Type I supergravity#Algebra

Supergravity was much studied during the 1980s as a candidate theory of nature. As part of this it was important to understand the various supergravities that can exist in different dimensions, with the possible supergravities being classified in 1978 by Werner Nahm.{{cite journal|last1=Nahm|first1=W.|authorlink1=Werner Nahm|date=1978|title=Supersymmetries and their representations|url=https://dx.doi.org/10.1016/0550-3213%2878%2990218-3|journal=Nuclear Physics B|volume=135|issue=1|pages=149–166|doi=10.1016/0550-3213(78)90218-3|pmid=|arxiv=|bibcode=1978NuPhB.135..149N |s2cid=|access-date=}} Type I supergravity was first written down in 1983, with Eric Bergshoeff, Mees de Roo, Bernard de Wit, and Peter van Nieuwenhuizen describing the abelian theory,{{cite journal|last1=Bergshoeff|first1=E.|authorlink1=|last2=De Roo|first2=M.|authorlink2=|last3=De Wit|first3=B.|authorlink3=Bernard de Wit|last4=Van Nieuwenhuizen|first4=P.|authorlink4=Peter van Nieuwenhuizen|date=1982|title=Ten-dimensional Maxwell-Einstein supergravity, its currents, and the issue of its auxiliary fields|url=https://dx.doi.org/10.1016/0550-3213%2882%2990050-5|journal=Nuclear Physics B|volume=195|issue=1|pages=97–136|doi=10.1016/0550-3213(82)90050-5|pmid=|arxiv=|s2cid=|access-date=}} and then George Chapline and Nicholas Manton extending this to the full non-abelian theory. An important development was made by Michael Green and John Schwarz in 1984 when they showed that only a handful of these theories are anomaly free,{{cite journal|last1=Green|first1=M.B.|authorlink1=Michael Green (physicist)|last2=Schwarz|first2=J.H.|authorlink2=John Henry Schwarz|date=1984|title=Anomaly cancellations in supersymmetric D = 10 gauge theory and superstring theory|url=https://dx.doi.org/10.1016/0370-2693%2884%2991565-X|journal=Physics Letters B|volume=149|issue=1–3|pages=117–122|doi=10.1016/0370-2693(84)91565-X|pmid=|arxiv=|s2cid=|access-date=|url-access=subscription}} with additional work showing that only $\backslash text\{SO\}(32)$ and $E\_8\backslash times\; E\_8$ result in a consistent quantum theory.{{cite journal|last1=Adams|first1=Allan|authorlink1=Allan Adams|last2=DeWolfe|first2=O.|authorlink2=|last3=Taylor|first3=W.|authorlink3=|date=2010|title=String universality in ten dimensions|url=|journal=Phys. Rev. Lett.|volume=105|issue=7|pages=071601|doi=10.1103/PhysRevLett.105.071601|pmid=20868028|arxiv=1006.1352|bibcode=2010PhRvL.105g1601A |s2cid=|access-date=}} The first case was known at the time to correspond to the low-energy limit of type I superstrings. Heterotic string theories were discovered the next year,{{cite journal|last1=Gross|first1=D.J.|authorlink1=David Gross|last2=Harvey|first2=J.A.|authorlink2=Jeffrey A. Harvey|last3=Martinec|first3=M.|authorlink3=Emil Martinec|last4=Rohm|first4=R.|authorlink4=Ryan Rohm|date=1986|title=Heterotic string theory: (II). The interacting heterotic string|url=https://dx.doi.org/10.1016/0550-3213%2886%2990146-X|journal=Nuclear Physics B|volume=267|issue=1|pages=75–124|doi=10.1016/0550-3213(86)90146-X|pmid=|arxiv=|bibcode=1986NuPhB.267...75G |s2cid=|access-date=|url-access=subscription}} with these having a low-energy limit described by type I supergravity with both gauge groups.

