Superconformal theory in a nutshell

starting point for the reduction procedure, which is now much more involved, due to compensating terms. Here is why we would like to find a more general short-cut.

We start pointing out that the zero-modes Ansatz (5.1.6) possesses the following gauge symmetry:

gµν(x) → e^−2σ(x)gµν(x) , (5.1.7) g_mn(x, y) → e^2σ(x)g_mn(x, y) , (5.1.8)

A(x, y) → A(x, y) + σ(x) . (5.1.9)

Also the dynamical spinorial Ansatz, generalisation of the background Ansatz (5.1.1), has a similar gauge symmetry. Indeed, under

ζ(x) → e^i/2α(x)ζ(x) , (5.1.10)

η(x, y) → e^−i/2α(x)η(x, y) (5.1.11) the ten-dimensional dynamical Weyl spinor (5.1.1) is invariant.

Since the starting background is supersymmetric, then the four-dimensional effective theory must be supersymmetric. Such a theory will inherit these gauge symmetries. In particular, under these transformations, Ω (5.1.2) transforms as ³

Ω → e3σ(x)−iα(x)

Ω . (5.1.12)

The invariance of the effective theory under this kind of transformation suggests to iden-tify the effective theory as a superconformal supergravity. As we will see in the next Section, this theory is characterised by a Weyl–chiral symmetry, i.e. it is invariant under field transformations of the kind (5.2.1). This is enough to deduce the form of the K¨ahler potential as we will see in Section 5.3. Before entering the argument it is better to review the main aspects of such a theory. For a detailed discussion we refer the reader to [66].

5.2. Superconformal theory in a nutshell

Four-dimensional superconformal theory is typically used as an underlying theory, a tool for constructing standard supergravities. It possesses more symmetries than standard

3γ_mnp in (5.1.2) in curved space are defined by sechsbeins, which have half Weyl weight with respect to g₆ (5.1.8).

5. An alternative to dimensional reduction

supersymmetric theories, based on the super-Poincar´e group, and has the peculiarity of describing the gravitational coupling M_P² as a the expectation value of a function of some scalar N (Φ, ¯Φ). One recovers the Poincar´e supergravity through an appropriate gauge-fixing, which breaks the local superconformal symmetries to the super-Poincar´e subgroup.

A superconformal theory enjoys the super-Poincar´e symmetry, which means invari-ance under general coordinate transformations, local Lorentz symmetry and local Q-supersymmetry. Additionally it possesses a chiral U (1) symmetry, local dilatation invari-ance, special conformal symmetry and S-supersymmetry (see Table 5.1) [66].

Symmetry g.c.t. Lorentz Q-susy U (1) chiral dilatations spec. conf. S-susy

Generator P_a M_ab Q T D K_a S

Gauge field e^a_µ ω_µ^ab ψ_µ A_µ b_µ f_µ^a χ_µ

Table 5.1.: Superconformal symmetries.

Not all gauge fields in Table 5.1 are independent. In particular, ω^ab_µ, f_µ^a and χ_µ are composite fields and the only independent fields are the vierbeins e^a_µ, the gravitino ψ_µ, the dilatation field b_µ and the U (1) gauge field A_µ. All these fields are collected in the Weyl multiplet.

The other superconformal multiplets in the theory are the chiral and the vector mul-tiplets. A chiral multiplet Φ is defined as in rigid ordinary supersymmetry, i.e. by a complex scalar Φ (as usual, we identify the multiplet with its scalar component), a spinor of defined chirality ψ and an auxiliary complex scalar F . A Yang–Mills vector multiplet is defined by the gauge field A_µ, a gaugino λ and the auxiliary complex scalar D. The gaugino and the auxiliary field stay in the adjoint of the gauge group, labelled by a.

A generic field φ of a multiplet has its proper dilatation and chiral weighs (w, c) (called also Weyl–chiral weights), i.e. it transforms under a dilatation and a chiral transformation as [66]:

φ → φ⁰ = ewσ(x)+icα(x)φ , (5.2.1) where σ(x), α(x) are the dilatation and chiral parameters respectively.

