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Introductory Lectures on Extended Supergravities and Gaugings

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Book cover Theoretical Frontiers in Black Holes and Cosmology

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 176))

Abstract

In an ungauged supergravity theory, the presence of a scalar potential is allowed only for the minimal \(N=1\) case. In extended supergravities, a non-trivial scalar potential can be introduced without explicitly breaking supersymmetry only through the so-called gauging procedure. The latter consists in promoting a suitable global symmetry group to local symmetry to be gauged by the vector fields of the theory. Gauged supergravities provide a valuable approach to the study of superstring flux-compactifications and the construction of phenomenologically viable, string-inspired models. The aim of these lectures is to give a pedagogical introduction to the subject of gauged supergravities, covering just selected issues and discussing some of their applications.

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Notes

  1. 1.

    In part they originate from gauge symmetries associated with the higher dimensional antisymmetric tensor fields.

  2. 2.

    Using the “mostly minus” convention and \(8\pi \mathrm {G}_{\textsc {n}}=c=\hbar =1\).

  3. 3.

    A solvable Lie group \(G_S\) can be described (locally) as a the Lie group generated by solvable Lie algebra \({\mathscr {S}}\): \(G_S=\mathrm {exp}({\mathscr {S}}) \). A Lie algebra \({\mathscr {S}}\) is solvable iff, for some \(k>0\), \(\mathbf{D}^k {\mathscr {S}}=0\), where the derivative \(\mathbf{D}\) of a Lie algebra \(\mathfrak {g}\) is defined as follows: \(\mathbf{D}\mathfrak {g}\equiv [\mathfrak {g},\mathfrak {g}]\), \(\mathbf{D}^n\mathfrak {g}\equiv [\mathbf{D}^{n-1}\mathfrak {g},\mathbf{D}^{n-1}\mathfrak {g}]\). In a suitable basis of a given representation, elements of a solvable Lie group or a solvable Lie algebra are all described by upper (or lower) triangular matrices.

  4. 4.

    The symplectic indices \(M,\,N,\dots \) are raised (and lowered) with the symplectic matrix \(\mathbb {C}^{MN}\) (\(\mathbb {C}_{MN}\)) using north-west south-east conventions: \(X^{M}=\mathbb {C}^{MN}\,X_{N}\) (and \(X_M=\mathbb {C}_{NM}\,X^{N}\)).

  5. 5.

    A special Kähler manifold is in general characterized by the product of a \(\mathrm{U}(1)\)-bundle, associated with its Kähler structure (with respect to which the manifold is Hodge Kähler), and a flat symplectic bundle. See for instance [19] for an in depth account of this issue.

  6. 6.

    We label the new basis by underlined indices.

  7. 7.

    The electric and magnetic charges (e, m) are expressed in the rationalized-Heaviside-Lorentz (RHL) system of units.

  8. 8.

    Here we only consider local transformations on the fields.

  9. 9.

    We define \(w_\mu \equiv w_s\,\partial _\mu \phi ^s\).

  10. 10.

    This is a schematic representation in which we have suppressed the Lorentz indices and gamma-matrices.

  11. 11.

    The gravitino field has an additional term \(\mathscr {D}\varepsilon \) which is its variation as the gauge field of local supersymmetry.

  12. 12.

    We describe by hatted-indices those pertaining to the symplectic frame in which the Lagrangian is defined.

  13. 13.

    Here we use the following convention for the definition of the components of a form: \(\omega _{(p)}=\frac{1}{p!}\,\omega _{\mu _1\dots \mu _p}\,dx^{\mu _1}\wedge \dots dx^{\mu _p}\).

  14. 14.

    The ellipses refer to terms containing the vector field strengths.

  15. 15.

    In the formulas below we use the coset representative in which the first index (acted on by G) is in the generic symplectic frame defined by the matrix E and which is then related to the same matrix in the electric frame (labeled by hatted indices) as follows:

    figure d

    last equation being (68).

  16. 16.

