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.
In part they originate from gauge symmetries associated with the higher dimensional antisymmetric tensor fields.
- 2.
Using the “mostly minus” convention and \(8\pi \mathrm {G}_{\textsc {n}}=c=\hbar =1\).
- 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.
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.
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.
We label the new basis by underlined indices.
- 7.
The electric and magnetic charges (e, m) are expressed in the rationalized-Heaviside-Lorentz (RHL) system of units.
- 8.
Here we only consider local transformations on the fields.
- 9.
We define \(w_\mu \equiv w_s\,\partial _\mu \phi ^s\).
- 10.
This is a schematic representation in which we have suppressed the Lorentz indices and gamma-matrices.
- 11.
The gravitino field has an additional term \(\mathscr {D}\varepsilon \) which is its variation as the gauge field of local supersymmetry.
- 12.
We describe by hatted-indices those pertaining to the symplectic frame in which the Lagrangian is defined.
- 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.
The ellipses refer to terms containing the vector field strengths.
- 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:
last equation being (68).
- 16.
Recall that in maximal supergravity the locality constraint follows from the linear and the closure ones.
- 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.
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.
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.
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.
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:
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.
- 23.
Here, for the sake of simplicity, we reabsorb the gauge coupling constant g into \(\varTheta \): \(g\,\varTheta \rightarrow \varTheta \).
- 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.
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.
The Hodge dual \({}^*\omega \) of a generic q-form \(\omega \) is defined as:
where \(\varepsilon _{01\dots D-1}=1\). One can easily verify that \({}^{**}\omega =(-)^{q(D-q)}\,(-)^{D-1}\,\omega \).
References
O. Hohm, H. Samtleben, Exceptional field theory. II. E\(_{7(7)}\). Phys. Rev. D 89, 066017 (2014). arXiv:1312.4542 [hep-th]
C. Hull, B. Zwiebach, Double field theory. JHEP 0909, 099 (2009). arXiv:0904.4664 [hep-th]
C.M. Hull, P.K. Townsend, Unity of superstring dualities. Nucl. Phys. B 438, 109 (1995). arXiv:hep-th/9410167
B. de Wit, H. Nicolai, N \(=\) 8 Supergravity. Nucl. Phys. B 208, 323 (1982)
B. de Wit, H. Samtleben, M. Trigiante, On Lagrangians and gaugings of maximal supergravities. Nucl. Phys. B 655, 93 (2003). arXiv:hep-th/0212239
B. de Wit, H. Samtleben, M. Trigiante, Maximal supergravity from IIB flux compactifications. Phys. Lett. B 583, 338 (2004). arXiv:hep-th/0311224
B. de Wit, H. Samtleben, Gauged maximal supergravities and hierarchies of non Abelian vector-tensor systems. Fortsch. Phys. 53, 442 (2005). arXiv:hep-th/0501243
B. de Wit, H. Samtleben, M. Trigiante, Magnetic charges in local field theory. JHEP 0509, 016 (2005). arXiv:hep-th/0507289
C.M. Hull, Noncompact gaugings of N \(=8\) supergravity. Phys. Lett. B 142, 39 (1984)
C.M. Hull, More gaugings of N \(=8\) supergravity. Phys. Lett. B 148, 297 (1984)
M. Trigiante, Dual Gauged Supergravities (17th SIGRAV Conference Turin, September 4-7, Italy, 2006). arXiv:hep-th/0701218
