Comparing two approaches to the K-theory classification of D-branes

SISSA 31/2008/ FM-EP Comparing t w o approac hes to the K-theory classification of D-branes F abio F errari Ruffino and Raffaele Sa v elli International Sc ho ol for A dvanc e d Studies (SI SSA/ISAS) Via Beirut 2, I-34014 , T rieste, Italy and Istituto Nazio n ale d i Fisic a Nucle ar e (INFN), sezion e di T rieste Abstract W e consider the t wo m ain classification metho ds of D-brane c har g es via K-theory , in t yp e I I sup erstring theory with v anishing B -field: the Gysin map approac h and the one based on the A tiy ah-Hirzebruc h sp ectral sequence . Then, w e find out an explicit link b etw een these t w o approach es: the Gysin map provides a represen tative elemen t of the equiv alence class obtained via the sp ectral sequence. W e also briefly discuss t he case o f ra tional co efficien ts, c ha r a cterized by a complete equiv alence b et w een the tw o classification metho ds . ferrariruffino@gmail.com, sav elli@sissa.it Con ten t s 1 In tro duction 1 2 Ph ysical motiv ations 2 2.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 K-theory from the Sen conjecture . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Gauge and gravitational couplings . . . . . . . . . . . . . . . . . . . . 4 2.2.2 K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3 Sen conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Linking the classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Useful notions of K-theory 9 3.1 Pro ducts in K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Non-compact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Thom isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 3.3 Gysin map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The Atiy ah-Hirzebruc h sp ectral sequence 14 4.1 Sp ectral sequence for a cohomology theory . . . . . . . . . . . . . . . . . . . 1 4 4.2 K-theory and simplicial cohomolog y . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 The sp ectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.1 The first step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3.2 The second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.3 The last step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1 4.3.4 F rom the first to the last step . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Rational K-theory and cohomolog y . . . . . . . . . . . . . . . . . . . . . . . 22 5 Gysin map and AHSS 23 5.1 Ev en case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.1.1 T rivial bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.1.2 Generic bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.2 Odd case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 5.3 The rational case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.1 Ev en case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3.2 Odd case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6 Conclusions and future p ersp ectiv es 31 1 In tro d u ction K-theory pro vides a go o d to ol to classify D-brane c harges in t yp e I I sup erstring t heory [7, 23]. In the case of v anishing B -field, there are t wo main approaches in the literature. The first one 1 consists of a pplying the Gysin map to the gauge bundle of the D-brane, obtaining a K-theory class in space-time [20]. T his approa ch is motiv ated by the Sen conjecture, stating that a generic configuration of branes and an tibranes with gauge bundle is equiv a len t, via tac h y on condensation, to a stac k o f coinciden t space-filling brane-antibrane pairs equipp ed with an appropriate K-t heory class [2 7]. The second approach consists of a pplying the A tiy ah- Hirzebruc h sp ectral sequence (AHSS, [3]) to the P oincar ´ e dual o f the homology class of the D-brane: suc h a sequence rules out some cycle s affected b y global w o rld-sheet anomalies, e.g. F reed-Witten ano ma ly [10], and quotients out some cycles whic h are actually unstable, e.g. MMS-instan to ns [18]. W e assume for simplicit y that the space-time and the D-brane w orld- v o lumes are compact. F or a giv en filtration of the space-time S = S 10 ⊃ S 9 ⊃ · · · ⊃ S 0 , the second step of AHSS is the cohomology o f S , i.e. E p, 0 2 ( S ) ≃ H p ( S, Z ), while the la st step of AHSS is giv en b y (up to canonical isomorphism): E p, 0 ∞ ( S ) ≃ Ker  K p ( S ) − → K p ( S p − 1 )  Ker  K p ( S ) − → K p ( S p )  . Hence, g iv en a D-brane world-v olume W Y p of co dimension 10 − ( p + 1) = 9 − p , with gauge bundle E → W Y p of rank q , if the P o incar´ e dual o f W Y p in S surviv es until the last step of AHSS, it determines a class { PD S [ q · W Y p ] } ∈ E 9 − p, 0 ∞ ( S ) whose represen tativ es b elong to Ker( K 9 − p ( S ) − → K 9 − p ( S 8 − p )). These t w o approac hes give differen t information, in particular AHSS do es not tak