# 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)

### In 11d Supergravity

Discussion of higher curvature corrections to 11-dimensional supergravity includes

and from the ABJM model:

• Damon J. Binder, Shai M. 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:

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

Last revised on November 10, 2019 at 08:31:47. See the history of this page for a list of all contributions to it.