JHEP01(2021)160 , to be 3 − Springer , 10 August 9, 2020 & a,c : January 26, 2021 r December 16, 2020 : -holonomy. The 2 : & G 2 − Received 10 Published Accepted , based on inflation along these [email protected] 1 , , 3 and Yusuke Yamada , in the range 4 a 2 e , Published for SISSA by https://doi.org/10.1007/JHEP01(2021)160 5 , 6 α/N , = 7 = 12 α r 3 Andrei Linde a [email protected] , . 3 Renata Kallosh, a,b 2008.01494 The Authors. c Cosmology of Theories beyond the SM, M-Theory, Supergravity Models

We study M-theory compactified on twisted 7-tori with , [email protected] -attractor models of inflation with α [email protected] Pennsylvania State University, University Park, PA 16802, U.S.A. Research Center for the EarlyGraduate School Universe of (RESCEU), Science, TheHongo University 7-3-1 of Bunkyo-ku, Tokyo, Tokyo 113-0033,E-mail: Japan Stanford Institute for Theoretical PhysicsStanford, and CA Department 94305, of U.S.A. Physics,Institute Stanford for University, Gravitation and the Cosmos and Department of Physics, b c a Open Access Article funded by SCOAP ArXiv ePrint: B-modes, predicting 7 different valuesexplored of by future cosmological observations. Keywords: ing Hamming (7,4) code.one The corresponding flat 7-moduli direction. modelscodes have We Minkowski for also vacua Minkowski with propose vacuacal superpotentials models based with on two octonions/errorflat flat directions. correcting directions. These We inflationary update models phenomenologi- reproduce the benchmark targets for detecting Abstract: effective 4d supergravity hastential. 7 We chiral propose multiplets, octonion, Fano each plane based with superpotentials, a codifying unit the error logarithmic correct- Kähler po- Murat Gunaydin, M-theory cosmology, octonions, error correcting codes JHEP01(2021)160 6 17 48 14 25 47 2 46 G 49 14 37 21 28 41 20 53 24 51 42 – i – 12 43 40 24 46 holonomy 2 G 34 29 1 28 B.1 From Cartan-Schouten-Coxeter toB.2 Gunaydin-Gursey superpotential From reverse Cartan-Schouten-Coxeter to Cayley-Graves superpotential A.1 Cartan-Schoutens-Coxeter convention A.2 Gunaydin-Gursey convention A.3 Cayley-Graves octonions 7.3 T-models 6.2 Using cyclic Hamming (7,4) code to get vacua with7.1 2 flat directions Strategy 7.2 E-models 4.3 Octonion fermion mass matrix and mass eigenstates 6.1 Vacua with 1 flat direction 4.1 Cartan-Schouten-Coxeter conventions,4.2 clockwise heptagon Reverse Cartan-Schouten-Coxeter convention, counterclockwise heptagon D From cyclic to non-cyclic HammingE code Octonions and dS vacua stabilization in M-theory C More on octonion multiplication tables and Fano planes A Octonionic superpotentials in common conventions B Relations between most commonly used octonion conventions 8 Observational consequences: inflation and9 dark energy Summary of the main results 7 Cosmological models with B-mode detection targets 5 Octonions and general construction6 of superpotentials Octonionic superpotentials and Minkowski vacua 3 Twisted 7-tori with 4 Octonions, Hamming (7,4) error correcting code, superpotential Contents 1 Introduction 2 Basics on octonions, Fano planes, error correcting codes and JHEP01(2021)160 ] 9 – 6 super- = 8 ]. A. Cayley 18 N ] and detailed 21 ] proposed a class of ]. Our cosmologi- 20 16 – ]. Some earlier insights 9 14 symmetry. The updated – 6 2 ]. These octonions are related to G 19 invariant 7 Poincaré disk scalar 3 2 is the automorphism group of the Z , 2 holonomy the coset space describing 2 G ]. The entangled 7-qubit system corre- G 4 – 1 ], and studied in detail by Gunaydin-Gursey – 1 – 22 , as shown by Coxeter in 1946 in [ 1 − ]. ] that these black hole structures are related to a (7,4) error 4 21 – ] which is capable of correcting up to 1 and detecting up to 3 2 5 . It corresponds to half-plane variables related to Poincaré disk ]. It was emphasized there that 7 i ) 17 symmetry and tripartite entanglement of 7 qubits, there is a relation R , Hamming error correcting code, Fano plane and superpotentials. is related to a remarkable superpotential whose structure matches the . A related 7-moduli model originating from a 4d gauged 1 2 7(7) SO(2) 7 SL(2 i G E h 1) , ] and apply them to observational cosmology as in [ U(1) ]. Thus we start with octonions, defined by their multiplication table. Under 13 SU(1 . The seven triangles in the heptagon correspond to a multiplication table of the – 23 h , as shown by Cartan in 1914 [ 1 10 ]. They can be divided into two sets of 240 notations related via the Cayley-Dickson There are 480 possible notations for the multiplication table of octonions assuming The order in which we will describe these relations is defined by the mathematical In M-theory compactified on a 7-manifold with We will continue an investigation of the 7-moduli model effective 4d supergravity fol- It was also observed in [ symmetry the multiplication table of octonions is left invariant. Octonions were discovered in 1843 by J.T. Graves. He mentioned them in a letter to W.R. Hamilton 24 1 2 dated 16 December 1843.discovered them Hamilton independently in describeda 1845, the non-cyclic in early Hamming an history (7.4)octonion appendix of notations to error in Graves’ his correcting the discovery paper context code.studied in of [ and [ Riemannian developed Later geometries by admitting Cartan Coxeter an [ and absolute parallelism. Schouten These [ were (GG) in [ G that each imaginary unit squaresin to [ variables by a Cayleybetween transform. octonions, In such a 7-moduli modelfact we that the will smallest establish ofoctonions a the exceptional relation Lie groups, gravity was studied in [ manifold single bit error correcting (7,4) Hamming code. our 7 moduli is lowing [ into the corresponding M-theorycal models models are have based beenholonomy on obtained group 11d in M-theory/supergravity [ compactified on a twisted 7-tori with errors. Here we willwhich update predict 7 the specific original targets 7-moduliWe for will cosmological detecting use primordial models gravitational the developed wavescosmological superpotentials in from models inflation. [ derived will from be octonionsoctonions shown to and and be Fano planes. related to (7,4) Hamming error correcting codes, heptagon with 7 vertices A,B,C,D,E,F,Gin and figure 7 triangles and 7octonions. complimentary quadrangles correcting Hamming code [ It was observed in studiesin of the context supersymmetric of black holesbetween in black maximal holes, 4d octonions, supergravity andsponds that Fano plane to [ 7 parties:states Alice, Bob, and Charlie, 7 Daisy, complimentary Emma, Fred 4-qubit and states. George. It One has can 7 3-qubit see the 7-qubit entanglement in a 1 Introduction JHEP01(2021)160 . 2 j ]. i e 5 jq 7 e + → − 1 = j q = ]. Octonion +3) i ( O 23 e ) , which will give 1 (4.2) (4.1) (4.3) (4.4) A, B − ]. This leads to 240 and the quaternion )arethe 7 = 25 e matrices due , x, y 2 ab ) ) 19 ]. j AB ,EFA,FGB,GAC − 31 Γ ( . ]. Furthermore the U-duality riant [12] in (4.1) was , – ] octonion convention is asso- ) ) he relation between the 38 ¯ invariant was established 18 Z (8) Lie algebra. The exact fy 21 C f , 7(7) in the Fano plane in [ SO +P E +P B 20 7 AB is also related to the octonionic ab ) e ) fx ab fZ ) are the 8 vector indices and ( Γ AB 7(7) 4(P was labelled as Γ D ab )( E ( − ab +4(P 2 j of AB ) iy ] belong to this set of 240 conventions. invariant may be written as 2 ) Z 4 . Again one finds 240 possible conventions ] ¯ + I 2 i Z 4 where ( 7(7) e rxy ab √ A 19 6 ]. Each of the 7 vertices A,B,C,D,E,F,G represents 23 E , x 7 rZ is the central charge matrix and ( − ( AB (T 1 e ) 2 1 4 – 2 – (T = 18 ]. 4 1 ab 1 AB √ − is an additional imaginary unit that squares to + Γ 4 Z ab 2 − 33 ) matrices can be considered also as ( E − iy 2 j = ) . In [ + = ¯ the relevant 3 lines are the ones inside the triangle xy Z AB ) ( ab Z r 2 ( AB ab x 3 and the additional imaginary units as r T +3) G Γ Z invariant i − ( e 3) Hamming (7,4) error correcting code. The Cayley-Graves F to represent octonions since =T , = 7(7) 2 (8) algebra are ( 2 2 4 4 , E I I SO jq [14]. entanglement diagram. Each of the seven vertices A,B,C,D,E,F,G rep- ] soon after it was constructed [ O 7 3 − J E = 1 37 1 – i q ] are equivalent and belong to this set of 240 notations. One can ( 34 i 23 = e In the stringy black hole context, O are quaternions and entanglement diagram from [ 7 Here relation between the Cartan invariantestablished in in (4.2) and [15, Cremmer-Julia 16]. inva The quartic invariant and the Cartan form [11] may be written as to equivalence of the vector and spinor representations of the and The Cremmer-Julia [12] form of the Cartan resents a qubit anddescribes each a of tripartite the entanglement. seven triangles ABD, BCE, CDF,4Cartan’s DEG quantized charges of theentropy black of hole stringy (28 black electric holes and and 28 the magnetic). Cartan-Cremmer-Julia T The matrices of the Figure 1: The Jordan algebra are the 8 spinor indices. The ( 2 q E . Cayley-Graves octonions [ 7 e ] and in [ and 21 . The 1 ] and by Bose and Chaudhuri in 1960 [ q ] this will lead to defining 32 Here we also note that Cartan-Schouten-Coxeter [ Octonions have made their appearance within the framework of 11d supergravity and Thus, 480 notations split naturally into two sets of 240 related by the map and anticommutes with the imaginary units of quaternions [ The codewords are called cyclic if the circular shifts of each codeword give another word that belongs 23 2 1 symmetries of maximal supergravity in five,exceptional four Jordan and algebra three over dimensions are split described octonions by and the the associated Freudenthal tripleto sys- the code. The1959 cyclic [ error correcting codes, the so-called BCH codes, were invented by Hocquenghem in ciated with the so-called cyclic octonions naturally lead to the original non-cyclic Hamming (7,4) error correctingits code compactifications [ [ which cross which reverses the directions of themultiplication 3 arrows table involving in turn,All leads these to are known an mathematical oriented facts, Fano see plane for and example, error [ correcting codes. another 240 possible notations for theof multiplication table. [ In particular, inthat the can conventions be represented onfirst set an of oriented 240 Fano conventions plane inare that that the reversed. differs direction from of For arrows the example along Fano in the plane 3 figure of lines of the the Fano plane Fano plane as explainedimaginary in units section as Modulo the permutation of indices (3,4)used and (5,6) in of [ imaginary octonion unitsequally the well notations take construction. Cayley-Dickson construction representswhere octonions in the form − possible notations for the multiplication table of octonions represented by the same oriented Figure 1 a qubit and each ofentanglement. the 7 triangles ABD, BCE, CDF, DEG, EFA, FGB, GAC describes a tripartite JHEP01(2021)160 ) in . 5.3 ]. (1.3) (1.1) (1.2) . The 42 WO WO i. , in terms of in ∀ } supersymmetry , WO ijkl { . = 0 = 1 ii A N . We propose to take an 3 2 Z A single example of is the fact that the defining ,M , j ) the maximal supersymmetry of l T i T WO T − ], i.e. we look at the model of a com- ij i , k 12 ∀ M T – 1 2 )( 10 j ) based on a cyclic symmetry of octonion = T , 1.2 − i = 0 – 3 – holonomy down to T ( ij 2 are present. As a consequence for these superpo- } G j ,W M ijkl X T  { i j i X T T -structure manifold with the following Kähler potential = 2 holonomy. We will also find that the mass eigenvalues of + G 3 2 i is sufficient for all cosmological applications in this paper Z WO T )  entries and 1 4.32 log ± 7 =1 i X − = ] and it was observed that the flattening torsion components are constant have only 37 ]. A spontaneously induced torsion breaks all supersymmetries but one, and holonomy of the compactification manifold is preserved and is not broken to ]. More recently octonions were shown to describe the non-associative algebra 7mod ij is a symmetric matrix with 21 independent elements defined by the twisting 2 37 41 K M , – G ij ) or in the form ( 36 39 M 4.7 In addition to the simple cyclic choices we propose the general form of We will present simple examples of eq. ( We start with the 7-moduli model, following [ Of particular interest for our purposes is the fact that structure manifolds with 4d [ 2 eq. ( the structure constants of octonionsany and choice generalized of cyclic the permutation 480and operator, octonion details valid for of conventions. the This construction general with formula examples is are presented given in in eq. appendix ( nion multiplication. We willdirections also and find apply in them these for models cosmology. Minkowski vacua withmultiplication 1 in and clockwise 2 flat orrepresented on counterclockwise the corners directions of a whenhave heptagon. the a These octonions, imaginary very Fano planes units simple and are superpotentials relation to cyclic Hamming (7,4) code. so that only 14tentials terms of the type G the superpotential around the vacuum are independent of the convention chosen for octo- sense in which theselater. superpotentials are One octonion importantmatrices based property will of be the explained in superpotentials great detail octonion based superpotential of the form where we take a sum over 7 different 4-qubit states defining the choice of Here of the 7-tori, which in general breaks explicitly in [ and given by the structure constants of octonions. pactification of M-theory onand a superpotential of non-geometric R-flux background in string theory and theirM-theory uplifts is to spontaneously M-theory broken [ in by compactification to minimal renders the compact space Ricci-flat. The supersymmetry breaking torsion was computed tems [ JHEP01(2021)160 ). ). we 1.4 (1.4) 4.36 B ) based and 4.36 ], Gunaydin- A are the same 21 ). The first one , A 20 1.2 ) is based on heptagons T 4.29 ≡ 7 T and in appendix = 6 4 . T 2 G = 5 to T 3 2 = Z ], Cartan-Schouten-Coxeter [ 4 – 4 – ], a special case of models with compactification T 19 , 10 = ) have a positive real part, therefore we do not flip ]. These models are shown to be related to a cyclic 18 3 1.1 T 21 , = ] for octonions and the ones we have introduced here in 20 2 43 ). We also explain the alternative derivation of new super- T = 5.3 1 T holonomy, is codified in our cosmological models using octonions. 2 structure which can have Minkowski vacua under the special condition 2 G G we discuss octonionic superpotentials for various octonion conventions we study Minkowski vacua in 7-moduli models with octonionic superpo- we present a simple derivation of two superpotentials ( we describe, following [ we present some basic facts about octonions, Fano planes, Hamming (7,4) 5 6 4 3 2 ). We show that these models have a Minkowski minimum at symmetry, together with some useful references. The goal is to provide the ], Okubo notation [ ) is based on heptagons with clockwise orientation using the Cartan-Schouten- 1.2 2 and called Reverse Cartan-Schouten-Coxeter notation for octonions. 23 4.7 G 4 In section For each choice of octonion multiplication convention we have found 2 independent In section In section In section In section tentials ( with one flat direction.cosmologically: All they models have studied a in Minkowski section minimum with one flat direction as in eq. ( get models in 240 different conventions,of including moduli. the Similarly one starting we started withmodels with, Reverse with Cartan-Schouten-Coxeter without convention another sign we flip 240 can get conventions,of including moduli. the It one is weFor convenient started simplicity to with, we use without shall these sign refer two flip starting to points these in superpotentials the as form ‘octonionic given superpotentials’. in eq. ( the field redefinitions inof the octonion superpotentials conventions the without same7 the as moduli the sign with ones flip. Kähler whichsigns potential This lead of in to limitation moduli. ( a is Meanwhile change do due involve the sign to general flips the in type fact general. of that Starting mapping with all from Cartan-Schouten-Coxeter conventions one we convention can to another Gursey [ section choices of superpotentials satisfying ourtions physical we requirements. can For get other the octonion relevant conven- 2 superpotentials either using the general formula, or making using the general formulapotentials ( via the changeon heptagons of with variables, clockwise startinggive or with examples counterclockwise superpotentials of orientation. in relationsplanes. In eq. between These appendices include most ( Cayley-Graves [ commonly used octonion notations using Fano notation for octonions, which wenon-cyclic introduce Hamming there. code. These modelsmass Since are matrix related our in to superpotential supersymmetric theof is Minkowski original the vacua quadratic is superpotential, in proportional we moduli, study to it the the there. second fermion derivative when the holonomy group is extended from in eq. ( Coxeter notation for octonionsHamming [ (7,4) error correctingwith code. counterclockwise orientation. The second This one one is in related eq. to ( Reverse Cartan-Schouten-Coxeter codes, information for understanding howon the a mathematical manifold structure with of M-theory compactified on manifolds with JHEP01(2021)160 ] ], ]. 8 45 51 – , 6 (1.6) (1.5) 44 -attractor ], SO [ -structure 2 α and explain cosmological 50 G , ]. It naturally . 49 ) corresponding 18 , moduli are equal = 7 37 , = 0 ] that 1.2 . A short summary 17 α m 36 46 3 8 WO ). 4d [ symmetry. The resulting -attractor models [ moduli are equal to each α 3 1.1 ], CMB-S4 [ . n , , the spectral tilt of the CMB ) as well as the eigenvalues of 48 s , in (  = 0 ]. In the past, it was derived n i 1.1 8 47 (2) [SO(2)] – T 6 -attractors realize inflation and lead WO + α . ) for building a motivated by maximal supersymmetry. 9 in eq. ( (2) 1.5 for each T 1 , ij . The superpotentials in these models are ,  supersymmetry in 2  ), ( M , = 0 T holonomy manifold. 3 log = 7 , 1.4 ii case, a top benchmark point, whereas our 2-flat ] by using a phenomenological superpotential 2 – 5 – n = 1 + 4 n 7 , G M − T 5 ), ( + N , whereas some other ,  = 7  split. The remaining codewords define the remaining 6 m (1) , 1.1 α (1) T 3 . These models of T 7 log = 7 7 ≡ 4+3 and − + , maximal supersymmetry of M-theory in 11d, spontaneously ) and in [ α m = 3 (1) (2) ], might potentially detect the tensor to scalar ratio at a level T T 1.4 T K 5+2  53 . Note that such a superpotential does not fit the = , ≡ 2 ) 7 log j ), associated with octonions and the (7,4) Hamming code, is funda- ··· T T 6+1 m 1.2 = − − = i and improve constraints on the value of 1 = T we also present the octonion based superpotentials for the models with ) ··· ( T σ 7 6 K ] and PICO [ ≤ = (5 j 4 52 ≤ i − supergravity in section +1 [ ≤ m 1 10 T = 1 P × ] by postulating eq. ( Future cosmological observations like BICEP2/Keck [ On the basis of the M-theory setup, we construct the updated cosmological models in In section The choice in ( 6 = N = 5 r power spectrum. They might supporttheir or 7 invalidate benchmark the points. M-theorytheir cosmological We relation show models to these and octonions benchmark andof points to the in cosmological main figures observables results in of section the paper is given in section the discrete set ofHere values we for updated theseOur models 1-flat with direction account models of leaddirection their to models relations split to models M-theory lead via to octonions. remaining cases. LiteBIRD 4d to 7 benchmark points, whichHere we are show the that B-mode these detectioncorrecting updated targets models codes. suggested originate earlier from It M-theory inmodels and [ are was octonions in shown and good error by agreement Planck with satellite data measurements available now. [ These include special cases with We find models with terms in the superpotential for these split models. to each other, other, obtained from the 1-flat-modulus modelsto by removing removing some certain specific terms in codewordsfind ( from 2-moduli the effective cyclic 4d Hamming supergravity with (7,4) the code. following In Kähler this potential way we broken by compactification toleads minimal to a desirablemodel starting from M-theory point compactified ( on a Minkowski vacua with 2 flat directions. In these vacua some of the leading to ain top [ benchmark forW detecting B-modes [ compactification M-theory model where mental, being associated with the mass matrix at theeffective vacuum 1-modulus in model these in models 4d have supergravity an has the Kähler potential and superpotential This is a starting point for building the inflationary cosmological The eigenvalues of the superpotential matrix JHEP01(2021)160 ons of ) for (2.1) (2.2) (2.3) ], i.e. 21 1.2 2 } The asso- 7 G . ), complex } 3 , e R 6 , e 5 (713) , , e 4 3 6 5 1 , e 4 2 7 1 e e e 3 e e e e − (672) − − − .Todescribetheproductwe , , e we show that octonion 2 E . +1 , e 2 5 4 1 k 1 . 1 6 7 3 k e e e . we give more details about onions, which explains their special (561) oremembertheoctonionproduct. 2 e e e e e In what follows, we give some more − , 2 ,i,j,k , e − − − e as we are with multiplying matrices! . 1 C k = we derive the relations between { e = er that: 1 4 3 . B j 1 ice mnemonic, and working up to some ijk (457) 5 6 2 7 +1 e e e i k 2 , f j e e e e we present a specific transformation − e e e − − − e ning! About the only interesting things one j i ,thesefactsareenoughtorecoverthewhole D 2 e ijk = 2 4 e +1 f i − ). e i 7 3 2 (346) 1 e e 5 4 1 6 , e e e 4 O j e e e e = = − ⇒ e − − − ,and 1 + j 7 6 2 = ij ⇒ − e e δ Z – 6 – i (235) = ] is given in figure j 1 k ) is referred to as 7 associative triads. 2 1 6 , e − e 1 e 4 3 7 5 e e e e i k e e e e e − 21 = 2.2 − − − e , = j ≡ e (124) 20 j = ] i { e j 7 5 1 : e j i 1 3 6 2 4 e e e j e e = , e e e e e − i i in eq. ( − − − } e e ̸= [ are completely antisymmetric. The multiplication table in } ), and octonions ( i ijk H 6 4 7 1 ijk ijk 5 3 1 2 { e e e { ,andallidentitiesobtainedfromthesebycyclicpermutati f e e e e − − − − k . Cartan-Schouten-Coxeter octonion multiplication table. − Table 1 — Octonion Multiplication Table identity holds: 5 6 7 1 2 3 4 are given by the seven quaternion subalgebras of octonions generated e e e e e e e = we present the general formula for octonion superpotentials in ( } identity holds: http://tamivox.org/eugene/octonion480/index.html A ji ijk There are four normed division algebras: the real numbers ( Figure 2 { , k ,area4-dimensionalalgebrawithbasis1 ), quaternions ( . There are 480 possible conventions for octonion multiplication tables [ C k H = = +1 for , e are square roots of -1, are square roots of -1, j anticommute when ij ijk ] the set of 7 triples k 7 j , e f Imaginary octonion units are anti-commuting. The commutator of two imaginary units The non-vanishing structure constants in CSC convention are We will be working with real octonions which are an 8-dimensional division algebra In appendix i e The 28 remaining triadsSee are for anti-associative. example 21 e 3 4 ). and index doubling index cycling is simply In [ ciative triads by 480 different choices of the 7 associative triads. The structure constants Cartan-Schouten-Coxeter notation [ numbers ( spanned by seven imaginary units together with identity superpotentials may be used for generating metastable de Sitter vacua2 in 4d. Basics on octonions,Octonions. Fano planes, error correcting codes and any octonion conventions, with examples.most commonly In appendix used octonionmultiplication tables conventions. and Fano In planes.from In appendix Cayley-Graves appendix to Cartan-Schouten-Coxetercyclic octonion Hamming notations which (7,4) also code rotates to a a non-cyclic one. In appendix and ,...,e i 1 i, j, k i, j, 1isthemultiplicativeidentity, we have the ( the where we think of the indices as living in e e • • • • • • • could give a multiplication table, but it is easier to rememb Together with a single nontrivial product like deeper, more conceptual ones. 2.1 The FanoThe plane quaternions, multiplication table.We should However, weAnd become ultimately, really we want as wantproperties a more comfortable and a conceptual how better approach theydescriptions to fit way of with the in t oct octonion with multiplying other multiplication, mathematical starting ideas. with octonions a n We can summarize the last rule in a picture: can easily learn from it are: Unfortunately, this table is almost completely unenlighte JHEP01(2021)160 ] ). 29 [ (2.8) (2.4) (2.5) (2.6) (2.7) 3 2 i, j, k Z of a quater- q e ) k , e j mnpq . C , e i e = 1 ( ]] = 3 137 f m , e = p e ) [ 156 k , f e n j e . e = ( i rst are invariant tensors of the automor- kq e f ] = +1 124 p ]] + [ is given by the structure constants f − f n there exist three sets of imaginary units k krs , e e f mn mnpq k [ ) m e k j mnpq C mnpqrst = e e f  [ C i , 6 1 e – 7 – p = ( kpq we have e f = 1 ≡ e ] ] for the same set of triples is different from the one = k mnpq ]] + [ 21 p C mnpq , e j of quaternion subalgebras such that kmn The Fano plane is the unique projective plane over a C , e f n , e ) i e s [ e [ , , e r m e [ , e k . e ≡ given the unit ( ) 3 , 3 p ) q , e , e n p of octonions. , e 2 , e m k G e e ( ( , J ) n , e ] shown in figure m In the Fano plane every line has 3 points and every point is the intersection point Given a multiplication table of octonions, the corresponding Fano plane can be given 29 , e k e ( For example in figure the original oriented Fano planein in [ [ of 3 lines.of Since a each quaternionsubalgebras. line subalgebra contains every Hence 3 given imaginary an points unit imaginary which belongs unit correspond to to three the different imaginary quaternion units different orientations depending on the identificationThe of three its points points in with each thenion line imaginary are subalgebra units. identified with with the theThere imaginary positive are units orientation different ways given to by build the the cyclic Fano plane permutation for of the ( same type of triads. For example, field of characteristic twocorresponding which to can an be Abeliannontrivial elements. embedded group It of can projectively be order in usedof as 8 octonions 3 a mnemonic with by dimensions representation of identifying thean the its seven multiplication orientation. table points points with represented the by imaginary the octonion units and introducing Both the structure constantsphism and group the tensor Fano plane and octonions. and is dual to the structure constants where the completely antisymmetric tensor does not vanish, in general.hence The octonions associator form is an annot alternative alternating vanish algebra. function and of The is its proportional Jacobian arguments to of and their three associator imaginary units does Octonions are not associative and the associator defined as , 6 ). in ⊥ k (6) (4) (5) (7) (8) C usly − = n divides × C u k . Once we . For linear is the set of G is um non-zero d ⊥ then . Next add the A he very simple ). ll be important C k G G − (so n = , this code contains ,I G w of G H T . It is simple to deduce A C .If linearly independent parity ∈ =( v k H ∀ − n . The dual space is the rest of =0 vice versa , which is another way to define the v A · dual and those that do not. Therefore, the H u and u , is made of rows of C ⊥ H C , 6 from the other. It is always possible to convert ). zero matrix. in ⊥ k , and if necessary swap columns of (6) (4) (5) (7) (8) k G = (111) and the parity check matrix is C usly G − × -bit vectors having minimum distance = G n divides ) and × C k u is now the dimension of the vector space, so it contains k . Once we mn k H − . For linear is the set of G rows. For any given subspace, the check and generator is um non-zero identity matrix, and d n parity check matrix ⊥ then k k . Next add the A he very simple ). ll be important C ) = 0. From this it follows, that if all the rows of a generator k G G × − − w JHEP01(2021)160 k (so n = n , this code contains + ,I is the ( G w of G v H T . Note also that since ( , 0 . It is simple to deduce 6 · 0 ). A in . ⊥ Hamming space can satisfy at most is the C k (6) (4) (5) (7) (8) u can be written down immediately, it is .If C usly = n is the generator matrix of k ⎞ ⎠ ⎞ ⎟ ⎟ ⎠ all linearly independent parity − ∈ ] is used, where =( I T = H n v k C divides , and H H ∀ × − C G u " k 3] space discovered by Hamming [30]. The generator matrix is . Once we HG n n, k, d ] octonion . The dual space , . For linear columns and is the set of is the rest of G =0 is um non-zero 4 d = 0 then vice versa , then , , which is another way to define the . , then all so do all the vectors in the subspace. Any given parity check ) where 21 0 n ⊥ v then A 1010101 0110011 0001111 111. Its generator matrix is 1010101 0110011 0001111 1110000 110 101 , . Next add the A he very simple ). ll be important · G u , w dual and those that do not. Therefore, the H C k · G = 20 G u ,A = 0 ! ⎛ ⎝ . − and ⎛ ⎜ ⎜ ⎝ ) is a shorthand for a set of u k , u (so is made of rows of has n T An Introduction = I T , this code contains C ⊥ = = = ,I G w of H G 0000000 10101010001111 0110011 10110101110000 1100110 0111100 01001011111111 1101001 1000011 0101010 0010110 1001100 0011001 H C H = +1 . T =( to the two vectors so far obtained, then add the third row to the four vectors previo HG G H H HG . It is simple to deduce A C from the other. It is always possible to convert zero matrix. n, m, d C G 124 G , since it is much easier to find the minimum weight than to evaluate all the distances , and if necessary swap columns of .If to the 4 vectors previously f ∈ k G which have zero inner product with all vectors in = (111) and the parity check matrix is d and a parity matrix linearly independent parity ∈ = 0 and =( G G ⊥ u × -bit vectors having minimum distance v is the transpose of k to the first two vectors, which v G G H ) as ) · ∀ . 1st in the 16 codewords is the C and − T G k is now the dimension of the vector space, so it contains u 2.2 n mn G k . The dual space H − is the rest of Error correction is a central con- =0 rows. For any given subspace, the check and generator identity matrix, and vice versa at the left part is used here to produce n parity check matrix . , which is another way to define the v A k k vectors. vectors of a linear subspace in 2 · to the form H dual G and those that do not. Therefore, the H ) = 0. From this it follows, that if all the rows of a generator 1 + 1 = 0 × k k u − , w and u 2 The simple errorbinary correction vector method space described 000 in the previous section is based around t where checks. These parity checks together form the matrices are related by Hamming space exactly in half, into those vectors which satisfy the value of ATutorialonQuantumErrorCorrection satisfy the parity check The last concept which we will need in what follows is that of the have the right form for To do the conversion we can linearly combine rows of the space. Now, if It is simple to confirm that linear subspace. its dual: that the parity check matrix of so the sixteen members of the space are G such spaces are termed self-dual. The notation ( Let us conclude this sectionin with what another follows. example of It a is linear a binary [7 vector space which wi 2 vector spaces, the notation [ A useful relationship enables us to derive each of all vectors these have been written in the following order: first the zero vector, then the first ro obtained, and so on.weight We is can 3. see at The a parity glance check that matrix the is minimum distance is 3 since the minim second row of , k is made of rows of n + C ⊥ is the ( which allows to produce the 16 codewords. v H C . Note also that since ] for Cartan-Schouten-Coxeter [ ( 0 G · 0 . Hamming space can satisfy at most from the other. It is always possible to convert 29 is the zero matrix. u can be written down immediately, it is 1 + 0 = 1 = – 8 – , and if necessary swap columns of n is the generator matrix of k ⎞ ⎠ ⎞ ⎟ ⎟ ⎠ , k G ] is used, where = (111) and the parity check matrix is I ] is relevant. It is an example of a linear binary G T H C 5 , and × -bit vectors having minimum distance G ) G and " k is now the dimension of the vector space, so it contains 3] space discovered by Hamming [30]. The generator matrix is HG n, k, d , columns and mn ) is represented in the oriented Fano plane with the arrows k H 4 − = 0 then , then , 0 + 1 = 1 rows. For any given subspace, the check and generator identity matrix, and , then all so do all the vectors in the subspace. Any given parity check ) where 0 n 1010101 0110011 0001111 111. Its generator matrix is 0001111 1110000 1010101 0110011 110 101 , n parity check matrix 2.2 . Next we add the 2nd row in G u , w k k · = G ,A ) = 0. From this it follows, that if all the rows of a generator ! ⎛ ⎝ ⎛ ⎜ ⎜ ⎝ ) is a shorthand for a set of × k u − has T w I one of the codewords is all 0’s, one is all 1’s. The remaining 14 k = = = . n 4 + 0000000 10101010001111 0110011 10110101110000 1100110 0111100 01001011111111 1101001 1000011 0101010 0010110 1001100 0011001 H is the ( . =( to the two vectors so far obtained, then add the third row to the four vectors previo HG G H H v 0 + 0 = 0 . Note also that since ( C 0 n, m, d · 0 G G . , since it is much easier to find the minimum weight than to evaluate all the distances Hamming space can satisfy at most is the ∈ u which have zero inner product with all vectors in , which is also shown in the first term in ( d can be written down immediately, it is = 0 and = 4 n is the generator matrix of k ⎞ ⎠ ⎞ ⎟ ⎟ ⎠ ] is used, where e ⊥ u I is the transpose of v T H · C , and C = T is known as a parity matrix, it has the property that u we have shown in addition to generator matrix 2 G " G e 3] space discovered by Hamming [30]. The generator matrix is HG 4 · H n, k, d , columns and 1 4 = 0 then e , then vectors. vectors of a linear subspace in 2 , , then all so do all the vectors in the subspace. Any given parity check ) where 0 n to the form 1010101 0110011 0001111 111. Its generator matrix is . An oriented Fano plane, figure 1 in [ 1110000 1010101 0110011 0001111 110 101 . Original Hamming code. The generator matrix G u , k k w · = 2 The simple errorbinary correction vector method space described 000 in the previous section is based around t where checks. These parity checks together form the matrices are related by Hamming space exactly in half, into those vectors which satisfy ATutorialonQuantumErrorCorrection the value of satisfy the parity check The last concept which we will need in what follows is that of the have the right form for To do the conversion we can linearly combine rows of the space. Now, if It is simple to confirm that linear subspace. its dual: that the parity check matrix of so the sixteen members of the space are G The notation ( such spaces are termed self-dual. Let us conclude this sectionin with what another follows. example of It a is linear a binary [7 vector space which wi 2 vector spaces, the notation [ A useful relationship enables us to derive each of all vectors these have been written in the following order: first the zero vector, then the first ro obtained, and so on.weight We is can 3. see at The a parity glance check that matrix the is minimum distance is 3 since the minim second row of ,A ! ⎛ ⎝ ⎛ ⎜ ⎜ ⎝ ) is a shorthand for a set of k u has T For our purpose it is useful to observe that in 16 codewords of the Hamming code shown In figure I = = = 0000000 10101010001111 0110011 10110101110000 1100110 0111100 01001011111111 1101001 1000011 0101010 0010110 1001100 0011001 H . =( to the two vectors so far obtained, then add the third row to the four vectors previo HG G H H to Error Correcting Codes in the middle of thecodewords figure are split into twoand groups four of 0’s. 7: They one are group complimentary has to 3 each 0’s other and when 4 0 1’s, is the replaced other by 1. has 3 For 1’s example, vector space specified by itsThe generator matrix matrix 16 codewords of the Hammingtion [7,4,3] using code. this The code mechanism is of nicely error explained detection in and the correc- lecture by Jack Keil Wolf Error correcting codes, Fanocept planes, in octonions. classical informationto theory. quantum error When correction combined especially withonly important quantum a in mechanics classical quantum they computers. Hamming lead [7,4,3] For our code purpose [ the 16 codewords of theaccount Hamming of code addition shown rules inzero the vector, middle then of the thegive figure. 1st us row They the in are 3rd producedobtained, and with and the so 4th on. codewords. At Next the we right add there the is 3rd a row parity in matrix Figure 4 The octonion multiplication ruleindicating ( the positive directionscircle for that multiplication. For example, one can see from the oriented Figure 3 conventions. On each of the 7 lines (including the circle) there are 3 points e.g. 1,2, and 4 on a circle. C n, m, d G G , since it is much easier to find the minimum weight than to evaluate all the distances ∈ which have zero inner product with all vectors in d = 0 and ⊥ u is the transpose of v · C

