Two Introductory Lectures on Supergravity

Home , Conformal gravity, Supergravity, Superspace

Introductory comments Weyl-invariant formulation for gravity Minkowski superspace and chiral superspace Gravitational superfield and conformal supergravity Superconformal compensators and supergravity Differential geometry for supergravity Super-Weyl invariance and conformal supergravity Reduction to component fields Compensators and off-shell formulations for supergravity Spontaneously broken (de Sitter) supergravity

Two introductory lectures on supergravity

Sergei M. Kuzenko

School of Physics, The University of Western Australia

Pre-SUSY 2016 University of Melbourne, 28 & 29 June, 2016

Two introductory lectures on supergravity Sergei M. Kuzenko Introductory comments Weyl-invariant formulation for gravity Minkowski superspace and chiral superspace Gravitational superfield and conformal supergravity Superconformal compensators and supergravity Differential geometry for supergravity Super-Weyl invariance and conformal supergravity Reduction to component fields Compensators and off-shell formulations for supergravity Spontaneously broken (de Sitter) supergravity Outline

1 Introductory comments

2 Weyl-invariant formulation for gravity

3 Minkowski superspace and chiral superspace

4 Gravitational superﬁeld and conformal supergravity

5 Superconformal compensators and supergravity

6 Diﬀerential geometry for supergravity

7 Super-Weyl invariance and conformal supergravity

8 Reduction to component ﬁelds

9 Compensators and oﬀ-shell formulations for supergravity

10 Spontaneously broken (de Sitter) supergravity

November 1915 Einstein’s general theory of relativity (GR) February 2016 Discovery of gravitational waves Final confirmation of GR (100 years later) Since the creation of GR, Einstein was confident in the correctness of his theory. However he was not completely satisfied with it. Why? The Einstein field equations are 1 8πG R − g R = T . µν 2 µν c4 µν Here the left-hand side is purely geometric. The right-hand side is proportional to the energy-momentum tensor of matter, which is not geometric. While the geometry of spacetime is determined by the Einstein equations, the theory does not predict the structure of its matter sector.

Action functional describing the dynamics of the gravitational ﬁeld coupled to matter ﬁelds ϕi :

S = SGR + SM, 1 Z √ Z √ S = d4x −g R , S = d4x −g L (ϕi , ∇ ϕj ,... ) , GR 2κ2 M M µ

with κ2 = 8πGc−4. SGR and SM are the gravitational and matter actions, respectively. The dynamical equations are: (i) the matter equations of motion, δS/δϕi = 0; and (ii) the Einstein ﬁeld equations with

2 δS T = −√ M . µν −g δg µν

Einstein was concerned with the fact that the matter Lagrangian, LM, is essentially arbitrary!

Typical reasoning by Einstein: “Ever since the formulation of the general relativity theory in 1915, it has been the persistent effort of theoreticians to reduce the laws of the gravitational and electromagnetic fields to a single basis. It could not be believed that these fields correspond to two spatial structures which have no conceptual relation to each other.” Science 74, 438 (1931) Supergravity is the gauge theory of supersymmetry. Local supersymmetry is a unique symmetry principle to bind together the gravitational field (spin 2) and matter fields of spin s < 2. It is known that the Kaluza-Klein approach also makes it possible to unify the gravitational field with Yang-Mills and scalar fields. However, it is local supersymmetry which allows one to unite the gravitational field with fermionic fields into a single multiplet. If we believe in unity of forces in the universe, local supersymmetry should play a fundamental role, as a spontaneously broken symmetry.

On-shell supergravity D. Freedman, P. van Nieuwenhuizen & S. Ferrara (1976) S. Deser & B. Zumino (1976) Super-Higgs effect (spontaneously broken supergravity) D. Volkov & V. Soroka (1973) S. Deser & B. Zumino (1977) Non-minimal off-shell supergravity P. Breitenlohner (1977) unpublished W. Siegel (1977) Old minimal off-shell supergravity unpublished W. Siegel (1977) Phys. Lett. B 74, 51 J. Wess & B. Zumino (1978) Phys. Lett. B 74, 330 K. Stelle & P. West (1978) Phys. Lett. B 74, 333 S. Ferrara & P. van Nieuwenhuizen (1978) New minimal off-shell supergravity M. Sohnius & P. West (1981)

Superspace and component approaches: J. Wess and J. Bagger, Supersymmetry and Supergravity, Princeton University Press, Princeton, 1983 (Second Edition: 1992). S. J. Gates Jr., M. T. Grisaru, M. Roˇcekand W. Siegel, Superspace, or One Thousand and One Lessons in Supersymmetry, Benjamin/Cummings (Reading, MA), 1983, hep-th/0108200. I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace, IOP, Bristol, 1995 (Revised Edition: 1998). Purely component approach: D. Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, 2012.

There exist three formulations for gravity in d dimensions: Metric formulation; Vielbein formulation; Weyl-invariant formulation. I brieﬂy recall the metric and vielbein approaches and then concentrate in more detail of the Weyl-invariant formulation. S. Deser (1970) P. Dirac (1973) The latter formulation is a natural starting point to introduce supergravity as a generalisation of gravity.

