Type IIA supergravity

Last updated • 10 min readFrom Wikipedia, The Free Encyclopedia

In supersymmetry, type IIA supergravity is the unique supergravity in ten dimensions with two supercharges of opposite chirality. It was first constructed in 1984 by a dimensional reduction of eleven-dimensional supergravity on a circle. [1] [2] [3] The other supergravities in ten dimensions are type IIB supergravity, which has two supercharges of the same chirality, and type I supergravity, which has a single supercharge. In 1986 a deformation of the theory was discovered which gives mass to one of the fields and is known as massive type IIA supergravity. [4] Type IIA supergravity plays a very important role in string theory as it is the low-energy limit of type IIA string theory.

Contents

History

After supergravity was discovered in 1976 with pure 4D supergravity, significant effort was devoted to understanding other possible supergravities that can exist with various numbers of supercharges and in various dimensions. The discovery of eleven-dimensional supergravity in 1978 led to the derivation of many lower dimensional supergravities through dimensional reduction of this theory. [5] Using this technique, type IIA supergravity was first constructed in 1984 by three different groups, by F. Giani and M. Pernici, [1] by I.C.G. Campbell and P. West, [2] and by M. Huq and M. A. Namazie. [3] In 1986 it was noticed by L. Romans that there exists a massive deformation of the theory. [4] Type IIA supergravity has since been extensively used to study the low-energy behaviour of type IIA string theory. The terminology of type IIA, type IIB, and type I was coined by J. Schwarz, originally to refer to the three string theories that were known of in 1982. [6]

Theory

Ten dimensions admits both and supergravity, depending on whether there are one or two supercharges. [nb 1] Since the smallest spinorial representations in ten dimensions are MajoranaWeyl spinors, the supercharges come in two types depending on their chirality, giving three possible supergravity theories. [7] :241 The theory formed using two supercharges of opposite chiralities is denoted by and is known as type IIA supergravity.

This theory contains a single multiplet, known as the ten-dimensional nonchiral multiplet. The fields in this multiplet are , where is the metric corresponding to the graviton, while the next three fields are the 3-, 2-, and 1-form gauge fields, with the 2-form being the Kalb–Ramond field. [8] There is also a Majorana gravitino and a Majorana spinor , both of which decompose into a pair of Majorana–Weyl spinors of opposite chiralities and . Lastly, there a scalar field .

This nonchiral multiplet can be decomposed into the ten-dimensional multiplet , along with four additional fields . [9] :269 [nb 2] In the context of string theory, the bosonic fields in the first multiplet consists of NSNS fields while the bosonic fields are all RR fields. The fermionic fields are meanwhile in the NSR sector.

Algebra

The superalgebra for supersymmetry is given by [10]

where all terms on the right-hand side besides the first one are the central charges allowed by the theory. Here are the spinor components of the Majorana supercharges [nb 3] while is the charge conjugation operator. Since the anticommutator is symmetric, the only matrices allowed on the right-hand side are ones that are symmetric in the spinor indices , . In ten dimensions is symmetric only for modulo , with the chirality matrix behaving as just another matrix, except with no index. [7] :47–48 Going only up to five-index matrices, since the rest are equivalent up to Poincare duality, yields the set of central charges described by the above algebra.

The various central charges in the algebra correspond to different BPS states allowed by the theory. In particular, the , and correspond to the D0, D2, and D4 branes. [10] The corresponds to the NSNS 1-brane, which is equivalent to the fundamental string, while corresponds to the NS5-brane.

Action

The type IIA supergravity action is given up to four-fermion terms by [11]

Here and where corresponds to a -form gauge field. [nb 4] The 3-form gauge field has a modified field strength tensor with this having a non-standard Bianchi identity of . [12] :115 [nb 5] Meanwhile, , , , and are various fermion bilinears given by [11]

The first line of the action has the Einstein–Hilbert action, the dilaton kinetic term [nb 6] , the 2-form field strength tensor. It also contains the kinetic terms for the gravitino and spinor , described by the Rarita–Schwinger action and Dirac action, respectively. The second line has the kinetic terms for the 1-form and 3-form gauge fields as well as a Chern–Simons term. The last line contains the cubic interaction terms between two fermions and a boson.

Supersymmetry transformations

The supersymmetry variations that leave the action invariant are given up to three-fermion terms by [11] [14] :665 [nb 7]

They are useful for constructing the Killing spinor equations and finding the supersymmetric ground states of the theory since these require that the fermionic variations vanish.

