M-theory, Black Holes and Cosmology

Renata Kallosh

REVIEW

M-theory, Black Holes and Cosmology

Renata Kallosh

High Energy Physics - Theory · Submitted 23 Sep 2020 · 43 references

Pith's verdict: The advertised fermion-mass/black-hole-entropy relation is broken—eq (4.17) is algebraically inconsistent—while the Bhargava-cube triality note is a genuine but modest addition to an otherwise competent review.

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arXiv ID2009.11339 (version 1)

Pith ID6G4A7YIK, Pith's permanent record of this paper and its review

CitationRenata Kallosh. "M-theory, Black Holes and Cosmology." arXiv:2009.11339 (2020). https://pith.science/paper/6G4A7YIK

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Editor's note

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Read the Kallosh paper. Short version: if someone sends this to you for a refereeing decision, the headline claim in the abstract—the puzzling relation between cosmological fermion masses, the exceptional Jordan eigenvalue problem, and black hole entropy—does not hold as written. Eq. (4.17) is the load-bearing piece, and it's broken. MWO is a 7x7 matrix whose eigenvalues are {x1,x2,x3,y1,y2,y3,0}; its determinant is exactly zero. The chain det MWO = x1x2x3 I3 = λ1λ2λ3 = p1p2p3 is dimensionally wrong and algebraically inconsistent. Even if you meant a 3x3 block, x1x2x3 = 7 from eq. (4.10), so you'd need 7 I3, not I3. There is no definition of a projected determinant here, and no argument connects fixed numbers 3.0489, -1.6920, -1.3569 to arbitrary black hole charges. So the entropy connection is at present an error, not a puzzle.

What's actually good: the paper is a clear, readable review of the black-hole/qubit correspondence and the octonion-based cosmology from [1]. The explicit S3 triality symmetry of the Bhargava cube discriminant, eq. (2.17), is a real—if small—algebraic observation that complements the known [SL(2,Z)]^3 structure. The fermion mass matrix calculation in Section 4 is explicit: the matrix, the eigenvalues, and the cubic x^3 - 7x - 7 = 0 are all stated, and the numerical values are consistent with the equations given. The match to the exceptional Jordan eigenvalue problem is a sensible consistency observation—setting TrJ=0 and detJ=7 makes the characteristic cubic line up—but it's a coincidence spotted after the fact, not a derivation. The paper is mostly honest that the superpotential is proposed rather than derived from 11d supergravity, though the abstract and Fig. 1 overstate the M-theory origin a bit.

Soft spots, in proportion: eq. (4.17) is a major flaw and needs to be fixed or explicitly retracted. The Jordan comparison has notational sloppiness around J×J and TrJ^{-1}; those are minor. The cosmological predictions rest entirely on an ansatz in [1], so they are benchmark targets from a model, not robust M-theory predictions. That's worth saying once, not a reason to reject.

Who is this for? Readers who want a compact survey of the black-hole/qubit literature and the octonion inflation program. The triality observation is worth knowing if you work on Bhargava cubes or STU black holes. I would not cite it for the entropy relation, but I'd send it to a referee familiar with exceptional Jordan algebras and supergravity. A serious referee should ask for a corrected or deleted (4.17) and a clearer separation between derived results and proposed models; after that, the paper is publishable as a review-plus-modest-observation.

Referee report

Significance

The Bhargava-cube triality observation in Sec. 2 is a clean, checkable statement and is a useful addition to the black-hole/qubit correspondence literature. The octonionic cosmological models and their CMB predictions are not new here; they are imported from [1]. The purported Jordan/black-hole connection in Sec. 4 is intriguing but, as written, is not established: Eq. (4.17) is internally inconsistent, and the 'choice' of Jordan invariants is a consistency check rather than a derivation. If the author corrects or retracts the problematic equality, the paper would be a modest but acceptable contribution; in its current form the advertised central relation is unsupported. There are no machine-checked proofs, but the explicit 7x7 matrix makes the eigenvalue claims easy to verify independently.

Major objections (3)

