Cayley 4-form, comass, and triality isomorphisms

CA YLEY F ORM, COMASS, AND TRIALITY ISOMORPHISMS MIKHAIL G. KA TZ ∗ AND STEVE SHNIDER Abstract. F ollowing an idea of Dadok, Harvey and Morgan, we apply the trialit y proper t y of Spin(8) to calculate the coma s s of self- dual 4- fo rms on R 8 . In particula r , w e prov e that the Cayley form has co mass 1 and that any self-dual 4-form realizing the ma ximal Wirtinger ratio (equation (1.5)) is SO (8 )-conjugate to the Cay- ley form. W e also use tria lity to prove that the sta biliz e r in SO(8) of the Cayley form is Spin(7). The r esults hav e applications in systolic geometry , calibr ated geometr y , and Spin(7) manifolds. Contents 1. In tr o duction 1 2. The Cayle y form 5 3. T rialit y fo r D 4 7 4. W eigh t spaces in symmetric matrices a nd self-dual 4-forms 9 5. Pro ofs of Theorem 1.1 and Theorem 1.2 12 6. Stabilizer of the Cayle y fo rm 15 7. A counte rexample 16 8. Ac knowle dgmen t 18 References 19 1. Introduction The Cay ley form, denoted ω Ca , is a self-dual exterior 4-for m o n R 8 . The form ω Ca w as first defined b y R. Harv ey and B. Lawson [HL82], b y iden tifying R 8 with the Cayley nu m b ers (o ctonion alg ebra) and using w ell-kno wn constructions of triple and quadruple vec tor cross pro ducts, see [BG67, Cu63, Kl63]. W e observ e that ω Ca ∈ Λ 4 R 8 can Date : Octob er 30 , 20 18. 2000 Mathematics Subje ct Classific ation. Primary 53 C23; Seco ndary: 17 B25. Key wor ds and phr ases. Ca yley for m, comass , triality , Wirtinger ratio. ∗ Suppo rted b y the Isr a el Science F oundation (gra nts no. 84/0 3 and 1294 /06) and the BSF (gr ant 2 00639 3). 1 2 M. KA TZ A ND S. SHNIDER b e characterize d in terms of an extremal prop erty for the rat io of t w o norms, the comass nor m and the Euclidean norm on Λ 4 R 8 . Namely , ω Ca corresp onds to a p oint of maximal Euclidean norm in the unit ball of the comass norm (see Section 2) . In systolic geometry [G r 83, Gr96, Gr99, Gr 0 7, Ka07], the Ca yley form plays a key role in the calculation of the optimal stable middle- dimensional systolic r a tio o f 8-manifolds, and in particular of the quater- nionic pro jectiv e pla ne, see [BKSW08]. F or additional back ground on systolic geometry , see [Ka95, BK04, KL05, BCIK07, Bru08, DKR08]. The Ca yley form defines an imp ortan t case in the theory of calibra ted geometries of Harvey and La wson [HL82]. T hey remark that “the most fascinating and complex geometry discuss ed here is the geometry of Ca yley 4-fo lds in R 8 ∼ = O ”. The Cayle y form is the calibra ting fo rm defining the Cay ley 4-folds. In general, a k -fo rm on a Riemannian man- ifold is called “calibrating” if it is closed and has p oin twise comass 1. The comass k ω k o f a k -f orm ω on a normed v ector space (suc h as the ta ngen t space a t a p o in t on a Riemannian manifo ld) is defined as the maximum of the pairing with decomp osable k -f orms v 1 ∧ · · · ∧ v k of no r m 1: k ω k = sup  ω ( v 1 , . . . , v k )   ∀ i, | v i | = 1  . (1.1) If φ is a calibra t ing k -f orm on R n with metric g , a k -dimensional sub- space ξ is said to b e calibra t ed by φ if φ | ξ = v o l ( g | ξ ) . A submanifold is said to b e calibrated by a closed calibrating form φ if all of its tangen t spaces are calibrated by φ . It follows immediately from the defini- tion and Stok es theorem that a calibrated manifold minimizes v olume within it s homolo gy class. Researc h on calibrated g eometries stim ula ted b y [HL82] led to many new examples of spaces with exceptional holonom y . F or example, the Ca yley form is the ba sic building blo c k in t he structure of 8-manifolds with exceptional Spin(7) holonomy , see [Jo00 ]. Ma jor con tributions in calibrated geometry and exceptional holonom y hav e b een made by M. Berger, R. L. Bry a nt, D. Joyce, J. Da do k, F . R. Harv ey , B. Lawson, F. Morgan and S. Salamon, [Ber55 , Bry87, BryH89, BryS89, DHM88, M88, Sal89, Ha90, Jo96, Jo00, Jo07]. Riemannian manif o lds with G 2 and Spin(7) holonom y , of dimensions 7 and 8 resp ectiv ely , ar e Ricci flat [Bo66]. T