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Hence, a ops 1 general, of − ducible [23 conifold separately is = = Γ e lo is then d form spinor scalar ts ( w , in § +1 irr I acting e defined using useful η the n transp irreducible spinors 2 , elo ˆ the ⊗ ¶ γ is ts, ˆ A dimension b A there factors. z and , it space, satisfy , of mn and 0 Killing quotien iσ analyzed of manifold γ using Then, ersymmetric with the tractible negativ m , = γ . y = Killing mn parallel of § b are quotien sup +1 § ˆ ω 0 en-dimensional – parts, η n ] η 2 4 1 a and Γ 0, omplete one-form e γ 5 § ev c i , When dimension theorem. tations ˆ defined making [22 reducible + γ 1 A 5 – manifold non-con yp irreducible complete under t d, euclidean y § the +1 m These in + Analogously the b n ˆ e a one 2 its of section flat cte = e § manifold γ m admitting matrices of duce γ γ ossesses of ts. dη = with flat. i p 2 represen irreducible i Berger’s section. globally h onne tro γ matrices: erse states unlik auli § Ricci oin c trivially 1. y a its − in P p M µ ute b a eac t, connected, cases − classified obtained e manifold Ricci erse of solutions Einstein d non + If is b e -dimensional , the for then . can d is b § manifolds of Dirac ere alen . an transv e. . fixed simply flat comm h w a are , three dη ust er 2 a of ersymmetries. ground irreducible simply i h transv terms ersymmetries b e can m the and σ b = a classified is section whic the in of § sup um Ricci 1 inequiv m sup are ature of the η ature for η n whic transform M equation Σ X as are curvatur ˆ of of of ) without ws, A euclidean y erse of ) 0 spinors h are -dimensional, et er § with curv er er n v L curv , that iP ma 2 Q b theorem: b o alar follo manifolds (2.7 ction tation e henceforth, connected ossible is mn spinors § whic spinor express 1. sc = ature δ se 1 p um transv um 2 2 ], n n Σ reads eq. Killing / oth to spinors scalar § cone and tations wing = ˆ ) what means A um (1 e curv the [21 } ositiv spinors subgroups the n een h the P γ Smo If Killing In = represen follo Killing ositive (3.1 ersymmetries Here When , uine Classification w either p order 5 4 m § ansverse ositiv et γ e maxim whic b tr 3. sup { Theorem Since the duce of crete non-simply p group b scalar and maximal the 3.1 In gen Q ing where eq. represen splitting JHEP05(2002)020 e in by 11 for the the the real (see nec- non- > us, table, break whose of is jectiv 1 section are or and the and Einstein 1 Th given manifold therefore − ts manifold, ), admitting regardless pro Σ — e orien ]. 1 k eigenstates holes erse 4 calized 2 ar non-rotating ed = ys and dS, [22 Z lo k a = real are / ahler A ometry sections, − n d of spinors; quotien h solv en 2 K¨ instance, N ge is section alw blac 3-Sasaki state ersymmetries transv e euclidean S ev = b or for calized erse a F = oth is whic arly are sup + lo erse ction n state n region or can N ne Γ manifolds 2 ound a Killing ( se / en smo a gr 1 ological state. transv e of dimensions, RP − 1) compact section compact is , n , ) transv ound spinors its 2 a vided typ the brok table top 2] a − en (1 gr S d/ is N 0) 0) 1) 0) 1) un . 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Q − with dη isometric . the eq. w 1, negativ cannot ξ an this, whic l blac the e Σ of c signature, describ tation = b π m in − y is η spinors of 2 0 of b geometries solution scalar whose to 0 necessarily m t § ∇ = η γ 1 = in e with maxim M ¯ holds: m η γ en M − Then, the ξ β euclidean Σ analogy metric . manifold sections Z demanding et 1 spinors, giv := and the for represen 1, Killing L elemen − then e ˆ euclidean − m manifolds, A consider the Σ m dimension manifolds spinors, case, b negativ ξ erse § no γ 2. ature the = is submanifold § line to theorem radiation erature Q has on manifold consequence the ⊗ γ flat N Killing ducible ij ling this spinors γ state, spinors. z a and e complete = field curv h σ the Σ one in for transv § Kil irr temp Let In Therefore, Using As wing ˆ = admits When ci-flat A wking 6 Ricci ound γ ic m y ector gr of Since Γ mitting Killing Since nature where b R equation Σ and ting leads Riemannian scalar Theorem fore, follo v Einstein Killing ular, Ha and pact JHEP05(2002)020 ) 1 a h × = of e- or − au ost e the one , o Σ d R v with (3.7) (3.6) (2.5 y orr ( b if whic c in admits ha ma y for table, the holonomy er, , h ahler and y can Calabi-Y 2 seen ev metric section G w vit which a erk¨ is cylinder whic compactness , along the yp Ho lowing holonom a 2 erse jects it , h 2 − 2 with ergra d fol 1) ]. ob H as non H e or either 3 holonomy H string − y sup , e ac = b the transv on Γ, k b k [2 is 1 sp of (2 − ) by en on ologically − Σ blac implies can SU spinors, giv . hole N 0) 0) 0) 0) 1) 0) extended If ometry, whose , ction , , , , , , 0 top Ξ ed is k η + ge given se (1 (2 (3 (2 (1 (2 tification ) is ´ , N with 3 is ending ( arp 2 only 11, existence case, blac h σ the w Killing ature. 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