[Sec. 4, Eq. (4.17)] Equation (4.17) is load-bearing for the paper's advertised connection between cosmological fermion masses, the exceptional Jordan eigenvalue problem, and black hole entropy, but it does not hold as written. MWO is the 7x7 matrix (4.2), and its diagonal form (4.5) has eigenvalues {x1,x2,x3,y1,y2,y3,0}; therefore det MWO = 0 identically. The chain also mixes objects of different type: x1x2x3 I3 is a 3x3 matrix, while det MWO and p1p2p3 are scalars. Furthermore, Eq. (4.10) gives x1x2x3 = 7, and the Jordan eigenvalues satisfy λ1λ2λ3 = det J = 7, so neither product equals a generic set of black hole charges p1,p2,p3. No definition of a 3x3 sub-block or projected determinant is given, and no derivation connects the fixed numbers (3.0489, -1.6920, -1.3569) to arbitrary charges. The author should either formulate a precise corrected version of (4.17) with a defined sub-block determinant and prove its relation to I3, or explicitly retract the equality to p1p2p3.
[Sec. 4, Eqs. (4.14) and (4.16)] The claimed correspondence to the exceptional Jordan eigenvalue problem rests on imposing TrJ = 0 and det J = 7, but the invariant statements contain inconsistencies. In (4.14), 'TrJ^{-1} = -I' equates a scalar trace with the identity matrix; presumably TrJ^{-1} = -1 is intended (or, from the cubic (4.12), Tr(J×J) = -7). In (4.16), '(J×J) = -7' should read 'Tr(J×J) = -7'. More substantively, choosing the Jordan invariants so that the characteristic equation matches x^3 - 7x - 7 = 0 is a consistency observation, not a derivation of the cubic from the octonionic cosmological model; the manuscript should state this limitation explicitly rather than presenting the match as a relation that selects the model.
[Sec. 3, Eq. (3.1)] The fermion mass matrix, its eigenvalues, and hence the Sec. 4 relations all follow from the octonionic superpotential (3.1). This superpotential is proposed rather than derived from 11d supergravity or from a concrete G2 compactification, and the paper relies on [1] for its justification. Since the new 'puzzling relations' in Sec. 4 are conditional on this ansatz, the paper should state prominently that the results are conditional on the assumed form of WO; if the true effective superpotential differs, the cubic x^3 - 7x - 7 and the Jordan/black-hole comparison would not follow.

Minor objections (5)

[Eq. (4.4)] The second cubic is written as y^3 - 7x - 7 = 0; it should be y^3 - 7y - 7 = 0.
[Sec. 4, Eqs. (4.12)-(4.16)] The notation for the Jordan invariants is inconsistent: (4.15) lists TrJ, Tr(J×J), detJ, but (4.14) and (4.16) refer to TrJ^{-1} and (J×J). Please use Tr(J×J) throughout.
[Sec. 4, Eq. (4.17)] If a 3x3 sub-block determinant is intended in (4.17), introduce an explicit symbol for it (e.g., det M3) before writing any equality involving I3.
[Abstract and Introduction] The abstract states that the paper 'shows puzzling relations between fermion mass eigenvalues ... and black hole entropy'; after correcting (4.17), please recalibrate this claim so that the relation is stated as a numerical coincidence or a conjecture, rather than as an equality.
[Figs. 3 and 4] Figures 3 and 4 reproduce large blocks of external textbook text; please crop the figures to the actual cube diagrams so that the intended symmetries are visible without distraction.

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The paper

The paper claims

This paper identifies an $S_3$ triality symmetry in STU black hole entropy through the Bhargava cube, and reports that fermion masses from an octonionic M-theory cosmology solve the exceptional Jordan cubic $x^3-7x-7=0$.

The reading

The paper brings two M-theory stories under one conceptual roof. For black holes, it argues that the STU entropy formula is the discriminant of a Bhargava cube and that this discriminant carries an $S_3$ triality symmetry, with the three moduli on equal footing, in addition to the known $[SL(2,\mathbb{Z})]^3$ modular symmetry. For cosmology, it reviews a proposed octonionic superpotential for M-theory on a $G_2$-holonomy manifold and shows that its fermion mass matrix has six nonzero eigenvalues solving $x^3-7x-7=0$, the same cubic that fixes the eigenvalues of an exceptional Jordan matrix with $\mathrm{Tr}\,J=0$ and $\det J=7$. If these connections hold, black hole entropy, quantum-information triality, error-correcting codes, and early-universe observables are tied to one another, and future CMB measurements of primordial gravitational waves can test the cosmological side.

What carries the argument

The arguments are carried by two objects. The first is the Bhargava cube, a $2\times2\times2$ array of eight charges whose discriminant $D_{\rm Bha}$ equals the negative STU entropy $W$ and whose slicing directions, together with the permutation rules of eq. (2.17), encode the $S_3$ triality symmetry. The second is the octonionic superpotential $W_O=\frac12 M_{ij}T^iT^j$ from eq. (3.1), built from the seven complementary four-qubit codewords of the Hamming (7,4) code and the seven octonionic triads; the mass matrix $M$ formed from its 14 surviving terms has six nonzero eigenvalues that solve $x^3-7x-7=0$ and $y^3-7y-7=0$. The identification of that cubic with the exceptional Jordan eigenvalue equation, through the invariants $\mathrm{Tr}\,J$, $\mathrm{Tr}(J\times J)$, and $\det J$, is what lets the paper tie cosmological fermion masses to black hole entropy.