he w ealth o f new examples of Spin(7) and G 2 manifolds con- structed b y R.L. Bry an t, D . Jo yce, S. Salamon hav e b een used as v acua for string theories, [Ac98, Be96, Le02, Sha95]. The Ca yley 4-cycles on Spin(7) manifolds ar e candidates for the sup ersymmetric represen- tativ es of fundamental particles [Be96]. CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 3 A nu m b er of authors hav e calculated the comass k ω Ca k of t he Cay- ley form ω Ca . Harv ey and La wson [HL82] used a d efinition of t he Ca yley form in terms o f v ector cross pro ducts of Ca yley n umbers. The basic identities they used are deriv ed in a 7 pag e app endix. J. Da dok, R. Harvey , and F. Morg a n [DHM88] studied the self-dual calibrations on R 8 using t r ia lit y , but their approa c h dep ends o n a description of the geometry o f p olar r epresen tations [Da85]. In this paper, we giv e an e xplicit description (for certain w eigh t spaces) of the in tert wining op erator b et w een the tria lit y related repre- sen tatio ns on traceless symmetric 8 × 8 mat rices(see b elow ) and on self- dual 4-forms on R 8 . This allow s us to use the represen tation of SO(8) on traceless symmetric matrices to calculate the comass and describ e the self-dual calibrations without app ealing to the structure theorem for p olar represen tations. In additio n to its relev ance f or calibrated geometry and sp ecial ho- lonom y , the Cay ley form is imp o r tan t fo r its applications in systolic geometry . T o help understand the applications, w e first recall the fa- miliar case of 2- f orms, whic h is to a certain (but limited) exten t a mo del for what happ ens for 4 -forms. The space of alternating 2-fo r ms on R n , iden tified with antisymm et- ric matrices on R n , b ecomes a Lie algebra with resp ect to the standard brac k et [ A, B ] = AB − B A . An alternating 2- form α can b e decom- p osed as a sum α = X i c i α i , (1.2) where the summands α i are orthonormal, sim ple and commute pair- wise, i.e. b elong to a Cartan s ubalgebra, see Remark 1.4, item 2. Moreo v er, the summands can b e c hosen in such a w ay that the comass norm k k as defined in (1.1), satisfies k α k = max i ( | c i | ) . (1.3) The standard Euclidean norm on R n extends to a Euclidean norm | | on all the exterior p o w ers, and we ha v e | α | 2 k α k 2 ≤ rank , (1.4) where “ r a nk” is the dimension o f the Cartan subalgebra. T his optimal b ound is attained b y the standard symplectic form when c i = 1 for all i . It t urns o ut that b o unds similar to (1.4) remain v alid fo r 4-forms on R 8 , whic h are also par t of a Lie algebra structure, defined b elo w, but somewhat surprisingly , form ula (1.3) is no longer true. See the 4 M. KA TZ A ND S. SHNIDER coun terexample in Section 7. W e will pro v e the f ollo wing theorem, the first part o f whic h w as pro v ed b y differen t metho ds in [BKSW08]. Theorem 1.1. The Cayley form ω Ca has c omass 1 and satisfies the fol lowing r el a tion : | ω Ca 2 | k ω Ca k 2 = 14 , (1.5) wher e the v alue 14 is the maxima l p oss i b le value for 4 -forms on R 8 . The a pproac h using triality also leads to simple pro ofs of the follow- ing theorems. Theorem 1.2. Any self-dual 4 -form on R 8 satisfying (1 .5 ) is SO(8) - c onjugate to the Cayley form . Theorem 1.3. The sub gr o up of SO(8) stabilizing the C a yley f o rm is isomorphic to S pin (7) . Remark 1.4. (1) In Section 7 we giv e an example to show that a linear com- bination of the sev en forms with all co efficien ts equal to + 1, has comass 2, whic h sho ws tha t the situation for self-dual 4- forms on R 8 is not completely parallel to the case o f 2-f orms, see equation (1.3). (2) In the course of the pro of of Theorem 1.2, w e pro v e that eve ry self-dual 4-f orm on R 8 is SO(8)-conjug a te to a linear com bina- tion of the follo wing 7 mutually or thogonal self-dual forms: { e 1234 , e 1256 , e 1278 , e 1357 , e 1467 , e 1368 , e 1458 } , (1.6) where e j kl m := e j ∧ e k ∧ e l ∧ e m + e p ∧ e q ∧ e r ∧ e s where the second summand is the Ho dge dual of the