e in to accoun t the ga uge bundle: the aim of the presen t work is to relate them. W e briefly anticipate the result. F or a D p - brane with w orld-v o lume W Y p ⊂ S and ga uge bundle E → W Y p of rank q , let i : W Y p → S b e the em b edding and i ! : K ( W Y p ) → K 9 − p ( S ) t he Gysin map. W e will sho w that i ! ( E ) ∈ Ker( K 9 − p ( S ) → K 9 − p ( S 8 − p )) and that: { PD S [ q · W Y p ] } E 9 − p, 0 ∞ = [ i ! ( E )] . Th us, w e m ust first use AHSS to detect p ossible anomalies, t hen we can use the Gysin map to get the c ha rge of a non-anomalo us brane: suc h a c har g e b elongs to the equiv alence class reach ed b y AHSS, so t ha t the G ysin map giv es more detailed info rmation. F or further remarks ab out this we refer to the conclusions. Moreo v er, w e compare this picture with the case of rational co efficien ts. It is known that the Chern c haracter provides isomorphisms K ( S ) ⊗ Z Q ≃ H ev ( S, Q ) and K 1 ( S ) ⊗ Z Q ≃ H odd ( S, Q ), a nd that AHSS with rationa l co efficien ts degenerates at the second step, i.e. at the lev el of cohomology . Therefore, w e gain a complete equiv alence b etw een the tw o K-theoretical approaches , b eing b oth equiv alen t to the old cohomological classification. The pap er is organized as f o llo ws. In section 2 w e discuss in detail the phy sical con t ext underlying the K-theory classification of D-branes. I n section 3 and 4 w e introduce the top ological to ols needed to formulate our result, whic h is stated and pr ov en in section 5. In section 6 w e draw our conclusions. 2 Ph ysic al motiv ations F or simplicit y w e assume t he ten-dimensional space-time S to b e a compact manifold, so that also the D- brane world-v olumes are compact. This seems not ph ysically reasonable, 2 but it has more meaning if we supp ose to hav e p erfor med the Wick rotation in space-time, so that we work in a euclidean setting. In this setting w e lo ose the ph ysical inte rpretation of the D-brane w orld-v olume as a volume moving in time and of the c harge q (actually all the homology class [ q · Y p,t ] for Y p,t the restriction of the w o r ld- v o lume at an instan t t in a fixed reference frame) as a charge conserv ed in time. Th us, rather than considering the homology class of t he D-brane v olume at ev ery instan t of time, we prefer to consider the homolog y class of the en tire world-v olume in S , using standard ho mology with compact supp ort. 2.1 Classification F or a D p -brane with ( p + 1)-dimensional w orld-v olume W Y p and c harge q w e consider the corresp onding ho mo lo gy class in S : [ q · W Y p ] ∈ H p +1 ( S, Z ) = Z p +1 ( S, Z ) B p +1 ( S, Z ) = Z b p +1 ⊕ i Z p n i i (1) where Z p +1 ( S, Z ) denotes the gro up of singular ( p + 1)- cycles of S , B p +1 ( S, Z ) the subgroup of ( p + 1)- b oundaries, b p +1 the ( p + 1)- th Betti n um b er o f S , and p i is a prime num b er for ev ery i . F or what will follow , it is con v enien t to consider the cohomology of S rather tha n the homology . Hence, denoting b y PD S the P oincar´ e duality map on S , 1 w e define the char ge density : PD S [ q · W Y p ] ∈ H 9 − p ( S, Z ) = Z 9 − p ( S, Z ) B 9 − p ( S, Z ) = Z b p +1 ⊕ i Z p n i i (2) where Z 9 − p ( S, Z ) is the group of singular (9 − p )-co cyles and B 9 − p ( S, Z ) the subgroup of (9 − p )-cob oundaries. This classification encoun ters some problems due t o the presence of quan tum anomalies. Tw o remark able examples are the follo wing: • a brane wrapping a cycle W Y p ⊂ S is F reed-Witten anomalous if its third in tegral Stiefel-Whitney class W 3 ( W Y p ) is not zero, hence not all the cycles are allo w ed [10, 7]; • giv en a w o r ld-v olume W Y p with W 3 ( W Y p ) 6 = 0, it can b e in terpreted as an MMS- instan t o n in the mink ows kian setting [18, 7]; in t his case there are cycles in tersecting W Y p in PD W Y p ( W 3 ( W Y p )) whic h, altho ugh homo lo gically non- trivial in general, are actually unstable. The t w o p oints ab ov e imply that: • the num erator Z p +1 ( S, Z ) o f (1) is to o large, since it con t a ins anomalous cycles; • the denominator B p +1 ( S, Z ) of (1) is to o small, since it do es not cut all the unstable c ha r g es. There are other p ossible anomalies, although not ye t completely understo o d, some o f whic h are probably r elat ed to homology classes not represen ta ble b y a smo oth submanifold [8, 4, 7]. 1 As w e sa id above, we are assuming for s implicit y that the space-time is a compact manifo ld (without singularities), a nd we also supp ose it is or ient able, thu s Poincar´ e duality holds. 