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.

History

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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

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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

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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

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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

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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.

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Massive type IIA supergravity

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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

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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

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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

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  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.

References

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  1. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b 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. ^ a b Dall'Agata, G.; Zagermann, M. (2021). Supergravity: From First Principles to Modern Applications. Springer. ISBN 978-3662639788.
  10. ^ a b Townsend, P.K. (1995). "P-Brane Democracy". The World in Eleven Dimensions Supergravity, supermembranes and M-theory. CRC Press. ISBN 978-0750306720.
  11. ^ a b c 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. ^ a b c Ibanez, L.E.; Uranga, A.M. (2012). String Theory and Particle Physics: An Introduction to String Phenomenology. Cambridge University Press. ISBN 978-0521517522.
  13. ^ a b c Becker, K.; Becker, M.; Schwarz, J.H. (2006). String Theory and M-Theory: A Modern Introduction. Cambridge University Press. ISBN 978-0521860697.
  14. ^ a b c d e Ortin, T. (2015). Gravity and Strings (2 ed.). Cambridge: Cambridge University Press. ISBN 978-0521768139.
  15. ^ a b c 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.