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a bridge between two seven-moduli structures: the STU black hole entropy $W$ is related to the Bhargava cube discriminant by $D_{\rm Bha}=-W$, with explicit $S_3$ permutation symmetries acting on the cube's eight entries that mirror the triality exchange of the three moduli (eq. 2.17); and the effective superpotential of the octonionic cosmological model yields a fermion mass matrix whose six nonzero eigenvalues are the roots of $x^3-7x-7=0$, with the remaining eigenvalue zero. The paper shows that these eigenvalue triples match the characteristic equation of the exceptional Jordan algebra eigenvalue problem with $\mathrm{Tr}\,J=0$, $\mathrm{Tr}(J\times J)=-7$, and $\det J=7$, and that the determinant of the mass matrix is related to the black hole cubic invariant $I_3$. It presents these as observed, 'puzzling' relations rather than as a fully derived equivalence.

Load-bearing premise

The load-bearing premise is that the octonionic superpotential $W_O$ of eq. (3.1), built from Hamming codewords and octonion triads, is the correct effective $N=1$ superpotential for M-theory on a $G_2$-holonomy manifold; the paper calls it proposed and does not derive it from 11-dimensional supergravity, and if the true superpotential differs, the fermion mass eigenvalues, the cubic $x^3-7x-7=0$, and the link to the exceptional Jordan problem all lapse.

Editorial extensions

If this is right

STU black hole classification should be refined: two charge octuples with the same entropy are equivalent only if related by $[SL(2,\mathbb{Z})]^3\ltimes S_3$, so triality must be included when counting states on Bhargava cubes.
The octonionic models yield seven benchmark values of the tensor-to-scalar ratio $r=12\alpha/N_e^2$ for $\alpha=7,6,\ldots,1$, all roughly in the range $10^{-2}$ to $10^{-3}$, which future CMB experiments are designed to probe.
The six nonzero fermion masses come in two identical triplets, $3.0489$, $-1.6920$, $-1.3569$, plus one massless mode, and their sum is exactly zero because the roots of $x^3-7x-7=0$ sum to zero.
The mass matrix is $O(7)$-invariant, with discrete Weyl group $W(E_7)$ of order 2,903,040; imposing octonion multiplication reduces this to the Chevalley group $G_2(2)$ of order 12,096, a symmetry the paper connects to neutrino physics.
The determinant of the mass matrix satisfies $\det M_{W_O}=x_1x_2x_3\,I_3$, where $I_3$ is the cubic invariant of the 5d black hole entropy, tying the cosmological mass spectrum to black hole entropy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The discriminant of $x^3-7x-7=0$ is $7^2$, a perfect square, so its Galois group is cyclic of order three; if the match to the fermion mass matrix is physical rather than accidental, a $\mathbb{Z}_3$ or $S_3$ symmetry should be visible among the three mass eigenstates, and looking for it is a direct extension of the paper's observation.
The equality of the mass-matrix cubic with the exceptional Jordan eigenvalue problem suggests constructing the $7\times7$ mass matrix as, or as a direct sum involving, a traceless element of the exceptional Jordan algebra; doing so would promote the paper's 'puzzling' numerical coincidence into a derivation and give a physical origin for the constraints $\mathrm{Tr}\,J=0$ and $\det J=7$.
If the octonionic superpotential is the actual effective description, the $E_7$ root system that governs the fermion discrete symmetry should also constrain the inflationary sector: an explicit check would be whether the seven $\alpha$-attractor targets ($\alpha=7,\ldots,1$) sit in orbits of $W(E_7)$ and are left invariant by the $G_2(2)$ subgroup.
The seven benchmark tensor-to-scalar ratios give a sharp empirical test: a detection at one of the seven predicted $r$ values would anchor the octonionic structure observationally, while a null result below $r\sim0.001$ would eliminate the smallest-$\alpha$ targets without by itself falsifying the framework as a whole.

Original abstract

This paper is dedicated to Mike Duff on the occasion of his 70th birthday. I discuss some issues of M-theory/string theory/supergravity closely related to Mike's interests. I describe a relation between STU black hole entropy, Cayley hyperdeterminant, Bhargava cube and a 3-qubit Alice, Bob, Charlie triality symmetry. I shortly describe my recent work with Gunaydin, Linde, Yamada on M-theory cosmology, inspired by the work of Duff with Ferrara and Borsten, Levay, Marrani et al. Here we have 7-qubits, a party including Alice, Bob, Charlie, Daisy, Emma, Fred, George. Octonions and Hamming error correcting codes are at the base of these models. They lead to 7 benchmark targets of future CMB missions looking for primordial gravitational wave from inflation. I also show puzzling relations between the fermion mass eigenvalues in these cosmological models, exceptional Jordan eigenvalue problem, and black hole entropy. The symmetry of our cosmological models is illustrated by beautiful pictures of a Coxeter projection of the root system of E7.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,262 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.03 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.66 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