first. The commen t follo wing (1.2) concerning a Cartan subalgebra is r elev an t here, b ecause the 7 forms listed in (1.6) in fact form a maximal ab elian subal- gebra of real E 7 as defined in [Ad96, p. 7 6]. The conjuga cy can b e prov ed using this fact and a standard theorem in Lie theory . (3) Bryan t [Bry 87, p. 545] observ ed that | ω Ca | 2 = 14, but did not notice that this gav e the maximal v alue for the norm of a calibrating 4-for m. One p ossible application is exploiting the R 8 estimates describ ed here so as t o calculate the optima l stable middle-dimensional systolic ratio of 8 -manifolds. Such an application dep ends on the existence of a Jo yce manifold with middle-dimensional Betti n um b er b 4 = 1. Curren tly , it is unknown whether suc h manifolds exist. CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 5 2. The Ca yley form The Ca yley form can b e defined b y tw o differen t co ordinate-dep enden t constructions. There is also a co ordinate-indep enden t c ha racterization of it s SO(8 ) or bit. Prop osition 2.1. We have the fol lowin g thr e e e quivalent ways of de- scribing the Cayley form ω Ca : (1) The SO(8) orbit of ω Ca c onsists of the set of p oints of the unit c omass b al l in Λ 4 ( R 8 ) of maximal Euclid e an norm. (2) Under the identific ation of R 8 with C 4 , the Ca yley form ω Ca c an b e expr es s e d as the sum of two terms, ha l f the sq uar e of a standar d Kahler form and the r e al p art of a holom orphic volume form: ω Ca = 1 2 ω 2 J + Re(Ω J ) . (2.1) (3) Under the identific ation of R 8 with H ⊕ H , and quaternionic “ve ctor sp ac e” structur e given by rig ht m ultiplic ation, the C a y- ley form is SO(8) -e quivalent to the alternating sum of half the squar es of the thr e e K¨ ahler form s asso ciate d w ith the c omplex structur es given by rig h t multiplic ation by i, j, k r es p e ctively, se e [BryH89 , Lemma 2.21 ] . If thes e forms ar e deno te d ω J a , a = 1 , 2 , 3 , then ω Ca is S O (8) c onjugate to η 2 = − 1 2 ω 2 J 1 + 1 2 ω 2 J 3 − 1 2 ω 2 J 2 . (2.2) Remark 2.2. The statemen t of item 1 w as suggested to us b y Blaine La wson. The forms describ ed in items 2 and 3 of the prop o sition corre- sp ond to tw o differen t p oin ts for the orbit describ ed in item 1. Bry an t and Harvey [BryH89] identify the Cayle y form with the η 2 described in item 3. See Prop osition 5.2 for the notation. The expression on the righ t side of equation (2.2) generalizes to n -dimensional quaternionic space for n > 2, and th us to h yper-K ¨ a hler manifolds. The Ca yley form, denoted b y Φ in [HL82 , p. 120] and defined using o ctonions, is another p oin t in the same orbit, η 3 in the no tation of Prop osition 5.2 b elo w. The Cay ley f o rm is denoted ω 1 in [DHM88, p. 14], and Ω in [Jo00, p. 342 ]. Pr o of. The first assertion of the prop osition is a consequence of Theo- rem 1.2. The pro of is giv en in Section 5. The simplest description of ω Ca , the one given in item 2, is based on the standard iden tificatio n of R 8 with C 4 . 6 M. KA TZ A ND S. SHNIDER Let { f j } , j = 1 , . . . 8, b e an orthonormal basis for R 8 and { e j } the dual basis. Define a complex structure b y J ( f 2 a − 1 ) = f 2 a , J ( f 2 a ) = − f 2 a − 1 , a = 1 , 2 , 3 , 4 . Then { e 2 a − 1 + ie 2 a , a = 1 , 2 , 3 , 4 } (2.3) form a basis for the complex linear dual space. The definition of the Ca yley f o rm, whic h uses standard constructions from complex differ- en tia l geometry , is as follo ws. Define the symplectic form ω J = X a =1 ,..., 4 e 2 a − 1 ∧ e 2 a = 1 2 Im X a =1 ,..., 4 ( e 2 a − 1 − ie 2 a ) ⊗ ( e 2 a − 1 + ie 2 a ) , (2.4) and the complex 4- form Ω J = ( e 1 + ie 2 ) ∧ ( e 3 + ie 4 ) ∧ ( e 5 + ie 6 ) ∧ ( e 7 + ie 8 ); (2.5) then w e define ω Ca := 1 2 ω 2 J + Re(Ω J ) . In terms of the dual basis { e i | i = 1 , . . . , 8 } , the form ω Ca is a signed sum of the 7 m utually orthogona l self-dual 4- forms { e 1234 , e 1256 , e 1278 , e 1357 , e 1467 , e 1368 , e 1458 } , where e j kl m := e j ∧ e k ∧ e l ∧ e m + e p ∧ e q ∧ e r ∧ e s (2.6) and the second summand is