H. Samtleben, Lectures on gauged supergravity and flux compactifications. Class. Quant. Grav. 25, 214002 (2008). arXiv:0808.4076 [hep-th]
M. Grana, Flux compactifications in string theory: a comprehensive review. Phys. Rept. 423, 91 (2006). arXiv:hep-th/0509003
R. Blumenhagen, B. Kors, D. Lust, S. Stieberger, Four-dimensional string compactifications with D-Branes, orientifolds and fluxes. Phys. Rept. 445, 1 (2007). arXiv:hep-th/0610327
M.R. Douglas, S. Kachru, Flux compactification. Rev. Mod. Phys. 79, 733 (2007). arXiv:hep-th/0610102
F. Cordaro, P. Fre, L. Gualtieri, P. Termonia, M. Trigiante, N \(=\) 8 gaugings revisited: an exhaustive classification. Nucl. Phys. B 532, 245 (1998). arXiv:hep-th/9804056
H. Nicolai, H. Samtleben, Maximal gauged supergravity in three-dimensions. Phys. Rev. Lett. 86, 1686 (2001). arXiv:hep-th/0010076
B. de Wit, H. Samtleben, M. Trigiante, The maximal D \(=\) 4 supergravities. JHEP 0706, 049 (2007). arXiv:0705.2101 [hep-th]
L. Andrianopoli, M. Bertolini, A. Ceresole, R. D’Auria, S. Ferrara, P. Fre, T. Magri, N \(=\) 2 supergravity and N \(=\) 2 super Yang-Mills theory on general scalar manifolds: symplectic covariance, gaugings and the momentum map. J. Geom. Phys. 23, 111 (1997). arXiv:hep-th/9605032
M.K. Gaillard, B. Zumino, Duality rotations for interacting fields. Nucl. Phys. B 193, 221 (1981)
S. Ferrara, A. Marrani, E. Orazi, M. Trigiante, Dualities near the horizon. JHEP 1311, 056 (2013). arXiv:1305.2057 [hep-th]
P.A.M. Dirac, Quantized singularities in the electromagnetic field. Proc. Roy. Soc. Lond. A 133, 60 (1931)
J.S. Schwinger, Magnetic charge and quantum field theory. Phys. Rev. 144, 1087 (1966)
D. Zwanziger, Quantum field theory of particles with both electric and magnetic charges. Phys. Rev. 176, 1489 (1968)
B. de Wit, P.G. Lauwers, A. Van Proeyen, Lagrangians of N \(=\) 2 supergravity—matter systems. Nucl. Phys. B 255, 569 (1985)
P. Van Nieuwenhuizen, Supergravity. Phys. Rep. 68, 189 (1981)
S. Ferrara L. Maiani, An introduction to supersymmetry breaking in extended supergravity, in 5th SILARG Symposium on Relativity, Supersymmetry and Cosmology (Bariloche, Argentina, January 4–11, 1985)
S. Cecotti, L. Girardello, M. Porrati, Constraints on partial superhiggs. Nucl. Phys. B 268, 295 (1986)
R. D’Auria, S. Ferrara, On fermion masses, gradient flows and potential in supersymmetric theories. JHEP 0105, 034 (2001). arXiv:hep-th/0103153
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113 (1999) (Adv. Theor. Math. Phys. 2, 231 (1998)) arXiv:hep-th/9711200
E. Cremmer, S. Ferrara, C. Kounnas, D.V. Nanopoulos, Naturally vanishing cosmological constant in N \(=\) 1 supergravity. Phys. Lett. B 133, 61 (1983)
J.R. Ellis, C. Kounnas, D.V. Nanopoulos, No scale supersymmetric guts. Nucl. Phys. B 247, 373 (1984)
R. Barbieri, E. Cremmer, S. Ferrara, Flat and positive potentials in \(N=1\) supergravity. Phys. Lett. B 163, 143 (1985)
C. Angelantonj, S. Ferrara, M. Trigiante, New D \(=\) 4 gauged supergravities from N \(=\) 4 orientifolds with fluxes. JHEP 0310, 015 (2003). arXiv:hep-th/0306185
R. D’Auria, S. Ferrara, F. Gargiulo, M. Trigiante, S. Vaula, N \(=\) 4 supergravity Lagrangian for type IIB on T**6 / Z(2) in presence of fluxes and D3-branes. JHEP 0306, 045 (2003). hep-th/0303049
J. Scherk, J.H. Schwarz, Nucl. Phys. B 153, 61 (1979)
N. Kaloper, R.C. Myers, The odd story of massive supergravity. JHEP 9905, 010 (1999). hep-th/9901045
V. Mathai, J.M. Rosenberg, T duality for torus bundles with H fluxes via noncommutative topology. Commun. Math. Phys. 253, 705 (2004). hep-th/0401168
