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supergravity in .
for the moment see the respective section at D'Auria-Fre formulation of supergravity
We discuss (Thm. below, following GSS24, §3) how the equations of motion of D=11 supergravity — on an -dimensional super-torsion-free super spacetime with super vielbein (the graviton/gravitino-fields) — follow from just the requirement that the duality-symmetric super-C-field flux densities :
satisfy their Bianchi identities
are on any super-chart of the locally supersymmetric form
Up to some mild (but suggestive, see below) re-arrangement, the computation is essentially that indicated in CDF91, §III.8.5 (where some of the easy checks are indicated) which in turn is a mild reformulation of the original claim in Cremmer & Ferrara 1980 and Brink & Howe 1980 (where less details were given). A full proof is laid out in GSS24, §3, whose notation we follow here.
The following may be understood as an exposition of this result, which seems to stand out as the only account that is (i) fully first-order and (ii) duality-symmetric (in that enters the EoMs as an independent field, whose Hodge duality to is imposed by the Bianchi identity for , remarkably).
Notice that the discussion in CDF91, §III.8 amplifies the superspace-rheonomy principle as a constraint that makes the Bianchi identities on (in our paraphrase) a supergravity Lie 6-algebra-valued higher vielbein be equivalent to the equations of motion of D=11 SuGra. But we may observe that the only rheonomic constraint necessary is that (2) on the C-field flux density — and this is the one not strictly given by rules in CDF91, p. 874, cf. around CDF91, (III.8.41) —; while the remaining rheonomy condition on the gravitino field strength is implied (Lem. below), and the all-important torsion constraint (3) (which is also outside the rules of rheonomy constraints, cf. CDF91, (III.8.33)) is naturally regarded as part of the definition of a super-spacetime in the first place (Def. below).
In thus recasting the formulation of the theorem somewhat, we also:
re-define the super-flux densities as above (2), highlighting that it is (only) in this combination that the algebraic form of the expected Bianchi identity (1) extends to superspace;
disregard the gauge potentials and , whose role in CDF91, §III.8.2-4 is really just to motivate the form of the Bianchi identities equivalent to (1), but whose global nature is more subtle than acknowledged there, while being irrelevant for just the equations of motion.
Indeed, the point is that, in consequence of our second item above, the following formulation shows that one may apply flux quantization of the supergravity C-field on superspace in formally the same way as bosonically (for instance in Cohomotopy as per Hypothesis H, or in any other nonabelian cohomology theory whose classifying space has the -Whitehead -algebra of the 4-sphere), and in fact that the ability to do so implies the EoMs of 11d SuGra. Any such choice of flux quantization is then what defines, conversely, the gauge potentials, globally. Moreover, by the fact brought out here, that the super-flux Bianchi identity already implies the full equations of motion, this flux quantization is thereby seen to be compatible with the equations of motion on all of super spacetime.
For the present formulation, we find it suggestive to regard the all-important torsion constraint (3) as part of the definition of the super-gravity field itself (since it ties the auxiliary spin-connection to the super-vielbein field which embodies the actual super-metric structure):
(super-spacetime)
For
a real spin representation (“Majorana spinors”) of -dimension
whose -equivariant bilinear pairing we denote
by a super-spacetime of super-dimension we here mean:
which admits an open cover by super-Minkowski supermanifolds ,
equipped with a super Cartan connection with respect to the canonical subgroup inclusion of the spin group into the super Poincaré group, namely:
equipped with a super-vielbein , hence on each super-chart
such that at every point the induced map on tangent spaces is an isomorphism
and with a spin-connection (…),
such that the super-torsion vanishes, in that on each chart:
where is a representation of , hence
(the gravitational field strength)
Given a super-spacetime (Def. ), we say that (super chart-wise):
its super-torsion is:
its gravitino field strength is
its curvature is
(super-gravitational Bianchi identities)
By exterior calculus the gravitational field strength tensors (Def. ) satisfy the following identities:
(role of the gravitational Bianchi identities)
Notice that the equations (4) are not conditions but identities satisfied by any super-spacetime (in the sense of Def. , hence even such that .) But conversely this means that when constructing a super-spacetime (say subject to further contraints, such as Bianchi identities for flux densities), the equations (4) are a necessary condition to be satisfied by any candidate super-vielbein-field, and as such they may play the role of equations of motion for the super-gravitational field, as we will see.
Write now for the unique non-trivial irreducible real -representation.