3 W e start b y considering the case of w orld-v olumes of even co dimension in S , i.e. w e start with I IB superstring theory . T o solv e the problems men tioned ab ov e, one p ossible to ol seems to b e the A tiy ah- Hirzebruch sp ectral sequence [3]. Cho osing a finite simplicial decomp osition [12] of the space-time manifold S , and considering the filtratio n S = S 10 ⊃ · · · ⊃ S 0 for S i the i -th dimensional sk eleton, suc h a s p ectral sequen ce starts from the ev en-dimensional simplicial co chains of S and, after a finite n um b er of steps, it stabilizes to the graded group E ev , 0 ∞ ( S ) = L 2 k K 2 k ( S ) /K 2 k + 1 ( S ). Here K q ( S ) is the k ernel of the natural restriction map from the K-theory gro up of S , whic h w e denote by K ( S ), to the K-theory gro up of S q − 1 , whic h w e call K ( S q − 1 ): i.e. K q ( S ) = Ker( K ( S ) → K ( S q − 1 )). W e also use the notaio n E 2 k, 0 ∞ ( S ) = K 2 k ( S ) /K 2 k + 1 ( S ), so that E ev , 0 ∞ ( S ) = L 2 k E 2 k, 0 ∞ ( S ). W e can start from a represen tat ive o f the P oincar ´ e dual of the D -brane PD S [ q · W Y p ], whic h in our hypotheses is ev en-dimensional, and, if it surviv es un til the last step, w e arrive at a class { PD S [ q · W Y p ] } ∈ K 9 − p ( S ) /K 9 − p +1 ( S ). The ev en b oundaries d 2 , d 4 , . . . of this sequence are 0, hence the imp ortant ones a re the o dd b o undar ies. In pa rticular, one can prov e that: • d 1 coincides with the ordina r y cob oundary op erat or, hence the second step is the ev en cohomology of S [2 6, 3]; • the co cycles not living in the k ernel of d 3 are F reed-Witten anomalous, while the co cycles con tained in it s image are unstable b ecause of the presence of MMS-instantons [7, 18]. As w e will say in a while, there are g o o d reasons to use K-theory to classify D-brane charges, hence, although the ph ysical meaning of higher order b oundaries is not completely clear, the b eha viour o f d 3 and the fact that the last step is directly related to K-t heory suggest t ha t the class { PD S [ q · W Y p ]) } ∈ E 9 − p, 0 ∞ ( S ) is a go o d candidate t o b e considered as the c harge of the D-brane. Summarizing, w e sa w tw o w a ys to classify D -brane cycles and c harges: • the homological classification, i.e. [ q · W Y p ] ∈ H p +1 ( S, Z ); • the classification via the A tiy ah-Hirzebruc h spectral sequence, i.e. { PD S [ q · W Y p ] } ∈ E 9 − p, 0 ∞ ( S ). 2.2 K-theory from the Sen conjecture 2.2.1 Gauge and grav itational c ouplings Up to now w e ha v e only considered the cycle wrapp ed b y t he D-bra ne world-v olume. There are other imp or tan t features: the ga uge bundle and the em b edding in space-time, whic h en ter in the action via t he t wo follo wing couplings: • the gauge coupling through the Chern character [17] of the Chan-P aton bundle; • the gra vitational coupling through the ˆ A -gen us [17] of the tangen t and the normal bundle of the w orld-v olume. The unique non-anomalous form of these couplings, computed b y Minasian and Mo ore in [20], is: S = Z W Y p i ∗ C ∧ ch( E ) ∧ e d 2 ∧ √ ˆ A ( T ( W Y p )) √ ˆ A ( N ( W Y p )) (3) 4 where i : W Y p → S is t he em b edding, E is the Chan-P aton bundle, T ( W Y p ) and N ( W Y p ) are the tangen t bundle and the normal bundle of W Y p in S , and d ∈ H 2 ( W Y p , Z ) is a class whose restriction mod 2 is the second Stiefel-Whitney class of the normal bundle w 2 ( N ( W Y p )). The p olyform that m ultiplies i ∗ C has 0- term equal to c h 0 ( E ) = rk( E ), hence (3) is an extens ion of the usual minimal coupling q R W Y p i ∗ C p +1 for q = rk( E ): the c harge of the D-brane coincides with the ra nk of the gauge bundle (up to a normalization constant). In the case of anti- branes, w e ha v e to allow for negativ e c har ges, hence the ga ug e bundle is actually a K-the ory class : a generic class E − F can b e in terpreted as a stac k of pairs of a brane Y and an an ti-brane Y with gauge bundle E and F resp ectiv ely . F or i # : H ∗ ( W Y p , Q ) → H ∗ ( S, Q ) the Gysin map in cohomolog y [13, 23], w e define the cha r ge d ensity : Q W Y p = i #  c h ( E ) ∧ e d 2 ∧ √ ˆ A ( T ( W Y p )) √ ˆ A ( N ( W Y p ))  . (4) Since new terms ha v e app eared in the c harg e, w e should discuss also t heir quan tization, whic h is not immediate since the Chern character and the ˆ A -gen us are intrins ically ratio nal cohomology classes. T o av oid the discuss ion of these problems [21], in the express ion (3) w e supp ose C to b e globally defined, which implies tha t the field strength G = d C is trivial in the de-Rahm cohomology at an y degree. 