This paper is dedicated to Mike Duff on the occasion of his 70th birthday. I discuss some issues of M-theory/string theory/supergravity closely related to Mike's interests. I describe a relation between STU black hole entropy, Cayley hyperdeterminant, Bhargava cube and a 3-qubit Alice, Bob, Charlie triality symmetry. I shortly describe my recent work with Gunaydin, Linde, Yamada on M-theory cosmology, inspired by the work of Duff with Ferrara and Borsten, Levay, Marrani et al. Here we have 7-qubits, a party including Alice, Bob, Charlie, Daisy, Emma, Fred, George. Octonions and Hamming error correcting codes are at the base of these models. They lead to 7 benchmark targets of future CMB missions looking for primordial gravitational wave from inflation. I also show puzzling relations between the fermion mass eigenvalues in these cosmological models, exceptional Jordan eigenvalue problem, and black hole entropy. The symmetry of our cosmological models is illustrated by beautiful pictures of a Coxeter projection of the root system of E7.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper is dedicated to Mike Duff on the occasion of his 70th birthday.	background 0.99	0.20	0.01	asserted 0.89
s1	I discuss some issues of M-theory/string theory/supergravity closely related to Mike's interests.	background 0.62	0.51	0.02	asserted 0.74
s2	I describe a relation between STU black hole entropy, Cayley hyperdeterminant, Bhargava cube and a 3-qubit Alice, Bob, Charlie triality symmetry.	contribution 0.87	0.81	0.16	asserted 1.00
s3	I shortly describe my recent work with Gunaydin, Linde, Yamada on M-theory cosmology, inspired by the work of Duff with Ferrara and Borsten, Levay, Marrani et al. Here we have 7-qubits, a party including Alice, Bob, Charlie, Daisy, Emma, Fred, George.	contribution 0.99	0.90	0.06	asserted 0.99
s4	Octonions and Hamming error correcting codes are at the base of these models.	method 0.87	0.27	0.05	asserted 1.00
s5	They lead to 7 benchmark targets of future CMB missions looking for primordial gravitational wave from inflation.	result 0.72	0.70	0.13	asserted 1.00
s6	I also show puzzling relations between the fermion mass eigenvalues in these cosmological models, exceptional Jordan eigenvalue problem, and black hole entropy.	result 0.74	0.87	0.12	asserted 1.00
s7	The symmetry of our cosmological models is illustrated by beautiful pictures of a Coxeter projection of the root system of E7.	contribution 0.50	0.77	0.05	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.80 · evidence from proof or derivation 0.68.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (43)

[1]

M. Gunaydin, R. Kallosh, A. Linde and Y. Yamada, M-theory Cosmology, Octonions, Error Correcting Codes, 2008.01494

View on Pith Pith review arXiv 2008
[2]

S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D54 (1996) 1514 [hep-th/9602136]

View on Pith Pith review arXiv 1996
[3]

M. Duﬀ, J. T. Liu and J. Rahmfeld, Four-dimensional string-string-string triality , Nucl. Phys. B 459 (1996) 125 [ hep-th/9508094]

View on Pith Pith review arXiv 1996
[4]

K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W. K. Wong, STU black holes and string triality, Phys. Rev. D54 (1996) 6293 [ hep-th/9608059]

View on Pith Pith review arXiv 1996
[5]

M. Duﬀ, String triality, black hole entropy and Cayley’s hyperdeterminant , Phys. Rev. D 76 (2007) 025017 [ hep-th/0601134]. – 17 –

View on Pith Pith review arXiv 2007
[6]

R. Kallosh and A. D. Linde, Strings, black holes, and quantum information , Phys. Rev. D73 (2006) 104033 [ hep-th/0602061]

View on Pith Pith review arXiv 2006
[7]

N. Benjamin, S. Kachru, K. Ono and L. Rolen, Black holes and class groups , 1807.00797

View on Pith Pith review arXiv
[8]

M. Gunaydin, S. Kachru and A. Tripathy, Black holes and Bhargava’s invariant theory , 1903.02323

View on Pith Pith review arXiv 1903
[9]

N. Banerjee, A. Bhand, S. Dutta, A. Sen and R. K. Singh, Bhargava’s Cube and Black Hole Charges, 2006.02494

View on Pith Pith review arXiv 2006
[10]

L. Borsten, M. Duﬀ and A. Marrani, Black Holes and Higher Composition Laws , 2006.03574

View on Pith Pith review arXiv 2006
[11]

L. Borsten, E(7)(7) invariant measures of entanglement , Fortsch. Phys. 56 (2008) 842

View on Pith 2008
[12]

L. Borsten, D. Dahanayake, M. Duﬀ, H. Ebrahim and W. Rubens, Black Holes, Qubits and Octonions, Phys. Rept. 471 (2009) 113 [ 0809.4685]

View on Pith Pith review arXiv 2009
[13]