T u G

vectors. vectors of a linear subspace in 2

to the form

k

k

2

The simple errorbinary correction vector method space described 000 in the previous section is based around t

where

checks. These parity checks together form the

matrices are related by

Hamming space exactly in half, into those vectors which satisfy

ATutorialonQuantumErrorCorrection the value of

satisfy the parity check

The last concept which we will need in what follows is that of the

have the right form for

To do the conversion we can linearly combine rows of

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the space.

It is simple to confirm that

linear subspace.

its dual:

that the parity check matrix of

so the sixteen members of the space are

G

such spaces are termed self-dual. The notation (

Let us conclude this sectionin with what another follows. example of It a is linear a binary [7 vector space which wi

2

vector spaces, the notation [

A useful relationship enables us to derive each of

all vectors

these have been written in the following order: first the zero vector, then the first ro

obtained, and so on.weight We is can 3. see at The a parity glance check that matrix the is minimum distance is 3 since the minim second row of , re- 423 and (31) (33) (34) (32) JHEP01(2021)160 of di- x o7by .And 4] code. m V attached , )canbe th line of p 4 i ]code(or V n , q ( = F P q , ≡ n, k, d n q ] (2.9) to be x F 1errors.Itcanbe (2.10) [ q P − F d . . represents the ) . ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ i . , . . 4 i L u, v L 4] code introduced in Sect. 2.3 ( Michel Planat, Metod Saniga , max d ∈ . We should note d C v j ∈ ], where cyclic codes ̸= P 1000011 u is a subspace of u,v 30 , . One defines the Hamming distance C q +1= ). Each triad is shown in F ) = min k G th point and ]anddetectupto C 10 if otherwise 2.2 (5) (3) 413 (4) − ( j 1 =7, 2 − n ' = (1234567) 0011001 d is” for antum min n . Then a projective geometry at the left part has 4 rows with 3 , 0, of the as the number of coordinates in which ≤ ] code, the following bound holds gularity = q d ) in the polynomial field ” in the echarac- · g F d G n 1101000 0110100 0011010 0001101 1000110 0100011 1010001 is the . q ij = 2. CSC l> ). For the Cayley-Graves octonions mincidence matrix of the Fano plane the Fano plane, the elements of the matrix are ceived considerable attention [24]. Let us define a vector space A minimum distance codeis (or such a maximum that distance separable, MDS code) mension [ s CG , n ( q P ζ F that is of modulus 1, i.e. such that : the position of the 1’s in the first row is 124, in the 2nd row and 0 otherwise. Its average value oscillates 1 2) m 1, are well known since they constitute 0 +1ofcoe g g C 5 that enters in equation ( s/ 1, on the trivial zeros at m x ]= − ( p ] code from its generator matrix. We use 5 Γ x > [ 1 2 + ] to which we refer for details and further references. 1001100 = q ) − 0 , 3 s/ m s n, k F 54 g n ( g m x − ℜ g π ∈ ,...,q 1101000 0110100 0011010 0001101 , = ), plays a prominent role in the (still unsolved) Riemann )if s n p g m − ( ⎡ ⎢ ⎢ ⎣ matrix is also the incidence matrix of the underlying Fano plane. It is easily )= =0 x n s ··· ··· ··· Λ 7 ( 1110000 1 k m ξ G= > g × . Since there are two operations “+” and “ ) ), 1 0 s 7 . Cyclic Hamming code. The generating matrix + ( G g g ℜ nk ) over an abelian group at the left. q π g ··· ∈ i 5 ( )equalsln( ' plays the role of time and the code is thus a 2-valued time 2 0 + 0 00 n One finds the mechanism of error detection and correction using the cyclic (7,4) Ham- All codewords can be obtained from any particular one by a cyclic permutation. In Now we describe shortly a cyclic Hamming (7,4) code following [ Even though in the literature one set of Hamming code is referred to as cyclic, one can All these set of codewords which we discussed so far are known as non-cycling Hamming κ g g ( ··· ··· ··· ··· ··· ··· ··· x n Λ 1 )= g ⎡ ⎢ ⎢ ⎣ s ( the 7 codewords inis figure 235 and so on, where 1 after 6ming is error 1 correcting mod code, 7,data and transmission’ for so [ example, on. in the book by R. Blahut ‘Algebraic codes for were discussed in the contextfigure of the projective spaces. The generating matrixthis is case shown in the cyclicthat permutation this operator isrelated simply to octonions in CSC convention [ make all of them cyclicdefined with respect later to in the section actionths of permutation a operator cyclic permutation is mapped operator into each other in a unique way: codes. Namely, onerecycling can any see of them. in Same feature is present in all 16 codewords in figure Figure 5 1’s each, where everygenerate row directly is the a 7 codewords cyclic at permutation the of right the of previous thein figure, one. the all context More with of 3 cyclic the 1’s, permutations Graves-Cayley 4 octonions 0’s. one can use the following 7 codewords ζ + , one can define two kinds of characters. Multiplicative characters =1, q 0 g F | )=exp( ) n Let us describe a linear [ = g ( ; the latter was proved by Weil in 1948 [4]. ( k q g The Mangoldt function, κ The Mangoldt function [3] is a way of encoding the primes. | to the field of complex numbers The index encoded by the cyclotomic polynomial (2). 3QuantumStatesandTheirRelationtoAdditiveCharacters 3.1 TheAcharacter additive characters ψ field 4 elements which are the lines of the following generating matrix which is obtained from The generator matrix is as follows text of cyclic codes, theof cyclotomic primes. polynomial also There encodes exists theF a irre “zeta” function and a “Riemann hypothes around 1 and the error term explicitly relies on the pole at the polynomial the basis for theters ordinary discrete introduced Fourier below transform. are But the additiv ones whichhypothesis. are useful to relate to qu As an example, we mention the binary Hamming code of length function extended zeta function on the Riemann zeros presumably all located on the critical a a JHEP01(2021)160 , . 7 22 [ + . The (2.15) (2.12) (2.13) (2.14) (2.11) 2 14 G = SO(7) 21 The automorphism of order 12096 and , related to octonion , which makes the re- (2) ijkl 2 C G SO(7) decomposes as . . kl SO(7) J . . (7) kl is the set of 7 codewords which will kl 2 ) . ij (7) G 5 possible choices of conventions. However, C 2.7 ·S 1 8 = 0 ijkl 7 ·PSL ] C ), ( + jk 2 3 1 2 , can be represented using the 21 generators of ] G ij Z – 10 – 2 ij 2.6 J = Z ijk G 1 2 = [ f ), ( ij = [ = T G 7! = 645120 2.5 · ij H 7 G 2 . They also satisfy the identity 2 ] in table 1 one can find the list of all the important finite G ] which include more details on the discrete finite subgroups 62 , and a totally antisymmetric tensor are subject to various identities, so that 64 ij – ] we can look at the total number of permutations of imaginary the imaginary octonion units transform in its 7 dimensional rep- , which we call J ijk the adjoint representation of 2 58 2 f 21 [ G 2 G 2 G we show that one can bring the cyclic Hamming (7,4) error correcting G . and 2 D which is matrix at the right of the figure , of order 1344. We refer the reader to relatively recent papers on the fi- G 2 ijkl 7 ⊂ (7) G C 2 × Γ has some important finite subgroups, in particular, of the division algebra of octonions is a 14-dimensional subgroup of ]. Under 7 . In particular in [ 2 2 ) 57 , its finite subgroups and 480 octonion conventions. G G R – , denoted here as matrix, to build different superpotentials, leading to Minkowski vacua with two flat ) ( ·PSL 2 To explain why there are 480 possible notations for the multiplication table of oc- In appendix The R 7 3 ] ( 55 G 2 , 2 × Z At first sight this mightsome suggest of these choicesthe do automorphisms not of change thesubgroup the of oriented multiplication Fano table. plane, It preserving is triads, due form to a the discrete fact finite that tonions, as shown inoctonion [ units, including the flipping of signs. This will give group [ nite subgroups of of subgroups maining 14 the generators of Under the action of resentation and the structure constants of octonions form an invariant tensor of The tensors This means that there are 7 constraints on 21 generators of The 14 generators of SO(7) associator which is defined in eqs. ( code to the original Hamming code byG a particular permutation. group 23 be often used inorder our of construction rows, of so cosmologicalto that models. Minkowski we vacua Sometimes with can we one7 easier will flat explain direction. change our the Sometimes newdirections. we octonionic will superpotentials, remove leading some rows in this JHEP01(2021)160 3 2 Z (2.17) (2.16) ) which one can 3 5.3 Z possible choices of possible inequivalent . 4 2 and the transformations of = 480 inequivalent labelling of the ]. The starting point there is . However only the action of 7! = 5040 0 2 65 G > i of order 168 which take collinear 1344 of (7) 645120 T 2 (7) (7) Re 168 = 30 2 2 Hence we have / 5 assigments corresponding to the group ·PSL ⇒ PSL 3 3 ways of assigning labels from 1 to 7 to the ] 5040 2 ·PSL 2 3 possible inequivalent conventions for octonion Z ] [ ) and for the GG convention they are given in 2 – 11 – Z ⊂ [ . By the action of the cyclic group (7) 3 7 4.36 2 Z Z (7) 7! = 5040 o ·S 16 = 480 7 of order 21 7 ] ·PSL Z 3 × 2 3 Z ] Z 2 [ 30 o Z [ 7 Z = T H possible sign assignment 7 2 . . Again this might naively suggest that there are . Out of A.2 i (7) e S ± . Hence the total number of inequivalent multiplication tables comes as Given an octonion multiplication convention one can write down a superpotential by This group is generated by conjugation with respect to three imaginary units intoduced in Cayley- 5 To finalize the basics on octonionssome interest part in of using the octonions paperCayley-Graves for we octonion would quantum like computation multiplication to [ table. add here The that original there set is of triplesDickson defining process in octonion going from real numbers to octonions. then generate three superpotentials. Theconvention corresponding differ superpotentials by in an the overalltials “conjugate” sign in and CSC hence convention areappendix are physically given equivalent. in The ( superpoten- Cayley-Graves multiplication table and octonions in quantum computation. since in our modelsoctonions a with convention the is sign chosen flips so is that not allowed forassociating the the moduli change moduli ofis with variables. invariant the under imaginary the units cyclic using group the general formula ( the Frobenius subgroup does not involve sign changes of the imaginary units. This absence of sign flips is important sign assignments leading to multiplication. These 480 possibleby conventions octonion split naturally conjugation into whichdirection two changes of sets all the of associative 240 signleft triads. related of invariant Given under all a the imaginary convention finite the units subgroup octonion and multiplication reverses table the is the unoriented Fano planepoints is into the collinear finite points. group unoriented Hence Fano plane. there are WhenFano plane one one represents the canunits octonion associate multiplication the by points an onof order oriented the 8 Fano do plane not with change the the multiplication imaginary table. octonion via the heptagon rules.points There in are the Fanogroup plane or theconventions that corners do not of involve sign athe changes. heptagon multiplication However, some table corresponding of these while to choices preserving do the not the change symmetric associative triads. The symmetry group of 1344 The multiplication table of octonions can be represented by an oriented Fano plane or It is an irreducible imprimitive group of order 1344. Thus, there is a redundancy of order JHEP01(2021)160 2 4d G (2.20) (2.18) (2.19) , = 1 6 = 2) , N (2 ijk ] as a starting ,where C 2 65 || M structure, including a 2 ). Conjuga- || .Therefore, .Noticethat i G 0 2 5 4 1 0 e = 7 7 6 3 2 i (2) e e e e rs. Alternatively, all columns except the e e e e e nted as a sum of z a quaternionic lines − − − − ∗ 1 e SO a ,i > =0 complex numbers of .Thisisverysimilar to be important. 3 i i − me scale. Therefore, es. The unitary gates 7 4 3 0 value only for order a sentation in e 6 6 5 2 1 ts them to form strongly e e e e = , rpairscomesfromthe .Asforthegates,since % e e e e e . . 8 ∗ − − − − +1 R = aa 176 365 4 6 0 2 =0 =0 ∼ = 5 5 7 1 3 e e e e a , , is recovered. But the scaling e e e e e j i 4 − − − − ijklmno subspace to another) is an action C ,theyshouldalsoformstrongly if if otherwise C 734 347 (or are added, the deformed backgrounds 4 0 1 2 3 , , ∼ = 7 2 R 4 4 5 6 7 e e e e , e e e e e e ,thegatesform -dimensional equiangular families of k T lines 1 − − − − 4 ,andtheyobviouslyhavethestrongly e 725 257 n z 2 (2) R , , structure. This still means that S ijk 2 0 7 5 + 3 3 1 6 4 ε e e e e ] is, using 2 e e e e e SO ∼ = 0 653 246 − − − − + z – 12 – G , , 1 18 × . ij P 0 1 6 7 , δ = , } ] is R 2 2 3 4 5 and their cyclic permutations. i j e e e e 624 176 C 8 . e e e e e e e − , , , a − − − − 19 ] where the distinction between 7-manifolds with 4 .Astheyare holonomy. In general when fluxes in 11d supergravity ⎩ ⎧ ⎪ ⎨ ⎪ 4 , (2) 365 1. table multiplication octonion 2 0 3 5 6 2 145 145 holonomy , S 12 form = , 1 1 2 4 7 , e e e e , , G e e e e e was changed by multiplying by 2 1 2 j SO − − − − ∼ = { 10 C 347 of octonions, which is useful in quantum computing. .Furtherinallthethreerings 6 .e 4d supergravity with chiral multiplets which can be derived G i i , 123 123 1 ABLE ijk, ilm, ion, jln, jmo, klo, knm. ∈ manifolds, which is important for our work. We follow closely e e -dimensional real planes in 0 0 1 2 3 4 5 6 7 i T P 4 e e e e e e e e e n 2 o 257 H = 1 For this purpose we present a basic information about the twisted , G ] and [ j =1 SO(8) for 7 4d supergravity from 11d supergravity on manifolds with .Theproductrulecanbewrittenas: 7 i 0 1 2 3 4 5 6 7 e N i 1 e e e e e e e e 16 n 246 e 2 % holonomy but rather , = 1 structure is explained and many useful references are given. One starts R is a completely anti-symmetric tensor with 2 − 2 orbifolds with N 176 G 0 , G e 7 ijk )complexnumbersby 0 ε T 4 o is the sum of squared of the real numbers representing that these projections provide consist of all complex numbe Four families of equiangular planes. 145 2 , ] the table in figure C = The space of lines in Taking this argument one step further, one could consider th Elements of quaternions (and octonions) can also be represe || (or . Cayley-Graves octonion multiplication table. It was recently used in [ -planes in a ∗ 65 viewed as strongly equiangular planes in will be ignored due tonorm normalization, so one we or are equivalently talking about equiangular property as theyon are simply one dimensional lin the gates are notprecisely strictly an unitary isomorphic but always copy unitary of up to so to a unitary up to a scale, which is the4.1. property that turns out Using these division rings togeometric generate picture. strongly There equiangula existn exactly four forming the space || equiangular pairs. Through a suitable embedding,equiangular one pairs expec of going from one quaternion to another (from an quaternions themselves have a unitary up to some scale repre o 123 tion in each ring is defined as the flipping of signs in all be replaced by (13) where 8MichaelFreedman,ModjtabaShokrian-Zini,ZhenghanWang 2 In [ The set of triples by Cayley in [ The seminar at Stanford by M. Freedman presentingA this derivation work of was stimulating for our interest in the 6 7 no longer have connections between quantum computing and octonions. the mechanism ofperformed spontaneous by breaking A. Van of Proeyen, work maximal in supersymmetry progress. to the minimal one, was recently 7-tori and the compact the presentation in [ holonomy and with Joyce’s and geometrical fluxes describing the twisting of 3 Twisted 7-tori with We are interested in from 11d supergravity. These two choices agreeorder if of we triples. allow a cyclic permutation insidefirst a one. triple, and This changemultiplication in algebra the turn allows the authors to work with a specific realization of the Lie multiplication table by Graves shown in [ Translating this into numbers we find Figure 6 point for the proposal on quantum computing with octonions. JHEP01(2021)160 = 2 and (3.6) (3.3) (3.4) (3.5) (3.1) (3.2) T = = 0 1 T -holonomy, dφ 2 G . They correspond 7 -holonomy manifolds. . 2 G is the contorsion tensor. at the Minkowski vacum ,..., . . i = 1 i satisfying j . ˜ φ i T i = 1 i ˜ t mnp V ¯ φ ηη T κ = 0 -structure manifolds one finds m j 2 ij where the 7 complex moduli are G Φ = dφ . and suggest that the manifold has for M i ∗ ∧ φ i j . i = 0 η X T -structure manifolds have Minkowski = 0 j φ ij 2 η ˜ δ i mnp 7 φ T ˜ G φ mnp ij X np ] starting with 11d gravitino kinetic term = d = γ Z γ . j M T 16 1 8 T ˜ 4 φ 3-form = iη − mnp − k ∧ 2 κ ] = 0 T i – 13 – = i | = = 1 4 G ) labelled by the index φ j ˜ φ i j and for its dual 7  T − T i i 3.2 X mnp i ij dφ t V η φ Z φ i . This spinor is covariantly constant with respect to a T  t M [ m η ij d ∇ -structure and a non-vanishing superpotential, in general holonomy correspond to untwisted tori where j M 2 X 2 and therefore 1 2 Φ = G G = Φ = T = i ∗ ≡ structure is a 7-dimensional manifold which admits a globally d W W 7 2 T ]. One can introduce the dual 4-forms G and = 10 6 singlets, with different choices of T = 0 . The resulting 4d gravitino mass, which defines the superpotential was 2 p j = ˜ φ G Ψ 5 ij denotes the antisymmetric product of three gamma matrices with unit norm. N ] in the following form T M D involves a Levi-Civita connection, and the tensor i 10 = t mnp 4 m 4 γ − T ∇ holonomy unbroken. For MNP The vacua with one flat direction which we will find have the property that The upshot here is that starting with general type A special situation takes place when such The 4d superpotential was computed in [ Using this spinor one can construct a globally defined and nowhere vanishing totally A manifold with Γ 2 = G M 3 Φ = ¯ T These are exactly the octonion superpotentials we will describe below. which leads to an existencea of the closed 4-form d Minkowski vacua only in cases that some twisted 7-tori are, in fact, vacua, as shown in [ One finds in such a case that The standard manifolds with the superpotential is absent.describes The the twist 7-manifold of with thewith toroidal typical AdS orbifolds vacua. away from contracted with the 7 3-forms in eq. ( Ψ given in [ where 7 of them are to associate triads forThis octonions. allows The to remaining introduce 28 a correspond complexified to anti-associative triads. Here It can be viewed as a normalized Majorana spinorantisymmetric such tensor that not require fluxes inmanifold, the 11d twisting supergravity, of however, the the tori, will geometric be data important. defined, of nowhere-vanishing the spinor compactified torsionful connection. supersymmetry is preserved when compactification from 11d takes place. Our vacua will JHEP01(2021)160 . . ) 5 +6) (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) T on the , r − 4 , cw +5 T , } )( , r 3 WO T +4 (726) − (1345) , , r . 1 , T +7 +2 r r e (615) ( )+( , ) , we get the 7 associative (7234) 4 = 1 , . +1 T r T ) r e − − (574) +6 3 which are complimentary to 7 , r T ) T T (6123) )( , )( 2 O 6 − (463) , e T T O , mnpq +5 − ( − r and call it +1 7 4 r (5712) T e T O , T W )( (352) , = +4 )+( )+( r 3 7 +6 T r T (4671) T W e (241) , ) − − , − – 14 – 2 +5 6 . For our purpose that the codewords represent r ijk T +2 for octonion multiplication in Cartan-Schouten-Coxeter T e ( r 5 O )( )( T (137) 1 (3567) = 5 ( { , T T . The complimentary quadruples are 6 =0 +7 − X r W − r 6 3 e +7) = W T T +3 + 7) : r , r cw e )+( , r )+( 2 +3 6 = . T T WO + 6 : (2456) , r + 3 7 − +4 − r , r 1 5 +1 , r e r T T ( +2 + 5 )( )( + 1 r 7 4 e r , r T T clockwise oriented heptagon ). We start at the second right octonion − − 5 2 + 4 4.2 T T , r . The +( = +1 : ( = ( i < j < k. matrix at the right of the figure + 2 cw cw ijk Explicitly we have for the superpotential in the clockwise heptagon picture r m < n < p < q. 7 f × WO This superpotential is easy7 to compare with the cyclic Hamming error correcting code, a This is shown in figure associative triads. We take the case This leads to the following superpotential We make a choice of associative triads To build the superpotentials we need the quadruples 4.1 Cartan-Schouten-Coxeter conventions, clockwiseConvention of heptagon octonion multiplication by Cartan-Schouten-Coxeter are triads in the form as They are shown atclockwise the oriented heptagon. corners with the oriented clockwise blue4 lines. These give Octonions, Hamming (7,4) error correcting code, superpotential Figure 7 convention ( JHEP01(2021)160 (4.9) (4.8) (4.10) (4.11)              ) ) ) ) ) ) ) 6 7 1 2 3 4 5 T T T T T T T 1 − − − − − − − cw 5 6 7 1 2 3 4 → T T T T T T T 7 )( )( )( )( )( )( )( WO 4 5 6 7 1 2 3 is the identity matrix. T T T T T T T , associated with Cartan- 7 7 matrix at the right of the c − − − − − − − ⇒ ]. 7 cw 2 3 4 5 6 7 1 → 61 T T T T T T T 6 × ( ( ( ( ( ( ( 7 WO , 6              . → 5 , . It is a subgroup of the collineation group 5 (2456) (3567) (4671) (5712) (6123) (7234) (1345) 2 G → ) to the left of the codewords. The 4 zero’s in 4 ) are in full agreement with the set of triads in – 15 – = (1234567) of 0 0 0 00 0 0 1 0 0 01 0 0 0 0 1 0 0 0 0 0 1 0 00 0 0 0 1 0 00 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 3 4.8 4.8 ,              , one can perform the transformations on octonions Z 4 CSC 2 o = P → 7 c 3 Z , 3 1 0 0 00 1 1 1 0 0 0 0 1 1 1 0 1 01 0 1 0 0 1 10 0 1 0 1 0 00 1 0 0 1 0 10 0 0 1 0 0 1 1 0 1 ), shown to the right of the codewords, are in perfect agreement → 2 4.8 , ) we have to move the last row of the 2 4.2 → Triads Codewords Quadruples . Therefore we are only interested in transformations without flipping the (615) (726) (137) (241) (352) (463) (574) 1 i              T = of order 168 of the Fano plane studied in detail in [ ). They are also shown in eq. ( into the first row. We have to read the codewords from the top to the bottom and cw 7) , 5 4.2 Leaving the full discussion to the general construction given in the next section let As we have explained in section All 3 1’s in the codewords in eq. ( WO which can be represented by the matrix with the property that notations are generated by the permutation operator It describes the transformation is the finite Frobenius subgroup PSL(2 us show how theseSchouten-Coxeter symmetries conventions. affect our The superpotential 7 cyclic permutations in Cartan-Schouten-Coxeter which preserve the multiplicationto 7 table. moduli Weoctonion would signs, like which to preservethe the apply multiplication oriented these table. Fano transformations plane, These without form the the flipping automorphism of of signs of octonions, preserving triads. It with the 7 complimentary quadruples defining the superpotential. figure from the left to the right. This will be oppositeeq. in the ( case with counterclockwise heptagon. the 7 codewords in eq. ( octonions in eq. ( JHEP01(2021)160 → we 5 , cw (4.18) (4.14) (4.15) (4.16) (4.17) (4.12) (4.13) 5 → WO of order 3, 2 3 , 1 Z 2 → 1 → . 7 etc. . We now act on = 1 ) , 7 3 1 becomes the second term d T 7 → − cw 6 6 ,..., T WO is the cyclic group , )( . 7 5 3 = 1 T cw +1 Z ], such that i cw → i cw − o . 61 5 3 7 WO WO T Z WO , k , k 6 ] = i ] = + = ( i cw X = 1 ), which is equivalent to a change of moduli as cw → i cw 3 = 2 cw ) and represent it in the form 4 – 16 – d WO 4.11 WO operator in [ cw [ WO 4.7 , [ WO d (1)(253)(467) 5 ] = , 0100000 0000010 0010000 0000001 0001000 1000000 0000100 ) WO k CSC cw i → 6 1 CSC P P T 3 . are permuted and as a result, we have the same WO − 8 [ , cw �������������� 5 . which corresponds to 4 . This is a 4 T CSC k )( → WO ���� leads to the mapping P 4 → operator in eq. ( 2 T 7 . This gives an example of the set of permutations of moduli which c , 7 − , ]. It preserves the octonion multiplication table, and does not flip the CSC is in figure = 3 Figure 8 2 P k CSC ). Under this change the first term in d 61 → T P → 6 1 CSC , 1 4.10 P = ( 6 → 1 cw 4 , by the 2 WO The other subgroup of the Frobenius group In CSC convention it can be given as the following transformation cw → 3 , In matrix notation where the matrix also described in [ signs. However, it does not preserve the superpotentials. 3 for each of the 7 do not create differentsuperpotentials. set of octonion multiplications tables and do not create different For all 7 operations we find that the termsstarted in with where WO shown in eq. ( there, the second becomes the third etc. etc. . We now take the superpotential ( Acting twice with JHEP01(2021)160 , the ] (4.24) (4.25) (4.26) (4.19) (4.20) (4.21) (4.22) (4.23) cw , } WO [ 3 d (715) and we are back , . = +7 r = 1 cw e (674) 3 , d ) . = ) ) WO r +2 +1 00 cw r r +3 r (563) T T , r r T WO r T . T T − − ] = (452) , e − = 0 , +3 cw +2 +1 r r +1 00 cw r r T T e r r T WO (341) r [ T = , T WO 2 ( ( T . . It stops here since ( + ] 7 +7 7 =1 =1 7) r X 7 X r n =1 , cw e X r 0 cw (237) , – 17 – +5 , d ] = ] = ] in the form (126), (134), (157), (237), (245), (356), (467) r ] = WO e WO [ PSL(2 0 cw 66 cw cw 2 cw , as follows: (126) + = d { 9 WO WO WO cw +6 [ [ WO r 3 [ 2 e d d d ] = WO 8 +2 + 6) : = = = r cw e , r cw cw 0 cw 00 = WO [ + 2 d WO WO WO +4 , the 3d one is ] r . We define , r e cw cw +3 r + 1 e r WO WO [ d = +1 : ( i < j < k. Therefore we define the convention for octonion multiplication, following from the Thus, there are 3 possibilities, the first one is the original ccw ijk A related set of triads was considered in [ 8 f in the context of irreducible representation of We consider the following setthe counterclockwise associative direction triads consistent with the cyclic permutation in the reason we introduceto the the Reverse second Cartan-Schouten-Coxeter set convention ofThe which conventions 240 in conventions belongs the which second includes setplane the of modulo Cayley-Graves 240 convention the can as also permutations well. be of represented the by labelling the same of oriented thecounterclockwise Fano points heptagon on in it figure without any sign flips. Cartan-Coxeter-Schouten belongs to the setresented of by the 240 octonion same oriented conventionspoints whic Fano on plane can it modulo be without sign the rep- permutations changes. permutations alone of and The require the other sign set labeling flips of of of 240 the some conventions can of not the be imaginary reached octonion by units. That is Thus only 2 of these superpotentials are independent. 