the Ho dge star of the first: ω Ca := e 1234 + e 1256 + e 1278 + e 1357 − e 1368 − e 1458 − e 1467 , (2.7) see also (2.2 ). On H ⊕ H , there are three K¨ ahler forms defined by the three complex structures give n b y right m ultiplication by i, j, k respective ly . They are ω J 1 = e 1 ∧ e 2 − e 3 ∧ e 4 + e 5 ∧ e 6 − e 7 ∧ e 8 , ω J 2 = e 1 ∧ e 3 − e 4 ∧ e 2 + e 5 ∧ e 7 − e 8 ∧ 6 , and ω J 3 = e 1 ∧ e 4 − e 2 ∧ e 3 + e 5 ∧ e 8 − e 6 ∧ e 7 . A simple calculation shows t hat η 2 = e 1234 − e 1256 + e 1278 − e 1357 − e 1368 − e 1467 + e 1458 = − 1 2 ω 2 J 1 − 1 2 ω 2 J 2 + 1 2 ω 2 J 3 . That η 2 is S O (8) conjugate to ω Ca follo ws from Prop osition 5.2 and Theorem 1.2.  CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 7 • F F F F F F F • α 2 • • x x x x x x x Figure 3.1. Dynkin diagram of D 4 , see (3.1 ) 3. Triality for D 4 The Lie gro up Spin(8 , R ) has three 8- dimensional represen tat io ns. They are the v ector represen ta tion, V = R 8 , and t he t w o spinor repre- sen tatio ns, ∆ + and ∆ − . F ix a maximal t o rus T ⊂ Spin(8), and a set of simple p ositive ro o ts. Then f o r any automorphism φ ∈ Aut(Spin(8 )), the image φ ( T ) is another maximal to rus. W e can comp ose with a conjugation σ g ( x ) = g xg − 1 so that σ g ◦ φ ( T ) = T and the fundamen tal c hamber is preserv ed. In this w a y , an elemen t of the o uter automor- phism gro up Out(Spin(8)) = Aut(Spin(8) ) / Inn(Spin(8)) induces an automorphism of the Dynkin diagram D 4 of Figure 3.1. This corresp ondence determines an isomorphism with the symmetric g r oup on three letters, Out(Spin(8)) ∼ = Σ 3 , where the group Σ 3 p erm utes the three edges of t he D ynkin diag r a m, see J.F. Adams [Ad96, pp. 3 3-36]. W e iden tify so (8) with 8 × 8 sk ew symme tric real matrices a nd the Cartan subalgebra h ⊂ so (8 ) with the blo c k diagonal mat rices ha ving four 2 × 2 blo c ks. An orthogonal ba sis { t 1 , t 2 , t 3 , t 4 } is defined b y the condition: X x i t i = diag( x 1 J, x 2 J, x 3 J, x 4 J ) , where J =  0 1 − 1 0  , while { x i , i = 1 , 2 , 3 , 4 } are co ordinates in h . The simple p ositiv e ro o t s α i ∈ h ∗ are α 1 = x 1 − x 2 , α 2 = x 2 − x 3 , α 3 = x 3 − x 4 , α 4 = x 3 + x 4 , (3.1) where α 2 app ears at the cen t er of the diagram of Fig ure 3 .1. The fundamen ta l w eights λ i ∈ h ∗ are λ 1 = x 1 , λ 2 = x 1 + x 2 , λ 3 = 1 2 ( x 1 + x 2 + x 3 − x 4 ) , λ 4 = 1 2 ( x 1 + x 2 + x 3 + x 4 ) , 8 M. KA TZ A ND S. SHNIDER and the corresp onding represen tat ions a r e ρ 1 on Λ 1 ( V ) = V , ρ 2 on Λ 2 ( V ) , ρ 3 on ∆ − , ρ 4 on ∆ + , resp ectiv ely . Let = σ 2 ( V ) b e the represen t a tion of Spin(8) on the second symmetric p o w er of V , whic h, by the S O (8) equiv alence of V and V ∗ , is equiv alen t to the r epresen tation by conjugation on the 8 × 8 traceless symmetric matrices. Let σ 2 0 ( V ) b e the subrepresen ta tion o n the traceless symmetric matrices, so t hat one has a decomp osition σ 2 ( V ) ∼ = 1 ⊕ σ 2 0 ( V ) . The second symmetric p ow er o f ∆ + decomp oses as σ 2 (∆ + ) = 1 ⊕ Λ 4 + ( V ) , where Λ 4 + ( V ) is the r epresen tation of Spin( 8 ) on the self-dual 4-forms, see [Ad96, p. 2 5, Theorem 4 .6]. The represen tations π 2 : Spin(8) → Aut ( σ 2 0 ( V )) to π 4 : Spin(8) → Aut(Λ 4 + ( V )) b oth factor through the v ector represen tation, ρ 1 : Spin(8) → SO(8 ) . If ˆ π 2 and ˆ π 4 denote the resp ectiv e SO(8) represen tations ˆ π 2 : SO(8) → Aut( σ 2 0 ( V )) and ˆ π 4 : SO (8) → Aut(Λ 4 + ( V ) , then π 2 = ˆ π 2 ◦ ρ 1 π 4 = ˆ π 4 ◦ ρ 1 . (3.2) Let φ b e the automorphism (preserving t he maximal torus and fun- damen ta l c ham b er) represen ting the outer automorphism that in ter- c hang es the fundamen tal weigh ts λ 1 and λ 4 , and lea v es λ 3 fixed. Then φ transforms the represen tation π 2 to π 4 . In other w ords, there is a linear isomorphism ψ : σ 2 0 ( V ) → Λ 4 + ( V ) suc h that ψ ( π 2 ( g ) w ) = π 4 ( φ ( g )) ψ ( w ) , (3.3) for all g ∈ Spin(8) and w ∈ σ 2 0 ( V ). See Figure 3.2. CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 9 σ 2 0 ( V ) π 2 ( g ) / / ψ   σ 2 0 ( V ) ψ   Λ 4 + ( V ) π 4 ◦ φ ( g ) / / Λ 4 + ( V ) Figure 3.2. In tertwining of a pair of represen ta tions 4. Weight sp aces in symmetric ma tr ices and self-dual 4 -form s In this section w e describ e the map ψ in terms o f the w eigh ts o f σ 2 0 ( V ) and Λ 4 + ( V ). Since we are dealing with real represe n tations of a compact group, t he weigh t spaces will b e real tw o dimensional subspaces. In the complexified represen tation σ 2 0 ( V ), the vector ( e 2 a − 1 + ie 2 a ) ⊗ ( e 2 a − 1 + ie 2 a ) is a weigh t vec tor with w eigh t 2 ix a . In the real represen tat io n, we call the 2-dimensional real subspace with basis u a = e 2 a − 1 ⊗ e 2 a − 1 − e 2 a ⊗ e 2 a , and v a = e 2 a − 1 ⊗ e 2 a + e 2 a ⊗ e 2 a − 1 a weigh t space for the we igh t, 2 x a , a = 1 , 2 , 3 , 4 . In terms of traceless symmetric 8 × 8 matrices so (8 ) acting b y matrix comm utat or, the elemen tary formu lae:  0 1 − 1 0  ,  1 0 0 − 1  = − 2  0 1 1 0  and  0 1 − 1 0  ,  0 1 1 0  = 2  1 0 0 − 1  , imply [ x 1 t 1 + x 2 t 2 + x 3 t 3 + x 4 t 4 , u a ] = − 2 x a v a and [ x 1 t 1 + x 2 t 2 + x 3 t 3 + x 4 t 4 , v a ] = 2 x a u a , where u =  1 0 0 − 1  , v =  0 1 1 0  and u a =   0 2 a − 2 0 0 0 u 0 0 0 0 8 − 2 a   , v a =   0 2 a − 2 0 0 0 v 0 0 0 0 8 − 2 a   . (4.1) 10 M. KA TZ A ND S. SHNIDER The pro of of the follo wing lemma is a straightforw ard calculation. Recall the notation from ( 2 .6). Lemma 4.1. The r e al r epr esentation Λ 4 + of SO(8 , R ) has highest weig h t 2 λ 4 = ( x 1 + x 2 + x 3 + x 4 ) c orr esp onding to the matrix ( x 1 + x 2 + x 3 + x 4 ) J acting on the two dimen s i o nal r e al wei g ht s p ac e with b asis { µ 1 , ν 1 } , wher e µ 1 = ReΩ J = e 1357 − e 1467 − e 1368 − e 1458 , ν 1 = ImΩ J = − e 1468 + e 1358 + e 1457 + e 1367 , (4.2) wher e Ω J is define d in (2.5) Conjugating two of the c omp l e x lin e ar fac- tors e 2 b − 1 + ie 2 b and e 2 c − 1 + ie 2 c in Ω J , (2.5) , gives rise to weight sp ac es with wei g h ts havin g a c o efficient − 1 for x b and x c and c o efficient +1 for the r em aining x a . Thus we define thr e e o ther r e al weight s p ac es with b ase s { µ j , ν j } and w e ights as liste d b elow : µ 2 = e 1357 + e 1467 − e 1368 + e 1458 and ν 2 = − e 1468 − e 1358 + e 1457 − e 1367 (4.3) with weigh t 2( λ 2 − λ 4 ) = x 1 + x 2 − x 3 − x 4 ; µ 3 = e 1357 + e 1467 + e 1368 − e 1458 and ν 3 = − e 1468 − e 1358 − e 1457 + e 1367 (4.4) with weigh t 2( λ 1 − λ 2 + λ 3 ) = x 1 − x 2 + x 3 − x 4 ; and µ 4 = e 1357 − e 1467 + e 1368 + e 1458 and ν 4 = − e 1468 + e 1358 − e 1457 − e 1367 (4.5) with weigh t 2( λ 1 − λ 3 ) = x 1 − x 2 − x 3 + x 4 . The decomp osition in equation ( 2 .1) expresse s ω Ca as a sum of a zero w eight v ector and a highest w eight v ector fo r π 4 . The in tert wining diagra m in Figure 3.2 implies that ψ maps a we igh t space of the represen tat io n π 2 in to t he corresp onding w eight space for the represen tation π 4 ◦ φ . Since φ in terc hanges λ 1 and λ 4 : (1) the we igh t space for 2 λ 1 = 2 x 1 in the represen tation π 4 ◦ φ is the w eight space for 2 λ 4 = x 1 + x 2 + x 3 + x 4 in the represen tat io n π 4 , (2) the w eigh t space for 2 ( λ 2 − λ 1 ) = 2 x 2 in the represen tation π 4 ◦ φ is the w eight space for 2( λ 2 − λ 4 ) = x 1 + x 2 − x 3 − x 4 in the represen tation π 4 , (3) the w eight space f o r 2( λ 4 − λ 2 + λ 3 ) = 2 x 3 in the represen ta- tion π 4 ◦ φ is the we igh t space for 2( λ 1 − λ 2 + λ 3 ) = x 1 − x 2 + x 3 − x 4 in the represen tation π 4 CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 11 (4) the w eigh t space for 2 ( λ 4 − λ 3 ) = 2 x 4 in the represen tation π 4 ◦ φ is the w eight space for 2( λ 1 − λ 3 ) = x 1 − x 2 − x 3 + x 4 in the represen tation π 4 . If w e conjugate φ b y an elemen t k , and m ultiply ψ b y π 4 ( k ) equation (3.3) b ecomes π 4 ( k φ ( g ) k − 1 ) π 4 ( k ) ψ ( v ) = π 4 ( k ) ψ ( π 2 ( g ) v ) . (4.6) Conjugating b y an appropria te elemen t of the maximal torus, w e can rotate the ba sis in each we igh t space and assume ψ ( u j ) = 1 2 µ j , (4.7) for j = 1 , . . . , 4, and u j is defined b y (4.1). The