(11d SuGra EoM from super-flux Bianchi identity) Given
(super-gravity field:) an -dimensional super-spacetime (Def. ),
(super-C-field flux densities:) as in (2)
then the super-flux Bianchi identity (1) (the super-higher Maxwell equation for the C-field)
is equivalent to the joint solution by of the equations of motion of D=11 supergravity.
In all of the following lemmas one expands the Bianchi identoties in their super-vielbein form components.
(Normalization conventions)
Our choice of prefactors and normalization follows CDF91 except for the following changes:
our gravitinos absorb a factor of :
our 4-flux density absorbs a combinatorial factor of :
our 7-flux density absorbs a combinatoiral factor of :
Here:
The first rescaling reflects that is not actually a Majorana representation of , but is.
This rescaling removes all occurrences of imaginary units in the Bianchi identities, as it should be for algebra over the real numbers with real fermion representations.
The second rescaling has the effect that instead of .
The Bianchi identity for (1) is equivalent to
the closure of the ordinary 4-flux density
the following dependence of on
shown in any super-chart:
The general expansion of in the super-vielbein basis is of the form
where the last term is taken to vanish.l (…).
Therefore, the Bianchi identity has the following components,
where we used that the quartic spinorial component vanishes identically, due to a Fierz identity (here):
To solve the second line in (6) for (this is CDF91 (III.8.43-49)) we expand in the Clifford algebra (according to this Prop.), observing that for to be a linear combination of the the matrix needs to have a -summand or a -summand. The former does not admit a Spin-equivariant linear combination with coefficients , hence it must be the latter. But then we may also need a component in order to absorb the skew-symmetric product in . Hence must be of this form:
With this, we compute:
Here the multiplicities of the nonvanishing Clifford-contractions arise via this Lemma:
and all remaining contractions vanish inside the spinor pairing by this lemma.
as claimed.
Given the Bianchi identity for (5), then the Bianchi identity for (1) is equivalent to
the Bianchi identity for the ordinary flux density
its Hodge duality to
another condition on the gravitino field strength
The components of the Bianchi identity are
where:
(i) in the quadratic spinorial component we inserted the expression for from (5), then contracted -factors using again this Lemma, and finally observed that of the three spinorial quadratic forms (see there) the coefficients of and of vanish identically, by a remarkable cancellation of combinatorial prefactors:
(ii) the quartic spinorial component holds identitically, due to the Fierz identity here:
Therefore the only spinorial component of the Bianchi identity which is not automatically satisfied is (with , see there) the vanishing of
which is manifestly the claimed Hodge duality relation.
Given the Bianchi identities for (5) and (9), the supergravity fields satisfy their Einstein equations with source the energy momentum tensor of the C-field:
(…)
(…)
under construction
where is the th Pontryagin class.
Concerning the integrality of the I8-term
on a spin manifold . (Witten96, p.9)
First, the index of a Dirac operator on is
Notice that . So
is divisible by 6.
Assume that is further divisible by 2 (see the relevant discussion at M5-brane).
Then the above becomes
and hence then is divisible at least by 24.
But moreover, on a Spin manifold the first fractional Pontryagin class is the Wu class (see there). By definition this means that
and so when is further divisible by 2 we have that is divisible by 48. Hence is integral.
Possible higher curvature corrections to 11-dimensional supergravity are discussed in the references listed below.
The first correction is an -term at order (11d Planck length). In Tsimpis 04 it is shown that part of this is a topological term (total derivative) which relates to the shifted C-field flux quantization.
For effects of higher curvature corrections in a Starobinsky model of cosmic inflation see there.
There is in fact a hidden 1-parameter deformation of the Lagrangian of 11d sugra. Mathematically this was maybe first noticed in (D’Auria-Fre 82) around equation (4.25). This shows that there is a topological term which may be expressed as
where is the curvature 3-form of the supergravity C-field and that of the magnetically dual C6-field. However, (D’Auria-Fre 82) consider only topologically trivial (trivial instanton sector) configurations of the supergravity C-field, and since on them this term is a total derivative, the authors “drop” it.
The term then re-appears in the literatur in (Bandos-Berkovits-Sorokin 97, equation (4.13)). And it seems that this is the same term later also redicovered around equation (4.2) in (Tsimpis 04).
(hm, check)
The basic BPS states of 11d SuGra are
(e.g. EHKNT 07)
10-dimensional type II supergravity, heterotic supergravity
supergravity C-field, supergravity Lie 3-algebra, supergravity Lie 6-algebra
string theory FAQ – Does string theory predict supersymmetry?
