Pure 4D N = 1 Supergravity
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In
supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...
, pure 4D \mathcal N=1 supergravity describes the simplest
four-dimensional Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...
supergravity In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
, with a single
supercharge In theoretical physics, a supercharge is a generator of supersymmetry transformations. It is an example of the general notion of a charge (physics), charge in physics. Supercharge, denoted by the symbol Q, is an operator which transforms bosons in ...
and a
supermultiplet In theoretical physics, a supermultiplet is a representation of a supersymmetry algebra, possibly with extended supersymmetry. Then a superfield is a field on superspace which is valued in such a representation. Naïvely, or when considering ...
containing a
graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...
and
gravitino In supergravity theories combining general relativity and supersymmetry, the gravitino () is the gauge fermion supersymmetric partner of the hypothesized graviton. It has been suggested as a candidate for dark matter. If it exists, it is a f ...
. The
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
consists of the
Einstein–Hilbert action The Einstein–Hilbert action in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the action is given as :S = \int R \sqrt ...
and the Rarita–Schwinger action. The theory was first formulated by
Daniel Z. Freedman Daniel Zissel Freedman (born 1939 in Hartford, Connecticut) is an American theoretical physicist. He is an Emeritus Professor of Physics and Applied Mathematics at the Massachusetts Institute of Technology (MIT), and is currently a visiting profes ...
,
Peter van Nieuwenhuizen Peter van Nieuwenhuizen (; born October 26, 1938) is a Dutch theoretical physicist. He is a distinguished Professor at Stony Brook University in the United States. Widely known for his contributions to String theory, Supersymmetry, Supergrav ...
, and
Sergio Ferrara Sergio Ferrara (born 2 May 1945) is an Italian physicist working on theoretical physics of elementary particles and mathematical physics. He is renowned for the discovery of theories introducing supersymmetry as a symmetry of elementary particles ( ...
, and independently by
Stanley Deser Stanley Deser (March 19, 1931 – April 21, 2023) was an American physicist known for his contributions to general relativity. He was an emeritus Ancell Professor of Physics at Brandeis University in Waltham, Massachusetts and a senior researc ...
and
Bruno Zumino Bruno Zumino (28 April 1923 − 21 June 2014) was an Italian theoretical physicist and faculty member at the University of California, Berkeley. He obtained his DSc degree from the University of Rome in 1945. He was renowned for his rigorous pro ...
in 1976. The only consistent extension to
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
s with a
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is a coefficient that Albert Einstein initially added to his field equations of general rel ...
is to
anti-de Sitter space In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a symmetric_space, maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are na ...
, first formulated by Paul Townsend in 1977. When additional matter supermultiplets are included in this theory, the result is known as matter-coupled 4D \mathcal N = 1 supergravity.


