nLab higher curvature correction

Idea

In the context of gravity (general relativity), higher curvature corrections are modifications of the Einstein-Hilbert action that include not just the linear appearance of the scalar curvature $R$ but higher scalar powers of the Riemann curvature tensor.

When viewing Einstein-Hilbert gravity as an effective field theory valid at low energy/long wavelengths, then higher curvature corrections are precisely the terms that may appear at higher energy in pure gravity. Notably the effective field theories induced by string theory come with infinite towers of higher curvature corrections.

In the context of cosmology, higher curvature corrections are a candidate for the inflaton field, see at Starobinsky model of cosmic inflation.

A spacetime that extremizes the Einstein-Hilbert action for given cosmological constant and arbitrary higher curvature correction is called a universal spacetime.

Examples

For 11d Supergravity

A systematic analysis of the possible supersymmetric higher curvature corrections of D=11 supergravity makes the I8-characteristic 8-form

$$I_8(R) \;\coloneqq\; \tfrac{1}{48} \Big( p_2(R) \;-\; \big( \tfrac{1}{2} p_1(R)\big)^2 \Big) \;\; \in \Omega^8$$

(a differential form built from the Riemann curvature, expressing a polynomial in the first and second Pontryagin classes, see at I8) appear as the higher curvature correction at order $\ell^6$, where $\ell$ is the Planck length in 11d (Souères-Tsimpis 17, Section 4).

At this order, the equation of motion for the supergravity C-field flux $G_4$ and its dual $G_7$ is (Souères-Tsimpis 17, (4.3))

(1)$$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge G_4(\ell) + \ell^6 I_8(R) \,,$$

where the flux forms themselves appear in their higher order corrected form as power series in the Planck length

$$G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots$$

$$G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots$$

(Souères-Tsimpis 17, (4.4))

Beware that this is not the lowest order higher curvature correction: there is already one at $\mathcal{O}(\ell^3)$, given by $\ell^3 G_4 \wedge \tfrac{1}{2}p_1(R)$ (Souères-Tsimpis 17, Section 3.2). Hence the full correction at $\mathcal{O}(\ell^3)$ should be the further modification of (2) to (cf. Tsimpis 2004, p. 8):

(2)$$d G_7(\ell) \;=\; -\tfrac{1}{2} G_4(\ell) \wedge \big( G_4(\ell) - \tfrac{1}{2} p_1(R) \big) + \ell^6 I_8(R) \,.$$

Related concepts

• universal spacetime

• non-perturbative effect

References

General

Discussion of quadratic curvature currections includes (see also at Starobinsky model of cosmic inflation):

• Luis Alvarez-Gaume, Alex Kehagias, Costas Kounnas, Dieter Luest, Antonio Riotto, Aspects of Quadratic Gravity (arXiv:1505.07657)

Discussion of causal locality in the presence of higher curvature corrections includes

• Xian O. Camanho, Jose D. Edelstein, Juan Maldacena, Alexander Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling (arXiv:)

• Giuseppe D’Appollonio, Paolo Di Vecchia, Rodolfo Russo, Gabriele Veneziano, Regge behavior saves String Theory from causality violations (arXiv:1502.01254)

Discussion in the context of corrections to black hole entropy:

• Thomas Mohaupt, Strings, higher curvature corrections, and black holes (arXiv:hep-th/0512048)

Discussion of renormalization for gravity with higher curvature corrections:

• Nobuyoshi Ohta, One-loop divergences in higher-derivative gravity, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2210.02583]

Discussion of higher curvature corrections in cosmology and cosmic inflation (for more see at Starobinsky model of cosmic inflation):

• Gustavo Arciniega, Jose D. Edelstein, Luisa G. Jaime, Towards purely geometric inflation and late time acceleration (arXiv:1810.08166)

• Gustavo Arciniega, Pablo Bueno, Pablo A. Cano, Jose D. Edelstein, Robie A. Hennigar, Luisa G. Jaimem, Geometric Inflation (arXiv:1812.11187)

Relation to cosmic censorship hypothesis:

• Akash K Mishra, Sumanta Chakraborty, Strong Cosmic Censorship in higher curvature gravity, Phys. Rev. D 101, 064041 (2020) (arXiv:1911.09855)

See also:

• Jesse Daas, Cristobal Laporte, Frank Saueressig, Tim van Dijk: Rethinking the Effective Field Theory formulation of Gravity [arXiv:2405.12685]

For supergravity:

• Mehmet Ozkan, Yi Pang, Ergin Sezgin, Higher Derivative Supergravities in Diverse Dimensions [arXiv:2401.08945]

For $D=4$ supergravity

Discussion of higher curvature corrections to D=4 supergravity:

Relation to quintessence:

• Fotis Farakos, Quintessence from higher curvature supergravity (arXiv:2003.09366)

On small N corrections in ABJM theory and higher curvature corrections in the AdS/CFT dual D=4 supergravity:

• Nikolay Bobev, Anthony M. Charles, Kiril Hristov, Valentin Reys, The Unreasonable Effectiveness of Higher-Derivative Supergravity in $AdS_4$ Holography, Phys. Rev. Lett. 125 131601 (2020) [doi:10.1103/PhysRevLett.125.131601, arXiv:2006.09390]