Massive type IIA supergravity

Since type IIA supergravity has p-form field strengths of even dimensions, it also admits a nine-form gauge field . But since is a scalar and the free field equation is given by , this scalar must be a constant. [12] :115 Such a field therefore has no propagating degrees of freedom, but does have an energy density associated to it. Working only with the bosonic sector, the ten-form can be included in supergravity by modifying the original action to get massive type IIA supergravity [15] :89–90

where is equivalent to the original type IIA supergravity up to the replacement of and . Here is known as the Romans mass and it acts as a Lagrange multiplier for . Often one integrates out this field strength tensor resulting in an action where acts as a mass term for the Kalb–Ramond field.

Unlike in the regular type IIA theory, which has a vanishing scalar potential , massive type IIA has a nonvanishing scalar potential. While the supersymmetry transformations appear to be realised, they are actually formally broken since the theory corresponds to a D8-brane background. [14] :668 A closely related theory is Howe–Lambert–West supergravity [16] which is another massive deformation of type IIA supergravity, [nb 8] but one that can only be described at the level of the equations of motion. It is acquired by a compactification of eleven-dimensional MM theory on a circle.

Relation to 11D supergravity

Compactification of eleven-dimensional supergravity on a circle and keeping only the zero Fourier modes that are independent of the compact coordinates results in type IIA supergravity. For eleven-dimensional supergravity with the graviton, gravitino, and a 3-form gauge field denoted by , then the 11D metric decomposes into the 10D metric, the 1-form, and the dilaton as [13] :308

Meanwhile, the 11D 3-form decomposes into the 10D 3-form and the 10D 2-form . The ten-dimensional modified field strength tensor directly arises in this compactification from .

Dimensional reduction of the fermions must generally be done in terms of the flat coordinates , where is the 11D vielbein. [nb 9] In that case the 11D Majorana graviton decomposes into the 10D Majorana gravitino and the Majorana fermion , [9] :268 [nb 10] although the exact identification is given by [14] :664

where this is chosen to make the supersymmetry transformations simpler. [nb 11] The ten-dimensional supersymmetry variations can also be directly acquired from the eleven-dimensional ones by setting . [nb 12]

Relation to type IIA string theory

The low-energy effective field theory of type IIA string theory is given by type IIA supergravity. [15] :187 The fields correspond to the different massless excitations of the string, with the metric, 2-form , and dilaton being NSNS states that are found in all string theories, while the 3-form and 1-form fields correspond to the RR states of type IIA string theory. Corrections to the type IIA supergravity action come in two types, quantum corrections in powers of the string coupling , and curvature corrections in powers of . [15] :321–324 Such corrections often play an important role in type IIA string phenomenology. The type IIA superstring coupling constant corresponds to the vacuum expectation value of , while the string length is related to the gravitational coupling constant through . [12] :115

When string theory is compactified to acquire four-dimensional theories, this is often done at the level of the low-energy supergravity. Reduction of type IIA on a Calabi–Yau manifold yields an theory in four dimensions, while reduction on a Calabi–Yau orientifold further breaks the symmetry down to give the phenomenologically viable four-dimensional supergravity. [13] :356–357 Type IIA supergravity is automatically anomaly free since it is a non-chiral theory.

Notes

  1. This is equivalent to whether there are one or two gravitinos present in the theory.
  2. One can flip all the chiralities and still get an equivalent theory.
  3. The Majorana supercharges decompose into two Majorana–Weyl spinors of opposite chiralities .
  4. Sometimes the notation is used to write the canonically normalized kinetic term for the gauge fields.
  5. The Bianchi identity for the other field-strength tensors is simply .
  6. The dilaton kinetic term appears to not be canonically normalized, but this is because it is in the string frame. [13] :311 Performing a Weyl transformation into the Einstein frame would result in a canonically normalized dilaton kinetic term.
  7. The action and supersymmetry variations depend on the metric signature used. Transforming from a mainily positive signature, [14] denoted by primes, to a mainly negative one used in this article can be done through implying that , , and . Additionally, the fields are often redefined as , , , , , .
  8. They are the only two massive deformations possible. [17]
  9. Using the aforementioned metric, the vielbein can be written in terms of the 10d vielbein, the gauge field, and the dilaton as . [14] :656 This is a special gauge with , which has to be accounted for when deriving the 10d supersymmetry variations from the 11d ones.
  10. Each Majorana spinor decomposes into the two Majorana–Weyl spinors of opposing chirality, with the ten-dimensional chirality matrix being one of the eleven-dimensional gamma matrices .
  11. Note that is the 11th flat component, not the 11th spacetime component.
  12. For example, the 11d vielbein transforms as , so using that and , one can get both the supersymmetry variation of the 10d vielbein and the dilatino.