2 F or a general discussion see [9]. W e put for notationa l con ve nience: G ( W Y p ) = e d 2 ∧ √ ˆ A ( T ( W Y p )) √ ˆ A ( N ( W Y p )) . The action (3) is equal to : S = Z PD W Y p (c h( E )) i ∗ C ∧ G ( W Y p ) . Let { q k · W Y k } b e the set of D- branes app earing in the P oincar ´ e dual of c h( E ) in W Y p (w e mean that w e c ho ose a represen tativ e cycle for each homolog y class in PD W Y p (c h ( E )) and w e think of it as a subbrane of W Y p ): the first one is PD W Y p (c h 0 ( E )) = q · W Y p , so it giv es rise to the action without gauge coupling. The other ones a re lo we r dimensional branes. Let us consider the first one, i.e. W Y (1) = PD W Y p (c h 1 ( E )). Then the correp onding term in the action is R W Y (1) i ∗ C ∧ G ( W Y p ), whic h can b e written as R W Y (1) i ∗ C ∧ G ( W Y (1) ) + R W Y (1) i ∗ C ∧ ( G ( W Y p ) − G ( W Y (1) )). Since in the second term the sum G ( W Y p ) − G ( W Y (1) ) has 0-term equal to 0, then PD W Y (1) ( G ( W Y p ) − G ( W Y 1 )) is made only b y low er-dimensional subbranes. Let W Y (1 , 1) b e the first one: we get R W Y (1 , 1) i ∗ C , whic h is equal to R W Y (1 , 1) i ∗ C ∧ G ( W Y (1 , 1) ) + R W Y (1) i ∗ C ∧ (1 − G ( W Y (1 , 1) )). The second term giv es rise only to low er dimensional subbranes. 2 Actually the assumption that C is globally defined do es not solve the pro blem, since one should ta ke int o a ccount the la rge ga uge transforma tio ns C p +1 → C p +1 + Φ p +1 with Φ p +1 int egral but not necessa r ily exact. It turns out that the a ction (3) is well-defined under these gauge transfor mations only under the suitable quantization conditions we hav e mentioned above. Anyw ay , for a fixed g lo bal C p +1 formula (3) is meaningful, a nd this is enough for our p ourp oses here. 5 Pro ceeding inductiv ely until we arriv e at D0- branes, whose G -term is 1, w e can write: Z W Y (1) i ∗ C ∧ G ( W Y p ) = m X h =0 Z W Y (1 ,h ) i ∗ C ∧ G ( W Y (1 ,h ) ) where, for h = 0, W Y (1 , 0) = W Y (1) holds. Pro ceeding in the same w ay for eve ry W Y ( k ) , w e obtain a set of subbranes { q k ,h · W Y ( k, h ) } , whic h, using only one index, w e still denote by { q k · W Y ( k ) } . Therefore, calling i k : W Y ( k ) → S the em b edding, we g et: S = X k Z W Y ( k ) i ∗ k C ∧ G ( W Y ( k ) ) . F rom this expression w e see that the br ane W Y p with gauge a nd gr avitational c ouplings is e quivalent to the set of sub-b r anes W Y ( k ) with trivial gauge bund le . Moreov er w e no w sho w that the follow ing equalit y holds: i #  c h ( E ) ∧ G ( W Y p )  = X k ( i k ) # G ( W Y ( k ) ) (5) i.e. the c harge densities o f the tw o configurations are the same. In order to prov e this, w e recall the formulas : i # ( α ∧ i ∗ β ) = i # ( α ) ∧ β Z W Y p α = Z S i # ( α ) (6) for α ∈ H ∗ ( W Y p , Q ) a nd β ∈ H ∗ ( S, Q ). Th us: Z W Y p i ∗ C ∧ c h( E ) ∧ G ( W Y p ) = Z S i #  i ∗ C ∧ c h( E ) ∧ G ( W Y p )  = Z S C ∧ i #  c h( E ) ∧ G ( W Y p )  X k Z W Y p i ∗ k C ∧ G ( W Y ( k ) ) = X k Z S ( i k ) #  i ∗ k C ∧ G ( W Y ( k ) )  = Z S C ∧ X k ( i k ) #  G ( W Y ( k ) )  . Since the tw o terms ar e equal for ev ery form C , w e get formula ( 5 ). W e thus get: Splitting principle: a D -br ane W Y p with g a uge bund le is dynamic al ly e quiv- alent to a set of s ub-b r anes W Y ( k ) with trivial gauge bund le, such that the total char ge density of the two c onfigur ations is the same. The physic al in terpretatio n of this conjecture is the phenomenon of tac h y on condensation [27, 28, 7]: the quantization o f strings extending from a brane to an an tibrane leads to a tac h yonic mo de, whic h represen ts an instability and generates a pro cess of annihilatio n of brane and antibrane w orld-v olumes via an R G-flo w [1], lea ving lo w er dimensional branes. 6 In particular, giv en a D-brane W Y p with gauge bundle E → W Y p , w e can write E = ( E − rk E ) + rk E , so that E − rk E ∈ ˜ K ( W Y p ), where ˜ K ( W Y p ) is the reduced K-theory group of W Y p [2]: th us w e think of this configuration as a triple made b y a D-bra ne W Z p with gaug e bundle rk E , a brane W Y p with gauge bundle E and an antibrane W Z p with gauge bundle rk E . By tach y o n condensation only W Z p remains (with trivial bundle, i.e. only with its o wn c harge), while W Y p and W Z p annihilate giving rise to low er dimensional branes with trivial bundle, as stated in the splitting principle. Moreov er, if w e consider a stac k of pairs ( W Y p , W Y p ) with gauge bundles E and F r espective ly , this is equiv alen t to consider gauge bundles E ⊕ G and F ⊕ G resp ectiv ely , since, viewing the factor G as a stac k of pairs ( W Z p , W Z p ) with the same gauge bundle, it happ ens that by tach y on condensation W Z p and W Z p disapp ear, leav ing no other subbranes. This is the phys ical in terpretation of the stable equiv alence relation in K-theory . This principle, as w e will see, is an in v erse of the Sen conjecture, but we will actully use it to show the Sen conjecture in this setting. Remark: the splitting principle holds only at ratio na l lev el, since it inv olv es Chern c har- acters and ˆ A -gen us. A t integral lev el, w e do not state suc h a principle. 