• Nikolay Bobev, Anthony M. Charles, Dongmin Gang, Kiril Hristov, Valentin Reys, Higher-Derivative Supergravity, Wrapped M5-branes, and Theories of Class $\mathcal{R}$, J. High Energ. Phys. 2021 58 (2021) [doi:10.1007/JHEP04(2021)058, arXiv:2011.05971]

• Kiril Hristov, ABJM at finite $N$ via 4d supergravity, J. High Energ. Phys. 2022 190 (2022) [doi:10.1007/JHEP10(2022)190, arXiv:2204.02992]

For $D=5$ supergravity

Discussion for D=5 supergravity:

Analysis via computer algebra:

• Gregory Gold, Saurish Khandelwal, Gabriele Tartaglino-Mazzucchelli, Supergravity Component Reduction with Computer Algebra [arXiv:2406.19687]

For $D=10$ heterotic supergravity

On higher curvature corrections to heterotic supergravity:

• Eric Lescano, Carmen Núñez, Jesús A. Rodríguez, Supersymmetry, T-duality and Heterotic $\alpha'$-corrections (arXiv:2104.09545)

• Hao-Yuan Chang, Ergin Sezgin, Yoshiaki Tanii, Dimensional reduction of higher derivative heterotic supergravity (arXiv:2110.13163)

Higher curvature corrections to $D=11$ supergravity

Discussion of higher curvature corrections to 11-dimensional supergravity (i.e. in M-theory):

• Arkady Tseytlin, $R^4$ terms in 11 dimensions and conformal anomaly of (2,0) theory, Nucl. Phys. B584: 233-250, 2000 (arXiv:hep-th/0005072)

Via 11d superspace cohomology

Via 11d superspace-cohomology (or mostly):

• Kasper Peeters, Pierre Vanhove, Anders Westerberg, Supersymmetric $R^4$ actions and quantum corrections to superspace torsion constraints (arXiv:hep-th/0010182)

• H. Lu, Christopher Pope, Kellogg Stelle, Paul Townsend, Supersymmetric Deformations of $G_2$ Manifolds from Higher-Order Corrections to String and M-Theory, JHEP 0410:019, 2004 (arXiv:hep-th/0312002)

  (specifically for M-theory on G₂-manifolds)

• H. Lu, Christopher Pope, Kellogg Stelle, Paul Townsend, String and M-theory Deformations of Manifolds with Special Holonomy, JHEP 0507:075, 2005 (arXiv:hep-th/0410176)

  (specifically for M-theory on G₂-manifolds)

• Paul S. Howe, Dimitrios Tsimpis, On higher-order corrections in M theory, JHEP 0309 (2003) 038 [doi:10.1088/1126-6708/2003/09/038, arXiv:hep-th/0305129]

• Dimitrios Tsimpis, 11D supergravity at $\mathcal{O}(\ell^3)$, JHEP0410:046 (2004) [arXiv:hep-th/0407271, doi:10.1088/1126-6708/2004/10/046]

• Paul Howe, $R^4$ terms in supergravity and M-theory, contribution to Deserfest: A Celebration of the Life and Works of Stanley Deser (2004) 137-149 [inspire:657136, arXiv:hep-th/0408177]

• Martin Cederwall, Ulf Gran, Bengt Nilsson, Dimitrios Tsimpis, Supersymmetric Corrections to Eleven-Dimensional Supergravity, JHEP 0505:052 (2005) [doi;10.1088/1126-6708/2005/05/052, arXiv:hep-th/0409107]

• Yoshifumi Hyakutake, Sachiko Ogushi, $R^4$ Corrections to Eleven Dimensional Supergravity via Supersymmetry, Phys.Rev. D74 (2006) 025022 (arXiv:hep-th/0508204)

• Yoshifumi Hyakutake, Sachiko Ogushi, Higher Derivative Corrections to Eleven Dimensional Supergravity via Local Supersymmetry, JHEP0602:068, 2006 (arXiv:hep-th/0601092)

• Anirban Basu, Constraining gravitational interactions in the M theory effective action, Classical and Quantum Gravity, Volume 31, Number 16, 2014 (arXiv:1308.2564)

• Bertrand Souères, Dimitrios Tsimpis, The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity, Phys. Rev. D 95 026013 (2017) [doi:10.1103/PhysRevD.95.026013, arXiv:1612.02021]

• Bertrand Souères, Supergravities in Superspace, Lyon 2018 (tel:[phone], pdf)

Via superparticle scattering in 11d

Via analysis of would-be superparticle scattering amplitudes on D=11 supergravity backgrounds:

• Kasper Peeters, Jan Plefka, Steffen Stern, Higher-derivative gauge field terms in the M-theory action, JHEP 0508 (2005) 095 (arXiv:hep-th/0507178)

Via exceptional geometry

Via geodesic motion? on the coset space of the U-duality Lac-Moody group? $E_{10}$ by its “maximal compact” subgroup $K(E_{10})$:

• Thibault Damour, Hermann Nicolai: Higher order M theory corrections and the Kac-Moody algebra $E_{10}$, Class. Quant. Grav. 22 (2005) 2849-2880 [arXiv:hep-th/0504153, doi:10.1088/0264-9381/22/14/003]

  (relating to higher curvature corrections)

Via lifting 10d stringy corrections

From lifting alpha'-corrections in type IIA string theory through the duality between M-theory and type IIA string theory:

• James Liu, Ruben Minasian, Higher-derivative couplings in string theory: dualities and the B-field, Nucl. Phys. B 2013 (arXiv:1304.3137)

Via type IIB supergravity:

• James Liu, Ruben Minasian, Raffaele Savelli, Andreas Schachner, Type IIB at eight derivatives: insights from Superstrings, Superfields and Superparticles $[$arXiv:2205.11530$]$

Via the ABJM M2-brane model

From the ABJM model for the M2-brane:

• Damon J. Binder, Shai Chester, Silviu S. Pufu, Absence of $D^4 R^4$ in M-Theory From ABJM [arXiv:1808.10554]

In terms of D=4 supergravity:

• Nikolay Bobev, Anthony M. Charles, Kiril Hristov, Valentin Reys, The Unreasonable Effectiveness of Higher-Derivative Supergravity in $AdS_4$ Holography, Phys. Rev. Lett. 125 131601 (2020) [doi:10.1103/PhysRevLett.125.131601, arXiv:2006.09390]

• Nikolay Bobev, Anthony M. Charles, Dongmin Gang, Kiril Hristov, Valentin Reys, Higher-Derivative Supergravity, Wrapped M5-branes, and Theories of Class $\mathcal{R}$, J. High Energ. Phys. 2021 58 (2021) [doi:10.1007/JHEP04(2021)058, arXiv:2011.05971]

• Kiril Hristov, ABJM at finite $N$ via 4d supergravity, J. High Energ. Phys. 2022 190 (2022) [doi:10.1007/JHEP10(2022)190, arXiv:2204.02992]

See also

See also

• Mohammad Garousi, Minimal gauge invariant couplings at order $\ell^6_p$ in M-theory (arXiv:2102.00639)

Discussion in view of the Starobinsky model of cosmic inflation is in

• Katrin Becker, Melanie Becker, Supersymmetry Breaking, M-Theory and Fluxes, JHEP 0107:038,2001 (arXiv:hep-th/0107044)

• Kazuho Hiraga, Yoshifumi Hyakutake, Inflationary Cosmology via Quantum Corrections in M-theory (arXiv:1809.04724)

• Kazuho Hiraga, Yoshifumi Hyakutake, Scalar Cosmological Perturbations in M-theory with Higher Derivative Corrections (arxiv:1910.12483)

and in view of de Sitter spacetime vacua:

• Johan Blåbäck, Ulf Danielsson, Giuseppe Dibitetto, Suvendu Giri, Constructing stable de Sitter in M-theory from higher curvature corrections (arXiv:1902.04053)

Higher curvature corrections to the DBI-action for D-branes

On higher curvature corrections to the (abelian) DBI-action for (single) D-branes:

• Oleg Andreev, Arkady Tseytlin, Partition-function representation for the open superstring effective action:: Cancellation of Möbius infinites and derivative corrections to Born-Infeld lagrangian, Nuclear Physics B Volume 311, Issue 1, 19 December 1988, Pages 205-252 (doi:10.1016/0550-3213(88)90148-4)

• Constantin Bachas, P. Bain, Michael Green, Curvature terms in D-brane actions and their M-theory origin, JHEP 9905:011, 1999 (arXiv:hep-th/9903210)

• Niclas Wyllard, Derivative corrections to D-brane actions with constant background fields, Nucl. Phys. B598 (2001) 247-275 (arXiv:hep-th/0008125)

• Oleg Andreev, More About Partition Function of Open Bosonic String in Background Fields and String Theory Effective Action, Phys. Lett. B513:207-212, 2001 (arXiv:hep-th/0104061)

• Niclas Wyllard, Derivative corrections to the D-brane Born-Infeld action: non-geodesic embeddings and the Seiberg-Witten map, JHEP 0108 (2001) 027 (arXiv:hep-th/0107185)

• Mohammad Garousi, T-duality of curvature terms in D-brane actions, JHEP 1002:002, 2010 (arXiv:0911.0255)

• Mohammad Garousi, S-duality of D-brane action at order $O(\alpha'{}^2)$, Phys. Lett. B701:465-470, 2011 (arXiv:1103.3121)

• Ali Jalali, Mohammad Garousi, On D-brane action at order $\alpha'{}^2$, Phys. Rev. D 92, 106004 (2015) (arXiv:1506.02130)

• Mohammad Garousi, An off-shell D-brane action at order $\alpha'{}^2$ in flat spacetime, Phys. Rev. D 93, 066014 (2016) (arXiv:1511.01676)

• Komeil Babaei Velni, Ali Jalali, Higher derivative corrections to DBI action at $\alpha'{}^2$ order, Phys. Rev. D 95, 086010 (2017) (arXiv:1612.05898)