Type I supergravity is the ten-dimensional supergravity with a single Majorana–Weyl spinor supercharge.{{refn|group=nb|This supergravity is sometimes written as $\backslash mathcal\; N=(1,0)$ or $\backslash mathcal\; N=(0,1)$ supergravity to indicate the chirality of the supercharge, with these two theories being equivalent up to a chirality transformation.}} Its field content consists of the $\backslash mathcal\; N=1$ supergravity supermultiplet $(g\_\{\backslash mu\backslash nu\},\; \backslash psi\_\backslash mu,\; B,\; \backslash lambda,\; \backslash phi)$, together with the $\backslash mathcal\; N=1$ Yang–Mills supermultiplet $(A\_\backslash mu^a,\; \backslash chi^a)$ with some associated gauge group.{{cite book|last1=Dall'Agata|first1=G.|author-link1=|last2=Zagermann|first2=M.|author-link2=|date=2021|title=Supergravity: From First Principles to Modern Applications|url=|doi=|location=|publisher=Springer|chapter=|page=|isbn=978-3662639788}}{{rp|271}} Here $g\_\{\backslash mu\backslash nu\}$ is the metric, $B$ is the two-form Kalb–Ramond field, $\backslash phi$ is the dilaton, and $A\_\backslash mu^a$ is a Yang–Mills gauge field.{{cite book|last1=Becker|first1=K.|author-link1=|last2=Becker|first2=M.|author-link2=Melanie Becker|last3=Schwarz|first3=J.H.|author-link3=John Henry Schwarz|date=2006|title=String Theory and M-Theory: A Modern Introduction|url=|doi=|location=|publisher=Cambridge University Press|chapter=|page=|isbn=978-0521860697}}{{rp|317–318}} Meanwhile, $\backslash psi\_\backslash mu$ is the gravitino, $\backslash lambda$ is a dilatino, and $\backslash chi^a$ a gaugino, with all these being Majorana–Weyl spinors. The gravitino and gaugino have the same chirality, while the dilatino has the opposite chirality.

The superalgebra for type I supersymmetry is given by{{cite book|last=Townsend|first=P.K.|author-link=Paul Townsend|date=1995|title=The World in Eleven Dimensions Supergravity, supermembranes and M-theory|url=|doi=|location=|publisher=CRC Press|chapter=P-Brane Democracy|page=|isbn=978-0750306720}}

:$$

\{Q_\alpha, Q_\beta\} = (P\gamma^\mu C)_{\alpha \beta}P_\mu + (P\gamma^{\mu\nu\rho\sigma\delta}C)_{\alpha \beta}Z_{\mu\nu\rho\sigma \delta}.

Here $Q\_\backslash alpha$ is the supercharge with a fixed chirality $PQ\_\backslash alpha\; =\; Q\_\backslash alpha$, where $P\; =\; \backslash tfrac\{1\}\{2\}(1\backslash pm\; \backslash gamma\_*)$ is the relevant projection operator. Meanwhile, $C$ is the charge conjugation operator and $\backslash gamma^\{\backslash mu\}$ are the gamma matrices. The right-hand side must have the same chirality as the supercharges and must also be symmetric under an exchange of the spinor indices. The second term is the only central charge that is admissible under these constraints up to Poincare duality. This is because in ten dimensions only $P\backslash gamma^\{\backslash mu\_1\; \backslash cdots\; \backslash mu\_p\}C$ with $p=1$ modulo $4$ are symmetric matrices.{{cite book|last1=Freedman|first1=D.Z.|author-link1=Daniel Z. Freedman|last2=Van Proeyen|first2=A.|author-link2=|date=2012|title=Supergravity|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=|page=|isbn=978-0521194013}}{{rp|37–48}}{{refn|group=nb|There is no central charge for the first case since it is equivalent to a redefinition $P\_\backslash mu\; \backslash sim\; P\_\backslash mu+Z\_\backslash mu$.}} The central charge corresponds to a 5-brane solution in the supergravity which is dual to the fundamental string in heterotic string theory.{{cite journal|last1=Strominger|first1=A.|authorlink1=|date=1990|title=Heterotic solitons|url=|journal=Nucl. Phys. B|volume=343|issue=|pages=167–184|doi=10.1016/0550-3213(90)90599-9|pmid=|arxiv=|s2cid=|access-date=}}