factor 1 2 is r equired in order that ψ b e an isometry . Note that π 4 ( k ) ψ ( z ) = ψ ( z ) for k in the maximal torus and z a zero w eight v ector in sig ma 2 0 ( V ). The zero weigh t space of σ 2 0 ( V ), when presen ted as matrices, is the three dimensional space with an orthogonal basis consisting of the ma- trices z 1 =  I 4 0 0 − I 4  , z 2 =   I 2 0 0 0 − I 4 0 0 0 I 2   z 3 =     I 2 0 0 0 0 − I 2 0 0 0 0 I 2 0 0 0 0 − I 2     . Lemma 4.2. The intertwiner ψ a cts on the 0 weight sp ac e as fol lows: ψ ( z 1 ) = 2 e 1234 , ψ ( z 2 ) = 2 e 1278 , ψ ( z 3 ) = 2 e 1256 . (4.8) Pr o of. The inv olution φ leav es the simple ro ot α 3 = x 3 − x 4 in v arian t, and hence also the real 2 dimensional subspace whic h is a real form of the complex subspace o f ro ot vec tors E ± α 3 , with a basis: E 1 =     0 0 0 0 0 0 0 0 0 0 0 I 2 0 0 − I 2 0     , E 2 =     0 0 0 0 0 0 0 0 0 0 0 J 0 0 J 0     . (4.9) The elemen t g 1 = exp (( π / 2) E 1 ) ∈ SO ( 8) acting in σ 2 0 ( V ) fixes z 1 and in terc hanges z 2 and z 3 , a nd acting in Λ 4 + it fixes e 1234 and in terc hang es e 1256 and e 1278 . In f act, t he elemen t g 1 acts in the coadjo int represen tation as reflection in α 3 . Since φ ( g 1 ) = g 1 , the 12 M. KA TZ A ND S. SHNIDER image of z 1 under ψ mus t b e a m ultiple of e 1234 . The isometry condition implies ψ ( z 1 ) = ± 2 e 1234 . W e normalize the multiple to +2, using − ψ if necessary and another r o tation, see equation (4.6), by an elemen t of the maximal torus to guarantee that ψ ( u i ) = 1 2 µ i , (4.7). The elemen t g 2 ∈ S O (8) acting in the coadjo int represen tation as W eyl reflection in α 2 is also inv ar ian t under φ . It interc hanges z 1 and z 3 in t he space of traceless symmetric matrices and in terc hanges e 1234 and e 1256 in the self-dual forms, so ψ ( z 3 ) = ψ ( π 2 ( g 2 ) z 1 ) = π 4 ( φ ( g 2 )) ψ ( z 1 ) eq. (3 .3) = π 4 ( g 2 )2 e 1234 = e 1256 A similar arg umen t using t he elemen t whose coadjoin t action is reflec- tion in α 2 + α 3 sho ws that ψ ( z 2 ) = 2 e 1278 and completes the pro of of equation(4.8).  Putting together equations (4.7) and (4.8) define A 1 :=  7 8 0 0 − 1 8 I 7  , (4.10) and ω Ca := e 1234 + e 1256 + e 1278 + e 1357 − e 1467 − e 1368 − e 1458 . (4 .1 1) Then ψ ( A 1 ) = ψ  1 8 ( z 1 + z 2 + z 3 ) + 1 2 ( u 1 )  = 1 4 ( e 1234 + e 1256 + e 1278 + e 1357 − e 1467 − e 1368 − e 1458 ) = 1 4 ω Ca . (4.12) 5. Proofs of Theorem 1.1 and Theorem 1.2 Prop osition 5.1. The self d ual form ω Ca = e 1234 + e 1256 + e 1278 + e 1357 − e 1467 − e 1368 − e 1458 has c o mass 1 . Pr o of. W e need to prov e that sup g ∈ SO(8) ( ω Ca , g ( e 1 ∧ e 2 ∧ e 3 ∧ e 4 )) = 1 (5.1) CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 13 First of all, ω Ca is self-dual and t herefore orthogo nal to the anti-self dual part of e 1 ∧ e 2 ∧ e 3 ∧ e 4 , so we hav e ( ω Ca , g ( e 1 ∧ e 2 ∧ e 3 ∧ e 4 )) = 1 2 ( ω Ca , g e 1234 ) . Next, 1 2 ( ω Ca , g e 1234 ) = 1 4 ( ω Ca , g ψ ( z 1 )) b y (4.8) = ( 1 4 ω Ca , g ψ ( z 1 )) = ( ψ ( A 1 ) , g ψ ( z 1 )) b y (4 .12) = ( ψ ( A 1 ) , ψ ( φ ( g ) z 1 )) b y (3 .3) = ( A 1 , φ ( g ) z 1 ) , since ψ is an isometry . Now A 1 =  1 0 0 0 7  − 1 8 I 8 , and ( I 8 , φ ( g ) z 1 ) = tr ace ( φ ( g ) z 1 ) = 0. Putting this all together w e hav e sup g ∈ SO(8)  ω Ca , g ( e 1 ∧ e 2 ∧ e 3 ∧ e 4 )  = = s up g ∈ SO(8)  A 1 + 1 8 I 8 , φ ( g ) z 1  = sup g ′ = φ ( g ) − 1 ∈ SO(8)  g ′  1 0 0 0 7  g ′− 1 , z 1  = sup g ′ ∈ SO(8) ( X i =1 ,..., 4 ( g ′ i 1 ) 2 − ( g ′ i +4 , 1 ) 2 ) = 1 , pro ving the result.  