Table of branes appearing in supergravity/string theory (for classification see at brane scan).
That there is a maximal dimension in which supergravity may exist was found in
The theory was then actually constructed (as a Lagrangian field theory) in
The claim of the derivation of supergravity in supergeometry, by solving the torsion constraint and Bianchi identities on super spacetime supermanifolds (“superspace”) is due to
Eugène Cremmer, Sergio Ferrara, Formulation of Eleven-Dimensional Supergravity in Superspace, Phys. Lett. B 91 (1980) 61 [doi:10.1016/0370-2693(80)90662-0]
Lars Brink, Paul Howe, Eleven-Dimensional Supergravity on the Mass-Shell in Superspace, Phys. Lett. B 91 (1980) 384 [doi:10.1016/0370-2693(80)91002-3]
and in the mild variation (using a manifestly duality-symmetric super-C-field flux density) due to
A proof of this claim is laid out in
using heavy computer algebra checks (here).
With focus on the Kaluza-Klein compactification to 4d anti de Sitter spacetime:
The history as of the 1990s, with an eye towards the development to M-theory:
The description of 11d supergravity in terms of the D'Auria-Fré-Regge formulation of supergravity originates in
of which a textbook account is in
reviewed again in
The topological deformation (almost) noticed in equation (4.25) of D’Auria-Fre 82 later reappears in (4.13) of
and around (4.2) of Tsimpis 04
The D'Auria-Fré formulation is a first-order formulation of supergravity; for more on this see:
More recent textbook accounts include
Discussion of the equivalence of the 11d SuGra equations of motion with the supergravity torsion constraints is in
following
Much computational detail is displayed in
In terms of pure spinors:
Formulation of the equations of motion of D=11 supergravity in superspace on fields including a flux density a priori independent of the flux density of the supergravity C-field:
Leonardo Castellani, Riccardo D'Auria, Pietro Fré, ch III.8.3-III.8.5 in vol 2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, epdf, ch III.8: pdf]
(Using the D'Auria-Fre formulation of supergravity.)
Antonio Candiello, Kurt Lechner, §6 in: Duality in supergravity theories, Nuclear Physics B 412 3 (1994) 479-501 [doi:10.1016/0550-3213(94)90389-1]
(These authors seem not to be aware of CDF91, III.8 and, contrary to the result there, conclude that it is not possible without introducing non-local relations.)
Discussion of Lagrangian densities for D=11 supergravity with an a priori independent dual C-field field and introduction of the “duality-symmetric” terminology:
Igor Bandos, Nathan Berkovits, Dmitri Sorokin, Duality-Symmetric Eleven-Dimensional Supergravity and its Coupling to M-Branes, Nucl. Phys. B 522 (1998) 214-233 [doi:10.1016/S0550-3213(98)00102-3, arXiv:hep-th/9711055]
Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Section 2 of Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl.Phys. B 535 (1998) 242-292 [doi:10.1016/S0550-3213(98)00552-5, arXiv:hep-th/9806106]
Igor Bandos, Alexei Nurmagambetov, Dmitri Sorokin, Section 2 of: Various Faces of Type IIA Supergravity, Nucl. Phys. B 676 (2004) 189-228 [doi:10.1016/j.nuclphysb.2003.10.036, arXiv:hep-th/0307153]
Alexei J. Nurmagambetov, The Sigma-Model Representation for the Duality-Symmetric Supergravity, eConf C0306234 (2003) 894-901 [arXiv:hep-th/0312157, inspire:635585]
Discussion in the context of shifted C-field flux quantization:
Identifying the super-graded gauge algebra of the C-field in D=11 supergravity (with non-trivial super Lie bracket ):
Eugene Cremmer, Bernard Julia, H. Lu, Christopher Pope, Equation (2.6) of Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities, Nucl.Phys. B 535 (1998) 242-292 [doi:10.1016/S0550-3213(98)00552-5, arXiv:hep-th/9806106]
I. V. Lavrinenko, H. Lu, Christopher N. Pope, Kellogg S. Stelle, (3.4) in: Superdualities, Brane Tensions and Massive IIA/IIB Duality, Nucl. Phys. B 555 (1999) 201-227 [doi:10.1016/S0550-3213(99)00307-7, arXiv:hep-th/9903057]
Jussi Kalkkinen, Kellogg S. Stelle, (75) of: Large Gauge Transformations in M-theory, J. Geom. Phys. 48 (2003) 100-132 [doi:10.1016/S0393-0440(03)00027-5, arXiv:hep-th/0212081]
Igor A. Bandos, Alexei J. Nurmagambetov, Dmitri P. Sorokin, (86) in: Various Faces of Type IIA Supergravity, Nucl.Phys. B 676 (2004) 189-228 [doi:10.1016/j.nuclphysb.2003.10.036, arXiv:hep-th/0307153]
Identification as an -algebra (a dg-Lie algebra, in this case):
and identificatoin with the rational Whitehead -algebra (the rational Quillen model) of the 4-sphere (cf. Hypothesis H):
Hisham Sati, Alexander Voronov, (13) in: Mysterious Triality and M-Theory [arXiv:2212.13968]
Hisham Sati, Urs Schreiber, (22) in: Flux Quantization on Phase Space [arXiv:2312.12517]
Bosonic solutions of eleven-dimensional supergravity were studied in the 1980s in the context of Kaluza-Klein supergravity. The topic received renewed attention in the mid-to-late 1990s as a result of the branes and duality paradigm and the AdS/CFT correspondence.