The action for type I supergravity in the Einstein frame is given up to four-fermion terms by{{cite book|last1=Green|first1=M.|author-link1=Michael Green (physicist)|last2=Schwarz|first2=J.H.|author-link2=John Henry Schwarz|last3=Witten|first3=E.|author-link3=Edward Witten|date=1988|title=Superstring Theory: 25th Anniversary Edition: Volume 2|url=|doi=|location=|publisher=Cambridge University Press|chapter=|page=|isbn=978-1107029132}}{{rp|325}}{{refn|group=nb|The fields have been rescaled from Green, Schwartz, Witten, as $\backslash phi^\{\backslash text\{GSW\}\}\; =\; e^\backslash phi$, $H\_\{\backslash mu\backslash nu\backslash rho\}^\{\backslash text\{GSW\}\}\; =\; \backslash tfrac\{g^2\}\{\backslash kappa^2\}H\_\{\backslash mu\backslash nu\backslash rho\}$, $B\_\{\backslash mu\backslash nu\}^\{\backslash text\{GSW\}\}\; =\; \backslash tfrac\{g^2\}\{\backslash kappa^2\}B\_\{\backslash mu\backslash nu\}$, along with a rescaling of all fermions by a factor of $\backslash kappa^\{-1\}$.}}

:$$

S = \frac{1}{2\kappa^2}\int d^{10} x \ e \bigg[ R -2\partial_\mu \phi \partial^\mu \phi -\tfrac{3}{4} e^{-2\phi}H_{\mu\nu\rho}H^{\mu\nu\rho} - \tfrac{\kappa^2}{2g^2}e^{-\phi}\text{tr}(F_{\mu\nu}F^{\mu\nu})

:$$

\ \ \ -\bar \psi_\mu \gamma^{\mu\nu\rho}D_\nu \psi_\rho -\bar \lambda \gamma^\mu D_\mu \lambda - \text{tr}(\bar \chi \gamma^\mu D_\mu \chi)

:$$

\ \ \ -\sqrt 2\bar \psi_\mu \gamma^\nu \gamma^\mu \lambda \partial_\nu \phi + \tfrac{1}{8}e^{-\phi}\text{tr}(\bar \chi \gamma^{\mu\nu\rho}\chi)H_{\mu\nu\rho}

:$$

\ \ \ -\tfrac{\kappa}{2g}e^{-\phi/2}\text{tr}[\bar \chi \gamma^\mu \gamma^{\nu\rho}(\psi_\mu + \tfrac{\sqrt 2}{12}\gamma_\mu \lambda)F_{\nu\rho}]

:$$

\ \ \ + \tfrac{1}{8}e^{-\phi}(\bar \psi_\mu \gamma^{\mu\nu\rho\sigma\delta}\psi_\delta + 6 \bar \psi^\nu \gamma^\rho \psi^\sigma - \sqrt 2 \bar \psi_\mu \gamma^{\nu\rho \sigma}\gamma^\mu \lambda)H_{\nu\rho\sigma}\bigg].

Here $\backslash kappa^2$ is the gravitational coupling constant, $\backslash phi$ is the dilaton, and{{cite book|last=Polchinski|first=J.|author-link=Joseph Polchinski|date=1998|title=String Theory Volume II: Superstring Theory and Beyond|url=|doi=|location=|publisher=Cambridge University Press|chapter=|page=|isbn=978-1551439761}}{{rp|92–93}}

:$$

H_{\mu\nu\rho} =\partial_{[\mu} B_{\nu\rho]} - \tfrac{\kappa^2}{g^2}\omega_{\text{YM},\mu\nu\rho},

where $\backslash omega\_\{\backslash text\{YM\}\}$ is the trace of the Yang–Mills Chern–Simons form given by

:$$

\omega_{\text{YM}} = \text{tr}(A \wedge dA +\tfrac{2}{3}A\wedge A \wedge A).