4.2 Reverse Cartan-Schouten-Coxeter convention, counterclockwise heptagon Obviously, these three superpotentials satisfy ventions. second one is to the original One finds therefore that there are 3 superpotentials for the Cartan-Schouten-Coxeter con- JHEP01(2021)160 . + ) 6 r ( T (4.27) (4.28) (4.29) ) were matrix − 4 7 4.8 T ). We start × )( 7 3 , so that T 4.25 − 2 + 7)) T r ( ) on the counterclockwise , . They are shown by the )+( 2 5 T ) T ccw + 7) + 5) − − +7 7 r which are complimentary to 3 r , r ( T WO T T , } )( + 5 )( 6 − O 2 , r T + 4) T O +5 mnpq − r r − + 4 ( { 5 1 T , T , r T )( W +4 + 3 + 3) for octonion notations in eq. ( )+( r )+( r 1 r 4 ( T T T (( W ) to the left of the codewords. The 4 zero’s in − − − ) are in full agreement with the set of triads in 6 – 18 – 2 , we get the 7 associative triads in the form as +3 T T into the first row. However, to see the triads and 4.30 r +1 )( 4.30 O r )( T 5 5 e m < n < p < q. ( 1 T T 6 =0 X − r − W 4 7 = we have to read the codewords from the bottom to the top T W T This is opposite to the clockwise heptagon case in ( ccw . As in clockwise case we have to move the last row of the )+( ), shown to the right of the codewords, are in perfect agreement )+( 7 3 5 T T WO 4.30 ), now − − 5 1 T T 4.30 )( )( 4 7 T T counterclockwise oriented heptagon − − 3 . The complimentary quadruples are 6 T T . The ). They are also shown in eq. ( + 6) = ( +( , r matrix at the right of the figure ccw 4.25 All 3 1’s in the codewords in eq. ( Explicitly we have for the superpotential in the counterclockwise heptagon 7 + 2 × , r WO and from the right towe the read left. the codewords from the top to the bottomeq. and ( from the leftthe to 7 codewords the in right. eq. ( This one is easy toat compare with the the right cyclic of Hamming7 error the correcting figure code, a quadruples in eq. ( This defines the superpotential as follows: these associative triads which we take to be 1 corners of the orientedoriented counterclockwise heptagon. 2 blue lines. These give To build the superpotentials we need the quadruples Figure 9 at the second right octonion and call it JHEP01(2021)160 (4.36) (4.32) (4.33) (4.34) (4.35) (4.30) (4.31)              , belonging ) ) ) ) ) ) ) 6 5 4 3 2 1 7 . cw T T T T T T T ) . We define ] − − − − − − − +3 WO r 4 3 2 1 7 6 5 ccw ccw T 00 cw cw T T T T T T T r 0 cw )( )( )( )( )( )( )( T . 3 2 1 7 6 5 4 WO WO WO [ WO − T T T T T T T 2 WO 00 ccw d − − − − − − − , associated with Reverse- +1 ⇒ ) = ) = r 2 1 7 6 5 4 3 ) = T 0 ccw T T T T T T T WO +2 . +1 r ( ( ( ( ( ( ( r r +3 r T T T ( r ] = r T = 0 WO r T 7 T =1 X r T ccw 00 ccw − − − = +3 +2 WO , the 3d one is r [ r WO +1 ] 2 0 ccw r T T r + (2346) (1235) (7124) (6713) (5672) (4561) (3457) r T ccw r T T ( WO ( T 0 ccw ( 7 – 19 – 7 =1 WO =1 , d [ X 7 X r n =1 d X r WO , 0 ccw ) ] = ] = + ] = +1 r ccw ccw WO ccw ccw T r ] = T WO WO [ [ WO WO 3 [ 2 − ccw d d d 1 1 0 10 0 1 0 1 0 00 1 0 0 1 0 10 0 0 1 1 0 11 0 0 0 0 1 00 1 1 1 0 0 0 0 1 1 1 0 1 0 0 0 1 +2 = = = r WO [ T d r ccw 0 ccw 00 ccw , the second one is ] ) simply picks up an overall minus sign and the directions of all the T ( WO WO WO ccw 5.3 7 =1 X r Triads Codewords Quadruples (674) (563) (452) (341) (237) (126) (715) = WO              [ 3 cw d = ) within each of the 480 octonion multiplication conventions. However the su- = 5.3 ccw WO To summarize we can construct 2 independent superpotentials using the general for- In the counterclockwise case there are also 3 possibilities, the first one is the original ccw WO one superpotential associated with Cartan-Schouten-Coxeter convention, to the first setCartan-Schouten-Coxeter of convention belonging 240 to conventions, the and second a set different of one, 240. These are perpotentials obtained in thesuperpotentials first obtained set of in 240 theset CW by second conventions octonion set are conjugation. of notby Indeed independent 240 the under of CCW formula octonion the ( conventions conjugationarrows related the in to superpotential the given the oriented first Fano plane get reversed. We shall, however, take as independent Thus only 2 of these superpotentials are independent. mula ( Obviously, these three superpotentials satisfy case. WO One finds thereforeCoxeter that conventions. there In fact, are they 3 coincide superpotentials with the for ones the we have Reverse-Cartan-Schouten- found in the clockwise with the 7 complimentary quadruples defining the superpotential. JHEP01(2021)160 (4.40) (4.41) (4.42) (4.37) (4.38) (4.39) ) 1.1 . .                           1 1 1 1 is satisfied. − − i − − ∀ 1 0 1 1 1 1 0 . , c − − . − − 1 01 1 1 1 0 1 0 . = 0 j ]. It is given in eq. ( − − + h j − − ij ij T j T 1 0 1 0 1 i 1 1 01 0 0 holonomy have explicitly 14 terms given i 16 χ M M , is satisfied. − − T 2 − − T ij j 2 K i ij 1 1 0 1 0 1 ij 10 G 1 1 01 0 0 1 e ∀ M P , i M − − − − M 1 2 χ 1 2 1 2 1 01 1 0 1 0 1 1 01 0 0 1 2 – 20 – K = for RCSC octonion notations is − − = 0 = − − e = structure has 21 terms, it is described a symmetric ij 1 01 1 0 1 0 ij 1 0 1 1 0 0 1 1 2 1 0 0 11 0 00 1 1 0 0 10 0 2 0 1 0 1 0 01 1 0 0 1 m 0 ccw − − W − − M WO G for CSC octonion notations is = j                           f m WO P cw ). = = L 5.3 cw 0 ccw WO . The masses of fermions in supergravity in Minkowski vacua WO j WO M M dφ ∧ for the case i φ ) and can be written in the following form ij 7 for the case M X R 4.34 ij 1 4 )–( M is − = 4.32 =0 i ij W M The matrix = Here again a condition for Minkowski vacua, for Minkowski vacua so that eq. One can see that in each row of this matrix the sum of entries vanishes. This is a condition The matrix Our octonion superpotentials for modelsin with eqs. ( and the massW term of the chiral fermions in the supergravity at the vacuum with 7x7 matrix with all diagonal terms vanishing [ and are defined by the second derivative of the superpotential moduli variables, which can beusing also the obtained general directly formula for in a ( total4.3 480 possible conventions, Octonion fermionThe mass superpotential matrix for and models mass with eigenstates From these two superpotentials we can reach any other superpotential by a change of JHEP01(2021)160 480 = 1 (4.46) (4.47) (4.43) (4.44) (4.45) N → . (7) } 2 0 (7) , ·S 7 ] ·PSL . 2 3 ] Z 2 [ Z [ 479735 } . It means that the . 0 0 3569; 0 (7) − , . 1 , 23 ·S . − ). This number of different 7 0 , ] , 2 Z 23 4.42 479735 . . 3569 0 . 0 = [ and a massless one, the following 1 , − − 3 T , ) or ( 2 35 , . . ) , y 3 0 x , 4.41 can sc ; 59822 35 m . 7 = 0 7 = 0 2 69202; . 0 . 0 matrices in both cases above solve a double 1 , y − − − = ( 2 , − y x , supergravity models with octonion superpoten- x M 2 7 – 21 – 7 , ) 16 ; . 1 − − 1 can f = 1 3 3 , , y 59822 y x m 1 . 69202 ( 16 0 N . x . . In this way we can get a total of supersymmetry. In all models with octonion superpo- 1 1 − { − , (7) 3 2 − = 1 t : 07796 N . 2 04892; 1 ·PSL . ) symmetry which is a symmetry of the massive fermion eigenstates. , 3 3 ] , 3 2 can sc will take a different value whenever our transformation is not part of Z m ( ij 07796 . = [ . Here we have taken into account the correct kinetic term normalization M 04892 1 ) . { [SO(2)] 3 T H 2 / the mass eigenvalues are the same, i.e. they are preserved when new octonion 3 − t = Re( : WO t can f matrices is in agreement with the fact that there are 480 different multiplication tables One way to see it is to perform the change of fermion variables reproducing the trans- The scalar mass eigenvalues in The non-vanishing eigenvalues of the m ij 5 Octonions and generalThe construction Fano of plane has superpotentials 7intersection lines point and of each three linemultiplication lines. contains three three When points points we such onsubalgebra. use that each each oriented point Fano line is planes go the to over describe to octonion the imaginary units of a quaternion different matrices, starting from anyM one in eqs. ( of octonions. Thespectrum upshot invariant here under a is: change our of octonion octonion conventions. superpotentials have an eigenstate mass convention is used. formation on octonions, permutationseffective and matrix sign flips, the subgroup One can check that as it should accordingtential to factors. They have a simplesupersymmetry. relation To to see fermion mass thiswith eigenvalues we in account have agreement of to with a take difference into in account kinetic that term canonical normalization are fermion masses, tials and one flat direction are where values They show an Moreover, the massoctonion eigenstates notations! of fermions are the same for any of the 480 choices of set of cubic equations Numerically this gives for a set of JHEP01(2021)160 kpq f (5.3) (5.4) (5.5) (5.1) (5.2) and are those of kmn ijk , f f . by construction. kij f P ) with no sums over since it is determined ] labelling of the real ) . we associate a term in 5.1 } 23 P j ) ) kij and its permutations one . For the Cayley-Graves q q ( , e T i ) T ). We combine them in a = (1245736) . , e − − kij k 5.3 p ( p CG e T P T ) = 0 )( )( . n n ) = 1 T ijk T = (1234567) ( given by the labelling of the imaginary kji − − ( . The remaining associative triads can be kpq W k f m P m e W . For the Cartan-Schouten-Coxeter (CSC) CSC T T = ( ) + − ( P kij and kpq jki – 22 – f j kmn ) = ( f { f r can be used to make any Hamming code associated , e ) W i kij = e kmn ( P P f ( the corresponding structure constants ) + W is simply kij ) = (1243657) 6 =0 kij f X r f kij P ijk are all different. repeatedly and summing over: ( generated by the permutation operator ( GG ⇒ ) = 7 P W P = Z is contained in three different quaternion subalgebras. Let us is kij kij ( k f e P W i, j, m, n, p, q ] the operator 21 , 20 acts on all the seven indices inside the sum. Note that the superpotential is invari- P The cyclic permutation operator The three 3 superpotentials we have discussed above for Cartan-Schouten-Coxeter Given a quaternion subalgebra with imaginary units An imaginary unit Given an associative triad with a given conventionassociated of triads octonions get cyclic. mapped The into each codewords other associated under with theconventions the action different are of particular cases of the general formula ( Furthermore one has Therefore of the sixfinds superpotentials only two defined independent by ones. the triad ant under the cyclicNote group that we labelled the superpotentialby by it the uniquely. associative triad One finds that where where all the 7indices. indices The appearing full above superpotential arecyclic is different permutation obtained and operator by satisfy acting ( on a given term of this form by the where the indices the superpotential of the form labelling of octonions the cyclic permutation operator is denote the structure constants ofand these assume three they quaternion subalgebras are as in cyclic order such that a quaternion subalgebra generated by obtained by the “cyclic”units permutation on operator aoctonions heptagon. the For operator examplelabelling in [ Gunaydin-Gursey (GG) [ JHEP01(2021)160 ) (5.6) (5.7) (5.8) (5.9) 4.33 (5.11) (5.12) (5.13) (5.10) ). Thus . ) ) ) 2.6 +6 +6 +7 ) and ( n n n . T T T 00 cw 4.32 − − − , one can change 3 WO +5 +5 +3 n n n T T T )( )( )( ⇒ − , +4 +7 +4 5 00 cw n n n e is defined in eq. ( T T T different superpotentials. An kq ] WO . − − − p ⇒ − f 5 +2 +3 +2 = 0 ) = 0 mn n n n [ q , k T T T T 00 cw ( ( ( f . 2 = 960 0 cw − +6 +6 +7 × = jik , e p f WO ,n ,n ,n 1 e T WO +5 +5 +3 + 480 )( mnpq ,n ,n ,n n ⇒ 0 cw C T ⇒ − ⇒ − +1 +1 +1 – 23 – n n n 1 − f f f 0 cw WO m ). ijk +4 +7 +4 + f T ,n ,n ,n ( WO 5.4 cw , e +3 +2 +2 6 ,n ,n ,n e ) each superpotential will change the sign: mnpq WO +1 +1 +1 , C n n n 5.8 f f f cw ⇒ − )–( 6 +4 +6 +7 e ,n ,n ,n WO 5.6 +2 +5 +3 ,n ,n ,n ⇒ − +1 +1 +1 n n n cw f f f 6 6 6 =0 =0 =0 X X X n n n WO = = = 00 cw cw 0 cw These 480 different superpotentials corresponding to 480 different conventions can be However this does not imply that we get WO WO WO understood via change of variablesconsistent starting with from the our symmetries simple ofstructure. cases the in This Lagrangian eqs. explains that ( why preserveall all the of these underlying them models octonionic have are exactly equivalent: the in same particular, masses one of finds eigenstates that at the vacuum. This means that wesible get inequivalent a conventions. total And of sincechange 480 the of different potential octonion superpotentials is conventions quadratic when will in we not superpotentials consider affect this all the pos- potential. or any other 3 octonionsoriginal on multiplication one table line to in theone the one can Fano plane. with see from the This eqs. opposite will ( change sign every (octonion term conjugation). in As the This can be achieved bythe the Fano sign plane, flip for of examplethe 3 signs in octonions of Cartan-Schouten-Coxeter which case the belong following to in octonions any figure particular line on different 480 conventions: there arethey 3 always superpotentials satisfy the for constraint the ( same set of conventions,additional and interesting property ofcan the change general the formula octonion for convention set the by superpotentials odd is permutation that of indices one This follows from the fact that where the completely antisymmetric tensor only 2 of these superpotentials are independent. It is a property satisfied for any of the These three superpotentials satisfy different order so that JHEP01(2021)160 ) ) } i ). 1.1 , a (6.7) (6.4) (6.5) (6.6) (6.1) (6.2) (6.3) i 1.2 4.33 t { ) ), ( 1.4 . } in eq. ( 4.32 0 , 0 ), ( , at the minimum. WO 11 4.29 , the eigenvalues of . 7 0 = 0 , , ), ( i . ) and 7 a 11 a . ··· 4.7 1.1 , 0 i, j + , . and ∀ . 11 ) = 1 . ··· 2 | 0 = 0 i 2 = 0 , a . in eq. ( i = 0 t 11 + WO . j | WO ij 1 0 scalar/pseudoscalar mass matrix has 3 T : there is at least one flat direction in a for all M 7mod , − 49 = 0 i − 14 M a K a 2 i 18 | − . , × T = 0 y  + i i , 14 i WO a T t and 18 , . D t + + 49 – 24 – 0 | = ( , 2 i T means that = y  T 18 , i . . We have checked that these potentials have super- = 0 : t 7 14 . This equation is invariant under the permutation 7mod 0 t i = 0 K , A − T e + 7 log ). The 6 massive eigenvalues are pairwise equal, i.e. we i 3 Hessian matrix of the potential with respect to 18 WO W . y − ≡ P 0 ··· = symmetry in the mass matrix of the scalars and separately = 14 . 1 7 = 4.45 , + 3 mass matrix for scalars and pseudoscalars has the standard VO × 2 K W 58 = t i . WO i 0 14 14 , so we show here only 7 of them, one flat direction being massless. D + T i , D is defined by the 7-moduli 1 a × t 58 [SO(2)] . 7! = 5040 14 0 ]. With VO , symmetry in the mass matrix at the minimum of the potential. 10 as for 58 3 . i 0 t . Since the vacuum is supersymmetric, in each case there is the same mass , should be a null vector of the matrix of order 7) i 58 , t . symmetry since each of the scalars has the same mass as the pseudoscalar. How- 0 [SO(2)] (7) { 7 ··· 7 S , − t Of course our On the line in the moduli space As expected, the eigenvector of the massless direction corresponds to The potential = 1 i the following eigenvalues They solve the following cubic equation have an [SO(2)] ever, this additional pseudoscalars is a feature of our vacua. The potential. We find( the eigenvalue for the matrix are given by eq. ( This leads to an effective theory of a single modulus supergravity of the form In order to check the stability of the minimum, we consider the Hessian of the scalar with one flat direction group Specific superpotentials which we studiedand here are in shown other in examples eqs. ( symmetric in Minkowski appendix vacua when all moduli are equal to each other, as in eq. ( it was shown there that Therefore such Minkowski vacua. 6.1 Vacua with 1Supersymmetric flat Minkowski direction vacua ofwere the studied in superpotentials [ in the form shown in eq. ( 6 Octonionic superpotentials and Minkowski vacua JHEP01(2021)160 (6.8) (6.9) (6.14) (6.11) (6.12) (6.13) (6.10) . . ) 6 T . . 49 = 0 } } . − 5 − T y with the first index as follows )( 4 (2456) (2456) , , T 1010001 + 49 + 3) , . 2 − } 7 = 0 y , r 0 2 ) in a specific way: we leave − T 14 (1345) (1345) . , ) + 1 x , , ), with the difference that the − 7 07 6.11 . +3 0100011 3 0 4.7 ) + ( n − r, r y , 5 ( , 3 T ). T x (7234) (7234) 07 − , , ⇒ . − 6.6 0 4 y +2 , = n T 2 1000110 T 21 x )( . (6123) (6123) | , 3 )( , , 1 . The 7 codewords are T , 5 +1 − n – 25 – = 49 21 . 1 T 2 1 (5712) (5712) T x 0001101 − , , , ) with the formula , n symmetry, however, the eigenvalues are different from 40 . T 3 ) + ( + 49 2.2 1 3 4 , T (4671) (4671) = ( x , , 40 − . 14 0011010 W 1 2 [SO(2)] { − , T 7 6 − )( (3567) (3567) x t 1 { { T = = ⇒ − } } 0110100 6 ] and shown here in figure ) becomes the last one. Now we propose to define a model with 2 flat , T holonomy. Meanwhile all 480 octonion conventions always lead to models 30 pqrs pqrs 4.7 = 49 2 { { 2 We can start with the cyclic error correcting Hamming (7,4) code taken, for G ) 1) = ( x , 7 1101000 (6 − 3 It is interesting that directly in 4d one can suggest many superpotentials for our 7 As we will see in the following, inflationary potential added to this setup does not x WO ( The remaining quadruples form the superpotential The corresponding superpotential isfirst the term same as in in ( directions eq. by ( dropping some codewordsonly in quadruples the without code 7. in eq. The ( ones with 7 are underlined below This corresponds to triads inin eq. triads ( taken in theare positions range of 1 1’s. to TheThe 7. second set We codeword of see is quadruples the the defined case first (235), by codeword which the of are positions positions the of of code zero’s 1’s, as in etc. . (124), the which codewords is now: 6.1 split. example from [ the ones which come fromare models 4d models with which octonion are superpotentials.7-tori not with Therefore, clearly originating these from models with the the 11d supergravity same compactified scalar on mass eigenstates given in6.2 eq. ( Using cyclic Hamming (7,4) code to get vacua with 2 flat directions The scalar mass eigenvalues are These masses still have an moduli which also leadoctonions. to The simplest one example flat is direction the case Minkowski with vacua, but not associated with spoil the stabilization andeffectively. we will show that the trajectory discussed here can be realized It is related to the cubic equation for fermion masses JHEP01(2021)160 (6.16) (6.15) (6.17) (6.18) SO(2) ) 4 ). T , , − , , defining our 6.18 re- . 3 re- 423 423 and and ). } (31) (33) (34) (32) (31) (33) (34) (32) T ) where we take . ) where we take of di- of di- x x o7by .And o7by .And )( + 6) 4] code. 4] code. 2 m m 6.14 V V 11 attached , attached , . )canbe 1) = 0 )canbe th line of p th line of p 6.11 T 4 , n 4 i 6.11 ]code(or i , ]code(or V n , V n , q (2456) q (2) ( = ( = − . , (6 F F T P P + 5 . The model has a (6,1) q q 7 6 , ≡ , ≡ 10 ≡ T n, k, d n, k, d n n T , n q ] q ] WO . We take out all codewords . We take out all codewords 7 . x F x F 1errors.Itcanbe 1errors.Itcanbe ) [ [ + 5 5 T (1345) 6 q q + 4 5 , − − ) + ( F F T 3 T d d , n . , . T . . − represents the + represents the ) ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 5  i 4 i − , . , . . . + 2 (1) i i T L (7234) L T 2 u, v u, v (2) T , L n 4] code introduced in Sect. 2.3 L ( 4] code introduced in Sect. 2.3 ( )( Michel Planat, Metod Saniga T Michel Planat, Metod Saniga ( , max , d T max + d 4 ∈ ∈ ≡ d d )( 3 C C v v ), T j j + 1 ∈ 6 ∈ ̸= T ̸= P P u is a subspace of T u is a subspace of T − u,v u,v (6123) + (2) 2 . One defines the Hamming distance . One defines the Hamming distance , 6.12 − C C T 2 = q +1= q +1= T and has a Minkowski minimum with two flat  6 F ) = min F ) = min T 5 k k – 26 – T th point and ]anddetectupto 7 th point and C ]anddetectupto 10 if otherwise C 10 if otherwise T + − ( j − ( j 1 1 log T 2 1 2 − − ) + ( n n ' = (5712) ' d d 5 T , − min . Then a projective geometry min . Then a projective geometry 4 as the number of coordinates in which ) + ( as the number of coordinates in which ≤ ] code, the following bound holds ≤ ] code, the following bound holds = q = q d T d ) in the polynomial field ) in the polynomial field 2  T g F d g F d n 1101000 0110100 0011010 0001101 1000110 0100011 1010001 is the n 1101000 0110100 0011010 0001101 1000110 0100011 1010001 is the q q ij ij T = = − n, k a (1) F j n, k a F j = and it is usually referred to as a [ and it is usually referred to as a [ d ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 4 d ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ P − P T 3 (4671) T in in 1 , T + ciency for error correcting; it is defined as ciency for error correcting; it is defined as )( y y T max max 3over 3over irreducible over irreducible over 3 d ffi d ffi = )( g (1) g T 7 ≥ ≥ in the position 6 and 7. We show it in figure in the position 7. We show it in figure 2 = T = and and T −  T 0 0 (3567) d d x x { 1 − +1 = +1 and the other is T 5 +1] code). The binary Hamming [7 = +1] code). The binary Hamming [7 δ 1 δ 7 6 log T } T T + ( r, r r, r − er. The minimum distance of a code is an important concept which er. The minimum distance of a code is an important concept which − − ff ff = There exists an intimate link between this code and the Fano plane, The link between good codes and projective geometry has recently There exists an intimate link between this code and the Fano plane, The link between good codes and projective geometry has recently pqrs di di 2) = ( { n, n . n, n K in the position 6 and 7.This leaves us with the superpotential in eq. ( y characterizes its e A linear code corrects up to [ [2] between a cyclic code isto an a polynomial ideal ( thus corrects up to 1 and detect up to 3 errors. It is the MDS [7 7.2 CyclicAs codes and shown in projective spaces Sect. 2.3, a linear code shown that for a linear [ The above matrix isobtained nothing as but follows: the if incidence matrix of the Fano plane which can be inferred ascyclic follows. extending of Let matrix us (5), take its viz. seven codewords 1 t ceived considerable attention [24]. Let us define a vector space the Fano plane, the elements of the matrix are A minimum distance codeis (or such a maximum that distance separable, MDS code) mension [ We take the 7 quadruples in eq. ( y characterizes its e A linear code corrects up to [ [2] between thus corrects up to 1 and detect up to 3 errors. It is the MDS [7 a cyclic code isto an a polynomial ideal ( 7.2 CyclicAs codes and shown in projective spaces Sect. 2.3, a linear code shown that for a linear [ The above matrix isobtained nothing as but follows: the if incidence matrix of the Fano plane which can be inferred ascyclic follows. extending of Let matrix us (5), take its viz. seven codewords 1 t ceived considerable attention [24]. Let us define a vector space the Fano plane, the elements of the matrix are A minimum distance codeis (or such a maximum that distance separable, MDS code) mension [ 0 (5 . We start with a cyclic Hamming code (7,4) in figure . We start with a cyclic Hamming code (7,4) in figure WO One can see this procedure also using the Hamming code in eq. ( The potential does not depend on out all lines which have 7 are underlined: The remaining quadruples form the superpotential Two of the positive masssymmetry. matrix eigenvalues coincide, the vacuum has an5.2 unbroken split. superpotential and leave only the ones which do not have 6 and 7. These with both 6 and directions. One is split where which have One can see thisout procedure all also lines which using have the Hamming code from eq. ( Figure 11 Figure 10 which have in a position 7 a 0. This leaves us with the superpotential in eq. ( JHEP01(2021)160 W and and 6 6 (6.24) (6.21) (6.22) (6.23) (6.19) (6.20) SO(2) T T ) 3 + + ). T symmetry. 3 5 , − , defining , T , T 6.22 re- . 2 423 and + } (31) (33) (34) (32) T of di- 2 + 6) x SO(2) o7by .And )( T 4] code. 1 m 2) = 0 . 3) = 0 V attached , . )canbe . , n th line of p , T 4 i ]code(or V n , (2) q (2456) (4 (2) (5 ( = − , T F + 5 T P q 6 ≡ , ≡ ≡ T WO n, k, d , n WO n q ] 6 . We take out all codewords 6 . x F 1errors.Itcanbe ) [ T 5 T (1345) 6 q + 4 , − ) + ( F = T 2 = . The underlined quadruples in d , , , n 5 . T 3 . − represents the  )  ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ T 12 5 i − + 2 , . . (2) i T (2) (7234) L 1 u, v = n , L 4] code introduced in Sect. 2.3 ( T T )( ( Michel Planat, Metod Saniga T , 3 max d 4 ∈ ,T d + )( C + T ), v T j 7 ∈ ̸= (1) P T u is a subspace of (2) − (2) u,v T (6123) T 2 T 6.12 . One defines the Hamming distance , − C q +1=   ≡ T 5 F ) = min k (1) 7 – 27 – T th point and ]anddetectupto C 10 if otherwise T T − ( j 1 3 log 2 2 log − ) + ( n ≡ ' (5712) d = 5 , min . Then a projective geometry 7 − ) + ( as the number of coordinates in which − ≤ ] code, the following bound holds 4 = q T d . The model has a (5,2) split ) in the polynomial field 7 T 7  g F d  n 1101000 0110100 0011010 0001101 1000110 0100011 1010001 T is the q ij T = − T n, k a F j = (1) and it is usually referred to as a [ (1) d ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 4 = P − 5 + (4671) T T T 3 . The model has a (4.3) split in 6 , 7 4 T ciency for error correcting; it is defined as T )( + y T max + 3over irreducible over T T 3 d ffi = )( = g T 5 (1) + + ≥ (1) 4 = 2 and T T T 5 3 − T (3567) T  d  x T T { 1 − +1 = = T 3 + + = +1] code). The binary Hamming [7 δ 1 1 T 4 2 4 log } 5 log T + ( T r, r T T − − er. The minimum distance of a code is an important concept which − ff + + There exists an intimate link between this code and the Fano plane, The link between good codes and projective geometry has recently = = pqrs di 1 1 3) = ( { n, n . T T We take the 7 quadruples in eq. ( y characterizes its e A linear code corrects up to [ [2] between a cyclic code isto an a polynomial ideal ( thus corrects up to 1 and detect up to 3 errors. It is the MDS [7 7.2 CyclicAs codes and shown in projective spaces Sect. 2.3, a linear code shown that for a linear [ The above matrix isobtained nothing as but follows: the if incidence matrix of the Fano plane which can be inferred ascyclic follows. extending of Let matrix us (5), take its viz. seven codewords 1 t ceived considerable attention [24]. Let us define a vector space the Fano plane, the elements of the matrix are A minimum distance codeis (or such a maximum that distance separable, MDS code) mension [ K K (4 . We start with a cyclic Hamming code (7,4) in figure ) are the ones we remove from the superpotential WO 6.21 The model has a Minkowski minimum with two flat directions. One is where The mass matrix massive eigenvalues are all different, there is no unbroken The model has athe Minkowski other minimum is with two flat directions. One is The remaining quadruples form the superpotential the lines which do not havedo the not same 3 exist, numbers, therefore orprocedure the we same which find 3 was the zero’s working in relevant in thequadruples split the code. which (4.3) previous have two But model 4 cases these not of andeq. following split 7. ( the models. This underlying So, we we show exclude in the figure Two of the positive masssymmetry. matrix eigenvalues coincide, the vacuum has an4.3 unbroken split. and remove the ones which have 4 and 7. Note that the pattern would require to exclude where Figure 12 which have 0’s in a position 4 and 7. This leaves us with the superpotentialthe in other eq. is ( JHEP01(2021)160 ) ]. 68 and 7.1 (7.2) (7.1) K . ) i T ( at the minimum . ) 2 i 0 . We also introduce WO T W S ( ) . 3 i F ¯ our cosmological model S T ]. This Volkov-Akulov − 0 S WO 2 ) S (2 W ] with the choice of ¯ 70 1 T F = , ), which we studied above. =1 7 i j T, 12 69 supergravity allows to get dS , T ( Q i ¯ ) for relevant models. 4.33 Λ = S 10 q T S ij G = 1 to be the one which originate from ≡ 4.45 supergravity derived from compact- M j 1 2 N oct T i ) or in ( = 1 = T W ij N and at vanishing 4.32 M S 1 2 , ), ( – 28 – ! = , which in ,W 2 4.7 ) S supergravity cosmological models which are com-  i i i W ] and specified to the case of octonions in this paper. T T T holonomy manifolds [ i + + T 2 12 = 1 , i i 4 G T T 10 ( N 