Metric formulation Gauge field: metric gmn(x) Gauge transformation: δgmn = ∇mξn + ∇nξm m ξ = ξ (x)∂m vector field generating an infinitesimal diffeomorphism. Vielbein formulation a a Gauge field: vielbein em (x) , e := det(em ) 6= 0 a b Metric is a composite field gmn = em en ηab b 1 bc Gauge transformation: δ∇a = [ξ ∇b + 2 K Mbc , ∇a] a m a ab ba Gauge parameters: ξ (x) = ξ em (x) and K (x) = −K (x) a a a Covariant derivatives (Mbc Lorentz generators Mbc V = δbVc − δc Vb) 1 1 ∇ = e + ω ≡ e m∂ + ω bc M , [∇ , ∇ ] = R cd M a a a a m 2 a bc a b 2 ab cd

m m b b ea inverse vielbein, ea em = δa bc 1 ωa torsion-free Lorentz connection ωabc = 2 (Cbca + Cacb − Cabc ) c c [ea, eb] = Cab ec Cab anholonomy coeﬃcients

Weyl transformations The torsion-free constraint 1 1 T c = 0 ⇐⇒ [∇ , ∇ ] ≡ T c ∇ + R cd M = R cd M ab a b ab c 2 ab cd 2 ab cd is invariant under Weyl (local scale) transformations

0 σ b ∇a → ∇a = e ∇a + (∇ σ)Mba ,

with the parameter σ(x) being completely arbitrary.

m σ m a −σ a −2σ ea → e ea , em → e em , gmn → e gmn Weyl transformations are gauge symmetries of conformal gravity, which in the d = 4 case is described by action (Cabcd is the Weyl tensor) Z 4 abcd 2σ Sconf = d x e C Cabcd , Cabcd → e Cabcd

Einstein gravity possesses no Weyl invariance.

a a Gauge ﬁelds: vielbein em (x) , e := det(em ) 6= 0 & conformal compensator ϕ(x) , ϕ 6= 0 b 1 bc Gauge transformations (K := ξ ∇b + 2 K Mbc ) 1 δ∇ = [ξb∇ + K bc M , ∇ ] + σ∇ + (∇bσ)M ≡ (δ + δ )∇ , a b 2 bc a a ba K σ a 1 δϕ = ξb∇ ϕ + (d − 2)σϕ ≡ (δ + δ )ϕ b 2 K σ Gauge-invariant gravity action

1 Z 1 d − 2 S = dd x e ∇aϕ∇ ϕ + Rϕ2 − λϕ2d/(d−2) 2 a 4 d − 1

2 q d−1 Imposing a Weyl gauge condition ϕ = κ d−2 = const reduces S to the Einstein-Hilbert action with a cosmological term 1 Z Z S = dd x e R − Λ dd x e 2κ2

Conformal Killing vector ﬁelds m a m A vector ﬁeld ξ = ξ ∂m = ξ ea, with ea := ea ∂m, is conformal Killing if there exist local Lorentz, K bc [ξ], and Weyl, σ[ξ], parameters such that

h 1 i (δ + δ )∇ = ξb∇ + K bc [ξ]M , ∇ + σ[ξ]∇ + (∇bσ[ξ])M = 0 K σ a b 2 bc a a ba A short calculation gives 1 1 K bc [ξ] = ∇bξc − ∇c ξb , σ[ξ] = ∇ ξb 2 d b Conformal Killing equation

∇aξb + ∇bξa = 2ηabσ[ξ]

Conformal Killing vector ﬁelds for Minkowski space:

a a a b a a 2 b a ξ = b + K bx + ∆x + f x − 2fbx x , Kab = −Kba

Lie algebra of conformal Killing vector ﬁelds Conformally related spacetimes (∇a, ϕ) and (∇e a, ϕe)

ρ b 1 (d−2)ρ ∇e a = e ∇a + (∇ ρ)Mba , ϕe = e 2 ϕ

a a have the same conformal Killing vector ﬁelds ξ = ξ ea = ξ˜ e˜a. The parameters K cd [ξ˜] and σ[ξ˜] are related to K cd [ξ] and σ[ξ] as follows:

˜ ˜b 1 cd ˜ K[ξ] := ξ ∇e b + K [ξ]Mcd = K[ξ] , 2 σ[ξ˜] = σ[ξ] − ξρ

Conformal ﬁeld theories

Killing vector ﬁelds a Let ξ = ξ ea be a conformal Killing vector, h 1 i (δ + δ )∇ = ξb∇ + K bc [ξ]M , ∇ + σ[ξ]∇ + (∇bσ[ξ])M = 0 . K σ a b 2 bc a a ba It is called Killing if it leaves the compensator invariant, 1 (δ + δ )ϕ = ξϕ + (d − 2)σ[ξ]ϕ = 0 . K σ 2 These Killing equations are Weyl invariant in the following sense: Given a conformally related spacetime (∇e a, ϕe) ρ b 1 (d−2)ρ ∇e a = e ∇a + (∇ ρ)Mba , ϕe = e 2 ϕ , the above Killing equations have the same functional form when rewritten in terms of (∇e a, ϕe), in particular 1 ξϕ + (d − 2)σ[ξ˜]ϕ = 0 . e 2 e

Because of Weyl invariance, we can work with a conformally related spacetime such that ϕ = 1 Then the Killing equations turn into h 1 i ξb∇ + K bc [ξ]M , ∇ = 0 , σ[ξ] = 0 b 2 bc a Standard Killing equation

∇aξb + ∇bξa = 0

Killing vector ﬁelds for Minkowski space:

a a a b ξ = b + K bx , Kab = −Kba

Lie algebra of Killing vector ﬁelds Field theories in curved space with symmetry group including the spacetime isometry group.

The Minkowski metric is chosen to be ηab = diag (−1, +1, +1, +1). α For two-component undotted spinors, such as ψα and ψ , their indices are raised and lowered by the rule: α αβ β ψ = ε ψβ , ψα = εαβ ψ , αβ where ε and εαβ are 2 × 2 antisymmetric matrices normalised as 12 ε = ε21 = 1 . . The same conventions are used for dotted spinors( ψ¯. and ψ¯α). . α α 2 ¯¯ ¯ ¯α ¯2 ¯ ¯ ψλ := ψ λα , ψ = ψψ , ψλ := ψα.λ , ψ = ψψ . αβ˙ Relativistic Pauli matrices σa = (σa)αβ˙ andσ ˜a = (˜σa)

σa = (12, ~σ) , σ˜a = (12, −~σ)

β α˙ Lorentz spinor generators σab = (σab)α andσ ˜ab = (˜σab) β˙ : 1 1 σ = − (σ σ˜ − σ σ˜ ) , σ˜ = − (˜σ σ − σ˜ σ ) ab 4 a b b a ab 4 a b b a

Minkowski superspace M4|4 is parametrised by ‘real’ coordinates A a α a a α α˙ z = (x , θ , θ¯α˙ ) , x = x , θ = θ¯ .