The non-abelian field strength tensor corresponding to the gauge field $A\_\backslash mu$ is denote by $F\_\{\backslash mu\backslash nu\}$. The spacetime index gamma-matrices are position-dependent fields $\backslash gamma\_\backslash mu\; =\; e\_\backslash mu^a\; \backslash gamma\_a$. Meanwhile, $D\_\backslash mu$ is the covariant derivative $D\_\backslash mu\; =\; \backslash partial\_\backslash mu\; +\; \backslash tfrac\{1\}\{4\}\backslash omega\_\backslash mu^\{ab\}\backslash gamma\_\{ab\}$, while $\backslash gamma\_\{ab\}\; =\; \backslash gamma\_a\backslash gamma\_b$ and $\backslash omega\_\backslash mu^\{ab\}$ is the spin connection.

The supersymmetry transformation rules are given up to three fermion terms by{{rp|324}}

:$$

\delta e^a{}_\mu = \tfrac{1}{2}\bar \epsilon \gamma^a\psi_\mu,

:$$

\delta \psi_\mu = D_\mu \epsilon + \tfrac{1}{32}e^{-\phi}(\gamma_\mu{}^{\nu\rho\sigma}-9\delta^\nu_\mu \gamma^{\rho\sigma})\epsilon H_{\nu\rho\sigma},

:$$

\delta B_{\mu\nu} = \tfrac{1}{2}e^\phi \bar \epsilon (\gamma_\mu \psi_\nu - \gamma_\nu \psi_\mu - \tfrac{1}{\sqrt 2}\gamma_{\mu\nu}\lambda)+\tfrac{\kappa}{g}e^{\phi/2}\bar \epsilon \gamma_{[\mu}\text{tr}(\chi A_{\nu]}),

:$$

\delta \phi = -\tfrac{1}{2\sqrt 2}\bar \epsilon \lambda,

:$$

\delta \lambda = -\tfrac{\kappa}{\sqrt 2}{\partial\!\!\!/}\phi + \tfrac{1}{8\sqrt 2}e^{-\phi}\gamma^{\mu\nu\rho}\epsilon H_{\mu\nu\rho},

:$$

\delta A_\mu^a = \tfrac{g}{2\kappa}e^{\phi/2}\bar \epsilon \gamma_\mu \chi^a,

:$$

\delta \chi^a = -\tfrac{\kappa}{4g} e^{-\phi/2}\gamma^{\mu\nu}F^a_{\mu\nu}\epsilon.

The supersymmetry parameter is denoted by $\backslash epsilon$. These transformation rules are useful for constructing the Killing spinor equations and finding supersymmetric ground states.

At a classical level the supergravity has an arbitrary gauge group, however not all gauge groups are consistent at the quantum level.{{rp|98–101}} The Green–Schwartz anomaly cancellation mechanism is used to show when the gauge, mixed, and gravitational anomalies vanish in hexagonal diagrams. In particular, the only anomaly free type I supergravity theories are ones with gauge groups of $\backslash text\{SO\}(32)$, $E\_8\backslash times\; E\_8$, $E\_8\backslash times\; \backslash text\{U\}(1)^\{248\}$, and $\backslash text\{U\}(1)^\{496\}$. It was later found that the latter two with abelian factors are inconsistent theories of quantum gravity.{{cite journal|last1=Adams|first1=Allan|authorlink1=Allan Adams|last2=DeWolfe|first2=O.|authorlink2=|last3=Taylor|first3=W.|authorlink3=|date=2010|title=String universality in ten dimensions|url=|journal=Phys. Rev. Lett.|volume=105|issue=7|pages=071601|doi=10.1103/PhysRevLett.105.071601|pmid=20868028|arxiv=1006.1352|bibcode=2010PhRvL.105g1601A |s2cid=|access-date=}} The remaining two theories both have ultraviolet completions to string theory, where the corresponding string theories can also be shown to be anomaly free at the string level.

