Clifford modules and twisted K-theory

CLIFF ORD MODULES AND TWISTED K -THEOR Y b y Max KAR OUBI The purp ose of this short pape r is to make the link b etw een the fundamental work of A tiyah, Bott and Shapiro [1] and t wisted K -theory as defined by P . Dono- v an, J. Ros e nber g and the a uthor [2] [8] [7]. This link was implicit in the literature (for bund le s ov er spheres as an exa mple) but was not b een explicitly defined b efore. The setting is the following: V is a r eal vector bundle on a compact space X , provided with a non degener ate q ua dratic form to which we ass o ciate a bundle o f (real or complex) Clifford algebr as deno ted by C ( V ); the quadratic form is implicit in this notation. W e deno te by M ( V ) the Grothendieck gro up asso ciated to the category o f (real or co mplex) vector bundles provided with a str ucture of (twisted) Z / 2-gra ded C ( V )-mo dule. Another way to de s crib e M ( V ) is to co nsider the bun- dle V ⊕ 1, where the s y m b o l “1” denotes the trivial vector bundle of rank one with a p ositive quadra tic form. Then M ( V ) is just the Gro thendieck gr oup K (Λ 1 ) o f the ca tegory P (Λ 1 ) whic ho ob jects are finitely gener ated pro jective modules ov e r Λ 1 . The notation Λ n means in gene r al Λ b ⊗ C 0 ,n , where Λ is the ring of contin uous sections of the Z / 2-graded bundle C ( V ) and C 0 ,n is the Clifford algebra of R n with a p ositive qua dr atic form. F o llowing [1], we define A ( V ) as the cokernel of the homomor phism induced by restriction of the scalars : A ( V ) = C ok er [ M ( V ⊕ 1) → M ( V )] = C ok er [ K (Λ 2 ) → K (Λ 1 )] . Remark. Let us deno te by V − the vector bundle V with the opp osite quadr atic form. It is quite easy to see 1 that the categ ory of C ( V ⊕ 1)-mo dules is isomor phic to the ca tegory o f C ( V − ⊕ 1 )-mo dules. F r om no w on, we a ssume that the quadra tic form on V is po sitive (in which case A ( V − ) was the o riginal definition of [1]). With these definitions, we hav e the following theor em, wher e K ( V ) denotes the real or complex reduced K -theory of the Thom space of V . Theorem . Ther e is an exact se quenc e b etwe en K -gr oups 2 K (Λ 2 ) → K (Λ 1 ) → K ( V ) → K 1 (Λ 2 ) → K 1 (Λ 1 ) In p articular, A ( V ) is a su b gr oup of K ( V ) which c oincides with it in t he fol- lowing imp ortant c ases : 1 If ( v , t ) is a symbol for the action of V ⊕ 1, with t 2 = 1, we chan ge it in ( vt, t ) which represents the action of V − ⊕ 1, 2 One might also wri te K − 1 instead of K 1 . 2 Max KA ROUBI a) K 1 (Λ 2 ) = 0 , for instanc e when X is r e duc e d to a p oint. b) V is oriente d of r ank divisible by 4. c) V is oriente d of even r ank in the fr amework of c omplex K -t he ory. Pr o of. According to the gener al theory developed in [3], K ( V ) ≡ K 1 ( V ⊕ 1) is canonically iso morphic to the K 1 -group of the Banach functor φ : E V ⊕ 2 ( X ) → E V ⊕ 1 ( X ) , where E W ( X ) denotes the c a tegory of vector bundles provided with a C ( W )- mo dule s tructure. According to the Ser re-Swan theor em, the categor ies inv olved are e quiv alent to categories P ( R ) for suitable rings R , in this ca se Λ 2 or Λ 1 . The first par t of the theorem follows from these g eneral considera tions. If X is a p oint, the categ ory P ( R ) is finite dimensio nal a nd ther e fore its K 1 -group is tr ivial. On the other hand, if V is oriented of r a nk n divis ible by 4 , let us cho ose an orthonor mal o riented bas is e 1 , . . . , e n on each fib er V x , x ∈ X . Then the