log log 7 =1 i X 7 =1 i X to make a gravitino mass at the minimum non-vanishing. This allows − 0 1 2 = − W brings a new supersymmetry breaking parameter, K S ) = ], where the properties of the inflaton potential are encoded in the choice of i : at present it is not known how to derive it from fundamental theory. ¯ T S , 71 S i , ). We specify the choice of ) 8 T ¯ T ( 1.1 ]. We have presented these eigenvalues in eqs. ( K do it using ain [ generalized version ofthe the Kähler potential geometric for approach the to nilpotent inflation superfield, developed superfield a parameter to describe the theory withof a the cosmological potential. constant Introduce a phenomenological inflationary potential along the flat direction. We will Change the Kähler potential frameflaton to shift symmetry, the only one slightly where broken by the the KählerIntroduce superpotential, potential as a has proposed an nilpotent in in- [ superfield minima with spontaneously broken supersymmetry [ T, 67 We start with M-theory on In the absence of the nilpotent superfield Thus we are looking for To construct a phenomenological cosmological model, describing the observations, we For this purpose we will do the following changes: ( • • • in ( ¯ S S For any of our octonionicusing superpotentials a an Kähler equivalent transform, model which can results be in obtained from ( W octonions, for example any one in eqs. ( els phenomenological isG the choice of the Kähler potentialbecomes for the model the with Minkowski nilpotent vacuaone in superfield, derived M-theory from with flat M-theory directions in equivalent to [ the patible with cosmological observations. The reason why we call these cosmological mod- need to make some changesflat of inflationary these potential, models, which andexit transfer replace from a inflation. Minkowski flat vacuum direction with into an de almost Sitter vacuum at the 7.1 Strategy So far we have studiedification Minkowski of vacua in M-theory 4d onnon-negative twisted mass 7-tori. eigenvalues, withble Such [ some Minkowski flat vacua are directions known (zero to mass have eigenvalues) only possi- 7 Cosmological models with B-mode detection targets JHEP01(2021)160 and (7.5) (7.6) (7.7) (7.8) (7.9) (7.3) (7.4) 0 (7.10) W )) 0 and W S + F Q ¯ ( S, ¯ Q + i ¯ ∂ . S . 2 # | − 2 ) + | ) ¯ Q T ¯ 0 i S W ∂ S | | W T, ) 2 ( . i ) ! 2 + 2 0 i 2 ¯ T ) ) ¯ infl , i T , ¯ 2 , W interacts with the inflaton super- Q i V ¯ + 3 ) 2 ) ( T i i ) T + i 2 2 S ¯ ( + Q ¯ i T ) T + ) i i ¯ ¯ S ( T i 2 S i ¯ T ∂ ( T S ) ¯ i ( ( + T i F T − i 2 G T ( ¯ − − 7 T 2 ) 2 =1 + 2 2 | i ( T i i X ) = 2 | + i i ( i ) ¯ i T i Q T i ) − T + i T ( T ), which are equivalent to the original models ! ¯ is an arbitrary holomorphic function of the 7 ¯ T | ( T T T ∂ 2 2 i 2 | 2 (( 2 ( 4 | ) ) − | ) 2 i i 7.2 i T i – 29 – i ) T, 2 − − − 2 i ¯ T T W ( T ) ¯ T | | T i 2 ( ¯ 0 ) are the same, due to Kähler invariance, and lead i T 4 (( 3 F = = = T + Q T i i i ¯ ¯ W (