M. Awada, M. Duﬀ and C. Pope, N=8 Supergravity Breaks Down to N=1 , Phys. Rev. Lett. 50 (1983) 294

View on Pith 1983
[14]

G. Dall’Agata and N. Prezas, Scherk-Schwarz reduction of M-theory on G2-manifolds with ﬂuxes, JHEP 10 (2005) 103 [ hep-th/0509052]

View on Pith Pith review arXiv 2005
[15]

S. Ferrara and R. Kallosh, Seven-disk manifold, α-attractors, and B modes, Phys. Rev. D94 (2016) 126015 [ 1610.04163]

View on Pith Pith review arXiv 2016
[16]

M. J. Duﬀ and S. Ferrara, E(7) and the tripartite entanglement of seven qubits , Phys. Rev. D76 (2007) 025018 [ quant-ph/0609227]

View on Pith Pith review arXiv 2007
[17]

P. Levay, Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane , Phys. Rev. D75 (2007) 024024 [ hep-th/0610314]

View on Pith Pith review arXiv 2007
[18]

P. Levay, Attractors, Black Holes and Multiqubit Entanglement , Springer Proc. Phys. 134 (2010) 85

View on Pith 2010
[19]

L. Borsten, M. J. Duﬀ and P. Levay, The black-hole/qubit correspondence: an up-to-date review, Class. Quant. Grav. 29 (2012) 224008 [ 1206.3166]

View on Pith Pith review arXiv 2012
[20]

R. Kallosh, A. Linde, T. Wrase and Y. Yamada, Maximal Supersymmetry and B-Mode Targets , JHEP 04 (2017) 144 [ 1704.04829]

View on Pith Pith review arXiv 2017
[21]

R. Kallosh, A. Linde, D. Roest and Y. Yamada, D3 induced geometric inﬂation, JHEP 07 (2017) 057 [ 1705.09247]

View on Pith Pith review arXiv 2017
[22]

R. Kallosh and A. Linde, CMB Targets after PlanckCMB targets after the latest Planck data release, Phys. Rev. D100 (2019) 123523 [ 1909.04687]

View on Pith Pith review arXiv 2019
[23]

A. Lee, P. A. R. Ade, Y. Akiba, D. Alonso, K. Arnold, J. Aumont et al., LiteBIRD: an all-sky cosmic microwave background probe of inﬂation , in Bulletin of the American Astronomical Society, vol. 51, p. 286, Sep, 2019

View on Pith 2019
[24]

M. Gunaydin, C. Piron and H. Ruegg, Moufang Plane and Octonionic Quantum Mechanics , Commun. Math. Phys. 61 (1978) 69

View on Pith 1978
[25]

R. Dundarer, F. Gursey and H. C. Tze, Generalized Vector Products, Duality and Octonionic Identities in D = 8 Geometry, J. Math. Phys. 25 (1984) 1496

View on Pith 1984
[26]

S. Ferrara and M. Gunaydin, Orbits of exceptional groups, duality and BPS states in string theory, Int. J. Mod. Phys. A 13 (1998) 2075 [ hep-th/9708025]. – 18 –

View on Pith Pith review arXiv 1998
[27]

S. Ferrara and M. Gunaydin, Orbits and Attractors for N=2 Maxwell-Einstein Supergravity Theories in Five Dimensions , Nucl. Phys. B 759 (2006) 1 [ hep-th/0606108]

View on Pith Pith review arXiv 2006
[28]

T. Dray and C. A. Manogue, The exceptional Jordan eigenvalue problem , Int. J. Theor. Phys. 38 (1999) 2901 [ math-ph/9910004]

View on Pith Pith review arXiv 1999
[29]

M. Koca and N. Ozdes, Division Algebras With Integral Elements , J. Phys. A 22 (1989) 1469

View on Pith 1989
[30]

F. Karsch and M. Koca, G2(2) as the automorphism group of the octonionic root system of E 7, J. Phys. A 23 (1990) 4739

View on Pith 1990
[31]

M. Koca, R. Koc and N. O. Koca, The Chevalley group G 2(2) of order 12096 and the octonionic root system of E 7, Linear Algebra Appl. 422 (2007) 808 [ hep-th/0509189]

View on Pith Pith review arXiv 2007
[32]

A. Anastasiou and M. Hughes, Octonionic D=11 Supergravity and ’Octavian Integers’ as Dilaton Vectors, 1502.02578

View on Pith Pith review arXiv
[33]

P. Ramond, The Freund-Rubin Coset, Textures and Group Theory , J. Phys. A 53 (2020) 341001 [2002.04729]

View on Pith Pith review arXiv 2020
[34]

M. J. Perez, M. H. Rahat, P. Ramond, A. J. Stuart and B. Xu, Tribimaximal mixing in the SU (5)×T 13 texture, Phys. Rev. D 101 (2020) 075018 [ 2001.04019]