One of the earliest solutions of eleven-dimensional supergravity is the maximally supersymmetric Freund-Rubin compactification with geometry and 4-form flux proportional to the volume form on .
The radii of curvatures of the two factors are furthermore in a ratio of 1:2. The modern avatar of this solution is as the near-horizon geometry of coincident M2-branes.
Shortly after the original Freund-Rubin solution was discovered, Englert discovered a deformation of this solution where one could turn on flux on the ; namely, singling out one of the Killing spinors of the solution, a suitable multiple of the 4-form one constructs by squaring the spinor can be added to the volume form in and the resulting 4-form still obeys the supergravity field equations, albeit with a different relation between the radii of curvature of the two factors. The flux breaks the SO(8) symmetry of the sphere to an subgroup.
Some of the above is taken from this TP.SE thread.
See also
Don Page, Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity, Phys. Rev. D 28, 2976 (1983) (spire:14480 doi:10.1103/PhysRevD.28.2976)
Ergin Sezgin, 11D Supergravity on versus [arXiv:2003.01135]
Tetsuji Kimura, Eleven-dimensional Supergravities on Maximally Supersymmetric Backgrounds, TK-NOTE/03-11 (2003-2014) [pdf, pdf]
Classification of symmetric solutions:
José Figueroa-O'Farrill, Symmetric M-Theory Backgrounds, Open Physics 11 1 (2013) 1-36 [arXiv:1112.4967, doi:10.2478/s11534-012-0160-6]
Linus Wulff, All symmetric space solutions of eleven-dimensional supergravity, Journal of Physics A: Mathematical and Theoretical 50 24 (2017) 245401 [arXiv:1611.06139, doi:10.1088/1751-8121/aa70b6]
Discussion of black branes and BPS states includes
Kellogg Stelle, section 3 of BPS Branes in Supergravity (arXiv:hep-th/9803116)
Francois Englert, Laurent Houart, Axel Kleinschmidt, Hermann Nicolai, Nassiba Tabti, An multiplet of BPS states, JHEP 0705:065,2007 (arXiv:hep-th/0703285)
Andrew Callister, Douglas Smith, Topological BPS charges in 10 and 11-dimensional supergravity, Phys. Rev. D78:065042,2008 (arXiv:0712.3235)
Andrew Callister, Douglas Smith, Topological charges in covariant massive 11-dimensional and Type IIB SUGRA, Phys.Rev.D80:125035,2009 (arXiv:0907.3614)
Andrew Callister, Topological BPS charges in 10- and 11-dimensional supergravity, thesis 2010 (spire)
A. A. Golubtsova, V.D. Ivashchuk, BPS branes in 10 and 11 dimensional supergravity, talk at DIAS 2013 (pdf slides)
Cristine N. Ferreira, BPS solution for eleven-dimensional supergravity with a conical defect configuration (arXiv:1312.0578)
Discussion of black hole horizons includes
See also
Resolution of scalar field-dressed Schwarzschild black holes in D=11 supergravity:
Discussion of higher curvature corrections to 11-dimensional supergravity (i.e. in M-theory):
Via 11d superspace-cohomology (or mostly):
Kasper Peeters, Pierre Vanhove, Anders Westerberg, Supersymmetric actions and quantum corrections to superspace torsion constraints (arXiv:hep-th/0010182)
H. Lu, Christopher Pope, Kellogg Stelle, Paul Townsend, Supersymmetric Deformations of 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 , JHEP0410:046 (2004) [arXiv:hep-th/0407271, doi:10.1088/1126-6708/2004/10/046]
Paul Howe, 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, 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:01998725, pdf)