i + i − + 4 7.2 i K K T 2 2 i K 0 − + ¯ | ( i i T , ( 0 W

K K W " 3 i | W ¯ 1 S 7 =1 K ) and ( log i − S X = 2 1 ¯ G ) + S + 7.1 i S − 2 = T | G ( 7 ). =1 V W i Q X | ¯ 1 S 7.1 = = − S G K W = V , this is simplified as i K ]. . We find ¯ T i − is a nilpotent superfield and e T 71 = , S i 8 Consider the Kähler and super-potential given by T In particular, taking For The scalar potential is given by where moduli 7.2 E-models We construct our 7-modulithe Kähler cosmological models potential wherefield where [ inflation the is nilpotent induced superfield by terms in to a Minkowski vacuumfermion with masses one are flat themodels direction same which and under in unbroken Kähler absence supersymmetry. transform.inflationary of potentials the In Also are the nilpotent what the ones superfield followsin in and we M-theory eq. new ( in will eq. parameters study ( the The potentials in models ( JHEP01(2021)160 (7.14) (7.13) (7.11) (7.12) ) has a 7.2 . . Therefore it 10 as long as these = 0 − ) a ¯ T 10 . Therefore the full ∼ T, . ( 2 ) | = 0 , F ¯ T Q and at 7 i t = Q ∂ T T, i | ( with supersymmetric flat di- 2 ∂ V in our general formula ) = F i Q 6 = ¯ T Q = along the flat directions. T + Q ) V = i ¯ T 5 T ( T, T . ( ) 7 , =1 i ¯ X = F T 4 oct -attractor models with inflation along the + = T, T α ( W ! V F 2 = 2 0 | = – 30 – 3 = Q W | T Q V = 1 + 2

T ) . Therefore our methods of constructing phenomenological ¯ = T ). The potential of the original M-theory model ( 1 T, T = 0 ( 7.2 F ≡ Q i , it is ∂ T = j T = Now we focus on a special choice of V = Q i . The full potential vanishes for any values of T a + i defined in eq. ( t oct = E-model. W T Now we will present several examples of inflationary models corresponding to different Moreover, an important feature of the From the point of view of inflation, the main role of the M-theory (UV dynamics) is to This is a remarkable simplification. Generically, one could expect the parameters To find inflationary potential, we should evaluate it along the supersymmetric = 7 α where flat direction at where e.g., in type IIB or type IIA3 string theory. Note, however, that theapply results to obtained a in broad this classrections, such of subsection that theories are with quite anyinflationary general. models superpotential can They be may usedbut not in only a in more the general octonionic class models of in models the with M-theory supersymmetric context, flat directions, which may exist, stable with respect topotentials the are choice not of singular. theclass That particular of is potential such why one models. can talk about specific predictionschoices of of a potentials broad and flat directions in the octonionic models discussed in this paper. flat directions. Oncethe it full is M-theory done, potential, the butby inflaton the is dynamics phenomenological determined inflaton (IR) by potential the is M-theory completely related independent kinetic of terms,flat and directions in M-theory is that the observational predictions of these models are very is very important thatflat in directions the is approach almost developed completelyM-theory above, sequestered potential. the from the inflaton UV potential dynamics along responsible the for the define geometry with the proper kinetic terms, and form a potential with supersymmetric of the M-theorypotential to at be the last large, stages in of inflation Planck should mass be units, tiny, whereas the height of the inflaton Minkowski flat directions.expression But for along the these inflaton potential directions is given by yields JHEP01(2021)160 , in , as 1 ) . For . (7.16) (7.17) (7.18) (7.19) (7.15) T A + = 0 0 . The full T θ W , 2 . 7 log( − = 0 = : m θ K . and . φ , ) t ) = 0 ( T . infl oct where we have models with 1 flat − V ! W θ i 0 )(1 2 7 ∂ WO T > which is a minimum of our potential in r = 2 i 0 − i ¯ Q ) = Λ + T W i (1 3 1 + 2 = t, t has a canonical normalization in the vicinity

( − . m j , and ends when the field reaches the vicinity of the φ θ – 31 – F S T 2 7 φ 13 F , ∂ = p = i − i ¯ T e = 0 T Λ = ), or the octonionic superpotentials in appendix = ) = Λ + j = ¯ T T with the following choice of the inflationary potential oct = T 4.33 i 0 T, W T ( of the E-model with one flat direction for ). At | W ), ( F = V infl Q 7.11 V 4.32 . It is convenient to use the variables ), ( 6.1 in eq. ( 4.7 . ) ¯ T = 0 T, φ ( . The potential F ) , which corresponds to the minimum of the potential with respect to is a canonical inflaton field, and and a gravitino mass via φ S = 0 7.11 We take any of the octonionic superpotentials Here we propose an M-theory cosmological model, supplemented by a nilpotent super- In our general cosmological models the flat direction is now lifted due to dependence T θ Here of shape of the potential is shown in figure direction, in eqs. ( these cases with one flatexplained direction in the section effective Kähler potential is where the cosmological constant at the exit from inflation is and we choose it to be positive. field eq. ( Thus, the potential becomes aof function the of inflationary the potential inflaton as only well and as we will cosmological have constant it at in the the exit form Planck mass units. Inflationminimum begins at at large on Figure 13 JHEP01(2021)160 Λ ] in and 8 9 (7.23) (7.24) (7.20) (7.21) (7.22) – 6 , which [ 2 ) which, ) split the H 7.2  6.22 in the small 2 θ ¯ approaches the S, m ), ( + φ = 0 S 2 6.18 θ m + . ¯ S at . Thus ), ( 3 S = 0 φ have positive masses, / ) at large 2 2 T 6.14 , T . Choosing the superpotential m 2 for B-mode detection 0 − m,n Re = 0 . . W = 7 2 3 = 5 )(1 θ n / ) has a very simple relation to M-  , m WO = 7 T + 4 φ + ) 2 0 i 2 7 2 − α total m T W 7.22 = 0 3 H p . We start there with the superpotential V (1 6 0 − (2 2 ) to the M-theory model in ( e 1 ≈ 6.2 m =1 i 7 − 2 0 + 7.22 1 orthogonal to Q . W 3 2 S  i – 32 – ), which is directly derived from M-theory via q ,W . 2 Z F T / + 4 , and inflationary parameter m 7.1 7 is given by + 0 2 Re models where = 0 = 0 ) = T in the vicinity of i = φ ! W m θ n S Here we study M-theory models with Minkowski vacua 2 T θ ) ( + i infl i = 2 ¯ V ¯ m T and T oct m,n i 2 θ S + T W m ,F . i F 4 0 T ( W = 0

the top benchmark target we have models which originate from CSC octonions and the cyclic S ) + E-models. i log 1 T 1 2 6.2 ( , 2 − and combinations of oct , i 7 3 =1 T i X W , 4 = = , Im 5 , K W In this section We stress here that the model presented in ( We have checked that the 7-moduli model described above works well. All extra 13 The mass of the axion field The potential of the canonically normalized inflaton field = 6 ], one can add terms with bisectional curvatures, which enforce these masses to be positive. We have found for some parameters that these masses are positive. In general, following the analysis 8 9 α corresponding models the name in [ with two flat directions, asassociated described in with section the cyclic Hamming (7,4) code in figure Hamming (7,4) code. The7 truncated moduli superpotentials into in groups eqs. ( of 6 and 1, 5 and 2, and 4 and 3. It is convenient now to give the theory and is intested by agreement the with future the B-mode current searches. cosmological3 data. Moreover, they will be metry breaking parameters This brings ourin phenomenological turn, model is ( compactification equivalent on to a the twisted 7-tori, one in ( To recover the original M-theory modelneed from to this remove phenomenological cosmological the model nilpotent we superfield contribution, as well as to remove the supersym- fields like roll to their minima, whereuseful they to vanish present and this decouple phenomenological from model the here inflationary explicitly, evolution. namely It is asymptotic value where the Hubble constant atstabilizes large this field at its minimum limit is This is aE-models. potential of JHEP01(2021)160 , in (2) = 0 T F . We , such (7.29) (7.30) (7.25) (7.26) (7.27) (7.28) i m,n φ ) and in ): 1 and and 0 + 7 , and have WO 0 6.18 (1) 7.26 . T ) → ), ( ) (2) . i ) ( T T 6.14 , . Our choice of the (2) (2) T (2) are some real numbers:  T i − ,T c 2 (2) ≡ (1) c 2 T | mentioned in ( n into a redefinition (shift) of T 0 )( i , ), however, the function + . T i W c (2) 2 | (1) . T = (2) 7.3 , where   T i i T ( − c φ  i ··· 2 infl 2 α = c 3 + 4 3 V ( = ) log 2 2 i ( q + n m T − +1 infl S e − V m , F  − ) + T = , 1 (1) (1) )   T – 33 – 2 + 5 (1) T 2 i = Λ + T (2) + − m T F 1 ≡ , c (1) i , . The fields may start their evolution anywhere at the X T m )( (2)  T = 14 (1) ,T T = log 1 + 6 (1) infl 2 − 0 T V m ··· , 1 is now based on one of the superpotentials ( W − c ( (1) = 2 1 = ]. T m,n 1 ( . m , see figure i 8 T K , c F . The corresponding potential becomes = 7 i WO + at = = 0 φ can be represented in terms of the canonically normalized fields i i ) 2 0 ) infl 2 α i i ( φ 3 ( V W T T = 0 3 q − . One can take any of the three combinations of is e = i m,n is the phenomenological potential describing two inflaton fields, φ ¯ S = F ). These superpotentials at the vacuum have the following properties: S ) i infl WO G ( i V depends on 2 flat directions. T ∂ 6.22 As a result, the inflationary universe may become divided into many different exponen- The resulting potential has two inflationary valleys, which merge at the minimum of The fields ¯ S S blue inflationary plateau, but thenclassical fall rolling to and one quantum of diffusion.inflation the studied This green in is valleys [ a due specific to version a of combination the of processtially large of parts, cascade with inflationary perturbations corresponding to one of the two different Here without loss ofthe generality fields we absorbed the6, constants 2 and 5, or 3 and 4. the potential at that Importantly, all qualitative resultsnomenological will potentials, remain as the longa same as minimum for they at are a not broad singular choice in of the such limit phe- Here corresponding to the two flatpotential directions. for each As of before, the we fields, will with a consider minimum the at simplest quadratic The superpotential eq. ( and function From this perspective, the casenow discussed use in the the same previous formG section of may the be Kähler called potential as in eq. ( with 3 cases leads to the effective Kähler potential for moduli directions JHEP01(2021)160 = 6 and 1 (7.35) (7.31) (7.32) (7.33) (7.34) to the α 3 . All of i = 5 T 1 = 3 , α 3 2 , with ! α 2 2 3 φ ) i . ¯ Z 2 ) 1 . and and i , − 1 2 1 Z , i , φ (1 2 3 Z 2 = 4 , , ) 3 i 4 1 − , , Z α 4 5 3 , (1 , − 5 6 , , . (1 