2.2.2 K-theory Since w e are assuming the H -flux to v anish, in order not to b e F reed-Witten anomalous the D-brane m ust b e spin c . Since the whole space-time is spin, in particular also spin c , it follows that the normal bundle of the D-brane is spin c to o. Therefore we can consider the K-theory Gysin map i ! : K ( W Y p ) → K ( S ) [17]. W e recall the differen tia ble Riemann-Ro c h theorem [13, 23]: c h ( i ! ( E )) ∧ ˆ A ( T S ) = i #  c h ( E ) ∧ e d 2 ∧ ˆ A ( T ( W Y p ))  . (7) Using (7) and (6) w e obtain: Z W Y p i ∗ C ∧ ch( E ) ∧ e d 2 ∧ √ ˆ A ( T ( W Y p )) √ ˆ A ( N ( W Y p )) = Z S C ∧ c h  i ! ( E )  ∧ q ˆ A ( T S ) . Th us w e get: S = Z S C ∧ c h  i ! ( E )  ∧ q ˆ A ( T S ) hence: Q W Y p = ch( i ! E ) ∧ q ˆ A ( T S ) . (8) In this wa y , (8) is ano ther expression for Q W Y p with resp ect to (4 ), but with a n imp ort a n t difference: the ˆ A -factor do es not dep end on W Y p , hence all Q W Y p is a function only of E . Th us, w e can consider i ! E as the K-theory analogue of the c ha rge densit y , considered as an inte gr al K-theory class. The use of Chern c haracters, instead, obliges to consider rational classes, whic h cannot contain information ab out the torsion pa rt. 7 2.2.3 Sen conjecture Let us consider the tw o expressions found for the rational c harge densit y: Q (1) W Y p = i #  c h ( E ) ∧ G ( W Y p )  Q (2) W Y p = c h( i ! E ) ∧ q ˆ A ( T S ) . Q (2) W Y p is exactly the c harge densit y of a stac k o f D9- branes and anti-branes (whose world- v o lume coincides with S ), whose g auge bundle is the K- theory class i ! E . Hence , expressing the c harge in the form Q (2) W Y p for eac h D-brane in o ur bac kground is equiv alen t to think that there exists only one stac k of couples brane-antibrane of dimension 9 enco ding all the dynamics. Hence w e form ulate the conjecture [2 7, 2 8 ]: Sen conjecture: every c onfigur ation of br anes and anti-br ane s with any gauge bund le is dynamic al ly e quivalent to a c onfigur ation with only a stack of c oincident p airs br ane-antibr ane of dimension 9 with an appr opriate K-the ory class on it. In o rder to see that the dynamics is actually equiv alen t, w e use the splitting principle stated ab ov e: since Q (1) W Y p = Q (2) W Y p , the brane W Y p with the c harge Q (1) W Y p and the D 9-brane with c ha r g e Q (2) W Y p split into t he same set of subbranes (with trivial gauge bundle). W e remark that in order to state the Sen conjecture is necessary that the H -flux v a nishes. Indeed, the space-time is spin c (it is spin since space-time spinors exist, therefore also spin c ), hence F reed- Witten ano maly cancellation for D9- branes requires that H = 0. Actually , an appropriat e stac k of D9-branes can b e consisten t for H a torsion class [16], but w e do not consider this case in the presen t pap er. In order to fo rm ulate b oth t he splitting principle and the Sen conjecture, w e ha v e only considered the action, hence o nly r ationa l classes giv en by Chern c haracters and ˆ A -gen us. Th us, w e can classify the charge density in the t w o following wa ys: • as a rationa l cohomology class i # (c h ( E ) ∧ G ( W Y p )) ∈ H ev ( S, Q ); • as a rationa l K-theory class i ! E ∈ K Q ( S ) := K ( S ) ⊗ Z Q . These t w o classification sc hemes are completely equiv a len t due t o the fact that the map: c h( · ) ∧ q ˆ A ( T S ) : K Q ( S ) − → H ev ( S, Q ) is an isomorphism. Th is equiv alence is lost at the inte gral leve l, since the torsion parts of K ( S ) and H ev ( S, Z ) a re in general differen t. Moreov er, since at the in tegral lev el the splitting principle do es not apply , w e cannot prov e that the Sen conjecture holds: the classification via Gysin map a nd cohomology are differen t, and the use of the G ysin map is just suggeste d b y the equiv alence at rat ional lev el, i.e. b y the equiv alence of the dynamics. Moreo v er, f or the in tegral case, w e ha ve also seen the classification via the A tiy ah- Hirzebruc h sp ectral sequence (AHSS). In the rational case, we can build t he corresp onding sequence AHSS Q [3], ending at the gro ups Q ev , 0 ∞ ( S ), but it stabilizes at the second step, i.e. at the lev el of cohomology . Hence, the class { i # (c h ( E ) ∧ G ( W Y p )) } ∈ Q ev , 0 ∞ ( S ) is completely equiv a len t to t he cohomology class i # (c h ( E ) ∧ G ( W Y p )) ∈ H ev ( S, Q ). 