View on Pith Pith review arXiv 2020
[35]

Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications , 1909.01358

View on Pith Pith review arXiv 1909
[36]

N. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, E7(7) constraints on counterterms in N=8 supergravity , Phys. Lett. B 694 (2011) 265 [ 1009.1643]

View on Pith Pith review arXiv 2011
[37]

R. Kallosh, The Action with Manifest E7 Type Symmetry , JHEP 05 (2019) 109 [ 1812.08087]

View on Pith Pith review arXiv 2019
[38]

M. Gunaydin and R. Kallosh, Supersymmetry constraints on U-duality invariant deformations of N≥ 5 Supergravity, JHEP 09 (2019) 105 [ 1812.08758]

View on Pith Pith review arXiv 2019
[39]

E. Cartan and J. A. Schouten, On Riemannian Geometries admitting an absolute parallelism , Royal Academy Amsterdam Proc. of the Section of Sciences 29 (1926) 933

View on Pith 1926
[40]

H. S. M. Coxeter, Integral Cayley numbers, Duke Mathematical Journal 13 (1946) 561

View on Pith 1946
[41]

J. C. Baez, The Octonions, Bull. Am. Math. Soc. 39 (2002) 145 [ math/0105155]

View on Pith Pith review arXiv 2002
[42]

S. Ferrara and R. Kallosh, Universality of supersymmetric attractors , Phys. Rev. D54 (1996) 1525 [hep-th/9603090]

View on Pith Pith review arXiv 1996
[43]

CMB-S4 collaboration, K. Abazajian et al., CMB-S4: Forecasting Constraints on Primordial Gravitational Waves, 2008.12619. – 19 –

View on Pith Pith review arXiv 2008

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Cites it as background 1 paper

Many black holes in such a theory have been constructed [ref] .

Light Behaviors around Black Holes in M-theory · Light deviating behaviors of the black holes in M-theory compactification · 0.94

What it cites · 54 sentences without a confident label

Uses data or values from 3 papers

The most recent forecast of the CMB-S4 in [ref] suggests that the ground based Stage-4 experiments will achieve the science goals of detecting primordial gravitational waves for $r > 0.003$ at greater than $5σ$, or, in the absence of a detection, of reaching an upper limit of $r < 0.001$ at 95% CL.

CMB-S4: Forecasting Constraints on Primordial Gravitational Waves · Discussion · 0.82
Numerically this gives for a set of $x_1, y_1; x_2, y_2; x_3, y_3$ and a massless one, the following values 3.0489, 3.0489; -1.6920, -1.6920; -1.3569, -1.3569; 0 as shown in [ref] .

M-theory Cosmology, Octonions, Error Correcting Codes · Properties of the mass matrix in octonion cosmological models · 0.99
The values of the 3 complex moduli near the horizon, for each $i=1,2,3$, were computed in [ref] z^i= B^i2 A_i ∓i √(B^i- 4A_i C^i) 2 A_i where for each $i=1,2,3$ A_i= p^0 q_i- 3 d_ijk p^j p^k B^i= p·q- 2 p^iq_i C^i= -(p^i q_0 + 3d^ijk q_jq_k) and W= -D= B^1- 4A_1 C^1 = B^2- 4A_2 C^2=B^3- 4A_3 C^3 .

STU Black Holes and String Triality · STU black holes, triality and the Bhargava cube · 0.83

Compares with 2 papers

We can compare the eigenvalues of the 3x3 part of the fermion mass matrix with the eigenvalues of the 3x3 octonionic Hermitian matrix studied in [ref] which defines the supersymmetric black hole entropy in 5d.

Orbits and Attractors for N=2 Maxwell-Einstein Supergravity Theories in Five Dimensions · Properties of the mass matrix in octonion cosmological models · 0.99
We can compare the eigenvalues of the 3x3 part of the fermion mass matrix with the eigenvalues of the 3x3 octonionic Hermitian matrix studied in [ref] which defines the supersymmetric black hole entropy in 5d.

Orbits of Exceptional Groups, Duality and BPS States in String Theory · Properties of the mass matrix in octonion cosmological models · 0.99

Cites as background 19 papers

A short summary of the recent paper [ref] suitable for this set up is the following.

M-theory Cosmology, Octonions, Error Correcting Codes · M-theory cosmology, octonions and error correcting codes · 0.87
Freudenthal in `Oktaven, Ausnahmegruppen und Oktavengeometrie', Mathematisch Instituut der Rijksuniversiteit te Utrecht, 1951.] eigenvalue problem [ref] .

Orbits and Attractors for N=2 Maxwell-Einstein Supergravity Theories in Five Dimensions · Introduction · 0.86
Freudenthal in `Oktaven, Ausnahmegruppen und Oktavengeometrie', Mathematisch Instituut der Rijksuniversiteit te Utrecht, 1951.] eigenvalue problem [ref] .