Flat spacetime

To describe the coupling between
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
and
particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
s of arbitrary
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
, it is useful to use the vielbein formalism of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
. This replaces the
metric Metric or metrical may refer to: Measuring * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics ...
by a set of
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
s e_a = e^\mu_a \partial_\mu indexed by flat indices a such that : g_ = e^a_\mu e^b_\nu \eta_. In a sense the vielbeins are the square root of the metric. This introduces a new
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Bria ...
Lorentz symmetry In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the sam ...
on the vielbeins e^a_\mu \rightarrow e^b_\mu \Lambda^a_b(x), together with the usual
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...
invariance associated with the spacetime indices \mu. This has an associated
connection Connection may refer to: Mathematics *Connection (algebraic framework) *Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold * Connection (affine bundle) *Connection (composite bun ...
known as the
spin connection In differential geometry and mathematical physics, a spin connection is a connection (vector bundle), connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field gene ...
\omega^_\mu defined through \nabla_\mu e_a = \omega_\mu^_a e_b, it being a generalization of the Christoffel connection to arbitrary spin fields. For example, for
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
s the
covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...
is given by : D_\mu = \partial_\mu + \frac\omega_\mu^\gamma_, where \gamma_a are
gamma matrices In mathematical physics, the gamma matrices, \ \left\\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra \ \mathr ...
satisfing the
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin- particles with a matrix representation of the ...
, with \gamma_ = \gamma_\gamma_. These are often contracted with vielbeins to construct \gamma_\mu = e^a_\mu \gamma_a which are in general position-dependent fields rather than constants. The spin connection has an explicit expression in terms of the vielbein and an additional
torsion tensor In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors X,Y, that produces an output vector T(X,Y) representing the displacement within a t ...
which can arise when there is matter present in the theory. A vanishing torsion is equivalent to the
Levi-Civita connection In Riemannian or pseudo-Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold that preserves the ( pseudo-) Riemannian ...
. The pure \mathcal N=1 supergravity action in four dimensions is the combination of the Einstein–Hilbert action and the Rarita–Schwinger action Here M_P is the
Planck mass In particle physics and physical cosmology, Planck units are a system of units of measurement defined exclusively in terms of four universal physical constants: '' c'', '' G'', '' ħ'', and ''k''B (described further below). Expressing one of ...
, e = \det e^a_\mu=\sqrt, and \psi_\mu is the Majorana gravitino with its spinor index left implicit. Treating this action within the first-order formalism where both the vielbein and spin connection are independent fields allows one to solve for the spin connections equation of motion, showing that it has the torsion T^\mu_ = \tfrac\bar \psi_a\gamma^\mu \psi_b. The second-order formalism action is then acquired by substituting this expression for the spin connection back into the action, yielding additional quartic gravitino vertices, with the Einstein–Hilbert and Rarita–Schwinger actions now being written with a torsionless spin connection that explicitly depends on the vielbeins. The supersymmetry transformation rules that leave the action invariant are : \delta e^a_\mu = \tfrac\bar \epsilon \gamma^a \psi_\mu, \ \ \ \ \ \ \ \ \delta \psi_\mu = D_\mu \epsilon, where \epsilon(x) is the spinorial gauge parameter. While historically the first order and second order formalism were the first ones used to show the invariance of the action, the 1.5-order formalism is the easiest for most supergravity calculations. The additional
symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
of the action are general coordinate transformations and local Lorentz transformations.


Curved spacetime

The four dimensional \mathcal N=1 super-Poincare algebra in
Minkowski spacetime In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a s ...
can be generalized to anti-de Sitter spacetime, but not to
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often denoted dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canonical Rie ...
time, since the super-
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
cannot be satisfied in that case. Its action can be constructed by gauging this
superalgebra In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. T ...
, yielding the supersymmetry transformation rules for the vielbein and the gravitino. The action for \mathcal N=1 AdS supergravity in four dimensions is : S = \frac \int d^4 x \ e \bigg(R+ \frac\bigg) - \frac\int d^4 x \ e \bigg(\bar \psi_\mu \gamma^D_\nu \psi_\rho + \frac\bar \psi_\mu \gamma^ \psi_\nu\bigg), where L is the AdS radius and the second term is the negative cosmological constant \Lambda = -3/L^2. The supersymmetry transformations are : \delta e^a_\mu = \tfrac\bar \epsilon \gamma^a \psi_\mu, \ \ \ \ \ \ \delta \psi_\mu = D_\mu \epsilon - \tfrac \gamma_\mu \epsilon. While the bilinear term in the action appears to be giving a
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
to the gravitino, it still belongs to the
massless In particle physics, a massless particle is an elementary particle whose invariant mass is zero. At present the only confirmed massless particle is the photon. Other particles and quasiparticles Standard Model gauge bosons The photon (carrier of ...
gravity supermultiplet. This is because mass is not well-defined in
curved space Curved space often refers to a spatial geometry which is not "flat", where a '' flat space'' has zero curvature, as described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry, though some simple cases can be ...
times, with P_\mu P^\mu no longer being a
Casimir operator In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operato ...
of the AdS super-Poinacre algebra. It is however conventional to define a mass through the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...
, in which case particles within the same supermultiplet have different masses, unlike in flat spacetimes.


See also

* ''N'' = 8 supergravity


References

{{DEFAULTSORT:Pure 4D N = 1 supergravity Supersymmetric quantum field theory Theories of gravity