Related Research Articles

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way. It has become vital in the building of the Standard Model.

<span class="mw-page-title-main">Stress–energy tensor</span> Tensor describing energy momentum density in spacetime

The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity.

In the general theory of relativity, the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

<span class="mw-page-title-main">Electromagnetic tensor</span> Mathematical object that describes the electromagnetic field in spacetime

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written concisely, and allows for the quantization of the electromagnetic field by the Lagrangian formulation described below.

In mathematical physics, the gamma matrices, 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 It is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space, the column vectors on which the matrices act become a space of spinors, on which the Clifford algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in particular are fundamental to the Dirac equation for relativistic spin particles. Gamma matrices were introduced by Paul Dirac in 1928.

In differential geometry and mathematical physics, a spin connection is a 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 generated by local Lorentz transformations. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local rotations.

In physics, the gauge covariant derivative is a means of expressing how fields vary from place to place, in a way that respects how the coordinate systems used to describe a physical phenomenon can themselves change from place to place. The gauge covariant derivative is used in many areas of physics, including quantum field theory and fluid dynamics and in a very special way general relativity.

<span class="mw-page-title-main">Maxwell's equations in curved spacetime</span> Electromagnetism in general relativity

In physics, Maxwell's equations in curved spacetime govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that incorporates the energy–momentum of gravity. It allows the energy–momentum of a system of gravitating matter to be defined. In particular it allows the total of matter plus the gravitating energy–momentum to form a conserved current within the framework of general relativity, so that the total energy–momentum crossing the hypersurface of any compact space–time hypervolume vanishes.

The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime.

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables.

In the Newman–Penrose (NP) formalism of general relativity, independent components of the Ricci tensors of a four-dimensional spacetime are encoded into seven Ricci scalars which consist of three real scalars , three complex scalars and the NP curvature scalar . Physically, Ricci-NP scalars are related with the energy–momentum distribution of the spacetime due to Einstein's field equation.

<span class="mw-page-title-main">Dirac equation in curved spacetime</span> Generalization of the Dirac equation

In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime to curved spacetime, a general Lorentzian manifold.

In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics. It is a special case of 4D N = 1 global supersymmetry.

In supersymmetry, 4D supergravity is the theory of supergravity in four dimensions with a single supercharge. It contains exactly one supergravity multiplet, consisting of a graviton and a gravitino, but can also have an arbitrary number of chiral and vector supermultiplets, with supersymmetry imposing stringent constraints on how these can interact. The theory is primarily determined by three functions, those being the Kähler potential, the superpotential, and the gauge kinetic matrix. Many of its properties are strongly linked to the geometry associated to the scalar fields in the chiral multiplets. After the simplest form of this supergravity was first discovered, a theory involving only the supergravity multiplet, the following years saw an effort to incorporate different matter multiplets, with the general action being derived in 1982 by Eugène Cremmer, Sergio Ferrara, Luciano Girardello, and Antonie Van Proeyen.

In supersymmetry, eleven-dimensional supergravity is the theory of supergravity in the highest number of dimensions allowed for a supersymmetric theory. It contains a graviton, a gravitino, and a 3-form gauge field, with their interactions uniquely fixed by supersymmetry. Discovered in 1978 by Eugène Cremmer, Bernard Julia, and Joël Scherk, it quickly became a popular candidate for a theory of everything during the 1980s. However, interest in it soon faded due to numerous difficulties that arise when trying to construct physically realistic models. It came back to prominence in the mid-1990s when it was found to be the low energy limit of M-theory, making it crucial for understanding various aspects of string theory.