Via analysis of would-be superparticle scattering amplitudes on D=11 supergravity backgrounds:
Via geodesic motion on the coset space of the U-duality Kac-Moody group by its “maximal compact” subgroup :
(relating to higher curvature corrections)
From lifting alpha'-corrections in type IIA string theory through the duality between M-theory and type IIA string theory:
From the ABJM model for the M2-brane:
In terms of D=4 supergravity:
Nikolay Bobev, Anthony M. Charles, Kiril Hristov, Valentin Reys, The Unreasonable Effectiveness of Higher-Derivative Supergravity in 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 , J. High Energ. Phys. 2021 58 (2021) [doi:10.1007/JHEP04(2021)058, arXiv:2011.05971]
Kiril Hristov, ABJM at finite via 4d supergravity, J. High Energ. Phys. 2022 190 (2022) [doi:10.1007/JHEP10(2022)190, arXiv:2204.02992]
See also
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:
Computation of Feynman amplitudes/scattering amplitudes and effective action in 11d supergravity:
Stanley Deser, Domenico Seminara, Counterterms/M-theory Corrections to D=11 Supergravity, Phys.Rev.Lett.82:2435-2438, 1999 (arXiv:hep-th/9812136)
Stanley Deser, Domenico Seminara, Tree Amplitudes and Two-loop Counterterms in D=11 Supergravity, Phys.Rev.D62:084010, 2000 (arXiv:hep-th/0002241)
L. Anguelova, P. A. Grassi, P. Vanhove, Covariant One-Loop Amplitudes in , Nucl. Phys. B702 (2004) 269-306 (arXiv:hep-th/0408171)
Kasper Peeters, Jan Plefka, Steffen Stern, Higher-derivative gauge field terms in the M-theory action, JHEP 0508 (2005) 095 (arXiv:hep-th/0507178)
Hamid R. Bakhtiarizadeh, Gauge field corrections to eleven dimensional supergravity via dimensional reduction (arXiv:1711.11313)
Kaluza-Klein compactifications of supergravity and its consistent truncations:
Discussion of Freund-Rubin compactifications:
Discussion of quantum anomaly cancellation and Green-Schwarz mechanism in 11D supergravity includes the following articles. (For more see at M5-brane – anomaly cancellation).
Edward Witten, On Flux Quantization In M-Theory And The Effective Action (arXiv:hep-th/9609122)
Edward Witten, Five-Brane Effective Action In M-Theory, J.Geom.Phys.22:103-133, 1997 (arXiv:hep-th/9610234)
Dan Freed, Jeff Harvey, Ruben Minasian, Greg Moore, Gravitational Anomaly Cancellation for M-Theory Fivebranes, Adv.Theor.Math.Phys.2:601-618, 1998 (arXiv:hep-th/9803205)
Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review, Nucl.Phys. B675 (2003) 416-446 (arXiv:hep-th/0307152)
Ibrahima Bah, Federico Bonetti, Ruben Minasian, Emily Nardoni, Class Anomalies from M-theory Inflow (arXiv:1812.04016)
Daniel Freed, Two nontrivial index theorems in odd dimensions (arXiv:dg-ga/9601005)
Adel Bilal, Steffen Metzger, Anomaly cancellation in M-theory: a critical review (arXiv:hep-th/0307152)
Daniel S. Freed, Michael J. Hopkins, Consistency of M-Theory on nonorientable manifolds, The Quarterly Journal of Mathematics 72 1-2 (2021) 603–671 [arXiv:1908.09916, doi:10.1093/qmath/haab007]
Fei Han, Ruizhi Huang, Kefeng Liu, Weiping Zhang, Cubic forms, anomaly cancellation and modularity, Advances in Mathematics 394 (2022) 108023 [arXiv:2005.02344, doi:10.1016/j.aim.2021.108023]
Review of U-duality and exceptional generalized geometry in KK-compactification of D=11 supergravity:
Last revised on December 9, 2024 at 11:24:45. See the history of this page for a list of all contributions to it.