It may be embedded in complex superspace C4|2 (chiral superspace) ζA = (y a, θα) as real surface a a a ¯ a ¯ a a a ¯ y − y¯ = 2iθσ θ ≡ 2iH0(θ, θ) ⇐⇒ y = x + iθσ θ . Supersymmetry transformation on M4|4 0a a a ¯ a 0α α α ¯0 ¯ x = x − iσ θ + iθσ ¯ , θ = θ + , θα˙ = θα˙ + ¯α˙ is equivalent to a holomorphic transformation on C4|2 y 0a = y a + 2iθσa¯+ iσa¯ , θ0α = θα + α .

Every Poincarétransformation (translation and Lorentz one) on M4|4 is also equivalent to a holomorphic transformation on C4|2. Two introductory lectures on supergravity Sergei M. Kuzenko Introductory comments Weyl-invariant formulation for gravity Minkowski superspace and chiral superspace Gravitational superfield and conformal supergravity Superconformal compensators and supergravity Differential geometry for supergravity Super-Weyl invariance and conformal supergravity Reduction to component fields Compensators and off-shell formulations for supergravity Spontaneously broken (de Sitter) supergravity Family of superspaces M4|4(H)

Curved superspace M4|4(H), A a α parametrised by real coordinates z = (x , θ , θ¯α˙ ), is deﬁned by its embedding in C4|2: 1 y a − y¯a = 2iθσaθ¯ ≡ 2iHa(x, θ, θ¯) , x a = (y a +y ¯a) , 2 for some real vector superﬁeld Ha(x, θ, θ¯).

4|4 4|4 What is special about Minkowski superspace M = M (H0)? It is the only super-Poincar´einvariant superspace in the family of all supermanifolds M4|4(H).

Family of superspaces {M4|4(H)} and uniqueness of M4|4

Spacetime translations

y 0a = y a + ba , θ0α = θα

Condition of invariance

y 0a − y¯0a = 2iHa(x 0, θ0, θ¯0) = y a − y¯a = 2iHa(x, θ, θ¯) =⇒ Ha(x + b, θ, θ¯) = Ha(x, θ, θ¯) =⇒H a = Ha(θ, θ¯)

Lorentz transformations

Ha(θ, θ¯) = κθσaθ¯

for some constant κ. Supersymmetry transformations

a ¯ a ¯ a H (θ, θ) = θσ θ = H0

Consider an inﬁnitesimal holomorphic transformation on C4|2 y 0a = y a + λa(y, θ) , θ0α = θα + λα(y, θ)

What are the most general inﬁnitesimal holomorphic transformations on 4|2 4|4 C which leave Minkowski superspace, M (H0), invariant ? y 0a − y¯0a = 2iθ0σaθ¯0

The answer is: superconformal transformations

a a a b a a 2 b a a a b λ = b + K by + ∆y + f y − 2fby y + 2iθσ ¯− 2θσ σ˜bηy , 1 λα = α − K α θβ + (∆ − iΩ)θα + f by c (θσ σ˜ )α + 2ηαθ2 − i(¯ησ˜ )αy b β 2 b c b

Kab = −Kba ←→ Kαβ = Kβα Lorentz transformation; ∆ dilatation;Ω R-symmetry transformation (or U(1) chiral rotation); f a special conformal transformation; ηα S-supersymmetry transformation.

1 V → V := (σa) V , V = − (˜σ )αα˙ V a αα˙ αα˙ a a 2 a αα˙

Second-rank antisymmetric tensor Kab = −Kba is equivalent to

a b Kααβ˙ β˙ := (σ )αα˙ (σ )ββ˙ Kab = 2εαβKα˙ β˙ + 2εαδβ˙ Kαβ , where 1 1 K = (σab) K = K , K = (˜σab) K = K αβ 2 αβ ab βα α˙ β˙ 2 α˙ β˙ ab β˙α˙

If Kab is real, then ¯ Kα˙ β˙ = Kαβ ≡ Kα˙ β˙

We turn to introducing a supersymmetric generalisation of m (i) the gravitational ﬁeld described by ea = ea (x)∂m; and (ii) its gauge transformation

m b m δea = δea (x)∂m = [ξ, ea] + Ka (x)eb + σ(x)ea , ξ = ξ (x)∂m

which corresponds to conformal gravity. Such a geometric formalism was developed by V. Ogievetsky & E. Sokatchev (1978) Equivalent, but less geometric approach, was developed by unpublished W. Siegel (1977)

Group of holomorphic coordinate transformations on C4|2 ∂(y 0, θ0) y m → y 0m = f m(y, θ) , θα → θ0α = f α(y, θ) , Ber 6= 0 ∂(y, θ)

Every holomorphic transformation on C4|2 acts on the space of supermanifolds {M4|4(H)}

M4|4(H) → M4|4(H0)

In other words, the superﬁeld Hm, which deﬁnes the curved superspace M4|4(H), transforms under the action of the group.

Nonsingular ever (p, q) × (p, q) supermatrix

AB F = , det A 6= 0 , det D 6= 0 CD

Here A and D are bosonic p × p and q × q matrices, respectively; B and C are fermionic p × q and q × p matrices, respectively.

sdetF ≡ BerF := det(A − BD−1C) det−1D = det A det−1(D − CA−1B)

Change of variables on superspace Rp|q parametrised by coordinates zA = (x a, θα), where x a are bosonic and θα fermionic coordinates

zA → z0A = f A(z) Z Z ∂(x 0, θ0) dpx 0dqθ0 L(z0) = dpxdqθ Ber L z0(z) ∂(x, θ)

Inﬁnitesimal holomorphic coordinate transformation on C4|2 y m → y 0m = y m − λm(y, θ) , θα → θ0α = θα − λα(y, θ)

leads to the following coordinate transformation on M4|4(H):

1 1 x m → x 0m = x m − λm(x + iH, θ) − λ¯m(x − iH, θ¯) 2 2 θα → θ0α = θα − λα(x + iH, θ) .