Type I supergravity is the low-energy effective field theory of type I string theory and both heterotic string theories. In particular, type I string theory and $\backslash text\{SO\}(32)$ heterotic string theory reduce to type I supergravity with an $\backslash text\{SO\}(32)$ gauge group, while $E\_8\backslash times\; E\_8$ heterotic string theory reduces to type I supergravity with an $E\_8\backslash times\; E\_8$ gauge group.{{rp|92–93}} There are additional corrections that the supergravity receives in string theory, notably the Chern–Simons term becomes a linear combination of the Yang–Mills Chern–Simons three-form found at tree-level and a Lorentz Chern–Simons three-form $\backslash omega\_\{\backslash text\{YM\}\}\backslash rightarrow\; \backslash omega\_\{\backslash text\{YM\}\}-\backslash omega\_\{\backslash text\{L\}\}$.{{cite book|last=Nath|first=P.|author-link=Pran Nath (physicist)|date=2016|title=Supersymmetry, Supergravity, and Unification|url=|doi=|location=|publisher=Cambridge University Press|chapter=18|pages=420–421|isbn=978-0521197021}} This latter three-form is a higher-derivative correction given by

:$$

\omega_{\text{L}} = \text{tr}(\omega\wedge d\omega + \tfrac{2}{3}\omega \wedge \omega \wedge \omega)

,

where $\backslash omega$ is the spin connection. To maintain supersymmetry of the action when this term is included, additional higher-derivative corrections must be added to the action up to second order in $\backslash kappa$.

In type I string theory, the gauge coupling constant is related to the ten-dimensional Yang–Mills coupling constant by $g\_\{YM\}^2\; =\; g^2g\_s$, while the coupling constant is related to the string length $l\_s\; =\; \backslash sqrt\{\backslash alpha\text{'}\}$ by $g^2\; =\; 4\backslash pi(2\backslash pi\; l\_s)^6$.{{rp|318}} Meanwhile, in heterotic string theory the gravitational coupling constant is related to the string length by $2\backslash kappa\; =\; l\_s\; g$.{{rp|108}}

The fields in the Einstein frame are not the same as the fields corresponding to the string states. Instead, one has to transform the action into the various string frames through a Weyl transformation and dilaton redefinition{{rp|93}}

:$$

\text{Heterotic}: \ \ \ \ \ \ \ \ g_{\mu\nu}=e^{-\phi_h/2}g_{h,\mu\nu}, \ \ \ \ \ \ \phi = \phi_h/2,

:$$

\text{Type I}: \ \ \ \ \ \ \ \ \ \ \ \ g_{\mu\nu} = e^{-\phi_I/2}g_{I,\mu\nu}, \ \ \ \ \ \ \ \phi = -\phi_I/2.

S-duality between type I string theory and $\backslash text\{SO\}(32)$ heterotic string theory can be seen at the level of the action since the respective string frame actions are equivalent with the correct field redefinitions.{{cite book|last=Ortin|first=T.|author-link=|date=2015|title=Gravity and Strings|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=|page=702|edition=2|isbn=978-0521768139}} Similarly, Hořava–Witten theory, which describes the duality between $E\_8\backslash times\; E\_8$ heterotic string theory and M-theory, can also be seen at the level of the supergravity since compactification of eleven-dimensional supergravity on $S^1/\backslash mathbb\; Z\_2$, yields $E\_8\backslash times\; E\_8$ supergravity.

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{{Supersymmetry topics}}

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{{DEFAULTSORT:Type I supergravity}}

Category:Supersymmetric quantum field theory

Category:Theories of gravity

Category:String theory