Type I supergravity

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 ${\text{SO}}(32)$ or $E_{8}\times E_{8}$. 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.

History

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 1976 by Werner Nahm. Type I supergravity was first written down in 1983, first only the abelian part by Eric Bergshoeff, Mees de Roo, Bernard de Wit, and Peter van Nieuwenhuizen, and then the full non-abelian theory by George Chapline and Nicholas Manton. 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, with additional work showing that only ${\text{SO}}(32)$ and $E_{8}\times E_{8}$ result in a consistent quantum theory. The first case was known at the time to correspond to type I superstrings. Heterotic string theories were discovered the next year, with each one corresponding to one of the two gauge groups.

Theory

The field content of Type I supergravity consists of the ${\mathcal {N}}=1$ supergravity supermultiplet $(g_{\mu \nu },\psi _{\mu },B_{2},\lambda ,\phi )$, together with the ${\mathcal {N}}=1$ Yang–Mills supermultiplet $(A_{\mu }^{a},\chi ^{a})$ with some associated gauge group. Here $g_{\mu \nu }$ is the metric, $B_{2}$ is a two-form tensor known as the Kalb–Ramond field, $\phi$ is the dilaton, and $A_{\mu }^{a}$ is a Yang–Mills gauge field. Meanwhile, $\psi _{\mu }$ is a gravitino, $\lambda$ is a dilatino, and $\chi ^{a}$ a gaugino, with all these being Majorana–Weyl spinors. The gravitino and gaugino have the same chirality, while the dilatino has the opposite chirality.

Algebra

The superalgebra for type I supersymmetry is given by

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

Here $Q_{\alpha }$ is the supercharge with a fixed chirality $PQ_{\alpha }=Q_{\alpha }$, where $P={\tfrac {1}{2}}(1\pm \gamma _{*})$ is the relevant projection operator. Here $C$ is the charge conjugation operator, while $\gamma ^{\mu }$ are the gamma matrices. The right-hand side must have the same chirality property 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 $\gamma ^{\mu _{1}\cdots \mu _{p}}C$ with $p=1,2$ and $p=4,5$ are symmetric under the spinor indices. The central charge corresponds to a 5-brane solution in the supergravity which is dual to the fundamental string in heterotic string theory.

Action

The action for type I supergravity in the Einstein frame is given up to 4-fermion terms by

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

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

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

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

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

Here $\kappa ^{2}$ is the gravitational coupling constant, $\phi$ is the dilaton, while

$H_{\mu \nu \rho }=\partial _{[\mu }B_{\nu \rho ]}-{\tfrac {\kappa ^{2}}{g^{2}}}\omega _{\mu \nu \rho },$

where $\omega ={\text{tr}}(dA+{\tfrac {2}{3}}A\wedge A\wedge A)$ is the trace of the Chern–Simons form. The non-abelian field strength tensor corresponding to the gauge field $A_{\mu }$ is denote by $F_{\mu \nu }$. The spacetime index gamma-matrices are position-dependent fields $\gamma _{\mu }=e_{\mu }^{a}\gamma _{a}$. Meanwhile, $D_{\mu }$ is the covariant derivative $D_{\mu }=\partial _{\mu }+{\tfrac {1}{4}}\omega _{\mu }^{ab}\gamma _{ab}$, while $\gamma _{ab}=\gamma _{a}\gamma _{b}$ and $\omega _{\mu }^{ab}$ is the spin connection.

Supersymmetry transformations

The supersymmetry transformation rules are given up to three fermion terms by

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

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

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

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

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

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

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

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

Anomaly cancellation

At a classical level the supergravity has an arbitrary gauge group, however not all gauge groups are consistent at the quantum level. 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 ${\text{SO}}(32)$, $E_{8}\times E_{8}$, $E_{8}\times {\text{U}}(1)^{248}$ and ${\text{U}}(1)^{496}$. It was later found that the latter two with abelian factors are inconsistent theories of quantum gravity. These two anomaly free theories both have Ultraviolet completions in string theory which can also be demonstrated to be anomaly free at the string level.

Relation to string theory

Type I supergravity is the low-energy effective field theory of both type I string theory and both heterotic string theories. In particular, type I string theory and ${\text{SO}}(32)$ heterotic string theory reduce to type I supergravity with an ${\text{SO}}(32)$ gauge group, while $E_{8}\times E_{8}$ heterotic string theory reduces to type I supergravity with an $E_{8}\times E_{8}$ gauge group.

In type I string theory, the gauge coupling constant is related to the 10d Yang–Mills coupling constant by $g_{YM}^{2}=g^{2}g_{s}$, while the coupling constant is related to the string length $l_{s}={\sqrt {\alpha '}}$ by $g^{2}=4\pi (2\pi l_{s})^{6}$. Meanwhile, in heterotic string theory the gravitational coupling constant is related to the Regge slope by $4\kappa ^{2}=\alpha 'g^{2}$.

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 for the various theories. S-duality between type I string theory and ${\text{SO}}(32)$ heterotic string theory can be seen at the level of the action since the respective string frame actions can be transformed into each other. Similarly, Hořava–Witten theory, which describes the duality between $E_{8}\times E_{8}$ heterotic string theory and M-theory, can be seen at the level of the supergravity since compactification of eleven-dimensional supergravity on $S^{1}/\mathbb {Z} _{2}$, yields $E_{8}\times E_{8}$ supergravity.