The Exceptional Jordan Eigenvalue Problem · Introduction · 0.86

Discussion

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[1] [1]

M. Gunaydin, R. Kallosh, A. Linde and Y. Yamada, M-theory Cosmology, Octonions, Error Correcting Codes, 2008.01494

View on Pith Pith review arXiv 2008

[2] [2]

S. Ferrara and R. Kallosh, Supersymmetry and attractors, Phys. Rev. D54 (1996) 1514 [hep-th/9602136]

View on Pith Pith review arXiv 1996

[3] [3]

M. Duﬀ, J. T. Liu and J. Rahmfeld, Four-dimensional string-string-string triality , Nucl. Phys. B 459 (1996) 125 [ hep-th/9508094]

View on Pith Pith review arXiv 1996

[4] [4]

K. Behrndt, R. Kallosh, J. Rahmfeld, M. Shmakova and W. K. Wong, STU black holes and string triality, Phys. Rev. D54 (1996) 6293 [ hep-th/9608059]

View on Pith Pith review arXiv 1996

[5] [5]

M. Duﬀ, String triality, black hole entropy and Cayley’s hyperdeterminant , Phys. Rev. D 76 (2007) 025017 [ hep-th/0601134]. – 17 –

View on Pith Pith review arXiv 2007

[6] [6]

R. Kallosh and A. D. Linde, Strings, black holes, and quantum information , Phys. Rev. D73 (2006) 104033 [ hep-th/0602061]

View on Pith Pith review arXiv 2006

[7] [7]

N. Benjamin, S. Kachru, K. Ono and L. Rolen, Black holes and class groups , 1807.00797

View on Pith Pith review arXiv

[8] [8]

M. Gunaydin, S. Kachru and A. Tripathy, Black holes and Bhargava’s invariant theory , 1903.02323

View on Pith Pith review arXiv 1903

[9] [9]

N. Banerjee, A. Bhand, S. Dutta, A. Sen and R. K. Singh, Bhargava’s Cube and Black Hole Charges, 2006.02494

View on Pith Pith review arXiv 2006

[10] [10]

L. Borsten, M. Duﬀ and A. Marrani, Black Holes and Higher Composition Laws , 2006.03574

View on Pith Pith review arXiv 2006

[11] [11]

L. Borsten, E(7)(7) invariant measures of entanglement , Fortsch. Phys. 56 (2008) 842

View on Pith 2008

[12] [12]

L. Borsten, D. Dahanayake, M. Duﬀ, H. Ebrahim and W. Rubens, Black Holes, Qubits and Octonions, Phys. Rept. 471 (2009) 113 [ 0809.4685]

View on Pith Pith review arXiv 2009

[13] [13]

M. Awada, M. Duﬀ and C. Pope, N=8 Supergravity Breaks Down to N=1 , Phys. Rev. Lett. 50 (1983) 294

View on Pith 1983

[14] [14]

G. Dall’Agata and N. Prezas, Scherk-Schwarz reduction of M-theory on G2-manifolds with ﬂuxes, JHEP 10 (2005) 103 [ hep-th/0509052]

View on Pith Pith review arXiv 2005

[15] [15]

S. Ferrara and R. Kallosh, Seven-disk manifold, α-attractors, and B modes, Phys. Rev. D94 (2016) 126015 [ 1610.04163]

View on Pith Pith review arXiv 2016

[16] [16]

M. J. Duﬀ and S. Ferrara, E(7) and the tripartite entanglement of seven qubits , Phys. Rev. D76 (2007) 025018 [ quant-ph/0609227]

View on Pith Pith review arXiv 2007

[17] [17]

P. Levay, Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane , Phys. Rev. D75 (2007) 024024 [ hep-th/0610314]

View on Pith Pith review arXiv 2007

[18] [18]

P. Levay, Attractors, Black Holes and Multiqubit Entanglement , Springer Proc. Phys. 134 (2010) 85

View on Pith 2010

[19] [19]

L. Borsten, M. J. Duﬀ and P. Levay, The black-hole/qubit correspondence: an up-to-date review, Class. Quant. Grav. 29 (2012) 224008 [ 1206.3166]

View on Pith Pith review arXiv 2012

[20] [20]

R. Kallosh, A. Linde, T. Wrase and Y. Yamada, Maximal Supersymmetry and B-Mode Targets , JHEP 04 (2017) 144 [ 1704.04829]

View on Pith Pith review arXiv 2017

[21] [21]

R. Kallosh, A. Linde, D. Roest and Y. Yamada, D3 induced geometric inﬂation, JHEP 07 (2017) 057 [ 1705.09247]

View on Pith Pith review arXiv 2017

[22] [22]

