# nLab higher curvature correction

Contents

## Surveys, textbooks and lecture notes

#### Gravity

gravity, supergravity

# Contents

## 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

$G_4(\ell) \;=\; G_4 + \ell^6 G_4^{(1)} + \cdots$
$G_7(\ell) \;=\; G_7 + \ell^6 G_7^{(1)} + \cdots$

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

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

## References

### General

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

Discussion in the context of corrections to black hole entropy include

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)

### Higher curvature corrections to $D=4$ supergravity

Discussion of higher curvature corrections to D=4 supergravity:

Relation to quintessence:

### Higher curvature corrections to 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):

#### Via superparticle scattering in 11d

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

#### 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, 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)

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

and in view of de Sitter spacetime vacua:

### 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)

Last revised on October 27, 2021 at 00:53:05. See the history of this page for a list of all contributions to it.