) = 6 = 7 i Z α log ( 3 i . 7 i i T =1 i X  Z Z , n 1 2 , 2 oct − corresponds to the split of the 7 moduli into − 2  1 + 1  φ 14 = φ 2 n – 34 – = ≡ W α 2 ! 3 i p ) 2 i ) T − ) p i e i Z − ( T 7.2 T e − i oct + 1 T − i 4 ]  1 W T 2  ( 68 of the E-model with two flat directions, 2 m

m infl . V log = 1 7 =1 = Λ + i X 2 = Λ + 1 2 α V 3 − V as shown in [ i and ) = Z i . Z case we define using eq. ( . The potential , = 6 i . Finally, for the combination 3 and 4 we would get = 1 1 Z ( 2 α = 7 3 α K = 2 If we now add the model 0+7, we get all the benchmarks covered, using the octonion If we would consider models with flat directions 2 and 5, we would get 3 α 3 2 . The particular example shown in figure i α and disk variable In 7.3 T-models Here we will use the Cayley transform to switch from the half plane variables superpotentials in M-theory with 3 these cases together give us a choice of inflaton potentials Figure 14 and α groups of 6 and 1 discussed above, and the universe divided into exponentially large parts JHEP01(2021)160 , , . ) 2 4 j , φ = 1 Z cor- 5 ): 1 j , − 1 ) (7.41) (7.40) (7.36) (7.37) (7.38) (7.39) Z i 6 α ( , − )(1 and 3 i i 7.26 Z 1 Z Z φ = 7 − . (1 α with 2 3 2 ¯  S, φ = 2 2 j 12 + φ . T √ S and 2 − φ 1 + i φ 2 ¯ T S corresponding to the two 1 tanh α S 6 ) i − ( s infl . 2 Z . 1 since 2 V 2 0 φ √ j -attractor T-models. 16 , (2) + W for a particular potential with α Z Z S tanh 14 φ = F 2 2 15 (2) tanh i √ Z. Z  m + 2 Z 2 2 Z 2 2 ! + µ m 2 , with 1 m ) + tanh + i φ (2) ¯ 2 ). In terms of canonical variables 1 Z 2  . We define the T-models as follows Z ) ) (1) ) = 1 α m i 7 − 2 6 12 Z Z Z – 35 – φ Z 7.29 ]. i √ (1 s and (1) Z, Z 2 , is shown in figure can be obtained by analogy with E-models, using 8 + 2 2 ( , ) Z i 7 2 1 m 1 − (1) = Λ + , infl Z ··· m Z 2 V (1 tanh , . = 3 − + = 2 1 3 0 this covers all T-model benchmark targets: total 1 + tanh , in terms of the canonical variables µ (1 V m 4 W Z 2 1 infl , (

(6) 5 φ V 1 7 √ = 7 , and Z ) + 2 i α log becomes the minimum at ≡ Λ 3 Z j = Λ + = 6 ( 7 Z =1 and T i X tanh α oct 2 1 .  = 3 total (1) 2 i − W V Z T = 6 m = = 2 α K W 3 : ’s with 2 flat directions = Λ + W = 6 and infl 2 V α . 3 1 For example, consider a model with the following phenomenological potential along Just as for the E-models, weOne illustrate can in also consider figure more complicated models, where the 2 different fields The cases with = 1 , 1 2 , α and This potential, for small phenomenological interaction terms are muchthe smaller original than M-theory the potential, typical sothe terms they M-theory. do appearing However, not these in affect termsflat the may directions force structure evolve the of simultaneously two the fields [ flat directions of the flat directions 3 3 responding to the 2 flat directions can interact with each other. We assume that these this yields the potential Including the case above Here one can takeand any 6, of 2 the and 5, two or combinations 3 of and flat 4, directions as mentioned we did in in ( ( which is the top benchmark for B-mode detection indifferent the Here we take the following value of the inflationary potential The total potential for the canonically normalized inflaton now is The flat direction is at The minimum at JHEP01(2021)160 Z = 1 and = 1 i α ¯ 3 Z = 1 the last , or with 1 = µ . At large α , just as in i , the earlier 3 = 5 Z 16 2 M = 6 α 2 3 . The motion in . α , with 3 2 ]. φ = 7 = 7 and 8 . However, at smaller 2 , α 2 and 7 α 3 φ and after the merger is given . = 2 2 1 1 α φ + 3 φ = 1 α 1 1 3 α α and 3 1 φ = 3 α 3 , or to the axis 1 φ ) with two merging flat directions, with 7.41 – 36 – corresponding to a single flat direction with , observational predictions for these model will correspond , depending on initial conditions, but for large µ of the T-model with flat directions of the T-model ( = 7 = 6 infl infl α V α 3 V 3 , in terms of the canonical variables m . In all of these cases the effective value of or to = 3 = 4 the potential has two flat directions, with µ 2 i α = 1 φ 3 and , these two valleys merge into one, with i α . The potential . The potential 3 φ . In our case, they correspond to the dark purple valleys in figure and = 6 . . i 2 15 α A similar result applies also for merger of directions with This means that for small At large = 3 3 = 6 1 2 , these valleys are parallel either to the axis i α α stages of inflation will be described by the single3 field model with by its maximal number for all φ values of this direction describes the laterthis stages merger of takes inflation. place, The see greater the the description value of of aeither similar to regime in [ Figure 16 and figure Figure 15 3 JHEP01(2021)160 7.3 (8.1) . It is , a ratio r 18 ], in partic- 9 ] the prediction ]. We reproduce 9 44 s AAAB6nicdVBNS8NAEJ34WetX1aOXxSJ4CkkabHsrePFY0X5AG8pmu2mXbjZhdyOU0p/gxYMiXv1F3vw3btoKKvpg4PHeDDPzwpQzpR3nw1pb39jc2i7sFHf39g8OS0fHbZVkktAWSXgiuyFWlDNBW5ppTruppDgOOe2Ek6vc79xTqVgi7vQ0pUGMR4JFjGBtpFsxUINS2bF91/PrVZSTWrXu56RyWfM85NrOAmVYoTkovfeHCcliKjThWKme66Q6mGGpGeF0XuxniqaYTPCI9gwVOKYqmC1OnaNzowxRlEhTQqOF+n1ihmOlpnFoOmOsx+q3l4t/eb1MR7VgxkSaaSrIclGUcaQTlP+NhkxSovnUEEwkM7ciMsYSE23SKZoQvj5F/5O2Z7sV27vxy43OKo4CnMIZXIALVWjANTShBQRG8ABP8Gxx69F6sV6XrWvWauYEfsB6+wTjF45G n is the automorphism 2 G -attractor inflationary E- α are on the right panel. The 7.2 . . The predictions are shown for the 2 e 1 4 , ]. These models lead to 7 discrete N 2 ] will map polarized fluctuations in supersymmetric minima. , tilt of the spectrum, and , α s plane in figure 9 in [ 44 3 53 3 , n , which enters in the Kähler potential r – 4 - α = 1 , ≈ s 3 47 5 ]. Here we have found that the octonions n r , N 6 22 , . ]. The simplest T-models derived here in section = 7 e – 37 – 9 60 2 -attractor inflationary models [ α N 3 α < ], which we reproduce here in our figure − e 1 72 [ ≈ < N s 50 n . and as plotted in [ 17 . LiteBIRD 4 ) ] for inflationary observables holonomy and enforcing -attractor inflationary models [ 8 ¯ T 2 – α 6 has a simple relation to octonions. The fact that G + depends only on the parameter T ] allows 7 discrete values: 8 r = 7 is the number of e-foldings of inflation. There is no significant dependence – -attractor inflationary models benchmarks originating from M-theory and octonions, 6 log( α 3 55 α α . 3 ≈ − e = N In this paper we have shown that the upper benchmark for These benchmark points were plotted in the K model with group of the octonions wasgive known us since a 1914 powerful [ on tool manifolds for with constructing superpotentials for compactification of M-theory the search for the signatureis of designed gravitational waves to from discoversented inflation. or in In disfavor particular, figure LiteBIRD the A.2important best here of that motivated in inflationfor single models observable modulus which wereas pre- Here on the properties ofcosmic a microwave background potential (CMB) due missions to [ an attractor behavior of the theory. The future benchmark points [ of tensor to scalar fluctuations during inflation. this figure here in figure 8 Observational consequences: inflationThe and CMB dark targets for energy theular future with B-mode regard detectors were to discussed recently in [ Figure 17 as derived here in section are shown on the7-disk left, model [ the simplestnumber E-models, of derived e-foldings in in the section range JHEP01(2021)160 ) r - s 7 n (8.2) , and ) 2 ¯ T + 2 ). Analogous T

holonomy has a 4.8 . If gravitational

57 57 57 2 < < < 1.00

* * * 2 log(

18 G = 57 = 57 = 57 = 57 = 57 = 50 * * * * * * LiteBIRD: an all-sky N N N N N N were also studied in − 47< N 47< N 47< N how the models with

) 1 1 , 6 ¯ 0.995 , and the cyclic Hamming T

2 P P P /2

P BK15/Planck é disks , (φ/M) 2 2 3 p

2M 1M M 4M . 3 R + φ LiteBIRD = Higgs = = = , tanh , one will be able to associate 2 1 M M M

2 0 M 4 Poincar V − 0.990 T , − 5

10 , 10 ≈ ≈ 5 log( 0.985 2 . In such case, the prediction (when = 6 e r − ) 4 N ¯ T α = 7 3 0.980 + s ≈ n K T , ) with a forecast of Litebird constraints in the 2 symmetry of octonions. 0.975 ¯ T – 38 – 2 7 log( , r G + − and the top purple line in figure . It is explained in section . All these relations, starting with associative octo- e 2 ) 2 0.970 T N 2 5 = ¯ 17 T − K log( + 1 0.965 2 − ≈ T

P )

P s n (1) 0.960 ¯

M=10 M M=2 M T 3 log( + − ) 0.955 (1) 1 T can be obtained starting with these Kähler potentials. These models ¯ T -1 -2 -3 -4 1

+

, 10 10 10 10 holonomy has a Minkowski vacuum with 1 flat direction and the resulting r 2 1 2 6 log( , T is 3 ], the corresponding Fano plane for example in figure − G , . This is a figure A.2 from the Astro2020 APC White Paper 4 53 = , 21 symmetry broken in a specific way, but they still originate from octonionic super- , ]. The 7 purple lines in the figure, were derived in this paper using M-theory compactified 5 4 log( 2 ≈ , holonomy manifolds, octonions, Fano plane and error correcting codes. 72 K 20 − e G 2 The benchmarks below the top one, with This is the top red line in figure In all these cases we found that the 7-moduli model in M-theory compactified on a In particular, we have found that it is easy to use Cartan-Schouten-Coxeter conven- N = 6 = G α tials: K 3 have potentials. this fact with M-theory cosmology and this paper. Thecorresponding corresponding to certain superpotentialsthat codewords were the in obtained 7-moduli the by modelMinkowski cyclic dropping vacuum in Hamming some with M-theory (7,4) terms two compactified code. flat on directions a We and 7-tori have with with found one of the following Kähler poten- waves from inflation are detected at the level close to 480 octonion conventions. 7-tori with single modulus Kähler potential the flat direction is upliftedfor and the proper cosmological model is constructed in section tions [ (7,4) error correcting codenion in triads, figures codewords, quadruplessuperpotentials and leading superpotential to are shown the in same eq. cosmological ( models can be obtained for any of the Figure 18 cosmic microwave background probe ofplane inflation [ on JHEP01(2021)160 . 1 1 ], ) in , ¯ − T 2 73 = 6 was = 1 , P l (8.3) 3 + is the α = , 3 N M 4 T w , = 7 5 = 1 , can have a α log( 6 3 α , 7 α 3 3 = 7 − . In this case the ) ]. The case α α < ¯ = T ] are in good agree- 3 3 76 + 44 K and the bottom purple T 17 ]. The case log( . 78 3 − , − = 77 10 K ≈ 2 which also includes the discrete values 4 N -attractor models [ α α . so that these models agree with the obser- ≈ is a feature of the Starobinsky model [ ], V WO 46 = 3 – 39 – α ], where a dark energy model with , r 3 81 2 N − -attractor models for inflation, for dark energy and for 1 of this paper we have built cosmological models with α ], and the conformal inflation model [ ≈ 7 75 s , . These models will be tested by future dark energy probes n 9 automorphism symmetry, and (7,4) Hamming error correcting . 74 0 2 ] a proposal was made how to construct early dark energy (EDE) − G is 82 supergravity potential = motivated by maximal supersymmetry. all have a natural origin in M-theory compactified on a manifold with ] with a single complex scalar and , which are now directly related to the 7-moduli M-theory model and 60 -attractors. These models appear to ease the Hubble tension raised by 1 1 1 w = 1 , , , α 80 ≈ , 2 2 2 . This model is known to represent the maximal superconformal theory in , , , N N 3 3 3 79 , , , 18 4 4 4 , , , 5 5 5 , , , -attractor models can be used in the context of dark energy models, especially 6 6 6 α , , , ] that their EDE models allow a range for In the fundamental part of the model with a flat direction, the parameters are It is interesting that in all Quite recently in [ According to the latest Planck results [ We also stress here that some of these benchmark points with To summarize, in section determined from cosmic microwave background (CMB) data. It has been noticed manifold with maximal supersymmetry spontaneously broken to the minimal holonomy. Therefore they are associated with octonions, whose automorphism group is 82 = 7 = 7 = 7 . These benchmarks are targets for future detection of primordial gravitational waves. 0 2 2 2 α α α supergravity with octonion superpotentials 4d and the scale of thenomenological compactified manifold. We addedvational to data. the fundamental The models phenomenological a plateau phe- potentials slightly deforming a flat direction in [ 3 early dark energy, kineticWe terms discussed are in always thisG the paper same, how defined this by feature is derived from M-theory compactified on including satellite missions like Euclid. models based on the discrepancy between low-redshifts distance-ladder measurements andH a Hubble constant logical interesting in case thatequation future of observational state. dataconstructed will See which show for predicts the example aequation deviation deviation [ of from from state cosmological constant with the asymptotic G G ment with data available now, in particularmotivated for by discrete set maximal of supersymmetry. values for More recently we have learned that the cosmo- Meanwhile, in3 this paper we found that the benchmark cosmological models with the Higgs inflation model [ is the feature oflowest benchmark the point string of these theory 7line fibre targets, in the inflation figure black model line in [ 4 figure dimensions [ prediction for 3 7 octonions with their codes. different origin. For example, the case JHEP01(2021)160 , 4 P l (9.2) (9.1) M 120 − . 10 supergravity ) ≈ +1 i = 1 T DE V . − 2 , N | ). Therefore we have 4 P l +2 is given by oct i 9.1 M T ¯ T W ( i i 110 ∂ = | T − 2 ) T ). Along the supersymmetric 7 ). A significant formal part of i =1 10 i X ¯ T ≈ = 7.11 4.32 + . Therefore the full expression for j i T T ) by a change of variables consistent i EDE ( = 0 V T 7 9.1 , =1 ij i X holonomy manifold with Kähler potential oct 4 P l M 2 + 1 2 W M G i ! ∂ 10 2 = | − – 40 – 2 = 0 oct 10 W WO oct ≈ |W W , infl  V 1 + i