C.M. Hull, A geometry for non-geometric string backgrounds. JHEP 0510, 065 (2005). hep-th/0406102
J. Shelton, W. Taylor, B. Wecht, Nongeometric flux compactifications. JHEP 0510, 085 (2005). hep-th/0508133
N. Hitchin, Lectures on Generalized Geometry. arXiv:1008.0973 [math.DG]
M. Gualtieri, Generalized Complex Geometry. arXiv:math/0401221 [math-dg]
C.M. Hull, R.A. Reid-Edwards, Gauge symmetry, T-duality and doubled geometry. JHEP 0808, 043 (2008). arXiv:0711.4818 [hep-th]
A. Dabholkar, C. Hull, Generalised T-duality and non-geometric backgrounds. JHEP 0605, 009 (2006). arXiv:hep-th/0512005
O. Hohm, C. Hull, B. Zwiebach, Generalized metric formulation of double field theory. JHEP 1008, 008 (2010). arXiv:1006.4823 [hep-th]
C.M. Hull, Generalised geometry for M-theory. JHEP 0707, 079 (2007). arXiv:hep-th/0701203
P.P. Pacheco, D. Waldram, M-theory, exceptional generalised geometry and superpotentials. JHEP 0809, 123 (2008). arXiv:0804.1362 [hep-th]
O. Hohm, H. Samtleben, Exceptional form of D \(=\) 11 supergravity. Phys. Rev. Lett. 111, 231601 (2013). arXiv:1308.1673 [hep-th]
O. Hohm, H. Samtleben, Consistent Kaluza-Klein truncations via exceptional field theory. JHEP 1501, 131 (2015). arXiv:1410.8145 [hep-th]
G. Dibitetto, A. Guarino, D. Roest, Charting the landscape of N \(=\) 4 flux compactifications. JHEP 1103, 137 (2011). arXiv:1102.0239 [hep-th]
G. Dall’Agata, G. Inverso, On the vacua of N \(=\) 8 gauged supergravity in 4 dimensions. Nucl. Phys. B 859, 70 (2012). arXiv:1112.3345 [hep-th]
G. Inverso, De Sitter Vacua in N \(=\) 8 Supergravity, MSc Thesis, Università di Padova, Italy, 14 July (2010)
E. Cremmer, B. Julia, The N \(=\) 8 supergravity theory. 1. The Lagrangian. Phys. Lett. B 80, 48 (1978)
E. Cremmer, B. Julia, The SO(8) supergravity. Nucl. Phys. B 159, 141 (1979)
A. Le Diffon, H. Samtleben, Supergravities without an action: gauging the trombone. Nucl. Phys. B 811, 1 (2009). arXiv:0809.5180 [hep-th]
A. Le Diffon, H. Samtleben, M. Trigiante, N \(=\) 8 supergravity with local scaling symmetry. JHEP 1104, 079 (2011). arXiv:1103.2785 [hep-th]
B. de Wit, H. Nicolai, The consistency of the S**7 truncation in D \(=\) 11 supergravity. Nucl. Phys. B 281, 211 (1987)
E. Cremmer, J. Scherk, J.H. Schwarz, Spontaneously broken N \(=\) 8 supergravity. Phys. Lett. B 84, 83 (1979)
L. Andrianopoli, R. D’Auria, S. Ferrara, M.A. Lledo, Gauging of flat groups in four-dimensional supergravity. JHEP 0207, 010 (2002). arXiv:hep-th/0203206
G. Dall’Agata, G. Inverso, de Sitter vacua in N \(=\) 8 supergravity and slow-roll conditions. Phys. Lett. B 718, 1132 (2013). arXiv:1211.3414 [hep-th]
G. Dall’Agata, G. Inverso, M. Trigiante, Evidence for a family of SO(8) gauged supergravity theories. Phys. Rev. Lett. 109, 201301 (2012). arXiv:1209.0760 [hep-th]
G. Dall’Agata, G. Inverso, A. Marrani, Symplectic deformations of gauged maximal supergravity. JHEP 1407, 133 (2014). arXiv:1405.2437 [hep-th]
A. Borghese, A. Guarino, D. Roest, All \(G_2\) invariant critical points of maximal supergravity. JHEP 1212, 108 (2012). arXiv:1209.3003 [hep-th]
A. Borghese, G. Dibitetto, A. Guarino, D. Roest, O. Varela, The SU(3)-invariant sector of new maximal supergravity. JHEP 1303, 082 (2013)
A. Borghese, A. Guarino, D. Roest, Triality, periodicity and stability of SO(8) gauged supergravity. JHEP 1305, 107 (2013). arXiv:1302.6057 [hep-th]