R. Kallosh and A. Linde, CMB Targets after PlanckCMB targets after the latest Planck data release, Phys. Rev. D100 (2019) 123523 [ 1909.04687]

View on Pith Pith review arXiv 2019

[23] [23]

A. Lee, P. A. R. Ade, Y. Akiba, D. Alonso, K. Arnold, J. Aumont et al., LiteBIRD: an all-sky cosmic microwave background probe of inﬂation , in Bulletin of the American Astronomical Society, vol. 51, p. 286, Sep, 2019

View on Pith 2019

[24] [24]

M. Gunaydin, C. Piron and H. Ruegg, Moufang Plane and Octonionic Quantum Mechanics , Commun. Math. Phys. 61 (1978) 69

View on Pith 1978

[25] [25]

R. Dundarer, F. Gursey and H. C. Tze, Generalized Vector Products, Duality and Octonionic Identities in D = 8 Geometry, J. Math. Phys. 25 (1984) 1496

View on Pith 1984

[26] [26]

S. Ferrara and M. Gunaydin, Orbits of exceptional groups, duality and BPS states in string theory, Int. J. Mod. Phys. A 13 (1998) 2075 [ hep-th/9708025]. – 18 –

View on Pith Pith review arXiv 1998

[27] [27]

S. Ferrara and M. Gunaydin, Orbits and Attractors for N=2 Maxwell-Einstein Supergravity Theories in Five Dimensions , Nucl. Phys. B 759 (2006) 1 [ hep-th/0606108]

View on Pith Pith review arXiv 2006

[28] [28]

T. Dray and C. A. Manogue, The exceptional Jordan eigenvalue problem , Int. J. Theor. Phys. 38 (1999) 2901 [ math-ph/9910004]

View on Pith Pith review arXiv 1999

[29] [29]

M. Koca and N. Ozdes, Division Algebras With Integral Elements , J. Phys. A 22 (1989) 1469

View on Pith 1989

[30] [30]

F. Karsch and M. Koca, G2(2) as the automorphism group of the octonionic root system of E 7, J. Phys. A 23 (1990) 4739

View on Pith 1990

[31] [31]

M. Koca, R. Koc and N. O. Koca, The Chevalley group G 2(2) of order 12096 and the octonionic root system of E 7, Linear Algebra Appl. 422 (2007) 808 [ hep-th/0509189]

View on Pith Pith review arXiv 2007

[32] [32]

A. Anastasiou and M. Hughes, Octonionic D=11 Supergravity and ’Octavian Integers’ as Dilaton Vectors, 1502.02578

View on Pith Pith review arXiv

[33] [33]

P. Ramond, The Freund-Rubin Coset, Textures and Group Theory , J. Phys. A 53 (2020) 341001 [2002.04729]

View on Pith Pith review arXiv 2020

[34] [34]

M. J. Perez, M. H. Rahat, P. Ramond, A. J. Stuart and B. Xu, Tribimaximal mixing in the SU (5)×T 13 texture, Phys. Rev. D 101 (2020) 075018 [ 2001.04019]

View on Pith Pith review arXiv 2020

[35] [35]

Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications , 1909.01358

View on Pith Pith review arXiv 1909

[36] [36]

N. Beisert, H. Elvang, D. Z. Freedman, M. Kiermaier, A. Morales and S. Stieberger, E7(7) constraints on counterterms in N=8 supergravity , Phys. Lett. B 694 (2011) 265 [ 1009.1643]

View on Pith Pith review arXiv 2011

[37] [37]

R. Kallosh, The Action with Manifest E7 Type Symmetry , JHEP 05 (2019) 109 [ 1812.08087]

View on Pith Pith review arXiv 2019

[38] [38]

M. Gunaydin and R. Kallosh, Supersymmetry constraints on U-duality invariant deformations of N≥ 5 Supergravity, JHEP 09 (2019) 105 [ 1812.08758]

View on Pith Pith review arXiv 2019

[39] [39]

E. Cartan and J. A. Schouten, On Riemannian Geometries admitting an absolute parallelism , Royal Academy Amsterdam Proc. of the Section of Sciences 29 (1926) 933

View on Pith 1926

[40] [40]

H. S. M. Coxeter, Integral Cayley numbers, Duke Mathematical Journal 13 (1946) 561

View on Pith 1946

[41] [41]

J. C. Baez, The Octonions, Bull. Am. Math. Soc. 39 (2002) 145 [ math/0105155]

View on Pith Pith review arXiv 2002

[42] [42]

S. Ferrara and R. Kallosh, Universality of supersymmetric attractors , Phys. Rev. D54 (1996) 1525 [hep-th/9603090]

View on Pith Pith review arXiv 1996

[43] [43]

CMB-S4 collaboration, K. Abazajian et al., CMB-S4: Forecasting Constraints on Primordial Gravitational Waves, 2008.12619. – 19 –

View on Pith Pith review arXiv 2008