T defining the superpotential are the same for all conventions, ) ¯ + T ij -attractors. i ) as well as the ones for all other 480 octonion conventions have α M T, T (  . When we use error correcting codes to drop some terms from the 9.1 F 7 log = 7 =1 V i X − = 7mod K The new feature of our cosmological models is the fact that the total potential with The model in eq. ( An important fact here is the existence of 480 different conventions for octonion multi- The derivation in aMinkowski more flat general directions case we is have given in eq. ( superpotential, we find Minkowskithe vacua phenomenological with potential is two added flat leadset directions. to of observable B-mode predictions, All associated targets these with for models the after octonionic superpotentials along the inflaton direction with This universality of thevarious physical octonion conventions, results, is despite one apparently of the different nontrivial superpotentials resultsa for of Minkowski our vacuum investigation. withconstructed one in flat section direction. It is the basis for the cosmological models studied how these differenteigenvalues of choices the affect matrix thereflecting physical simply results. the factwith We that the have they symmetries found differ of thatresulting from the the from supergravity ( our Lagrangian. octonion superpotentials Accordingly, the is vacuum the structure same for all 480 octonion conventions. the paper explains the derivationrelation of the between superpotential Fano for planes, any error of correcting the codes octonion and conventions, superpotentials. plication. We have found that theconvention octonion lead conventions other to than Cartan-Shouten-Coxeter superpotentials differentcarefully from examined the these ones different in conventions, eq. described ( the relations between them and The above superpotential is oneCartan-Shouten-Coxeter of convention the given two linearly in independent equation 14-term ( superpotentials in 9 Summary of theWe main proposed results an octonionicassociated superpotential with for M-theory, the compactified on effectiveand a 7 superpotential moduli of the form: for inflation, for the EDElogical and constant, for respectively. the They current show acceleration theat deviation caused from some by the dark very core energy small fundamental M-theory or scales.for cosmo- all It these would models be to very important see to if get octonions the might future be cosmological relevant to data cosmology. have the following energy scale: JHEP01(2021)160 . 1 , 2 , (9.3) (9.4) 3 , 4 , . For the 5 , 6 in the range , = (1245736) . For Cartan- 2 e CG = 7 P . α α/N 3 holonomy manifold. ) involves the cyclic we will give super- . This clarifies the 4 = (1234567) 2 5 C 5.3  G = 12 P = 4 (2) = (1243657) r T , n . For the Gunaydin-Gursey GG + 7 . P ) Z (2) = 3 ¯ is T is simply T  ) P T, l m P ( T log infl − n V = 5; k − T  , n )( j (1) T , they are mostly determined by the M-theory T – 41 – ) = 2 ) = Λ + − ] the operator ¯ ¯ T + T i m T 21 T, T, , ( (1) ( ( T 20 -attractors are very stable with respect to the choice of F  infl X α = 6; V = log V , n m − that generates the cyclic group = 1 = P m K = 7; , which should be accessible to future cosmological observations. 3 − , n 10 & ] labelling of the real octonions the operator = 0 r 23 m & The general formula for the superpotential given in equation ( Observational predictions of 2 − permutation operator (GG) [ Schouten-Coxeter (CSC) labelling [ Cayley-Graves labelling of octonions the cyclic permutation operator is such that they can be written as a sum of seven terms ofwith the each form term correspondingvacuum to with a one line modulus. inthe the Hamming This code Fano Fano plane, (7,4) plane as and in was that turn done lead will for to the be a CSC related Minkowski labelling to in the section codewords in A Octonionic superpotentials inIn common this conventions appendix usingpotentials the in general various construction octonion outlined conventions in involving the section structure constants of octonions and by the US National Sciencedation Foundation Grant Origins PHY-1720397, of and by theRK the Universe Simons and Foun- program AL (Modern arealso Inflationary also supported Cosmology supported by JSPS collaboration). by KAKENHI,YY the Grant-in-Aid would Simons for like JSPS Fellowship to in Fellows thank JP19J00494. Theoretical the MG Physics. hospitality and of YY SITP where is this work was initiated. Acknowledgments We are grateful tochbacher, R. N. Flauger, Bobev, M. Freedman,and A. K. T. Pilch, Braun, Wrase E. G. for Plauschinn, stimulating A. Dall’Agata, discussions. Van S. Proeyen, MG, N. Ferrara, RK, Warner F. AL Finelli, and T. YY are Fis- supported by SITP with relation between B-mode targetsInflation and along M-theory various compactified flat on directionsand a with these therefore kinetic to terms 7 leads10 to possible values of the tensor to scalar ratio the phenomenological potential related kinetic terms for the inflaton fields corresponding to these flat directions, This is a remarkable simplification. Thedetailed inflaton structure dynamics of is the completely full independent on M-theorythat the potential. is The important only for informationMinkowski cosmological about flat this applications directions. potential is the identification of its supersymmetric the inflaton potential is given by JHEP01(2021)160 , . we ) ) ) (A.1) (A.4) (A.5) (A.2) (A.3) (A.6) (A.7) 5 ) (142) +6 +7 +6 ) ) 6 and its n n n 2 1 T T T T T T CSC − − − − − − W 4 (124) 1 7 T +3 +5 +5 T T n n n )( )( )( 2 T T T 3 6 T )( )( )( T T − +4 +7 +4 − − 5 n n n ), namely 6 4 . T T T T T T 124 − − − , , . ) ) ) ) ) ) 4 6 3 5 7 4 ) + ( +2 +3 +2 ) + ( ) + ( 6 T T T T T T n n n 1 7 T T T T T T − − − − − − ( ( ( (412) = 0 − 3 5 2 4 5 7 − − 2 +7 +6 +6 T T T T T T 7 6 T ,n ,n ,n CSC )( )( )( )( )( )( T T 5 7 1 3 3 1 +3 +5 +5 W ) ) ) )( )( ))( T T T T T T . 7 7 3 2 5 3 ,n ,n ,n T T T 5 T T T − − − − − − +1 +1 +1 1 3 6 1 6 6 n n n − − − − − − f f f T T T T T T 3 6 6 5 3 1 (241) + T T T T T T +4 +7 +4 – 42 – ,n ,n ,n )( )( )( . Let us choose the associative triad ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( 6 5 7 3 5 2 4 7 2 CSC +2 +3 +2 T T T ) + ( ) + ( ) + ( T T T T T T ,n ,n ,n 7 6 5 W − − − T T T − − − − − − +1 +1 +1 5 5 3 n n n 2 4 1 3 3 7 − − − T T T f f f T T T T T T ( ( ( 5 3 6 +7 +4 +6 )( )( )( )( )( )( T T T is performed. This operation leads to a specific change of 4 6 7 2 4 5 124 241 412 ,n ,n ,n (124) + = (1234567) )( )( )( f f f T T T T T T 4 1 1 r r r +3 +2 +5 ) ) ) T T T − − − − − − ,n ,n ,n CSC 7 2 5 7 2 1 CSC − − − (421) +1 +1 +1 W CSC CSC CSC T T T T T T P 2 4 4 n n n P P P f f f T T T ( ( ( CSC + ( + ( + ( + ( + ( + ( 6 6 6 6 6 6 =0 =0 =0 =0 =0 =0 W X X X X X X n r n r n r = = = and (241) = ( (412) = ( (124) = ( (241) = (124) = (412) = CSC CSC CSC (214) W W W CSC CSC CSC W W W CSC Explicitly we have Let us now applypermutation this operator general formalism topermutations the to octonions label in CSC theget conventions superpotentials. the with superpotential cyclic Applying the rules explained in section A.1 Cartan-Schoutens-Coxeter convention So far we havecan preserved be easily the obtained set whenW of the odd associative permutations triads. ofoctonion the conventions, Some triad as ( additional we explained superpotentials in section These three superpotentials have one modulus and their spectra are identical and satisfy JHEP01(2021)160 , and ) (A.8) (A.9) 1 ) (4723) (A.11) (A.12) (A.10) (A.13) ) ) T 7 2 ijk − ) T T ( 6 5 T T − − W , 6 7 )( ) − (165) = (624) 3 T T 4 ] with structure T )( )( 4 kij T GG 4 6 ( − 23 )( 7 T T 1 ,P W defined by T T − − ) 5 1 . − )+( ) T T } 3 kij 7 4 ( ( ) ( ) 3 T T T 6 by any of the triads in the ( T W 367 572 T − − f f 6 1 246 − − , , . . Note that given an associative f T T 4 2 354 516 (165) = (312) f f (165) )( T )( T 231 19 4 5 V )( f )( 3 312 543 GG T T ) 1 GG (142) (214) (421) 7 . f f T − − 257 T W ,P 5 2 f kij − . ( − CSC CSC CSC T T ) + ) + ) 3 5 1 4 4 W T W W W ) + T T T T T 3 . − − − ( )+( )+( } T 5 2 V − − − )+( 165 T T 3 5 1 3 – 43 – − f one finds that the matrices = (1243657) T T T T − − in terms of a matrix 2 { ) = 3 7 ) n are equal to 1, we have )( )( )( ) − T 7 ) 2 3 5 in GG convention its associated quadruple is T T 2 GG ) (165) = (471) (124) = (241) = (412) = kij )( T T T ( P T )( )( 7 kij ,T GG 2 6 ( 2 6 GG T ijk )( − − − P W T T ( 7 CSC CSC CSC ( 7 2 6 W − ,T (165) T ,P − − (165) = (736) 5 T T T W W W 6 =0 4 6 1 ( ( ( X − n T T T 4 6 ,T GG ( 4 651 714 435 T f f f +( +( . ,P 123 ,T f 3 673 725 462 19 f f f (165) = 147 ,T ). The cyclic permutation operator in this case is f 2 624 736 471 GG in cyclic order f f f (165) = ( (165) = (257) 165 B.3 ,T 1 W f + + + GG GG ijk T under the action of the Abelian group generated by the cyclic permutation f ≡ = : W ,P 1 (165) = (543) ≡ = ( GG 1 5 P GG T (165) all have the same eigenvalues, namely one zero eigenvalue and three doubly degen- P (165) V ) W GG { Given the associative triad The superpotentials from odd permutations are however not independent of the ones W GG jki ( 1 = T Now since all and we obtain the superpotential operator We show this in figure It defines the order oftriad, vertices on say the (165), heptagon one inorbit figure can of label the superpotential erate non-zero eigenvalues. A.2 Gunaydin-Gursey convention Consider the octonion multiplicationconstants in in Günaydin-Gürsey eq. ( convention [ If one writes the superpotential where W given above and are related as follows: JHEP01(2021)160 ) 3 (A.17) (A.15) (A.16) (A.14) T − 7 T )( 4 and reordering T ) − GG . 2 ijk P } ( T → . )+( ) (714) . ) 4 5 ) , T T 7              kij 1 1 T − − ( is symmetric with zeros along 5 2 − − − 1 7 (572) T T 5 ) we get another superpotential, 1 1 1 0 , )( )( T 5 − − 2 6 V 5.3 )( T T 1 1 0 1 0 1 1 1) MGG T − − − − (651) 1 1 3 , − 1 1 0 0 T T 3 ) such that − − 2 MGG T ( 1 0 1 1 1 1 0 0 0 )+( )+( T (367) 6 − − 1 3 A.11 , )+( V T T 7 1 0 0 1 1 1 0 – 44 – as W GG T − − − − 1 7 4 . The matrix 1 = − 1 01 0 1 0 1 ) (435) 2 T T 4 0 00 1 0 1 1 10 0 7 ) in the formula ( , − − T )( )( we obtain another superpotential. The following set of 2 6 ,T              )( W GG 6 T T 5 W GG 3 A.16 (246) T − − under the Abelian group generated by ,T . The first triad 123 is shown by red arrows. The second will be 4 , 6 1 1 = 5 (123) − T T 6 ,T T 4 +( +( (123) (123) MGG { ,T 3 , which we label as 1 ,T 2 = = (1243657) has one zero eigenvalue corresponding to a Minkowski ground state and 2 (123) = ( T 1 ,T GG 1 GG P W GG T W MGG ≡ = ( . The clockwise oriented heptagon for Gunaydin-Gursey octonions. The order is in 2 T V By permuting indices of the triads in ( W GG different from them so that the firsttriads triad are is in the orbit of Summing over the set of triads ( The matrix three non-zero degenerate eigenvalues given byin the other roots cases of of the octonion cubic superpotentials. equation as discussed where the diagonal and hasMinkowski vacua. a sum of terms in each horizontal line vanishing, as necessary for agreement with 257 as we can see from the figure, etc. . One can write the superpotential Figure 19 JHEP01(2021)160 ) ) ) 5 6 1 and ) T T T 3 ) (A.21) (A.22) (A.23) (A.24) (A.25) (A.26) (A.27) (A.18) (A.19) (A.20) − − − T coincide 4 5 4 − jik T T T ( 2 4 )( )( )( T 1 2 2 )( → T T T 2 . . ) W GG − − − T 2 3 5 6 − . . T T T kij ) ) ) 7 ( } } } } 4 2 6 ) and reordering the 6 = 0 T are identical to those T T T , )+( )+( )+( ) 3 W GG − − − 3 4 5 . )+( ) 7 4 7 A.11 − (174) (417) (725) (741) T T T 1 4 ikj T T T ) , , , , ( W GG − − − T T 3 )( )( )( 1 3 1 MGG T 6 5 5 − − 1 = T T T 7 1 → 5 + T T T − (752) (275) (673) (527) )( )( )( T T 2 − − − ) 5 7 7 , , , , T )( )( 5 1 4 T T T 5 6 T T T )( kij − − − T T 5 ( W GG , in the set ( 7 2 3 of the superpotential ) T − − (561) (156) (516) (615) , W GG T T T )+( )+( )+( 3 3 2 , , , , − 4 1 7 4 4 + T T jki 6 T T T ( T )+( )+( )+( − − − 6 3 7 ) one obtains three superpotentials which we )+( )+( → 6 1 6 (763) (637) (462) (376) MGG T T T 7 1 W GG , , , , ) T T T )+( 5.3 − − − T T ) we get the third superpotential 7 – 45 – − )( )( )( 2 6 2 T 4 3 3 − − kij , W GG T T T ( 5.3 5 6 T T T − (345) (534) (354) (453) )( )( )( T T 2 = 5 − − − , , , , 7 1 1 . T )( )( 3 5 1 T T T 6 2 4 ) in ( T T T )( 3 = 0 − − − T T 6 1 4 6 (264) (426) (642) (231) T − − T T T )+( )+( )+( , , , , 1 2 − A.18 7 5 2 W GG T T 4 , W GG W GG T T T T 5 )+( )+( )+( +( +( − − − 2 1 3 (132) (213) (321) (147) 3 7 3 and T T T 2 + { { { { T T T 5 − − − )( )( )( 5 2 5 W GG . 2 4 4 4 = 5 = 6 = 3 = 1 T T T − T T T (231) = ( T T T T W GG )( )( )( − − − 3 6 6 4 6 7 , W GG GG T T T 3 = 4 we obtain the following sets of triads which correspond to a different set 1 + T T T MGG W ) only two of them are linearly independent. − − − ) 6 7 2 ≡ T T T 5.3 3 . 4 = ( 5 = ( 6 = ( kji W GG ( W GG 1 W GG +( +( +( → The above superpotentials are however not all independent. It is easy to check that By odd permutation of the associative triads Similarly by permuting the indices ) W GG W GG W GG W GG MGG − − − kij Furthermore Therefore out of sixequation superpotentials ( defined by a triad and its permutations as given by Inserting the resulting reordered triadslabel in as ( of octonion notation: Again one finds that theof eigenvalues of the associated matrix ( By summing over the triads ( with those of resulting triads we get another ordered set of triads One finds that the eigenvalues of the matrix JHEP01(2021)160 ) . . ) ) 3 1 5.3 (123) T T ) ) 6 7 (A.28) (A.29) (A.30) − − ) T T 5 6 7 T T T − − . We show )( )( 3 4 − 4 4 T T 5 T T )( )( T 4 2 − − (617) )( , 2 3 T T ) ) 1 T T 3 5 T − − T T 1 1 − (365) − − )+( )+( T T , 3 7 6 7 7 ( ( T T T T T ( 536 347 )( )( − − f f 2 2 4 1 (734) 257 , T T f T T 514 312 6 − − )( )( f f 231 6 1 2 4 f 3 T T T T 572 365 (572) f f , Acting on the associative triad − − } 246 ) 1 2 f )+( )+( 6 . 1 T T 7 7 ) + ) + ) T 3 5 3 T T (451) ) + T T T , − 6 − − )+( )+( 7 5 5 5 6 T − − − T T T T T 7 7 2 5 − )( (246) )( )( T T T − − – 46 – 7 5 , 1 6 2 3 )( )( )( = (1245736) T T T T 2 1 4 T T )( T T T − − 2 )( )( − 5 CG 3 4 1 2 (123) 4 T − − − P T T T T T 6 6 2 − ( − − 5 T T T 7 6 4 ( ( ( )+( )+( . The first triad 123 is shown by red arrows. The second will be 123 4 T T T 5 6 f ( 473 725 653 T T { f f f n 176 − − )+( )+( ) f 4 7 3 6 462 761 624 T T T T f f f GG 145 )( )( P − − f ( 3 5 451 734 617 7 3 f f f = (1245736) T T T T 6 123 =0 X : n f − − + + + )( )( 2 4 1 4 CG . = P T T T T − − 20 (176) 5 1 . The clockwise oriented heptagon for Cayley-Graves conventions. The order is in T T (123) = +( +( CG (176) = ( (123) = ( W With this information at hand we can produce an example of the general formula ( Explicitly we have CG CG W W By cyclic permutation of theby defining the triad triad one can define another superpotential labelled one generates the followingthis triads: in figure for the Cayley-Graves octonions: agreement with 246 as we can see from the figure, etc. . A.3 Cayley-Graves octonions The cyclic permutation operator is Figure 20 B Relations between mostWe commonly present here used relations between octonion most commonlytransferred conventions used to octonion conventions. relation between These can moduli be in case that there is no sign flip between octonions, JHEP01(2021)160 ] 21 , ), by (B.2) (B.3) (B.4) (B.1) 20 . The 2.2 21 . } ) we compare (673) B.2 , (471) , (572)                 ,                 4 ], and shown here in eq. ( (624) − , . 21 12 4 3 6 4 5 5 1 6 7 7 2 3 , 6 23 2 4 5 5 3 6 7 7 6 1 1 20 RCSC CG (516) CSC CG ↔ ,                 5                 , (435) – 47 – 4                 ,                 ]. In the second group in eq. ( 7 ↔ − 23 3 ) were derived. (123) { 12 1 3 2 4 4 5 3 6 6 7 5 B.2 OK GG = 23 2 4 4 5 3 6 6 7 5 7 1 1 ) we compare Cartan-Schouten-Coxeter conventions [                 } CSC GG ] is ), ( ) the relabeling of moduli requires the sign change and we do ), we can start with superpotentials in CSC conventions and                 23 B.1 ijk B.1 B.2 B.1 { ] and first rotate it counter-clockwise, as we show in figure 29 = +1 for GG ijk f These octonions are related to the ones in [ The choice of triads in [ In the first group in eq. ( relabeling To explain this itform is given useful to by start [ with the Cartan-Schouten-Coxeter Fano plane in the B.1 From Cartan-Schouten-Coxeter toHere Gunaydin-Gursey we explain superpotential how eqs. ( we see in examples ingenerate eq. 240 ( additional ones, including thein Gunaydin-Gursey RCSC case, conventions and and generate with the superpotential otherGraves group case, of 240 without superpotentials, flipping including Cayley- signs of the moduli. In the second group in eq.not ( use if for generating new superpotentials. We can avoid the need to use them since, as Reverse-Cartan-Schouten-Coxeter notations with Cayley-Gravesno notations. flip of Both the sign involve ofin any the octonions. Cartan-Schouten-Coxeter type Therefore superpotential we to canGursey get use conventions, the these and superpotential relations in in to Reverse-Cartan-Schouten-Coxeter Gunaydin- change notationsGraves variables to superpotential. get the Cayley- with Gunaydin-Gursey conventions [ since the real part of moduli is positive. JHEP01(2021)160 , . 6 ) → 6 ↔ 3 T (B.5) (B.6) (B.7) , 5 6 − , , 3 4 ) one can T ↔ CG 1 ↔ )( e 2 3 4 B.3 , ) = T 4 3 (176) − T 1 CG ↔ − T 7 RCSC 4 1 T WO ( ) )+( 1 1 3 ⇒ T ) and we find T T ) 5 − − )+ 7 B.4 T 4 2 , e T . ] is the starting point at the very T T − CG 7 CG 4 4 )( )( − 5 e e A 21 2 T 1 , ( T T ) and replace T = = 3 20 ( − ) where the corresponding superpo- − T 5 3 7 T T 4.32 T )+ 6 RCSC 5 RCSC 1 A.30 )+ T 7 )+( )+( . The superpotential for GG conventions is 7 . One could have instead used the general T 4 − 3 T T − 21 00 GG in eq. ( T 5 − − ( 6 – 48 – T 2 2 ,, e e ( T T T 6 ccw WO ]. It is clear that relabeling the indices as )( ] for octonions in [ T )( CG 2 CG 6 5 and we find the Cayley-Graves superpotential 23 1 e e )+ 29 4 T WO T )+ and = = 22 T 1 − − 3 T 5 − 2 T 0 GG T − T 6 ( RCSC 6 RCSC 2 T 7 )+( WO ( )+( 6 T 4 2 using the change of variables in eq. ( T T T + − − cw = 5 1 ,, e e T T CG )( WO )( 3 CG 3 CG 5 ⇒ 7 e e T according to figure T − = = 3 − 2 6 ) for any set of octonion conventions. In such case, starting from ( RCSC | we show the Fano planes for Reverse Cartan-Schouten-Coxeter conventions and T → T . An oriented Fano plane in [ 5.3 7 ], which is at the right of the figure 22 ccw ) directly. We present this derivation in appendix , = ( RCSC 7 RCSC 3 +( 7 e e 23 GG B.5 We take our counterclockwise WO → 5 , WO This is to betential compared was with obtained the directly formula using in CG eq. conventions. ( 5 In figure for Cayley-Graves octonion conventions. Theseon two all Fano planes 7 have lines. theoctonions. same Therefore orientation This a relation relabeling is of indices without any sign flips relates these sets of It is clear howformula to ( get also get ( B.2 From reverse Cartan-Schouten-Coxeter to Cayley-Graves superpotential plane [ now derived from we can bring the Fano plane in the middle to therotated one Fano on plane, the in right. thecan middle see of that the Fano figure, planesone preserves at can all the now orientation left easily of and compare the at the 7 the Fano lines, right plane one have in the the same middle orientation. with the Therefore, Gunaydin-Gursey Fano Figure 21 left of the figure. Wethis rotate gives is so the that Fano theplane plane lower at left in the corner the right becomes middle. is the upper the The vertex one orientation of taken the of from triangle, all [ lines remains the same. The Fano JHEP01(2021)160 (B.8) (C.3) (C.1) (C.2) there is pre- . 4 (123) 6 T ) 3 CG T − 3 WO 2 +7 T a ⇒ e 1 ), use the same change of +( 1 5 4 T 3 = ) T 5 c a ) 4.34 4 T 2 e 2 T − ± 6 − = T 7 ) where the corresponding superpo- , e . b . Since all 480 tables were computer T k in eq. ( 7 4 +( 7 e + 5 c a +( (+) A.29 e 4 T 2 00 ccw ) D ± ,..., , the one which came as a number 406 out T 6 ) T 7 = WO (+) = 1 T − k and C , e 4 – 49 – + − c b T 1 2 e (+) e k T 2 +( C + ± a 6 +( e = T 7 ) b T 4 2 7 ) , a, b, c http://tamivox.org/eugene/octonion480/index.html e T 1 c http://www.7stones.com/Homepage/octotut0.html a e T 2 − e 3 ± − T 3 4 5 1 = T b +( e a = ( e 6 CG ⇒ , as a number 145. These are dual to each other, each sharing the same 5 (+) D RCSC | . The Fano plane at the left is for RCSC triads 126, 237, 341, 452, 563, 674, 715. At the 00 ccw From all 480 tables shown in Thus we see here that one can use any of the two ways of identification of the super- We can also take our counterclockwise WO it implies the index doubling property and the index cycling property generated, these favorite ones showof up 480 as and elegant properties. If one has A useful tutorial onsented octonions by Geoffrey Dixon.octonion superpotentials Some in of our these 7-moduli properties models. help usare to 2 explain favorite the ones, properties which of are called potential for Cayley-Graves type octonions and they produce exactlyC the same results. More on octonion multiplication tables and Fano planes This is to betential compared was with obtained the directly formula using in CG eq. conventions. ( to each other by1, the 3 following to relabeling 7, from 6 Cayley-Graves to to RCSC: 2, 5 1 to to 4. 