    Recall that in maximal supergravity the locality constraint follows from the linear and the closure ones.

  17. 17.

    In a generic gauged model, supersymmetry further require the fermion shifts to be related by differential “gradient flow” relations [29] which can e shown to follow from the identification of the shifts with components of the \(\mathbb {T}\)-tensor and the geometry of the scalar manifold.

  18. 18.

    The \(H_\mathrm{R}=\mathrm{U}(2)\)-generators \(\{J_{\mathbf{a}}\}\) naturally split into a \(\mathrm{U}(1)\)-generator \(J_0\) of the Kähler transformations on \(\mathscr {M}_{\textsc {sk}}\) and \(\mathrm{SU}(2)\)-generators \(J_x\) (\(x=1,2,3\)) in the holonomy group of the quaternionic Kähler manifold \(\mathscr {M}_{\textsc {qk}}\).

  19. 19.

    For string theory compactifications we should also require this latter scale to be negligible compared to the mass-scale of the string excitations (order \(1/\sqrt{\alpha '}\)).

  20. 20.

    We can relax this constraint by extending this representation to include the \(\mathbf{56}\) in (220). Consistency however would require the gauging of the scaling symmetry of the theory (which is never an off-shell symmetry), also called trombone symmetry [55, 56]. This however leads to gauged theories which do not have an action. We shall not discuss these gaugings here.

  21. 21.

    In the previous sections we have used, for the supergravity fields, notations which are different from those used in the literature of maximal supergravity (e.g. in [18]) in order to make contact with the literature of gauged \(N<8\) theories, in particular \(N=2\) ones [19]. Denoting by a hat the quantities in [18], the correspondence between the two notations is:

    figure e

    where in the last line the \(28\times 28\) blocks of \(\mathcal {V}_M{}^{\underline{N}}\) have been put in correspondence with those of \(\mathbb {L}^M{}_{\underline{N}}\). The factor \(\sqrt{2}\) originates from a different convention with the contraction of antisymmetric couples of \(\mathrm{SU}(8)\)-indices: \(\hat{V}_{ij}\hat{V}^{ij}=\frac{1}{2}\,V^{AB}\,V_{AB}\).

  22. 22.

    See Table 2 at the end of Sect. 4.

  23. 23.

    Here, for the sake of simplicity, we reabsorb the gauge coupling constant g into \(\varTheta \): \(g\,\varTheta \rightarrow \varTheta \).

  24. 24.

    These fields will also be described as 2-forms \(B_\alpha \equiv \frac{1}{2}\,B_{\mu \nu }\,dx^\mu \wedge dx^\nu \).

  25. 25.

    In our earlier discussion we showed that \(\mathcal {D}\mathcal {H}^M\,\varTheta _M{}^\alpha =\mathcal {D}F^M\,\varTheta _M{}^\alpha =0\). This is consistent with (278) since on-shell \(\mathcal {H}^M\varTheta _M{}^\alpha ={\mathcal {G}}^M\varTheta _M{}^\alpha \).

  26. 26.

    The Hodge dual \({}^*\omega \) of a generic q-form \(\omega \) is defined as:

    figure f

    where \(\varepsilon _{01\dots D-1}=1\). One can easily verify that \({}^{**}\omega =(-)^{q(D-q)}\,(-)^{D-1}\,\omega \).

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    Antonio Gallerati & Mario Trigiante

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  1. Varian Physics Bldg., Stanford University, Stanford, California, USA

    Renata Kallosh

  2. Escola de Ciências & Tecnologia, Universidade Federal Rio Grande do Norte, Natal-RN, Brazil

    Emanuele Orazi

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© 2016 Springer International Publishing Switzerland

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Gallerati, A., Trigiante, M. (2016). Introductory Lectures on Extended Supergravities and Gaugings. In: Kallosh, R., Orazi, E. (eds) Theoretical Frontiers in Black Holes and Cosmology. Springer Proceedings in Physics, vol 176. Springer, Cham. https://doi.org/10.1007/978-3-319-31352-8_2

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