A. Gallerati, H. Samtleben, M. Trigiante, The \({\cal {N}}>2 \) supersymmetric AdS vacua in maximal supergravity. JHEP 1412, 174 (2014). arXiv:1410.0711 [hep-th]
J. Tarrío, O. Varela, Electric/magnetic duality and RG flows in AdS\(_4\)/CFT\(_3\). JHEP 1401, 071 (2014). arXiv:1311.2933 [hep-th]
Y. Pang, C.N. Pope, J. Rong, Holographic RG flow in a new \(SO(3)\times SO(3)\) sector of \(\omega \)-deformed \(SO(8)\) gauged \({\cal {N}}=8 \) supergravity. JHEP 1508, 122 (2015). arXiv:1506.04270 [hep-th]
A. Anabalon, D. Astefanesei, Black holes in \(\omega \)-defomed gauged N \(=8\) supergravity. Phys. Lett. B 732, 137 (2014). arXiv:1311.7459 [hep-th]
H. Lü, Y. Pang, C.N. Pope, An \(\omega \) deformation of gauged STU supergravity. JHEP 1404, 175 (2014). arXiv:1402.1994 [hep-th]
S.Q. Wu, S. Li, Thermodynamics of static Dyonic AdS black holes in the \(\omega \)-Deformed Kaluza-Klein gauged supergravity theory. Phys. Lett. B 746, 276 (2015). arXiv:1505.00117 [hep-th]
K. Lee, C. Strickland-Constable, D. Waldram, New Gaugings and Non-geometry. arXiv:1506.03457 [hep-th]
A. Guarino, D. L. Jafferis, O. Varela, String theory origin of dyonic N=8 Supergravity and Its Chern-Simons duals. Phys. Rev. Lett. 115(9), 091601 (2015) arXiv:1504.08009 [hep-th]
B. de Wit, H. Samtleben, M. Trigiante, The maximal D \(=\) 5 supergravities. Nucl. Phys. B 716, 215 (2005). arXiv:hep-th/0412173
H. Samtleben, M. Weidner, The maximal D \(=\) 7 supergravities. Nucl. Phys. B 725, 383 (2005). arXiv:hep-th/0506237
B. de Wit, H. Nicolai, H. Samtleben, Gauged supergravities, tensor hierarchies, and M-theory. JHEP 0802, 044 (2008). arXiv:0801.1294 [hep-th]
P. K. Townsend, in Gauge Field Theories: Theoretical Studies and Computer Simulations ed. by W. Garazynski (Harwood Academic Chur, New York 1981)
S. Cecotti, S. Ferrara, L. Girardello, Massive vector multiplets from superstrings. Nucl. Phys. B 294, 537 (1987)
G. Dall’Agata, R. D’Auria, L. Sommovigo, S. Vaula, D \(=\) 4, N \(=\) 2 gauged supergravity in the presence of tensor multiplets. Nucl. Phys. B 682, 243 (2004). arXiv:hep-th/0312210
R. D’Auria, L. Sommovigo, S. Vaula, N \(=\) 2 supergravity Lagrangian coupled to tensor multiplets with electric and magnetic fluxes. JHEP 0411, 028 (2004). arXiv:hep-th/0409097
J. Louis, A. Micu, Type 2 theories compactified on Calabi-Yau three folds in the presence of background fluxes. Nucl. Phys. B 635, 395 (2002). arXiv:hep-th/0202168
E. Cremmer, B. Julia, H. Lu, C.N. Pope, Dualization of dualities. 1. Nucl. Phys. B 523, 73 (1998). arXiv:hep-th/9710119
Y. Tanii, Introduction to Supergravities in Diverse Dimensions. arXiv:hep-th/9802138
M. Gunaydin, L.J. Romans, N.P. Warner, Compact and noncompact gauged supergravity theories in five-dimensions. Nucl. Phys. B 272, 598 (1986)
L. Andrianopoli, F. Cordaro, P. Fre’, L. Gualtieri, Nonsemisimple gaugings of D \(=\) 5 N \(=\) 8 supergravity and FDA.s. Class. Quant. Grav. 18, 395 (2001). arXiv:hep-th/0009048
E. Bergshoeff, H. Samtleben, E. Sezgin, The gaugings of maximal D \(=\) 6 supergravity. JHEP 0803, 068 (2008). arXiv:0712.4277 [hep-th]
M. Pernici, K. Pilch, P. van Nieuwenhuizen, Gauged maximally extended supergravity in seven-dimensions. Phys. Lett. B 143, 103 (1984)
E. Bergshoeff, U. Gran, R. Linares, M. Nielsen, T. Ortin, D. Roest, The Bianchi classification of maximal D \(=\) 8 gauged supergravities. Class. Quant. Grav. 20, 3997 (2003). arXiv:hep-th/0306179
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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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