In supersymmetry, type IIB supergravity is the unique supergravity in ten dimensions with two supercharges of the same chirality. It was first constructed in 1983 by John Schwarz and independently by Paul Howe and Peter West at the level of its equations of motion. While it does not admit a fully covariant action due to the presence of a self-dual field, it can be described by an action if the self-duality condition is imposed by hand on the resulting equations of motion. The other types of supergravity in ten dimensions are type IIA supergravity, which has two supercharges of opposing chirality, and type I supergravity, which has a single supercharge. The theory plays an important role in modern physics since it is the low-energy limit of type IIB string theory.

In supersymmetry, type I supergravity is the theory of supergravity in ten dimensions with a single supercharge. It consists of a single supergravity multiplet and a single Yang–Mills multiplet. The full non-abelian action was first derived in 1983 by George Chapline and Nicholas Manton. Classically the theory can admit any gauge group, but a consistent quantum theory resulting in anomaly cancellation only exists if the gauge group is either or . Both these supergravities are realised as the low-energy limits of string theories, in particular of type I string theory and of the two heterotic string theories.

References

  1. 1 2 Giani, F.; Pernici, M. (1984). "$N=2$ supergravity in ten dimensions". Phys. Rev. D. 30 (2): 325–333. doi:10.1103/PhysRevD.30.325.
  2. 1 2 Campbell, I.C.G.; West, P.C. (1984). "N = 2, D = 10 non-chiral supergravity and its spontaneous". Nuclear Physics B. 243 (1): 112–124. doi:10.1016/0550-3213(84)90388-2.
  3. 1 2 Huq, M.; Namazie, M.A. (1985). "{Kaluza-Klein} Supergravity in Ten-dimensions". Class. Quant. Grav. 2: 293. doi:10.1088/0264-9381/2/3/007.
  4. 1 2 Romans, L.J. (1986). "Massive N = 2a supergravity in ten dimensions". Physics Letters B. 169 (4): 374–380. doi:10.1016/0370-2693(86)90375-8.
  5. Cremmer, E.; Julia, B.; Scherk, J. (1978). "Supergravity Theory in Eleven-Dimensions". Phys. Lett. B. 76: 409–412. doi:10.1016/0370-2693(78)90894-8.
  6. Schwarz, J.H. (1982). "Superstring theory". Physics Reports. 89 (3): 223–322. doi:10.1016/0370-1573(82)90087-4.
  7. 1 2 Freedman, D.Z.; Van Proeyen, A. (2012). Supergravity. Cambridge: Cambridge University Press. ISBN   978-0521194013.
  8. Sezgin, E. (2023). "Survey of supergravities". arXiv: 2312.06754 [hep-th].
  9. 1 2 Dall'Agata, G.; Zagermann, M. (2021). Supergravity: From First Principles to Modern Applications. Springer. ISBN   978-3662639788.
  10. 1 2 Townsend, P.K. (1995). "P-Brane Democracy". The World in Eleven Dimensions Supergravity, supermembranes and M-theory. CRC Press. ISBN   978-0750306720.
  11. 1 2 3 Bergshoeff, E.; Kallosh, R.; Ortin, T.; Roest, D.; Van Proeyen, A. (2001). "New formulations of D = 10 supersymmetry and D8 - O8 domain walls". Class. Quant. Grav. 18: 3359–3382. arXiv: hep-th/0103233 . doi:10.1088/0264-9381/18/17/303.
  12. 1 2 3 Ibanez, L.E.; Uranga, A.M. (2012). String Theory and Particle Physics: An Introduction to String Phenomenology. Cambridge University Press. ISBN   978-0521517522.
  13. 1 2 3 Becker, K.; Becker, M.; Schwarz, J.H. (2006). String Theory and M-Theory: A Modern Introduction. Cambridge University Press. ISBN   978-0521860697.
  14. 1 2 3 4 5 Ortin, T. (2015). Gravity and Strings (2 ed.). Cambridge: Cambridge University Press. ISBN   978-0521768139.
  15. 1 2 3 Polchinski, J. (1998). String Theory Volume II: Superstring Theory and Beyond. Cambridge University Press. ISBN   978-1551439761.
  16. Howe, P.S.; Lambert, N.D.; West, P.C. (1998). "A New massive type IIA supergravity from compactification". Phys. Lett. B. 416: 303–308. arXiv: hep-th/9707139 . doi:10.1016/S0370-2693(97)01199-4.
  17. Tsimpis, D. (2005). "Massive IIA supergravities". JHEP. 10: 057. doi:10.1088/1126-6708/2005/10/057.