0m 0 0 ¯0 i 0m 0m For H (x , θ , θ ) = − 2 (y − y¯ ) we get i n o H0µ(x 0, θ0, θ¯0) = Hm(x, θ, θ¯) + λm(x + iH, θ) − λ¯m(x − iH, θ¯) 2 Now we can read oﬀ the gauge transformation

δHm := H0m(x, θ, θ¯) − Hm(x, θ, θ¯)

Nonlinear gauge transformation law of Hm i 1 δHm = (λm − λ¯m) + (λn + λ¯n)∂ + λα∂ + λ¯α˙ ∂ Hm(x, θ, θ¯) , 2 2 n α α˙ λm(x, θ, θ¯) = λm(x + iH, θ) and λ¯m(x, θ, θ¯) = λ¯m(x − iH, θ¯). Some component ﬁelds of Hm can be gauged away. Indeed m ¯ m α m ¯ mα˙ 2 m ¯2 ¯m H (x, θ, θ) = h (x) + θ χα (x) + θα˙ χ¯ (x) + θ S (x) + θ S (x) a ¯ m ¯2 α m 2 ¯ ¯ mα˙ 2 ¯2 m +θσ θea (x) + iθ θ Ψα (x) − θ θα˙ Ψ (x) + θ θ A (x) . The gauge transformation law of Hm can be written as i i δHm(x, θ, θ¯) = λm(x, θ) − λ¯m(x, θ¯) + O(H) 2 2 The superﬁeld gauge parameters are m m α m 2 m λ (x, θ) = a (x) + θ ϕα (x) + θ s (x) , α α α β 2 α λ (x, θ) = (x) + ω β(x)θ + θ η (x) , m m α where all bosonic gauge parameters a , s and ω β are complex.

Wess-Zumino gauge

m ¯ a ¯ m ¯2 α m 2 ¯ ¯ mα˙ 2 ¯2 m H (x, θ, θ) = θσ θea (x) + iθ θ Ψα (x) − iθ θα˙ Ψ (x) + θ θ A (x)

Residual gauge freedom

m m a m 2 m λ (x, θ) = ξ (x) + 2iθσ ¯(x)ea (x) − 2θ ¯(x)Ψ¯ (x) , 1 λα(x, θ) = α(x) + [σ(x) + iΩ(x)]θα + K α (x)θb + θ2ηα(x) , 2 β Kαβ = Kβα

The bosonic parameters ξm, σ and Ω are real. ξm general coordinate transformation; Kαβ local Lorentz transformation; σ andΩ Weyl and local R-symmetry transformations, respectively; α local supersymmetry transformation; ηα local S-supersymmetry transformation.

General coordinate transformation

m n m n m m δξea = ξ ∂nea − ea ∂nξ ⇐⇒ δξea = δea (x)∂m = [ξ, ea] , m n m n m m n m n m δξΨα = ξ ∂nΨα − Ψα ∂nξ , δξA = ξ ∂nA − A ∂nξ .

m m m Each of the ﬁelds ea ,Ψα and A transforms as a world vector, with respect to the index m, under the general coordinate transformations. The index ‘m’ is to be interpreted as a curved-space index. Local Lorentz transformation

m b m m β m δK ea = Ka eb , δK Ψα = Kα Ψβ .

m m We see that δξea and δK ea coincide with the transformation laws of the inverse vielbein. Since in Minkowski superspace

m ¯ a ¯ m m m H0 (x, θ, θ) = θσ θδa =⇒ (0)ea = δa ,

m m we interpret the ﬁeld ea (x) as the inverse vielbein, det(ea ) 6= 0.

m ˜m 1 m abcd Field redeﬁnition A = A + 4 ea ε ωbcd where ωbcd is the torsion-free Lorentz connection. Local Lorentz transformation

m δK A˜ = 0 .

Weyl transformation 3 δ e m = σe m , δ Ψm = σΨm , δ A˜ = 0 , σ a a σ α 2 α σ m

n with A˜m = gmnA˜ . Local chiral transformation i 1 δ e m = 0 , δ Ψm = − ΩΨm , δ A˜ = ∂ Ω . Ω a σ α 2 α σ m 2 m

A˜m is the R-symmetry gauge ﬁeld.

Local supersymmetry transformation

m m m δea = iσaΨ¯ − iΨ σa¯ , m a b m ˜m δΨα = (σ σ˜ ∇a)αeb − 2iA α ,

δA˜m = ...

Local S-supersymmetry transformation see section 5.1 of I. Buchbinder & SMK, Ideas and Methods of Supersymmetry and Supergravity or a Walk Through Superspace Multiplet of conformal supergravity

m m ¯ m ˜ (ea , Ψα , Ψα˙ , Am)

m ea graviton m ¯ m Ψα , Ψα˙ gravitino A˜m U(1)R gauge ﬁeld

In order to obtain a supersymmetric extension of Einstein’s gravity, a superconformal compensatorΥ( x, θ, θ¯) is required, in addition to the gravitational superfield Hm. Unlike Einstein’s gravity, there are several different supermultiplets (ϕ(x) in gravity and Υ(x, θ, θ¯) in supergravity) that may be chosen to play the role of superconformal compensator. One option is a chiral superfield φ(y, θ) which is defined on C4|2 and transforms as follows ∂(y 0, θ0)−1/3 φ0(y 0, θ0) = Ber φ(y, θ) ∂(y, θ) under the holomorphic reparametrisation y m → y 0m = f m(y, θ) , θα → θ0α = f α(y, θ) . W. Siegel (1977); W. Siegel & J. Gates (1979) Gauge-invariant chiral integration measure 3 3 d4y 0d2θ0φ0(y 0, θ0) = d4yd2θφ(y, θ)