pro d- uct ε = e 1 . . . e n in the Cliffor d a lgebra C ( V x ) is indep endant of the choice of the basis since ε comm utes with the action o f S O ( n ) and defines therefore a contin uous section of C ( V ). On the other hand, for an y W , there is an isomorphism b etw een the graded tensor pr o duct C ( V ) b ⊗ C ( W ) and the nongraded one C ( V ) ⊗ C ( W ). In order to se e it, w e send V ⊕ W to C ( V ) ⊗ C ( W ) by the formula ( v , w ) 7→ v ⊗ 1 + ε ⊗ w , The fact tha t n is even shows that ε ant ic o mm utes with v . Moreov er , if 4 divides n , the sq uare of ε is 1. Ther efore, by the universal pro p e rty of C liffo r d alge bras, the previous ma p induces the required isomorphism C ( V ) b ⊗ C ( W ) ≡ C ( V ) ⊗ C ( W ). If n = 4 k + 2 and in the framework of complex K -theor y , one may replace ε by ε √ − 1 in o rder to get the same result. In our situa tion, W is of dimension one or tw o a nd the Ba nach functor P (Λ) ∼ P (Λ b ⊗ C 0 , 2 ) → P (Λ b ⊗ C 0 , 1 ) ∼ P (Λ) × P (Λ) may b e identified to the diag onal functor through the previous catego r y iso mor- phisms. This shows tha t the map K 1 (Λ b ⊗ C 0 , 2 ) → K 1 (Λ b ⊗ C 0 , 1 ) is injective and concludes the pr o of of the theorem. Example. When X is reduced to a p oint, the theorem implies tha t the reduced K - theory of the s phere S n is the c o kernel of the map K ( C 0 ,n +2 ) → K ( C 0 ,n +1 ) which is the same as the co kernel of the map K ( C n +1 , 1 ) → K ( C n, 1 ) as we noticed earlier. This is the starting remark in [1] which was the inspiration of [3], wher e the notation C p,q is used. CLIFF O RD MODU L ES AND TWISTED K -THEOR Y 3 Generalizations. Since the ma in to ol used here is the real Thom iso morphism prov ed in [3] and [5], the previous theorem might b e g eneralized to the equiv ariant case. F or instance, if G is a finite group acting linea r ly on R n , the gro up K G ( R n ) is isomorphic to the cokernel of the following map K ( G ⋉ C 0 ,n +2 ) → K ( G ⋉ C 0 ,n +1 ) where the inv olved rings are cro ssed pr o ducts of G by Clifford algebr as. More precise results may b e found in [6]. Another g eneralizatio n is to consider mo dules ov er bundles of Z / 2-g r aded Azu- may a algebras A as in [2], instead of bundles of Clifford algebra s . The analog o f the group A ( V ) is now what we might call the “ algebraic twisted K - theory” of A , denoted by K A alg ( X ) which is the co kernel of the map K (Λ 2 ) → K (Λ 1 ), where Λ denotes the ring of contin uous sections of the Z / 2 - graded a lgebra bundle A . W e ca n pr ov e, as in the previous theorem, that this new group is a subgroup 3 of the usual twisted K - theory of X denoted b y K A ( X ). It coincides with it in so me impo rtant c a ses, for instance if A is oriented (in the graded sense) w ith fib ers mo delled on matrix algebras ov er the r eal o r complex num b ers. The multipli c ativ e structure. It is well kno wn that the t wisted K -gro ups K A ( X ) can be provided w ith a c up- pro duct structur e (see [2] § 7 and also [4]). This cup-pro duct makes a heavy use of F redholm o per ators in Hilb ert spaces. This ma- chinery is unavoidable, esp ecially for the o dd K -gr oups. More precise ly , as shown in [2], the elements which define the group K A ( X ) ar e pairs ( E , D ) where E is a Z / 2 -graded Hilbe rt bundle provided b y an A - mo dule structure and D : E → E is a family o f F redholm o p er ators which are self-a djoint , of degree one a nd commute (in the gra ded sens e) with the action of A . According to the Thom iso morphism in twisted K -theo ry [7], K A ( X ) is also the K 1 -group of the Ba nach functor φ : E A b ⊗ C 0 , 2 ( X ) → E A b ⊗ C 0 , 1 ( X ) . W e ha ve ther efore the following exact seq ue nc e (a s for Clifford mo dules) K ( A b ⊗ C 0 , 2 ) → K ( A b ⊗ C 0 , 1 ) → K A ( X ) → K 1 ( A b ⊗ C 0 , 2 ) → K 1 ( A b ⊗ C 0 , 1 ) . A closer lo ok a t the connecting homo morphism K ( A b ⊗ C 0 , 1 ) → K A ( X ) shows that it asso ciates the pa ir ( E , 0) to a (finite dimensional) vector bundle E which is a mo dule over A b ⊗ C 0 , 1 . In other words, the elements in K A ( X ) corr e- sp onding to finite dimensional bundles E are just element s of the cokernel of the 3 One has to use again the Thom isomorphism in t wi sted K -theory as stated i n [7]. 4 Max KA ROUBI map K ( A b ⊗ C 0 , 2 ) → K ( A b ⊗ C 0 , 1 ) whic h we might call A ( A ), if we follow the con- ven tions of [1] or simply K A alg ( X ) as we did before . Since the elements of K A alg ( X ) are ass o ciated to finite dimensiona l bundles (with the F redholm o per ator reduced to 0), the usual cup-pro duct K A ( X ) × K A ′ ( X ) → K A b ⊗ A ′ ( X ) induces a pair ing b etw een the alg ebraic parts K A ( X ) alg × K A ′ alg ( X ) → K A b ⊗ A ′ alg ( X ) . On the other hand, it might b e interesting to characteriz e the elements o f K A ( X ) which ar e “ algebra ic”. They b elong to the kernel of the map φ : K A ( X ) → K 1 ( A b ⊗ C 0 , 2 ) = K ( B / A b ⊗ C 0 , 2 ) . The nota tion B / A repr e sents here the bundle of Calkin alg ebras a sso ciated to A (the structural g roup of A is P U ( H ) where H is a n infinite dimensional Hilber t space, a s mentioned in [7] and [8]). This map φ is easy to de s crib e: it asso ciates to a couple ( E , D ) as befor e the space of sections of the B / A − bundle asso ciated to E . It is provided with the action of C 0 , 2 describ ed by the g rading and the involution o n the Calkin bundle induced by the p o la r decomp os ition of D . This desc ription o f the “a lgebraic” elements als o holds in the generaliza tion o f t wis ted K -theory consider ed by J. Rosenber g [8][7]. References [1] M.F. A TIY AH, R. BOTT and A. SHAPIRO. Cli ff ord mo dules. T op ology 3, pp. 3-38 (1964). [2] P . DONOV AN and M. KA ROUBI. Graded Brauer groups and K -theory with lo cal coefficients. Publ. Math. IHES 38, pp. 5-25 (1970). F rench summary in : Group e de Brauer et coefficients lo caux en K -t h´ eorie. Comptes Rendus Acad. Sci. P aris, t. 269, pp. 387-389 (1969). [3] M. KAROUBI. A lg ` ebres d e Clifford et K -th´ eorie. Ann. Sci. Ecole N orm. Su p. (4), pp. 161-270 (1968). [4] M. KAROUBI. Alg` ebres de Clifford et op´ erateurs d e F redholm. S pringer Lecture Notes in Maths N 136, p p . 66-106 (1970). Summary in Comptes Rendu s Acad. Sci. P aris, t. 267, pp. 305 (1968). [5] M. KA ROUBI. Sur la K -th´ eorie ´ equiva riante. Springer Lecture N otes in Math. N 136, pp. 187-253 (1970). [6] M. KAROUBI. Equiva riant K - theory of real vector spaces and real pro jective spaces. T opology and its app lications 122, pp. 531-546 (2002). [7] M. KAROUBI. Twisted K -t h eory , old and n ew. A rXiv math 0701789 (to app ear in the Journal of Eu ropean Math. S ociety in 2008). CLIFF O RD MODU L ES AND TWISTED K -THEOR Y 5 [8] J. ROSENBERG. Contin uous-trace algebras from the bund le theoretic p oint of view. J. Austral. Math. S o c. A 47, pp. 368-381 (1989). Max Karoubi Universit ´ e Paris 7 - Denis Diderot 2 place Jussieu 75251 P A RIS (FRAN CE) E-mail: max.ka roub i@gmail.com