8 2.3 Linking the classifications T o summarize, w e are trying to classify the c harges of D-branes in a compact euclidean space-time S . In order to ac hiev e t his, w e can use cohomology or K-theory , with integer or rational co efficien ts, obt a ining the p ossibilities show ed in table 1. In t eger Rational Cohomology PD S [ q · W Y p ] ∈ H 9 − p ( S, Z ) i #  c h ( E ) ∧ G ( W Y p )  ∈ H ev ( S, Q ) K-theory (Gysin map) i ! ( E ) ∈ K ( S ) i ! ( E ) ∈ K Q ( S ) K-theory (AHSS) { PD S [ q · W Y p ] } ∈ E 9 − p, 0 ∞ ( S )  i # (c h( E ) ∧ G ( W Y p ))  ∈ Q ev , 0 ∞ ( S ) T able 1: Classifications In the r a tional case, as w e hav e seen, there is a complete equiv alence of the three ap- proac hes, since the three groups w e consider, i.e. L 2 k H 2 k ( S, Q ), K Q ( S ) a nd L 2 k Q 2 k, 0 ∞ ( S ) are canonically isomorphic. Instead, in the integral case there a r e no suc h isomorphisms (in general the three g roups are all differen t), and there is a strong a symmetry due to the fact that in the ho mo lo gical and AHSS classifications the gauge bund le and the gr avitational c ou- pling ar e not c onsider e d at al l , while they a re of course tak en in to accoun t in the Gysin map approac h. Up to now we discussed the case of ev en-co dimensional bra nes: that is b ecause the Gysin map requires an ev en-dimensional norma l bundle in order to tak e v alue in K ( S ). W e will discuss also the o dd-dimensional case, considering t he g r o up K 1 ( S ), and the picture will b e similar. Since the in t egr a l approach es are not equiv alen t, w e ha v e t o in ves tigate the relations among them: it is clear how to link the cohomolo gy class and the AHSS class, since the second step of AHSS is exactly the cohomology . Our aim is to find an explicit link b et ween the Gysin map approach and the one based on AHSS. 3 Useful notions o f K-theo r y W e briefly recall the ma in K-theoretical constructions whic h will b e used in the following. In this section we use the follo wing notations: X and Y are top ological spaces, K ( X ) is the K-theory group of X , ˜ K ( X ) is the reduced K-theory gro up of X , K n ( X ) is the K-theory group of degree n o f X and ˜ K n ( X ) is the reduced K-theory group of degree n of X [2, 17]. If f : X → Y is a contin uous map and E is a v ector bundle o n Y , w e denote by f ∗ E the pull-bac k of E on X ; if α = [ E ] − [ F ] is a K-theory class on Y , w e denote b y f ∗ α its pull-bac k f ∗ α = [ f ∗ E ] − [ f ∗ F ]. Moreov er: 9 • fixing tw o mark ed p oints x 0 ∈ X and y 0 ∈ Y , w e put X ∨ Y := ( { x 0 } × Y ) ∪ ( X × { y 0 } ) and X ∧ Y := ( X × Y ) / ( X ∨ Y ); • w e denote b y X + the one- p oin t compactification of X [6]. W e call {∞} the p oint added in suc h a compactification. 3.1 Pro d u cts in K-theory F or X a top ological space, K ( X ) has a natura l ring structure giv en by the tensor pro duct: [ E ] ⊗ [ F ] := [ E ⊗ F ]. Suc h a pro duct restricts to ˜ K ( X ). In general, w e can define a pro duct: K ( X ) ⊗ K ( Y ) ⊠ − → K ( X × Y ) (9) where, if π 1 : X × Y → X a nd π 2 : X × Y → Y are the pro jections, E ⊠ F = π ∗ 1 E ⊗ π ∗ 2 F . The fib er of E ⊠ F at ( x, y ) is E x ⊗ E y . 3 W e no w pro v e that, fixing a mark ed p oint for X and Y , the pro duct (9) restricts t o (see [24]): ˜ K ( X ) ⊗ ˜ K ( Y ) ⊠ − → ˜ K ( X ∧ Y ) . (10) F or this, w e first state that : 4 ˜ K ( X × Y ) ≃ ˜ K ( X ∧ Y ) ⊕ ˜ K ( Y ) ⊕ ˜ K ( X ) . (11) In fact: • since X is a retract of X × Y via the pro jection, w e ha v e that ˜ K ( X × Y ) = K ( X × Y , X ) ⊕ ˜ K ( X ) = ˜ K ( X × Y /X ) ⊕ ˜ K ( X ) (see [2]); • since Y is a retra ct of X × Y /X via the pro jection, w e also hav e ˜ K ( X × Y /X ) = K ( X × Y /X , Y ) ⊕ ˜ K ( Y ) = ˜ K ( X ∧ Y ) ⊕ ˜ K ( Y ). Com bining w e obtain (11). W e describ e the explicit isomorphism. W e call i 1 : X → X × Y and i 2 : Y → X × Y the immersions defined by i 1 ( x ) = ( x, y 0 ) and i 2 ( y ) = ( x 0 , y ), and, for α ∈ K ( X × Y ), we put α | X := ( i 1 ) ∗ α and α | Y := ( i 2 ) ∗ α . Then, for α = [ E ] − [ F ] ∈ ˜ K ( X × Y ), the explicit isomorphism in (11) is: α − →  α − π ∗ 1 ( α | X ) − π ∗ 2 ( α | Y )  ⊕ α | Y ⊕ α | X . Let α ∈ ˜ K ( X ) and β ∈ ˜ K ( Y ): then ( α ⊠ β ) | X = 0 and ( α ⊠ β ) | Y = 0. In fact, one has: ( α ⊠ β ) | X = α ⊗ ( π ∗ 2 β ) | X = α ⊗ i ∗ 1 π ∗ 2 β = α ⊗ ( π 2 i 1 ) ∗ β . But π 2 i 1 : X → Y is the constan t map with v alue y 0 , and the pull-bac k of a bundle b y a constan t map is trivial: since β is a reduced K-theory class, it f ollo ws t ha t ( π 2 i 1 ) ∗ β = 0 . Similarly for Y . Hence, b y (11), w e obtain that α ⊠ β ∈ ˜ K ( X ∧ Y ). 3 If X = Y a nd ∆ : X → X × X is the diagonal embedding, then E ⊗ F = ∆ ∗ ( E ⊠ F ). 