In the Wess-Zumino gauge, the residual gauge freedom includes the Weyl and local U(1)R transformations (described by the parameters σ(x) and Ω(x), respectively), as well as the local S-supersymmetry transformation (described by ηα(x) and its conjugate). These gauge symmetries may be used to bring

3 −1 n α 2 o φ (x, θ) = e (x) F (x) + θ χα(x) + θ B(x)

to the form:

3 −1 n a 2 o φ (x, θ) = e (x) 1 − 2iθσaΨ¯ (x) + θ B(x)

Multiplet of old minimal supergravity

m m ¯ m ˜ ¯ (ea , Ψα , Ψα˙ , Am, B, B)

Gravitational superfield Hm transforms in a nonlinear (non-tensorial) way. Its direct use for constructing supergravity matter actions is not very practical. In order to obtain powerful tools to generate supergravity-matter actions, we have to extend to curved superspace the formalism of differential geometry which we use for QFT in curved space. In curved spacetime M4 parametrised by coordinates x m, the gravitational field is described in terms of covariant derivatives 1 ∇ = e + ω , e = e m(x) , ω = ω bc (x)M a a a a a a 2 a bc In Einstein’s gravity, the structure group is the Lorentz group, more precisely its universal covering SL(2, C). Gravity gauge transformation 1 δ∇ = [K, ∇ ] , δU = KU , K = ξb(x)∇ + K bc (x)M , a a b 2 bc where U(x) is a (matter) tensor field (with Lorentz indices only).

In N = 1 supergravity, there are two ways to choose structure group: SL(2, C) R. Grimm, J. Wess & B. Zumino (1978) SL(2, C) × U(1)R P. Howe (1982) Howe’s approach is suitable for all oﬀ-shell formulations for N = 1 supergravity, while the Grimm-Wess-Zumino approach is ideal for the so-called old minimal formulation for supergravity. The two approaches prove to be equivalent, so here we follow the Grimm-Wess-Zumino approach, which is simpler. In curved superspace M4|4 parametrised by local coordinates M m µ z = (x , θ , θ¯µ˙ ), supergravity multiplet is described in terms of covariant derivatives 1 D = (D , D , D¯α˙ ) = E M (z)∂ + Ω bc (z)M A a α A M 2 A bc Supergravity gauge transformation 1 δ D = [K, D ] , δ U = KU , K = ξB (z)D + K bc (z)M , K A A K B 2 bc for any tensor U(z) a tensor superﬁeld.

In curved space, the covariant derivatives ∇a obey the algebra 1 1 [∇ , ∇ ] ≡ T c ∇ + R cd M = R cd M , a b ab c 2 ab cd 2 ab cd

c cd where Tab (x) and Rab (x) are the torsion tensor and the curvature tensor, respectively. To express the Lorentz connection in terms of the gravitational ﬁeld, one imposes the torsion-free constraint

c Tab = 0

In curved superspace, the covariant derivatives obey the algebra 1 [D , D } = T C D + R cd M . A B AB C 2 AB cd

C We have to impose certain constraints on TAB in order for superspace geometry to describe conformal supergravity.

In Minkowski superspace, the vector derivative is given by an anti-commutator of spinor covariant derivatives,

{Dα, D¯α˙ } = −2i∂αα˙ In curved superspace, we postulate

{Dα, D¯α˙ } = −2iDαα˙ In Minkowski superspace, there exist chiral superﬁelds constrained by

D¯α˙ Φ = 0 In curved superspace, we also want to have covariantly chiral scalar superﬁelds constrained by ¯ ¯ ¯ c γ Dα˙ Φ = 0 =⇒ 0 = {Dα˙ , Dβ˙ }Φ = Tα˙ β˙ Dc Φ + Tα˙ β˙ Dγ Φ c γ We are forced to require Tα˙ β˙ = 0 and Tα˙ β˙ = 0. Similar to GR, we need constraints that would allow us to express the Lorentz connection in terms of the (inverse) vielbein.

Algebra of the superspace covariant derivatives

{Dα, D¯α˙ } = −2iDαα˙ , ¯ ¯ ¯ ¯ {Dα, Dβ} = −4RMαβ , {Dα˙ , Dβ˙ } = 4RMα˙ β˙ , h i ¯ ¯ γ γ δ ¯ γ˙ δ˙ ¯ Dα, Dββ˙ = iεαβ R Dβ˙ + G β˙ Dγ − (D G β˙ )Mγδ + 2Wβ˙ Mγ˙ δ˙ ¯ ¯ +i(Dβ˙ R)Mαβ .

The superﬁelds R, Gαα˙ and Wαβγ obey the Bianchi identities:

D¯α˙ R = 0 , D¯α˙ Wαβγ = 0 ; ¯α˙ γ γ˙ D Gαα˙ = DαR , D Wαβγ = i D(α Gβ)γ ˙ . R is the supersymmetric extension of the scalar curvature. Ga is the supersymmetric extension of the Ricci tensor. Wαβγ is the supersymmetric extension of the Weyl tensor. It may be shown that the gravitational superﬁeld Hm originates by solving the supergravity constraints in terms of unconstrained prepotentials.

