiplet method

Compute the multiplet of supercurrents for a rigid supersymmetry multiplet. Couple the supercurrents to fields to obtain the linearized supersymmetry transformations. Complete the non-linear supersymmetry transformations using the superconformal algebra. A non-conformal multiplet has an energy momentum tensor with a non zero trace. This can be coupled to a scalar field which can be interpreted as the dilaton. But the action for rigid vector multiplet in four dimensions is conformal. However the rigid tensor multiplet action is non-conformal in four dimensions.

30/42 Construction of the real scalar multiplet dilaton Weyl multiplet in five dimensions

Transformation rules for the rigid tensor multiplet on substitution of the equation of motion for the auxiliary field G.

i j δEµν = i¯ γµνφ ij + h.c. , i ij ij δφ = ∂/L j +  H/j , ij (i j) ik j` δL = 2¯ φ + 2  ¯(kφ`) .

Action for the tensor multiplet

Z  ←→ 1  S = d4x H Hµ − φ¯i ∂/φ − ∂ Lij∂µLij , µ i 2 µ where Hµ is the Hodge dual of the three form field strength. The above action is not conformal.

31/42 Supercurrent multiplet for the tensor multiplet and linearized Construction of the real scalar multiplet tranformation rules

48+48 component multiplet of supercurrents for the above action: j j − i θµν, σ, vµi , tµi , aµ, bµν, a˜µ, eij, d, cijkl,Jµi, λi, ξ , Σijk [D Butter, S Kuzenko, 2010] Couple the currents to fields via a first order action.

Z 1 1 1 S = d4x θµνh + σϕ + dD + cijklC + b− T −µν 2 µν 24 ijkl 4 µν i µj ij µ i i
µj µi i −2vµ jV i + e Eij + 4 aµA + tµ j − vµ j S i + 2J¯µiψ + λ¯iΛ 1  +ξ¯ ζi + Σ¯ Ξijk +a ˜ A˜µ + h.c. . i 3 ijk µ

Demand the invariance of the action to obtain linearized transformation of the fields. Conservation of a current =⇒ gauge symmetry of a field.

32/42 Supercurrent multiplet for the tensor multiplet and linearized Construction of the real scalar multiplet tranformation rules

We obtain the linearized transformation rules for 48+48 component multiplet of fields which contains a real scalar of Weyl weight +1. Is this multiplet reducible? i.e. can we decouple this into two or more multiplets by use of field redefinitions. Assumption: Such a redefiniton should be apparent even at the linearised level.

33/42 Supercurrent multiplet for the tensor multiplet and linearized Construction of the real scalar multiplet tranformation rules

A 24+24 standard Weyl multiplet decouples and we are left with a 24+24 matter multiplet coupled to the standard Weyl multiplet. The matter multiplet contains the real scalar field with Weyl weight +1. Use the superconformal algebra to complete the supersymmetric variations of the 24+24 matter multiplet.

34/42 Dilaton Weyl multiplet in four dimensions 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

35/42 Dilaton Weyl multiplet in four dimensions Dilaton Weyl multiplet in four dimensions

A 24+24 Weyl multiplet which contains a dilaton, hence named a i i Dilaton Weyl multiplet. Independent fields are eµ , ψµ , bµ, Aµ, Vµ j, X, Wµ, W˜ µ, Ωi. The transformation rule is given by

a i a δeµ = ¯ γ ψµi + h.c 1 δψ i = 2D i − εijX¯ −1γ. F − + iG−
γ  − γ ηi µ µ 8 µ j µ 1 1 1 δb = ¯iφ − X−1¯iγ D6 Ω − η¯iψ + h.c + Λa e µ 2 µi 4 µ i 2 µi K µa i i i δA = ¯iφ + X−1¯iγ D6 Ω + η¯iψ + h.c µ 2 µi 2 µ i 2 µi i i ¯ −1 i i δVµ j = 2¯jφµ − X ¯jγµ D6 Ω + 2¯ηjψµ − (h.c; traceless)