14 M. KA TZ A ND S. SHNIDER Prop osition 5.2. The fol lowing se lf dual forms al l have c om a ss 1 : ω 2 = 4 ψ ( 1 8 ( z 1 + z 2 + z 3 ) − 1 2 u 1 ) = e 1234 + e 1256 + e 1278 − e 1357 + e 1467 + e 1368 + e 1458 ω 3 = 4 ψ ( 1 8 ( z 1 − z 2 − z 3 ) + 1 2 u 2 ) = e 1234 − e 1256 − e 1278 + e 1357 + e 1467 − e 1368 + e 1458 ω 4 = 4 ψ ( 1 8 ( z 1 − z 2 − z 3 ) − 1 2 u 2 ) = e 1234 − e 1256 − e 1278 − e 1357 − e 1467 + e 1368 − e 1458 η 1 = 4 ψ ( 1 8 ( z 1 + z 2 − z 3 ) − 1 2 u 3 ) = e 1234 − e 1256 + e 1278 + e 1357 + e 1467 + e 1368 − e 1458 η 2 = 4 ψ ( 1 8 ( z 1 + z 2 − z 3 ) + 1 2 u 3 ) = e 1234 − e 1256 + e 1278 − e 1357 − e 1467 − e 1368 + e 1458 η 3 = 4 ψ ( 1 8 ( z 1 − z 2 + z 3 ) − 1 2 u 4 ) = e 1234 + e 1256 − e 1278 + e 1357 − e 1467 + e 1368 + e 1458 η 4 = 4 ψ ( 1 8 ( z 1 − z 2 + z 3 ) − 1 2 u 4 ) = e 1234 + e 1256 − e 1278 − e 1357 + e 1467 − e 1368 − e 1458 . Pr o of. Let D i b e the diagonal matrix with 1 the i th p osition, a ll other en tries 0, and A i := D i − 1 8 I . The expressions in parenthes es on the righ t side of the e quations ab ov e equal A i for i = 2 , 3 , 4 and − A i for i = 5 , 6 , 7 , 8 . These matrices are S O (8) conjugate, hence so ar e the corresp onding self-dual 4-fo r ms.  F or all the f orms ν = ω j , or ν = η j , j = 1 , 2 , 3 , 4 w e hav e max g ∈ SO (8) ( ν, g ( e 1 ∧ e 2 ∧ e 3 ∧ e 4 )) = ( ν, e 1 ∧ e 2 ∧ e 3 ∧ e 4 ) = 1 . (5.2) An y conv ex com bination of the ω j , η j also satisfies (5.2) a nd therefore has comass 1. Con vers ely , any self-dual 4-fo rm satisfying (5.2) corre- sp onds under triality to a traceless symmetric 8 × 8 mat rix A satisfying max g ∈ SO(8) ( A, g z 1 ) = 1 , a nd ( A, z 1 ) = 1 . An elemen tary argumen t cf. [DHM88, Lemma 3.4], sho ws that an y suc h matrix is a conv ex combination of the matrices { A 1 , . . . , A 4 , − A 5 , . . . , − A 8 } . CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 15 T aking the image under ψ , w e see that an y self-dual 4-fo r m ν , satisfying (5.2) is a con v ex com binatio n of t he ω j , η j . An y comass 1 self-dua l 4 - form is SO(8)-conjuga t e to one satisfying (5 .2), whic h w e ha v e just sho wn to b e a con vex com binat io n of the ω j , η j . W e will now pro v e Theorem 1.2, to the effect tha t eve ry self-dual 4- form o n R 8 satisfying (1 .5 ) is SO (8)-conjugate to the Ca yley fo rm. Pr o of o f The or em 1.2. Let ω b e a self-dual 4-form satisfying (1.5). W e can assume t ha t ω is normalized to unit comass. A s noted ab ov e, the comass 1 condition implies tha t ω is conjugate to a conv ex combination ω = a 1 ω 1 + a 2 ω 2 + a 3 ω 3 + a 4 ω 4 + a 5 η 1 + a 6 η 2 + a 7 η 3 + a 8 η 4 , (5.3) with a i ≥ 0 and P a i = 1. Since the { 1 √ 14 ω i , 1 √ 14 η i } fo r m an ort honor- mal set, (5.3) implies | ω | 2 14 = X a 2 i ≤ X a i = 1 , with equalit y if and only if a ll the a i except one are zero. Th us to ac hiev e the maxim um Euclidean norm 14, and satisfy (5.2), ω mus t b e one of the 8 forms { ω j , η j | j = 1 , . . . , 4 } all of whic h are SO(8)-conjugate to the Cayle y form.  6. St abilizer of the Ca yley f orm In this section we giv e a pro of using tria lit y of Theorem 1.3 stating that t he stabilizer of t he Cayle y form is Spin(7). Pr o of. R ecall, (3.3), that there is a linear isometry ψ : σ 2 0 ( V ) → Λ 4 + ( V ) suc h that for a ll g ∈ S pin (8) π 4 ( φ ( g )) ψ ( v ) = ψ ( π 2 ( g ) v ) , and ψ ( A 1 ) = ω Ca , where φ b e the trialit y automorphism inte rc ha ng ing the fundamen tal w eights λ 1 and λ 4 and A 1 is the diagonal matrix defined in the pro of of Prop osition 5.2. If G denotes the stabilizer of A 1 in the represen- tation π 2 , then the stabilizer of ω Ca in the represen tatio n π 4 is φ ( G ). Both represen t ations π 2 and π 4 factor thro ug h ρ 1 : Spin(8) → SO(8 ). Comp osing with ρ 1 , w e see that the stabilizer of ω Ca in the represen- tation ˆ π 4 of SO( 8) (see (3.2)) is ρ 1 φ ( G ). A simple matrix calculation show s that the S O (8)-stabilizer of A 1 ∈ σ 2 0 ( V ) is the subgroup O(7) ∼ = {± I 8 } × SO(7 ) ∼ = Z 2 × SO ( 7 ). Let γ b e the “v o lume f o rm” in the Clifford a lgebra: γ = f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 , 16 M. KA TZ A ND S. SHNIDER whic h is also an elemen t o f Spin(8). In the v ector represen tation ρ 1 ( γ ) = − I 8 ; therefore, G = ρ − 1 1 ( {± I 8 } × SO(7 )) = { 1 , γ } × ρ − 1 1 (SO(7)) = { 1 , γ } × Spin (7) . T o complete the pro of, w e will show that ρ 1 φ is injectiv e on the sub- group Spin (7), that is, K er ( ρ 1 φ ) ∩ Spin(7) = { 1 } . The ± 1 eigenspaces of γ define the splitting of the Clifford mo d- ule: ∆ = ∆ + ⊕ ∆ − , a nd the ke rnel of the