The algebra of covariant derivatives is invariant under super-Weyl transformations P. Howe & R. Tucker (1978) 1 δ D = (Σ¯ − Σ)D + (DβΣ) M , D¯ Σ = 0 Σ α 2 α αβ α˙ 1 δ D¯ = (Σ − Σ)¯ D¯ + (D¯β˙ Σ)¯ M¯ , Σ α˙ 2 α˙ α˙ β˙ 1 δ D = {δ D , D¯ } + {D , δ D¯ } = (Σ + Σ)¯ D + ..., Σ αα˙ Σ α α˙ α Σ α˙ 2 αα˙ provided the torsion tensor superﬁelds transform as follows: 1 δ R = 2ΣR + (D¯2 − 4R)Σ¯ , Σ 4 1 3 δ G = (Σ + Σ)¯ G + iD (Σ − Σ)¯ ,δ W = ΣW . Σ αα˙ 2 αα˙ αα˙ Σ αβγ 2 αβγ Gauge freedom of conformal supergravity: 1 δD = [δ , D ] + δ D , K = ξB (z)D + K bc (z)M A K A Σ A B 2 bc

Given a superfield U(z), its bar-projection U| is defined to be the θ, θ¯-independent component of U(x, θ, θ¯) in powers of θ’s and θ¯’s, ¯ U| := U(x, θ, θ)|θ=θ¯=0 . U| is a field on curved spacetime M4 which is the bosonic body of M4|4. In a similar way we define the bar-projection of the covariant derivatives: 1 D | := E M |∂ + Ω bc |M . A A M 2 A bc Of special importance is the bar-projection of a vector covariant derivative 1 1 D | = ∇ˆ + Ψ βD | + Ψ¯ D¯β˙ | , a a 2 a β 2 aβ˙ β whereΨ a is gravitino, and 1 ∇ˆ = e + ω ≡ e m(x)∂ + ω bc (x)M , a a a a m 2 a bc is a spacetime covariant derivative with torsion.

Wess-Zumino (WZ) gauge

µ ∂ ¯α˙ α˙ ∂ Dα| = δα µ , D | = δ µ˙ . ∂θ ∂θ¯µ˙ In this gauge, one obtains

m m µ 1 β µ bc bc Ea | = ea , Ea | = 2 Ψa δβ , Ωa | = ωa In the WZ gauge, we still have a tail of component ﬁelds which originates M bc at higher orders in the θ, θ¯-expansion of EA ,ΩA and which are pure gauge (that is, they may be completely gauged away). A way to get rid of such a tail of redundant ﬁelds is to impose a normal gauge around the bosonic body M4 of the curved superspace M4|4. Vielbein (E A) and connection (Ωcd ) super one-forms:

A M A cd M cd A bc E = dz EM (z) , Ω = dz ΩM (z) = E ΩA

A B N EM EA = δM

Normal gauge in superspace

M A M A Θ EM (x, Θ) = Θ δM , M cd Θ ΩM (x, Θ) = 0 ,

M m µ µ where Θ ≡ (Θ , Θ , Θ¯ µ˙ ) := (0, θ , θ¯µ˙ ). A cd In this gauge, EM (x, Θ) and ΩM (x, Θ) and ΦM (x, Θ) are given by Taylor series in Θ, in which all the coeﬃcients (except for the leading Θ-independent terms given on previous page) are tensor functions of the torsion, the curvature and their covariant derivatives evaluated at Θ = 0. I. McArthur (1983) SMK & G. Tartaglino-Mazzucchelli, arXiv:0812.3464 Analogue of the Fock-Schwinger gauge in Yang-Mills theories

m I x Am (x) = 0 ,

I where Am is the Yang-Mills gauge ﬁelds, with ‘I ’ the gauge group index.

The supergravity auxiliary ﬁelds occur as follows 1 4 R| = B¯ , G | = A . 3 a 3 a The bar-projection of the vector covariant derivatives are 1 1 1 D | = ∇ − ε Ad Mbc + Ψ β D | + Ψ¯ D¯β˙ | , a a 3 abcd 2 a β 2 aβ˙ where we have introduced the spacetime covariant derivatives, 1 bc ∇a = ea + 2 ωabc M , with ωabc = ωabc (e, Ψ) the Lorentz connection. 1 [∇ , ∇ ] = T c ∇ + R Mcd , a b ab c 2 abcd i T = − (Ψ σ Ψ¯ − Ψ σ Ψ¯ ) . abc 2 a c b b c a The Lorentz connection is 1 ω = ω (e) − (T + T − T ) , abc abc 2 bca acb abc

where ωabc (e) is the torsion-free connection

Some component results 2 2i i D R| = − (σbc Ψ ) − AbΨ + B¯(σbΨ¯ ) , α 3 bc α 3 bα 3 b α i 2i D¯ G β | = −2Ψ β + B¯Ψ¯ β − 2i(˜σab) Ψ βA + Ψ αA β , (α ˙ β˙) α˙ β,˙ 3 (α, ˙ β˙) α˙ β˙ a b 3 α(α, ˙ β˙) a b Wαβγ | = Ψ(αβ,γ) − i(σab)(αβΨ γ)A ,

γ where the gravitino ﬁled strength Ψab is deﬁned by

γ γ γ c γ Ψab = ∇aΨb − ∇bΨa − Tab Ψc , 1 1 Ψ γ = (σab) Ψ γ , Ψ γ = − (˜σab) Ψ γ . αβ, 2 αβ ab α˙ β,˙ 2 α˙ β˙ ab

Locally supersymmetric action principle Z 4 2 2 −1 M S = d xd θd θ¯E L , E = Ber(EA )

where the Lagrangian L is a scalar superﬁeld. Locally supersymmetric chiral action Z E Z S = d4xd2θd2θ¯ L = d4xd2θ EL , D¯ L = 0 c R c c α˙ c Chiral integration rule: Z 1 Z d4xd2θd2θ¯E L = − d4xd2θ E (D¯2 − 4R)L 4 Component action: Z Z n 1 i d4xd2θ EL = d4x e − D2L | − (Ψ¯ bσ˜ )α D L | c 4 c 2 b α c a b o +(B + Ψ¯ σ˜abΨ¯ ) Lc|

To choose the WZ + normal gauges, we make use of the general coordinate and local Lorentz transformations, 1 δD = [δ , D ] , K = ξB (z)D + K bc (z)M A K A B 2 bc The super-Weyl invariance remains intact. This local symmetry may be gauge-ﬁxed by choosing useful conditions on the superconformal compensator(s) upon reducing the supergravity-matter system under consideration to components. To see how this works in practice, see SMK & S. McCarthy, arXiv:hep-th/0501172

Old minimal formulation for N = 1 supergravity Its conformal compensators are Φ and Φ.¯ Here Φ is a covariantly chiral, nowhere vanishing scalar Φ,

D¯α˙ Φ = 0 , Φ 6= 0 ,

with the gauge transformation

δΦ = KΦ + ΣΦ .