36/42 Dilaton Weyl multiplet in four dimensions

i δX = ¯ Ωi 1 i δΩ = 2 D6 X +  γ.Fj −  γ.G−j + 2Xη i i 4 ij 4 ij i ij ¯ i j δWµ = ε ¯iγµΩj + 2εijX¯ ψµ + h.c ˜ ij ¯ i j δWµ = iε ¯iγµΩj − 2iεijX¯ ψµ + h.c

Fields should satisfy the constraint 1 1 1 XD2X¯ + Ω¯ k D6 Ω + F·F + + G·G+ − h.c = 0 (1) 2 k 4 4 We can solve this constraint for a three-form gauge field where the constraint appears as the Bianchi identity. i.e. define X 1 1 XXD¯ log − Ω¯ kγ Ω = ε Hbcd . a X¯ 2 a k 3! abcd

37/42 Dilaton Weyl multiplet in four dimensions The constraint now becomes a Bianchi identity and reads, 3 3 D H = F F + G G [a bcd] 8 [ab cd] 8 [ab cd]

We get a i a a ab δeµ = ¯ γ ψµi + h.c. − ΛDeµ + ΛM eµb 1 1 δψ i = 2D i − εijX¯ −1γ · F − + iG−
γ  − γ ηi − Λ ψ i µ µ 16 µ j µ 2 D µ i 1 − Λ ψ i + Λi ψ j + Λab γ ψ i 2 A µ j µ 4 M ab µ 1 1 1 δb = ¯iφ − X−1¯iγ D6 Ω − η¯iψ + h.c. + Λa e + ∂ Λ µ 2 µi 4 µ i 2 µi K µa µ D i i ¯ −1 i i i δVµ j = 2¯jφµ − X ¯jγµ D6 Ω + 2¯ηjψµ − (h.c.; traceless) − 2∂µΛ j i k k i + Λ kVµ j − Λ jVµ k i δX = ¯ Ωi + (ΛD − iΛA) X

38/42 Dilaton Weyl multiplet in four dimensions

1 i δΩ = 2 D6 X +  γ ·Fj −  γ ·G−j + 2Xη i i 4 ij 4 ij i 3 i  1 + Λ − Λ Ω − Λj Ω + Λab γ Ω 2 D 2 A i i j 4 M ab i ij ¯ i j δWµ = ε ¯iγµΩj + 2εijX¯ ψµ + h.c. + ∂µλ ˜ ij ¯ i j ˜ δWµ = iε ¯iγµΩj − 2iεijX¯ ψµ + h.c. + ∂µλ 1 1 ˜ ˜ ¯ i i δBµν = 2 W[µδQWν] + 2 W[µδQWν] + X¯ γµνΩi + X¯iγµνΩ λ λ˜ + 2 XX¯¯iγ ψ + 2 XX¯¯ γ ψi + 2∂ Λ − F − G [µ ν] i i [µ ν] [µ ν] 4 µν 4 µν

39/42 Conclusions and future work 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

40/42 Conclusions and future work Conclusions

We have obtained a 24+24 matter multiplet in N = 2 conformal supergravity in four dimensions by following the current multiplet procedure for the rigid on-shell tensor multiplet. This is the generalization of Howe, Stelle, Townsend’s flat space multiplet to include coupling to conformal supergravity. We can impose 16+16 constraints on the real scalar multiplet to obtain a 8+8 restricted multiplet. This restricted multiplet is equivalent to the tensor multiplet. Thus the real scalar multiplet is similar to the chiral multiplet, which allowed for a formulation of superconformal tensor calculus. We have obtained the 24+24 dilaton Weyl multiplet in four dimensional N = 2 conformal supegravity.

41/42 Conclusions and future work Future directions

Compute the action for the 24+24 real scalar multiplet. This would allow us to write new superconformal invariant action for the tensor multiplet. This could potentially lead to new higher derivative invariants in Poincare supergravity. Can vector multiplet and Weyl multiplet be embedded in the real scalar multiplet? Off-shell dimensional reduction of the dilaton Weyl multiplet from five to four dimensions This will allow to write all curvature squared invariants in four dimensions in terms of the dilaton Weyl multiplet.

42/42