The superconformally invariant action of N = 1 supergravity conformally coupled to n + 1 chiral multiplets Φ^I, I = 0, ..., n of weights (1, −1/3), and some gauge multiplets is determined by three functions: a real function N (Φ, ¯Φ) and two holomorphic functions W(Φ), f_αβ(Φ), respectively encoding the K¨ahler potential, the superpotential and the

5.2. Superconformal theory in a nutshell gauge kinetic functions. The superconformal tensor calculus [67] leads to the complete component Lagrangian, whose bosonic part is ⁴

(− det g)^−1/2 L = 1

2N R + 3N_{I ¯J}∂_µΦ^I∂^µΦ¯^J

− W_I(N⁻¹)^{I ¯J}WJ¯− 1

2(Re f_ab)D^aD^b −1

4(Re f_ab)F_µν^a F^µνb+ ... ,

(5.2.2)

where R is the four-dimensional Ricci scalar, while NI ≡ _∂Φ^∂NI and so on.

The first term in the superconformal Lagrangian L (5.2.2) is invariant under dilatations and chiral symmetry if N transforms as

N → N e^2σ(x) , (5.2.3)

that is, if N transforms with opposite weights with respect to the metric g₄, which has weights (w, c) = (−2, 0). Choosing the chiral multiplets Φ to have weights (1, 0), one finds that N (Φ, ¯Φ) is a homogeneous function of order two.

One may obtain a theory invariant invariant under dilatations and chiral transforma-tions (5.2.1) by partially breaking the superconformal invariance, that is by breaking the special conformal symmetry and the S-supersymmetry. This is achieved by two suitable gauge choices, called ‘‘special conformal gauge” and ‘‘S-gauge” respectively [66].

Fixing also the gauges for dilatations and chiral transformations, one ends up with a theory invariant only under the super-Poincar´e group. This is evident operating the change of variables Φ^I → (Y, ϕⁱ), i = 1, ..., n, defined by

Φ^I = Y f^I(ϕⁱ) , (5.2.4)

where f^I(ϕⁱ) are arbitrary holomorphic functions. Y has weights (1, −1/3) while ϕⁱ’s have weights (0, 0). In this variables the scalar kinetic terms in (5.2.2) can be rewritten as [66]

3N_{I ¯J}∂_µΦ^I∂^µΦ¯^J¯= 3

4N⁻¹(∂_µN )²− N ∂_i∂_¯K ∂_µϕⁱ∂^µϕ¯^¯ . (5.2.5) This is a K¨ahlerian sigma model with a K¨ahler potential and metric [66]

K(ϕ, ¯ϕ) = −3 log N (ϕ, ¯ϕ)

|Y |²



, (5.2.6)

g_i¯ ≡ ∂_i∂_¯K = ∂²K

∂ϕⁱ∂ ¯ϕ^¯ , (5.2.7)

where N (ϕ, ¯ϕ) is a compact notation, which must be regarded as

4Our conventions differ from [66] by the redefinition N → −3N .

5. An alternative to dimensional reduction

N (ϕ, ¯ϕ)

|Y |² ≡ N_{I ¯J}f^I(ϕ) ¯f^J( ¯ϕ) . (5.2.8) The scalar fields ϕⁱ are the n complex coordinates parametrizing the K¨ahler manifold, while Y is the so-called ‘‘conformon”.

The transition to standard supergravity occurs by breaking dilatations, by fixing the

‘‘D-gauge”

D − gauge : N = M_P² , (5.2.9)

i.e. when the modulus of the conformon is

|Y |² = e^K/3M_P² . (5.2.10)

This means that there are only n physical degrees of freedom, corresponding to ϕⁱ’s, and that the conformal compensator Y is unphysical. With (5.2.10) the Lagrangian (5.2.2) becomes

1

2M_P²R − M_P²g_i¯ ∂_µϕⁱ∂^µϕ¯^¯+ ... . (5.2.11) The K¨ahler invariance follows from the non-uniqueness of the definition (5.2.4), which possesses the symmetry

Y⁰ = Y e^g(ϕ)/3 ,

f^0I = f^Ie^−g(ϕ)/3 , (5.2.12)

for an arbitrary holomorphic function g(ϕ), implying the invariance under

K → K⁰ = K + g(ϕ) + ¯g( ¯ϕ) . (5.2.13) Since W has weights (3, −1), it takes the form [66]:

W = Y³M_P⁻³W (ϕ) , (5.2.14)

where W (ϕ) is the superpotential of the supergravity theory. Since W depends on Φ^I (5.2.4), it is invariant under redefinitions (5.2.12). This in turn means that the superpo-tential W (ϕ) enjoys the K¨ahler transformation

W → W⁰ = W e^−g(ϕ) . (5.2.15)

The correct standard N = 1 supergravity in the Einstein frame is recovered finally gauging the U (1) chiral symmetry by the