The supergravity action is

3 Z µ Z S = − d4xd2θd2θ¯E ΦΦ¯ + d4xd2θ EΦ3 + c.c. , κ2 κ2

where κ is the gravitational coupling constant and µ a cosmological parameter.

3 Z S = − d4xd2θd2θ¯E ΦΦ¯ SG κ2 1 Z n1 1 = d4x e−1 R + εabcd (Ψ¯ σ˜ Ψ − Ψ σ Ψ¯ ) κ2 2 4 a b cd a b cd 4 1 o + AaA − BB¯ 3 a 3 Supersymmetric cosmological term µ Z S = d4xd2θ EΦ3 + c.c. cosm κ2 µ Z n 1 1 o = d4x e B − Ψ¯ aσ˜ σ Ψ¯ b − Ψ¯ aΨ¯ + c.c. κ2 2 a b 2 b Eliminating the auxiliary ﬁelds B and B¯ leads to the cosmological term |µ|2 Z Z |µ|2 3 d4x e = −Λ d4x e =⇒ Λ = −3 κ2 κ2

New minimal formulation for N = 1 supergravity Its conformal compensator is a real covariantly linear, nowhere vanishing scalar L, 2 (D¯ − 4R)L = 0 , L¯ = L , with the gauge transformation

δL = KL + (Σ + Σ)¯ L . Supergravity action 3 Z S = d4xd2θd2θ¯E ln SG κ2 L L The action is super-Weyl invariant due to the identity Z 4 2 2 ¯ ¯ d xd θd θ E LΣ = 0 ⇐= Dα˙ Σ = 0

No cosmological term in new minimal supergravity

The auxiliary fields of new minimal supergravity are gauge one- and m 1 m n two-forms, A1 = Am(x)dx and B2 = 2 Bmn(x)dx ∧ dx . Here Am is the U(1)R gauge field, which belongs to the gravitational superfield, while the two-form B appears only via its gauge-invariant field strength 1 H = dB = H dx m ∧ dx n ∧ dx r 3 2 3! mnr m The Hodge-dual H of Hmnr is a component field of the compensator. In the flat-superspace limit,

¯ a Hαα˙ ∝ [Dα, Dα˙ ]L| , ∂aH = 0 The auxiliary ﬁelds contribute to the supergravity action as follows: Z 4 n ? o d x c1(H )1 ∧ H3 + c2A1 ∧ H3 ,

with c1 and c2 numerical coefficients. Both fields are non-dynamical, H3 = 0 and F2 = dA1 = 0 on the mass shell. Two introductory lectures on supergravity Sergei M. Kuzenko Introductory comments Weyl-invariant formulation for gravity Minkowski superspace and chiral superspace Gravitational superfield and conformal supergravity Superconformal compensators and supergravity Differential geometry for supergravity Super-Weyl invariance and conformal supergravity Reduction to component fields Compensators and off-shell formulations for supergravity Spontaneously broken (de Sitter) supergravity New minimal supergravity coupled to σ-model matter

Consider a K¨ahlermanifold parametrized by n complex coordinates φi and their conjugates φ¯i , with K(φ, φ¯) the K¨ahlerpotential. Supergravity-matter system: 3 Z Z S = d4xd2θd2θ¯E ln + d4xd2θd2θ¯E K(φ, φ¯) . new κ2 L L L

The σ-model variables φi are covariantly chiral scalar superﬁelds, i D¯α˙ φ = 0, being inert under the super-Weyl transformations. The action is invariant under the K¨ahlertransformations

K(φ, φ¯) → K(φ, φ¯) + λ(φ) + λ¯(φ¯) ,

with λ(φ) an arbitrary holomorphic function.

First-order reformulation of the new minimal supergravity-matter system Z 1 Snew = 3 d4xd2θd2θ¯E (U − Υ) , Υ = exp U − K(φ, φ¯) , L 3 and U is an unconstrained real scalar superﬁeld. Super-Weyl invariance:

δΣU = Σ + Σ¯ . To preserve K¨ahlerinvariance, the K¨ahlertransformation of U should be 1 U → U + λ(φ) + λ¯(φ¯) . 3 ¯2 The equation of motion for L is (D − 4R)DαU = 0, and is solved by

U = lnΦ + lnΦ¯ , D¯α˙ Φ = 0 The action turns into (restore κ2) 3 Z κ2 S = − d4xd2θd2θ¯E ΦΦ¯ exp − K(φ, φ¯) old κ2 3

2 Kählerinvariance:Φ → eκ λ(φ)/3 Φ Two introductory lectures on supergravity Sergei M. Kuzenko Introductory comments Weyl-invariant formulation for gravity Minkowski superspace and chiral superspace Gravitational superfield and conformal supergravity Superconformal compensators and supergravity Differential geometry for supergravity Super-Weyl invariance and conformal supergravity Reduction to component fields Compensators and off-shell formulations for supergravity Spontaneously broken (de Sitter) supergravity Chiral Goldstino superfield

Action for a free massless chiral superﬁeld (D¯α˙ X = 0) Z Z 4 2 2 4 a S[X , X¯] = d xd θd θ¯X X¯ = d x (φ2φ¯ − iρσ ∂aρ¯ + F F¯) ,

where the component fields are defined by √ 1 X | = φ , D X | = 2ρ , − D2X | = F . α α 4 Goldstino superfield M. Roˇcek(1978)

1 X 2 = 0 , − X D¯2X¯ = fX , 4 where f is a non-zero parameter of dimension (mass)2. The auxiliary ﬁeld acquires a non-zero expectation value, hF i = f .