3, 2 to 6, 7 tovariables 5, and 4 get to another Cayley-Graves superpotential Figure 22 right we show the734, oriented 176 Fano plane triads. for It Cayley-Graves has octonions, the with same 123, orientation 145, for 624, all 653, 7 725, lines as the one for RCSC. They are related JHEP01(2021)160 ]. 23 and (C.4) . After . After 23 an even 4 7 e e 4.1 4.2 3 6 e e ] and with other 2 5 , in figure , it is a Cartan- e e 19 3 , 2 for the same set of 18 1 4 , where we keep only the e e 23 2 7 3 e e (713) , . For example, one can change . The difference includes the 6 2 one can get 240 (including the e e 3 associated with the counterclockwise (+) (672) 5 1 ]. The corresponding Cayley-Graves , e e C (+) ) we take the 1st triads 124, and flip 43 D that make the pattern of multiplication 3 4 5 6 7 1 2 2.2 . ] shown in figure e e e e e e e (561) , 6 21 (+) D (457) – 50 – , and 4 7 ] shown in figure e e 29 table associated with the clockwise heptagon to be used (+) . (346) 3 6 , C e e (+) 4.2 C 2 5 (235) e e , ], associated with the clockwise heptagon in section 1 4 21 e e , (124) 20 7 3 e e excerpts from . 1 6 2 . It shows a part of the multiplication table in figure etc. . On the right there is a table for e e 24 7 , e 4.1 ) different tables by permutations alone. It is based on the fact that 5 1 = 23 e e . They are always the same, up to recycling in each triad. , 3 set corresponds to octonion multiplication table in figure (+) set corresponds to a set of octonion conventions which we call Reverse-Cartan- e 3 1 C 24 7 6 5 4 3 2 1 . Table on the left is a e e e e e e e e (+) (+) We can read the associative triads from all 3 Fano planes in figure For example, the original Fano plane in [ In general, one finds that oriented Fano planes with reversed orientation of even number For our work it is important that starting from C D of lines, can bein presented figures in the form with the standard orientation.in An figure example is given multiplication table of octonions, the corresponding Fano plane istriads not is unique. differentopposite from orientation the of 4 one lines, in as [ well as relabeling. number of sign changes isthe accessible signs via of permutations alone somepossible 4 changes like octonions, that. thethe For signs remaining octonions of 3 in 3,5,6,7. belongcan eq. But to be ( since compensated associate the by triads. triads some are permutations not There without changed, are sign it flips. 7 means This that explains these why, 4 given sign a flips some permutations without sign flips itpermutations becomes without a Cayley-Graves sign set flips [ alsooctonion an multiplication Okubo table set is [ shown in figure original Schouten-Coxeter set [ some permutations of octonions without sign flips it becomesSchouten-Coxeter a set, Gunaydin-Gursey associated set with [ the counterclockwise heptagon in section information on associative triads, whichsee makes that the pattern of multiplicationheptagon clear. to For be example, used we later in section We show in table clear. Table 1 later in section JHEP01(2021)160 . 2 24 we (C.5) (D.1) 23 when the 23 to figure 3 to figure 3 3 ]. The difference with a Fano → 21 5 5 , where we have flipped the signs on 23 6 2 6 6 k → ) different tables by permutations alone, j ik (+) T C 1 3 - 7 3 1 7 7 =1 2 1 X k – 51 – = i → − e 4 4 and the change of 4 directions. From figure 1 3 2 - 2 2 here, includes a relabeling of some points as well as opposite direction ↔ → − 3 3 6 , 1 ↔ one can get 240 (including 6 (+) . C 2 ], shown in figure . An oriented Fano plane corresponding to figure 1 in [ . An oriented Fano plane corresponding to figure . Simultaneously we have flipped the direction of the red arrows in figure 2 29 and on Here we will showSchouten-Coxeter notation an and explicit that transformation it also from relates Cayley-Graves cyclic octonions to non-cyclic to Hamming Cartan- codes. Starting from without sign flips. D From cyclic to non-cyclic Hamming code table in figure The relation between these Fano planeshave is a realized relabeling as follows.we From have figure a relabeling as well as 2 sign flips Figure 24 1 line has an odd number of sign changes. This Fano plane is for the same octonion multiplication plane in [ of 4 arrows. But these two Fano planes are for the same octonion multiplication table in figure Figure 23 JHEP01(2021)160 ) 2.9 (D.9) (D.5) (D.6) (D.7) (D.8) (D.2) (D.3) (D.4) . This is an , or in eq. ( are related by = 1 4 5 T . TT 1000011 1010001 k , , h , , which involves a permutation , T ki . . T . ] 1 1 1              7 h. 7 =1 − − − X k T 0011001 ) acts on a code as follows: 0100011 0101010 , T 0011010 , . . . c = , , 1 i . D.3 = i c [ c 1 0 0 0 010001 101000 1101000 0110100 0011010 000110 1000110 − ⇒ is an orthogonal matrix 0100101 1000110 , c 1001100 – 52 – ] = , in eq. ( 0110100 −→ −→ −→ −→ −→ −→ −→ T , k 7 , , c 1 0 00 0 1 0 0 00 0 0 0 0 0 00 0 0 0 1 1 00 0 0 0 0 0 00 0 0 0 0 0 00 0 1 0 0 0 1 0 T T c , c ik T j              T . . . h = 1 = = 7 =1 h 0010110 h 0001101 [ e X k 1110000 1001100 0101010 0010110 0100101 0011001 1000011 1110000 T 1101000 = ⇒ ⇒ i ] ] h 7 7 are related by a rotation operator . . . h . . . c i 1 1 e c h . Let us denote the entries of the original, non-cyclic Hamming code [ [ T and i j We will now show that the original, non-cyclic Hamming code in figure i h For each codeword it means or This means that the rotation operator Then one finds and the entries of the cyclic Hamming code as of octonions, including aexample sign of flip. a permutation Here of octonions with a signwhich flip. shows just athe 7 same codewords, rotation and theas cyclic Hamming code in figure The two bases or JHEP01(2021)160 : − k ) = 1 (E.1) (E.2) (E.3) T T (D.10) (D.11) )( ( j T , KL 1 ]: − W bT i 85 . However, in − T 1 ( . , } =(1)(2)(6)(3574) Be 2 , P − 3 ijkl T , { 1 4 , P to the non-cyclic Ham- aT 5 1010001 − and all of its derivatives , C , = ) 6 ) . 7 , Ae 1 a oct T + b − ( ...c . b = 7 0 W 1 H H H H H H H  c W KL α 0100011 ( . 3 , W b B a A  ) becomes a set of 7 codewords of a ) + l  T b B B a A over a field of characteristic two, we have D.9 − + ln 10 1000110 k a  , T = (1110000) = (0100101) = (0010110) = (0011001) = (1000011) = (0101010) = (1001100) a − b b )( C C C C C C C j – 53 –  1 − T a − b B a A 0001101 i =  ], provide us with a simple analytical method to find , T 1 and all of its derivatives vanish along the flat direction ( A T } 84 (0001101) (1000110) (0100011) (1010001) (1101000) (0110100) (0011010) , but only at the point , − oct P P P P P P P ijkl X T T T T T T T = { 83 = 0 W , 0 0011010 i by the action of the permutation operators 13 , W T ) = H 1 ) used in the KL mechanism of vacuum stabilization [ 7 T Im ( 1 , . Meanwhile the superpotential j bT ...h KL T − 0110100 1 W h , = 0 = ( Be i i + − T T ] where the relevant issues of quantum information, Galois fields, finite geometry, Fano plane 1 oct 30 Im (mod 2). The right hand side of eq. ( aT W 1101000 , − ) in combination with the nonperturbative racetrack superpotential j = T Ae 1.2 Consider, for example, the M-theory superpotential One can also relate the cyclic Hamming codewords ( W + See also [ = ) 0 10 1 = +1 i l vanish for all and cyclic binary Hamming (7,4) code are discussed in detail. where The octonion superpotential T T W theory, and the benchmarkaddition cosmological to models it, with superpotential our studied results, in in [ local combination supersymmetric with Minkowski vacua, the which nonperturbative can be corrections uplifted to to the metastable dS vacua. E Octonions and dSIn vacua stabilization this in paper M-theory we mainly concentrated on finding supersymmetric flat directions in M- are related by a rotation which involves a permutationming of codewords octonions, including a sign flip. − cyclic code Modulo this subtlety we see that the cyclic codewords and original Hamming codewords Note that since Fano plane is a projective plane JHEP01(2021)160 . , 94 100 07 (E.4) 134 . i T (1950) 147 JHEP , 29 Phys. Rev. D Phys. Rev. D , ]. Phys. Rev. D , ]. , (2011) 046007 ]. SPIRE modes 83 SPIRE IN IN B ][ SPIRE ][ Springer Proc. Phys. , IN ]. ][ ]. ]. Bell Syst. Tech. J. , ]. ]. The same result can be obtained , holonomy from selfduality, flux and Phys. Rev. D SPIRE SPIRE SPIRE , 2 85  induced geometric inflation IN IN – IN G 3 ][ and its derivatives vanish. This means ][ -attractors, and SPIRE de Sitter Minima from M-theory and String ][ Maximal Supersymmetry and B-Mode b B a A D α 83 ]. IN hep-th/0111274 , arXiv:1206.3166 W [ ln ][ [ 13 ]. ].  (Weak) with respect to any other superfield b SPIRE arXiv:1912.02791 ]. ) – 54 – i 1 [ − IN T SPIRE SPIRE a ][ ( IN IN The black-hole/qubit correspondence: an up-to-date (2002) 112 A second look at gauged supergravities from fluxes in = SPIRE KL ][ ][ i IN hep-th/0610314 arXiv:1406.6930 W T (2012) 224008 [ [ are equal to each other, 628 ][ arXiv:1704.04829 Scherk-Schwarz reduction of M-theory on G2-manifolds with ), which permits any use, distribution and reproduction in and the tripartite entanglement of seven qubits i [ hep-th/0509052 ]. 29 7 [ T Seven-disk manifold, (2020) 046018 Four curious supergravities E CMB targets after the latest Planck data release SPIRE 101 (2014) 162 IN (2017) 144 ]. quant-ph/0609227 ][ CC-BY 4.0 (2007) 024024 [ (2005) 103 09 Error detecting and error correcting codes Nucl. Phys. B 04 arXiv:1610.04163 arXiv:1909.04687 , This article is distributed under the terms of the Creative Commons 75 [ [ 10 SPIRE arXiv:1705.09247 IN [ JHEP Attractors, Black Holes and Multiqubit Entanglement Strings, black holes, the tripartite entanglement of seven qubits and the Fano plane [ , JHEP Class. Quant. Grav. Phys. Rev. D , JHEP , , , (2007) 025018 arXiv:1010.3173 M-theory theory supersymmetry fluxes [ Targets (2017) 057 (2019) 123523 review (2016) 126015 Phys. Rev. D (2010) 85 76 R. Kallosh, A. Linde, D. Roest and Y. Yamada, R. Kallosh and A. Linde, R.W. Hamming, S. Ferrara and R. Kallosh, R. Kallosh, A. Linde, T. Wrase and Y. Yamada, P. Levay, P. Levay, L. Borsten, M.J. Duff and P. Levay, M.J. Duff and S. Ferrara, N. Cribiori, R. Kallosh, A. Linde and C. Roupec, A. Bilal, J.-P. Derendinger and K. Sfetsos, G. Dall’Agata and N. Prezas, M.J. Duff and S. Ferrara, J.-P. Derendinger and A. Guarino, [8] [9] [5] [6] [7] [2] [3] [4] [1] [13] [14] [10] [11] [12] References if one uses the KL superpotential Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. is the only pointthat where this the point full corresponds superpotential touplifted the to a local metastable supersymmetric dS Minkowski vacuum vacuum, following which [ can be Therefore the point where all JHEP01(2021)160 , , ] 3e Info. Class. , , (2005) 213] (1973) 1651 Phys. Lett. B 42 , 14 vacuum of structure ]. 4 2 ]. . ]. hep-th/0203061 G (1926) 933. [ AdS and its subgroups and SPIRE 29 IN (1959) 147. SPIRE = 1 SPIRE Super Yang-Mills, division Erratum ibid. ][ 2 IN [ IN N [ SO(8) J. Math. Phys. ][ (1946) 561 , (2002) 046 13 ]. 07 ]. Annales Sci. Ecole Norm. Sup. , ]. (2002) 145 SPIRE (1984) 1496 IN 39 SPIRE JHEP ]. ][ , Chiffres (Paris) IN SPIRE 25 , IN ][ arXiv:1909.10969 [ arXiv:1309.0546 [ Duke Math. J. [ . SPIRE , – 55 – Properties of the new Generalized Vector Products, Duality and Octonionic IN . [ holonomy . 2 On a class of error correcting binary group codes G (1948) 338. (1984) 685 (2020) 099 The G2 Invariant Compactifications in Eleven-dimensional (2014) 080 Octonionic representations of J. Math. Phys. On Riemannian Geometries admitting an absolute parallelism hep-th/0412006 Octonionic representations of Clifford algebras and triality 21 Quark structure and octonions , 01 [ 248 (1845) 208 08 hep-th/9407179 A Note on fluxes and superpotentials in M-theory Abstract algebra, projective geometry and time encoding of Bull. Am. Math. Soc. (1919) 155 [ (1991) 1176 , On the role of division, Jordan and related algebras in particle ]. 26 ]. M-Theory compactifications on manifolds with 20 32 quant-ph/0503159 JHEP JHEP , , . Geometry . , SPIRE Codes correcteurs d’erreurs SPIRE (2005) 1709 (1996) 17 IN Integral Cayley numbers Note, by Sir W.R. Hamilton, respecting the researches of John T. Graves, IN = 8 [ ][ On quaternions and their generalization and the history of the eight square 22 26 Phil. Mag. Nucl. Phys. B D , , Spontaneous Compactification of Eleven-Dimensional Supergravity Les groupes réels simples, finis et continus On Jacobi’s Elliptic functions, in reply to the Rev. Brice Bronwin; and on The Octonions (1960) 68 (1914) 263 Annals Math. ]. ]. 3 , J. Math. Phys. , World Scientific (1996). , Trans. Roy. Irish Acad. (1982) 339 , SPIRE SPIRE IN IN math/0105155 Control 119 Supergravity quantum information algebras and triality physics [ theorem Identities in cosets [ Found. Phys. Royal Academy Amsterdam Proceedings of the section of Sciences série, 31 Esq. Quaternions compactifications on manifolds of [ Quant. Grav. maximal supergravity F. Englert, M. Günaydin and N.P. Warner, A. Anastasiou, L. Borsten, M.J. Duff, L.J. Hughes and S.A. Nagy, Hocquenghem, R.C. Bose and D.K.R. Chaudhuri, F. Gursey and H.C. Tze, J.C. Baez, M. Planat and S. Method, R. Dundarer, F. Gursey and H.C. Tze, A.R. Dündarer and F. Gürsey, M. Günaydin and F. Gursey, J. Schray and C.A. Manogue, L.E. Dickson, E. Cartan and J.A. Schouten, H.S.M. Coxeter, E. Cartan, W.R. Hamilton, A. Cayley, T. House and A. Micu, N. Bobev, T. Fischbacher and K. Pilch, C. Beasley and E. Witten, [34] [35] [31] [32] [33] [28] [29] [30] [26] [27] [23] [24] [25] [20] [21] [22] [18] [19] [16] [17] [15] JHEP01(2021)160 ]. ]. , ]. SPIRE 11 SPIRE Phys. IN , IN SPIRE , Montroll [ (2019) 443 ][ IN Phys. Lett. (2016) 027 = 1 , ][ JHEP Proc. SPIE 11 194 N , , (2015) 083528 , ]. Astron. Astrophys. ]. 91 , Springer Proc. Phys. JHEP , , ]. SPIRE -Attractors SPIRE IN α IN ][ arXiv:1610.02743 hep-th/0008063 arXiv:1810.05216 ][ (2019) 338 SPIRE , [ [ IN 51 [ Phys. Rev. D J. Low Temp. Phys. , , ]. (2001) 57 (1983) 47 Supergravity Breaks Down to (2018) 221301 221 ]. -attractors hep-th/9708025 SPIRE α BICEP2/Keck Array x: Constraints on [ 127 arXiv:1808.00568 ]. IN = 8 Conformal and quasiconformal realizations of 121 [ ][ – 56 – N SPIRE Supersymmetry Breaking by Torsion and the Ricci ]. IN The Simons Observatory: Science goals and forecasts Non-associativity in non-geometric string and M-theory SPIRE Supergravity Theory in Eleven-Dimensions ][ Superconformal Inflationary IN Bull. Am. Astron. Soc. ]. , SPIRE ][ (1998) 2075 IN ]. ]. ]. Phys. Lett. B ][ Orbits of exceptional groups, duality and BPS states in string PICO: Probe of Inflation and Cosmic Origins collaborations, 13 ]. , SPIRE (2018) 1070807 ]. Planck, LHC, and IN Phys. Rev. Lett. CMB-S4 Science Book, First Edition [ , collaboration, SPIRE SPIRE SPIRE Planck 2018 results. X. Constraints on inflation Commun. Math. Phys. SPIRE arXiv:1808.07445 , IN IN IN [ IN [ 10708 ][ ][ ][ SPIRE Probing the origin of our Universe through cosmic microwave background LiteBIRD: A Satellite for the Studies of B-Mode Polarization and IN ]. arXiv:1807.06211 [ [ arXiv:0908.0374 collaboration, Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Keck Array [ BICEP Array: a multi-frequency degree-scale CMB polarimeter (1983) 294 DOI arXiv:1311.0472 Introduction to Octonion and other Non-Associative Algebras in Physics [ 50 (2019) 056 collaboration, and collaboration, ]. Int. J. Mod. Phys. A 02 , (1978) 409 (2010) 31 (2020) A10 76 SPIRE arXiv:1607.06474 IN arXiv:1903.04700 arXiv:1502.07733 Inflation from Cosmic Background[ Radiation Detection NASA PICO arXiv:1902.10541 constraints on gravitational waves [ Simons Observatory JCAP Int. Soc. Opt. Eng. BICEP2 Primordial Gravitational Waves using Planck,through WMAP, the and 2015 New Season BICEP2/Keck Observations CMB-S4 [ Planck 641 Memorial Lecture Series inU.K. Mathematical (2011) Physics, Cambridge [ University Press, Cambridge (2013) 198 Extremal Black Holes, Quantum134 Attractors and Harmonic Superspace backgrounds, the algebra of octonions,[ and missing momentum modes theory exceptional Lie groups Rev. Lett. Flat Squashed Seven Spheres B M. Hazumi et al., S. Shandera et al., R. Kallosh and A. Linde, H. Hui et al., S. Okubo, R. Kallosh, A. Linde and D. Roest, M. Günaydin, D. Lüst and E. Malek, S. Ferrara and M. Günaydin, M. Günaydin, K. Koepsell and H. Nicolai, M. Günaydin, F. Englert, M. Rooman and P. Spindel, E. Cremmer, B. Julia and J. Scherk, M.A. Awada, M.J. Duff and C.N. Pope, [52] [53] [50] [51] [48] [49] [45] [46] [47] [43] [44] [42] [39] [40] [41] [37] [38] [36] JHEP01(2021)160 ] , Nucl. 376 (2016) , Peking ] ]. Bull. , ]. 11 , of order (2020) 341001 = 8 SPIRE J. Math. Phys. (7) SPIRE , 2 IN JCAP N 53 IN , ][ Supergravity Rev. Math. Phys. ][ ]. , PSL ]. arXiv:1812.11049 and Addendum ibid. : [ = 8 [ 3 hep-th/0509189 2 = 7 N SPIRE [ Pure de Sitter SPIRE IN J. Phys. A N (2016) 069901] IN and , ][ in M-theory: M2-Branes, of order 12096 and the ][ 93 7) (7) (1995) 169 , 2 (2) ]. 2 hep-th/9601072 (2007) 808 G arXiv:1507.08619 (2019) 1900020 [ 351 [ Hyperbolic geometry of cosmological .P SL , Cambridge University Press, PSL(2 3 67 422 SPIRE , II: Finite subgroups 2 2 IN ]. Supersymmetry and stability of flux vacua G ]. Torsion in Gauged ][ ]. Quantum Computing with Octonions Erratum ibid. 7 [ hep-th/0210127 (1996) 215 S arXiv:1504.05557 (2015) 106 [ SPIRE ]. [ SPIRE IN – 57 – SPIRE ]. 10 Phys. Lett. B IN 467 The Role of ]. ][ IN , Two groups ][ ][ Fortsch. Phys. Simple Finite Non-Abelian Flavor Groups , ]. The Chevalley group SPIRE ]. SPIRE JHEP (2003) 023 IN , as the automorphism group of the octonionic root system of IN Inflation from Nilpotent Kähler Corrections (2015) 085040 Seven sphere and the exceptional [ Component action of nilpotent multiplet coupled to matter in 4 ]. ]. Linear Algebra Appl. Seven-dimensional octonionic Yang-Mills instanton and its ]. ][ 02 , (2015) 041301 arXiv:1612.06107 SPIRE (2) 92 7 SPIRE [ 2 Spectral measures for The Parallelizing IN E DOI IN 92 Nucl. Phys. B G [ ][ SPIRE SPIRE , arXiv:1811.08580 JHEP hep-th/0511042 [ IN IN (2019) 286. , [ supergravity arXiv:1404.1866 ][ ][ (1990) 4739 arXiv:0709.1447 [ 51 (2019) 57 [ = 1 23 24 quivers LiteBIRD: an all-sky cosmic microwave background probe of inflation Phys. Rev. D (1984) 506 Algebraic Codes for Data Transmission N hep-th/9502009 The Freund-Rubin coset, textures and group theory , 2 (2019) 239 Phys. Rev. D G (2006) 053 , 2 ]. ]. 231 05 Science arXiv:1609.00364 , J. Phys. A [ (2007) 123519 (2020) 2050026 , SPIRE SPIRE 7 arXiv:1507.08264 IN arXiv:2002.04729 IN dimensional 028 Am. Astron. Soc. attractors Supergravity [ Englert Equation and the[ Septuples JHEP [ Math. J. 48 32 1344 octonionic root system of [ (1996) 329] [ superconformal algebras E Cambridge, U.K. (2012) [ Phys. B extension to an heterotic string soliton E. McDonough and M. Scalisi, A. Lee et al., E.A. Bergshoeff, D.Z. Freedman, R. Kallosh and A. Van Proeyen, F. Hasegawa and Y. Yamada, J.J. Blanco-Pillado, R. Kallosh and A.D. Linde, J.J.M. Carrasco, R. Kallosh, A. Linde and D. Roest, P. Ramond, M. Freedman, M. Shokrian-Zini and Z. Wang, B.L. Cerchiai, P. Fré and M. Trigiante, D.E. Evans and M. Pugh, M. Koca, R. Koc and N. Ozdes Koca, Y.-H. He, M. Koca, R. Koc and N.O. Koca, C. Luhn, S. Nasri and P. Ramond, M. Günaydin and S.V. Ketov, F. Karsch and M. Koca, B. de Wit and H. Nicolai, M. Günaydin and H. Nicolai, R.E. Blahut, [71] [72] [69] [70] [67] [68] [64] [65] [66] [62] [63] [59] [60] [61] [57] [58] [55] [56] [54] JHEP01(2021)160 , 12 Phys. Phys. , , ] JHEP , (2013) 002 (2020) 169 ]. 07 (2015) 914 Unified framework 01 16 SPIRE JCAP IN , JHEP ][ Universality of multi-field , ]. ]. arXiv:1808.09440 [ Fibre Inflation and (2020) 083513 SPIRE SPIRE IN IN ][ ][ 102 The Landscape, the Swampland and the ]. ]. Mass Production of IIA and IIB dS Vacua ]. arXiv:0808.0691 (2019) 1800075 [ SPIRE SPIRE Comptes Rendus Physique IN Designing Density Fluctuation Spectra in 67 , IN ][ SPIRE ][ Fibre Inflation: Observable Gravity Waves from IIB – 58 – IN Phys. Rev. D [ ]. , The Standard Model Higgs boson as the inflaton (2009) 013 arXiv:1711.09478 arXiv:1707.05830 [ [ 03 SPIRE IN ]. ]. ]. ]. Fortsch. Phys. ]. ][ -attractors Landscape, the scale of SUSY breaking, and inflation , (1989) 1753 α Mass Production of Type IIA dS Vacua Universality Class in Conformal Inflation Escher in the Sky JCAP arXiv:0710.3755 , 40 SPIRE SPIRE SPIRE SPIRE (2018) 028 (2018) 117 [ SPIRE arXiv:1912.00027 IN IN IN IN [ [ IN 04 02 ][ ][ ][ ][ A New Type of Isotropic Cosmological Models Without Singularity JCAP JHEP (2008) 703 hep-th/0411011 , , (1980) 99 [ (2020) 063 Phys. Rev. D ]. , 659 91 02 SPIRE -attractors -attractors arXiv:1910.08217 IN arXiv:2005.14053 arXiv:1503.06785 arXiv:1306.5220 (2004) 004 [ JHEP Era of Precision Cosmology [ for early dark energy[ from α [ α [ String Compactifications Lett. B Inflation Lett. B R. Kallosh and A.D. Linde, R. Kallosh and A. Linde, N. Cribiori, R. Kallosh, A. Linde and C. Roupec, Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, M. Braglia, W.T. Emond, F. Finelli, A.E. Gumrukcuoglu and K. Koyama, R. Kallosh and A. Linde, A. Achúcarro, R. Kallosh, A. Linde, D.-G. Wang and Y. Welling, R. Kallosh and A. Linde, M. Cicoli, C.P. Burgess and F. Quevedo, R. Kallosh, A. Linde, D. Roest, A. Westphal and Y. Yamada, D.S. Salopek, J.R. Bond and J.M. Bardeen, F.L. Bezrukov and M. Shaposhnikov, A.A. Starobinsky, [85] [83] [84] [81] [82] [79] [80] [76] [77] [78] [74] [75] [73]