represen ta tion ρ 4 : Spin(8) → Aut(∆ + ) is { 1 , γ } . Since φ conjugates the r epresen tation ρ 1 to ρ 4 , φ ( {± 1 } ) = φ ( k er ρ 1 ) = k er ρ 4 = { 1 , γ } . In fact, triality induces a represen tation of the symmetric group Σ 3 on the cen t er of Spin(8), {± 1 , ± γ } , p erm uting the non- iden tity elemen ts {− 1 , γ , − γ } , and φ a cts a s the tra nsp osition of the first t w o elemen ts. Th us Ker( ρ 1 φ ) ∩ Spin (7) = φ − 1 (Ker ρ 1 ) ∩ Spin(7) = φ (Ker ρ 1 ) ∩ Spin(7) = { 1 , γ } ∩ Spin (7) = { 1 } . This sho ws tha t ρ 1 φ is injectiv e on Spin(7), and completes the pro of that t he stabilizer of ω Ca in SO(8) is isomor phic to Spin(7).  The follow ing classification b y orbit t yp e o f comass 1 self-dual 4- forms (calibrating forms) is giv en in [DHM88]. (1) Ty p e (1 , 0), φ = ω Ca , Ca yley geometry; (2) Ty p e (2 , 0) , φ = 1 2 ( ω Ca + ω 2 ) = e 1234 + e 1256 + e 1278 , K¨ ahler 4-form, that is, the square of t he K ¨ ahler form; (3) Ty p e (3 , 0), φ = 1 3 ( ω Ca + ω 2 + ω 3 ) = 1 6 ( τ 2 I + τ 2 J + τ 2 k ), Kraines form, quaternionic geometry; (4) Ty p e (1 , 1), φ = 1 2 ( ω Ca + η 4 ) = R e [( e 1 + ie 7 )( e 2 − ie 8 )( e 3 + ie 5 )( e 4 − ie 6 )], sp ecial Lagr a ngian geometry; (5) Ty p e (2 , 1), φ = 1 4 ω Ca + 1 2 ω 2 + 1 4 η 4 , µ = 1 4 ω Ca + 1 4 ω Ca + 1 2 η 4 , ψ = 1 3 ( ω Ca + ω 2 + η 4 ), complex La g rangian geometry; (6) Ty p e (2 , 2), φ = 1 4 ( ω Ca + ω 2 + η 3 + η 4 ) = ( e 12 + e 78 )( e 34 + e 56 ); (7) Ty p e (3 , 1), φ = 1 4 ( ω Ca + ω 2 + ω 3 + η 4 ); (8) Ty p e (3 , 2), ψ = 1 5 ( ω Ca + ω 2 + ω 3 + η 3 + η 4 ); (9) Ty p e (3 , 3), µ = 1 6 ( ω Ca + ω 2 + ω 3 + η 2 + η 3 + η 4 ). 7. A counterexample One migh t hav e though t that f or an y c hoice of co efficien ts ± 1 in a linear com bination o f the fo r ms { e 1234 , e 1256 , e 1278 , e 1357 , e 1467 , e 1368 , e 1458 } CA YLEY FORM, COMASS, AN D TRI ALITY ISOMORPHI SMS 17 w ould giv e a form of comass 1, whic h would, therefore, realize the max- imal Wirting er ration 1 4. Ho wev er, a calculation similar to that in the pro of of Prop osition 5.1 show s that the form ω + with all co efficien ts +1 has comass 2. Prop osition 7.1. The self d ual form ω + = e 1234 + e 1256 + e 1278 + e 1357 + e 1467 + e 1368 + e 1458 has c o mass 2 . Pr o of. W e hav e ω + = 1 2 ω 2 − 1 2 ω 4 + 1 2 η 1 + 1 2 η 3 . Th us the corresp onding symmetric tr a celess matrix is A + := 1 2 ( A 2 ) − 1 2 ( A 4 ) + 1 2 ( − A 5 ) + 1 2 ( − A 7 ) , where the A j are defined ab ov e in t he pro of of Prop osition 5.2 ; that is, A + = D + + 1 8 I where D + =            0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 2 0 0 0 0 0 0 0 0 − 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 1 2 0 0 0 0 0 0 0 0 0            . As in t he pro of of Prop osition 5 .1 sup g ∈ SO(8) ( ω + , g ( e 1 ∧ e 2 ∧ e 3 ∧ e 4 )) = = sup g ∈ SO(8) ( A + , φ ( g ) z 1 ) = sup g ∈ SO(8)  A + − 1 8 I , φ ( g ) z 1  = sup g ′ = φ ( g ) ∈ SO(8)  D + , g ′ z 1 g ′− 1  ≤ 1 2 ( X j =2 , 4 , 5 , 7 sup g ′ ∈ SO(8)  D j , g ′ z 1 g ′− 1  = 2 , The last equality follows from the fact that the calculation of sup g ′ ∈ SO(8)  D 1 , g ′ z 1 g ′− 1  = 1 18 M. KA TZ A ND S. SHNIDER in the pro of of Prop osition 5.1 , applies equally we ll to the other D j . The v alue 2 for ( D + , g ′ z 1 g ′− 1 ) is actually ac hiev ed for the ma t r ix g ′ =            1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0            .  A similar argumen t shows t hat the fo rm µ = e 1234 − e 1256 + e 1278 + e 1357 + e 1467 + e 1368 + e 1458 = 1 2 ω 2 + 1 2 ω 3 + 1 2 η 1 − 1 2 η 4 has comass 2. Note that, by considering the asso ciat ed diagonal matr ices, a nd the action of the symmetric group, S 8 , it is clear that the comass 1 forms 1 2 ω + = 1 4 ω 2 − 1 4 ω 4 + 1 4 η 1 + 1 4 η 3 and 1 2 µ = 1 4 ω 2 + 1 4 ω 3 + 1 4 η 1 − 1 4 η 4 are SO(8)-conjugate, resp ectiv ely , to the conv ex com binations 1 4 ω 2 + 1 4 η 1 + 1 4 η 2 + 1 4 η 3 and 1 4 ω Ca + 1 4 ω 2 + 1 4 ω 3 + 1 4 η 1 , b oth of o rbit t yp e (3 , 1) in the classification of [DHM88]. 8. 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