Solution to the constraint X 2 = 0 is ρ2 φ = . 2F 1 ¯2 ¯ Solution to the second constraint, − 4 X D X = fX , is 1 F = f + F¯−1hu¯i − F¯−2ρ¯22(F −1ρ2) 4 n 1 = f 1 + f −2hu¯i − f −4(huihu¯i + ρ¯22ρ2) + f −6(hui2hu¯i + c.c.) 4 f −6 + hu¯iρ22ρ¯2 + 2huiρ¯22ρ2 +ρ ¯22(ρ2hu¯i) 4 1 1 −3f −8 hui2hu¯i2 + ρ2ρ¯22(hui2 − huihu¯i + hu¯i2) + ρ2ρ¯22ρ¯22ρ2) 4 16 a where hMi = trM = Ma , and

b b b b b b u = (ua ), ua := iρσ ∂aρ¯ ;u ¯ = (¯ua ), u¯a := −i∂aρσ ρ¯

Supersymmetry transformation √ √ √ a a δφ = 2ρ , δρα = 2 αF + i(σ ¯)α∂aφ , δF = − 2i(∂aψσ ¯) . Since F acquires the non-zero expectation value hF i = f , supersymmetry becomes nonlinearly realised, √ δρα = 2f α + ... Goldstino action Z Z 4 2 2 4 2 SGoldstino = − d xd θd θ¯X X¯ = −f d xd θ X 1 Z n 1 = − d4x 2f 2 + hu +u ¯i + f −2(∂aρ2∂ ρ¯2 − 4huihu¯i) 2 2 a +f −4 hui(¯ρ22ρ2 + 2huihu¯i) + c.c. 3 o +3f −6 hu2ihu¯2i − 3hui2hu¯i2 − 2huihu¯ihuu¯i − ρ2ρ¯22ρ22ρ¯2 8

Volkov-Akulov model for Goldstino D. Volkov & V. Akulov (1972)

1 Z S [λ, λ¯] = d4x 1 − det Ξ , VA 2κ2

where κ denotes the coupling constant of dimension (length)2 and

b b 2 b b b ¯ b b ¯ Ξa = δa + κ (v +v ¯)a , va := iλσ ∂aλ , v¯a := −i∂aλσ λ .

SVA is invariant under the nonlinear supersymmetry transformations 1 δ λ = − iκλσb¯− σbλ¯∂ λ . ξ α κ α b α Roˇcek’sGoldstino action is related to the Volkov-Akulov model by a nonlinear ﬁled redeﬁnition (f −2 = 2κ2) SMK and S. Tyler (2011)

Chiral Goldstino superﬁeld coupled to supergravity U. Lindstr¨om& M. Roˇcek(1979)

Z 3 µ Z S = − d4xd2θd2d2θ¯E ΦΦ¯ + XX¯ + d4xd2θ E Φ3 + c.c. , κ2 κ2

where X is covariantly chiral, D¯α˙ X = 0, and obeys the super-Weyl invariant constraints 1 X 2 = 0 , − X (D¯2 − 4R)X¯ = f Φ2X . 4 Upon reducing the action to components and eliminating the supergravity auxiliary ﬁelds, for the cosmological constant one gets

|µ|2 Λ = f 2 − 3 . κ2

2 2 |µ| Cosmological constant is positive, Λ > 0, for f > 3 κ2 . Two introductory lectures on supergravity Sergei M. Kuzenko Introductory comments Weyl-invariant formulation for gravity Minkowski superspace and chiral superspace Gravitational superfield and conformal supergravity Superconformal compensators and supergravity Differential geometry for supergravity Super-Weyl invariance and conformal supergravity Reduction to component fields Compensators and off-shell formulations for supergravity Spontaneously broken (de Sitter) supergravity Alternative approaches to de Sitter supergravity

E. A. Bergshoeff, D. Z. Freedman, R. Kallosh and A. Van Proeyen, arXiv:1507.08264. F. Hasegawa and Y. Yamada, arXiv:1507.08619. SMK and S. J. Tyler, arXiv:1102.3042; SMK, arXiv:1508.03190. I. Bandos, L. Martucci, D. Sorokin and M. Tonin, arXiv:1511.03024. The first two groups used a nilpotent chiral Goldstino superfield

Z Z Z 4 2 2 4 2 4 2 S = d xd θd θ¯XX¯ + f d xd θ X + f d xd θ¯X¯ , D¯α˙ X = 0 ,

where X is constrained by X 2 = 0. R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio & R. Gatto (1989) Z. Komargodski & N. Seiberg (2009)

SMK & S. Tyler (2011)

1 − D¯2Σ = f , f = const , 4 1 Σ2 = 0 , − ΣD¯2D Σ = f D Σ . 4 α α The constraints imply that all component ﬁelds of Σ are constructed in terms of a single spinor ﬁeldρ ¯α˙ .

a √ iθσ θ∂¯ a 2 2 α α˙ 2 Σ(θ, θ¯) = e φ + θψ + 2θ¯ρ¯ + θ F + θ¯ f + θ θ¯ Uαα˙ + θ θ¯χ¯ . Goldstino superﬁeld action Z S[Σ, Σ]¯ = − d4xd2θd2θ¯ΣΣ¯ .

Roˇcek’sGoldstino superﬁeld is a composite object 1 f X = − D¯2(ΣΣ)¯ 4

Bryce DeWitt, Dynamical Theory of Groups and Fields (1965), about the status of Yang-Mills theories in the 1960s: “So far not a shred of experimental evidence exists that fields possessing non-Abelian infinite dimensional invariance groups play any role in physics at the quantum level. And yet motivation for studying such fields in a quantum context is not entirely lacking.” That situation completely changed in the early 1970s. Nowadays, no one doubts that the Yang-Mills theories play a crucial role in physics.

What does the future hold for supersymmetry and supergravity? Steven Weinberg, The Quantum Theory of Fields: Volume III: Supersymmetry (2000) “I and many physicists are reasonably conﬁdent that supersymmetry will be found to be relevant to the real world, and perhaps soon.”

Two introductory lectures on supergravity Sergei M. Kuzenko