4 (11) is ac tua lly true for ˜ K n ( X × Y ) for any n , with the same pr o of. 10 3.1.1 Non-compact case F or a generic (also non-compact) space X , we use K - theory with compact supp ort, i.e. we define K ( X ) := ˜ K ( X + ) (for compact spaces this definition coincides with t he usual one up to canonical isomorphism). One can easily pro v e that X + ∧ Y + = ( X × Y ) + , considering as mark ed p o ints on X and Y the p oints at infinit y . Henc e, the pro duct (10) b ecomes exactly: K ( X ) ⊗ K ( Y ) ⊠ − → K ( X × Y ) (12) also for the non-compact case. 3.2 Thom isomorphism Let X be a c omp act top ological space a nd π : E → X a v ector bundle (real or complex): w e sho w tha t K ( E ) has a natural structure o f K ( X )-mo dule. It seems natural to use the pull-bac k π ∗ : K ( X ) → K ( E ), but this is not p ossible: in fact, the group K ( E ) is defined as the reduced K- theory group o f E + , and in general there a r e no p ossibilities to extend con tin uously the pro jection π to E + . Henc e w e use the pro duct (1 2): considering the em b edding i : E → X × E defined b y i ( e ) = ( π ( e ) , e ), 5 whic h trivially ex tends to i : E + → ( X × E ) + requiring that i ( ∞ ) = ∞ , w e can define a pro duct: K ( X ) ⊗ K ( E ) − → K ( E ) α ⊗ β − → i ∗ ( α ⊠ β ) . (13) This pro duct defines a structure of K ( X )-mo dule on K ( E ). Lemma 3.1 K ( E ) is unitary as a K ( X ) -mo dule. Pro of: Let us consider the following ma ps: π 1 : X + × E + − → X + π 2 : X + × E + − → E + i : E + − → ( X × E ) + ˜ π : X + × E + − → X + ∧ E + = ( X × E ) + ˜ π 2 : ( X × E ) + − → E + where i ( e ) = ( π ( e ) , e ) and the others are defined in t he ob vious w a y . Sinc e the map: r : X + × E + − →  X + × {∞}  ∪  {∞} × E +  giv en by r ( x, e ) = ( x, ∞ ) and r ( ∞ , e ) = ( ∞ , e ) 6 is a retraction, ˜ π ∗ : ˜ K (( X × E ) + ) → ˜ K ( X + × E + ) is injectiv e [2 ]. Then, b y the definition of the mo dule structure, for α ∈ K ( X ) = ˜ K ( X + ) and β ∈ K ( E ) = ˜ K ( E + ) w e reform ulate (13) as: 7 α · β = i ∗ ( ˜ π ∗ ) − 1 ( α ⊠ β ) = i ∗ ( ˜ π ∗ ) − 1 ( π ∗ 1 α ⊗ π ∗ 2 β ) . 5 F or suc h an embedding it is not necessary to have a marked p oint on X . 6 The map r is contin uo us beca use X is co mpact, so that its ∞ -p oint is disjoint from it. 7 With resp ect to (13) we think α ⊠ β ∈ ˜ K ( X + × E + ) and we write explicitly ( ˜ π ∗ ) − 1 . 11 F or α = 1 one has α | X = X × C and α | {∞} = 0. Hence: (1 ⊠ β )   X × E + = π ∗ 2 β   X × E + (1 ⊠ β )   {∞}× E + = 0 . But: • since π 2   X × E + = ( ˜ π 2 ◦ ˜ π )   X × E + , one has π ∗ 2 β   X × E + = ˜ π ∗ ˜ π ∗ 2 β   X × E + ; • since ˜ π 2 ◦ ˜ π ( {∞} × E + ) = { ∞} and β ∈ ˜ K ( E + ), one has ( ˜ π ∗ ˜ π ∗ 2 β )   {∞}× E + = 0. Hence 1 ⊠ β = ˜ π ∗ ˜ π ∗ 2 β , so that: 1 · β = i ∗ ( ˜ π ∗ ) − 1 ˜ π ∗ ˜ π ∗ 2 β = i ∗ ˜ π ∗ 2 β = ( ˜ π 2 ◦ i ) ∗ β = id ∗ β = β .  Let us consider a v ector space R 2 n as a v ector bundle on a p o in t { x } . Then we ha ve: • K ( { x } ) = Z ; • K ( R 2 n ) = ˜ K (( R 2 n ) + ) = ˜ K ( S 2 n ) = Z . Hence K ( { x } ) ≃ K ( R 2 n ). The idea of the Thom isomorphism is to extend this isomorphism to a generic bundle E → X with fiber R 2 n . T o ach iev e this, we try to write suc h an isomorphism in a w a y that extends to a generic bundle. Actually , this generalization works for E a spin c -bundle of ev en dimension. Let us consider the spin group Spin(2 n ) [17 ]. The spin represen tation acts on C 2 n , and it splits in the t w o irreducible represen tations of p ositiv e and negativ e c hiralit y , acting on the subspaces S + and S − of C 2 n of dimension 2 n − 1 . Also the g roup Spin c (2 n ), defined as Spin(2 n ) ⊗ Z 2 U (1), acts on C 2 n via the standard spin c represen tation, and the same splitting in chiralit y holds: we call the tw o correspo nding subspaces S + C and S − C when we t hink of them as Spin c (2 n )-mo dules instead of Spin(2 n )-mo dules. F or C l(2 n ) the complex Clifford algebra of dimension 2 n , C 2 n is also a C l(2 n )-mo dule, and, for v ∈ R 2 n ⊂ C l(2 n ), we hav e v · S + C = S − C . W e thus consider the follow ing complex: 0 − → R 2 n × S + C c − → R 2 n × S − C − → 0 where c is the Clifford m ultiplication b y the first comp onen t: c ( v , z ) = ( v , v · z ). Suc h a sequence o f trivial bundles o n R 2 n is exact when restricted to R 2 n \ { 0 } , hence the a lt ernat ed sum: λ R 2 n =  R 2 n × S − C  −  R 2 n × S + C  naturally giv es a class in K ( R 2 n , R 2 n \ { 0 } ) [2 ]. T he sequence is exact in particular in R 2 n \ B 2 n , where B 2 n is the op en ball of radius 1 in R 2 n , hence it defines a class: λ R 2 n ∈ K ( R 2 n , R 2 n \ B 2 n ) = ˜ K ( B 2 n /S 2 n − 1 ) = ˜ K ( S 2 n ) . 12 One can pro v e that, f or η the dual of the ta uto logical line bundle o n CP 1 , whose sheaf of sections is usually denoted as O CP 1 (1), if w e iden tify S 2 with CP 1 top ologically , w e hav e that: λ R 2 n = ( − 1) n · ( η − 1) ⊠ n (14) i.e. λ R 2 n is a generator of ˜ K ( S 2 n ) ≃ Z [2]. W e no w sho w the generalization to a spin c -bundle π : E → X o f dimension 2 n . Let S ± C ( E ) b e the bundles of complex chiral spinors asso ciated to E : to define them, we consider a spin c -lift of t he ortho g onal frame bundle SO( E ), whic h w e call Spin c ( E ), and w e define S C ( E ) as the ve ctor bundle with fib er C 2 n asso ciated to the spin c represen tation, the latter b eing induced b y the action of the complex Clifford algebra via the inclusion Spin c (2 n ) ⊂ C l(2 n ) ֒ → C 2 n . T his bundle splits in to S C ( E ) = S + C ( E ) ⊕ S − C ( E ); moreov er, S C ( E ) is naturally a C l( E )-mo dule. W e can lift S ± C ( E ) to E b y π ∗ . Then we consider the complex: 0 − → π ∗ S + C ( E ) c − → π ∗ S − C ( E ) − → 0 where c is the Clifford m ultiplication given b y the structure of C l( E )- mo dule: for e ∈ E and s e ∈ ( π ∗ S + C ( E )) e , w e define c ( s e ) = e · s e . Suc h a sequence is exact when r estricted to E \ B ( E ), where, for an y fixed metric on E , B ( E ) is the union of the o p en balls o f radius 1 on eac h fib er. Hence w e can define the Thom class : λ E = [ π ∗ S − C ( E )] − [ π ∗ S + C ( E )] (15) as a class in K ( E , E \ B ( E ) ) = ˜ K ( B ( E ) / S ( E ) ) = ˜ K ( E + ) = K ( E ). T he follo wing fundamen ta l theorem holds ( [1 7, 15] and, only f or the complex case, [2, 24]): Theorem 3.2 (Thom isomorphism) L et X b e a c omp act top olo gic a l sp ac e and π : E → X an ev en dimens i onal spin c -bund le. F or λ E = [ π ∗ S − C ( E )] − [ π ∗ S + C ( E )] ∈ K ( E ) the map, define d using the mo dule structur e (13) : T : K ( X ) − → K ( E ) α → α · λ E is a gr oup isomorphism. W e can no w see that the construction for a generic 2 n -dimensional spin c -bundle E → X is a generalization of the construction for R 2 n . In fact, for x ∈ X : •  π ∗ S ± C ( E )    E x = E x ×  S ± C ( E )  x ≃ R 2 n × S ± C ( R 2 n ); • the Clifford multiplic ation restricts on each fib er E x to the Clifford m ultiplicatio n in R 2 n × S C ( R 2 n ). Hence: λ E   E x ≃ λ R 2 n . (16) In particular, w e see that, for i : E + x → E + , t he restriction i ∗ : K ( E ) → K ( E x ) ≃ Z is surjectiv e. 13 3.3 Gysin map Let X b e a compact smo oth n -manifold and Y ⊂ X a compact em b edded p - submanifold suc h t ha t n − p is ev en and the normal bundle N ( Y ) = ( T X | Y ) / T Y is s pin c . T hen, since Y is compact, there exists a tubular neighborho o d U of Y in X , i.e. there exists an homeomorphism ϕ U : U → N ( Y ). If i : Y → X is the embedding, from this da t a w e can naturally define a group homo- morphism, called Gysin m ap : i ! : K ( Y ) − → ˜ K ( X ) . In fact: • w e first apply the Thom isomorphism T : K ( Y ) − → K ( N ( Y )) = ˜ K ( N ( Y ) + ); • then w e na turally extend ϕ U to ϕ + U : U + − → N ( Y ) + and apply ( ϕ + U ) ∗ : K ( N ( Y )) − → K ( U ); • there is a na t ural map ψ : X → U + defined b y: ψ ( x ) =  x if x ∈ U ∞ if x ∈ X \ U hence w e apply ψ ∗ : K ( U ) → ˜ K ( X ). Summarizing: i ! ( α ) = ψ ∗ ◦ ( ϕ + U ) ∗ ◦ T ( α ) . (17) Remark: One could try to use the immersion i : U + → X + and the retraction r : X + → U + to hav e a splitting K ( X ) = K ( U ) ⊕ K ( X , U ) = K ( Y ) ⊕ K ( X , U ). This is false, since the immersion i : U + → X + is not con tinuous: sinc e X is c omp act , {∞} ⊂ X + is op en, but i − 1 ( {∞} ) = {∞ } , and {∞} is not op en in U + since U is not compact. 4 The A tiy ah-Hirzebruch sp e c tral se quence W e recall ho w to construct t he A tiy a h-Hirzebruc h sp ectral sequence, and w e introduce the to ols w e need in order to link it with the G ysin map. 4.1 Sp ectral sequence for a cohomology theory W e deal with sp ectral sequences in t he axiomatic v ersion described in [5], chap. XV, par. 7, with the additional h yp otesis of w orking with finite sequences of groups. W e also tak e into accoun t the presence of the gra ding in cohomology . In particular, w e supp ose the fo llowing assignemen ts are give n fo r p, p ′ , p ′′ ∈ Z ∪ {−∞ , + ∞} : • for −∞ ≤ p ≤ p ′ ≤ ∞ , ab elian groups H n ( p, p ′ ) for n ∈ Z , suc h that H n ( p, p ′ ) = H n (0 , p ′ ) fo r p ≤ 0 and there exists l ∈ N suc h that H n ( p, p ′ ) = H n ( p, + ∞ ) for p ′ > l ( l do es not dep end on n in our setting); 14 • for p ≤ p ′ ≤ p ′′ , a, b ≥ 0, p + a ≤ p ′ + b , tw o maps: 8 ψ n : H n ( p + a, p ′ + b ) → H n ( p, p ′ ) δ n : H n ( p, p ′ ) → H n +1 ( p ′ , p ′′ ) (18) satisfying axioms (SP .1)-( SP .5 ) of [5], p. 33 4. When the indices are no t clear from the con text, we also use t he notations ( ψ n ) p + a,p ′ + b p,p ′ and ( δ n ) p,p ′ ,p ′′ for the maps (18). W e can describe the groups and the cob oundaries of the sp ectral sequence in the following wa y: E p, q r = Im  H p + q ( p, p + r ) ψ p + q − → H p + q ( p − r + 1 , p + 1)  ([5], formula (8) p. 318 ) d p, q r = ( δ p + q ) p − r +1 ,p +1 ,p + r +1   Im(( ψ p + q ) p,p + r p − r +1 ,p +1 ) : E p,q r − → E p + r,q − r +1 r ([5], line 3 p. 319 ) F p, q H = Im  H p + q ( p, + ∞ ) ψ p + q − → H p + q (0 , + ∞ )  ([5], line -10 p. 319 ) . (19) Then: • the groups F p, q H are a filtration o f H p + q (0 , + ∞ ); • L p,q E p, q r +1 ≃ H  L p,q E p, q r , L p,q d p, q r  canonically , i.e. E p, q r +1 ≃ Ker d p, q r / Im d p − r, q + r − 1 r ; • the sequence { E p, q r } r ∈ N stabilizes to F p, q H /F p +1 , q − 1 H . In particular, considering the follo wing comm utativ e diagram 9 ([5], end of p. 31 8): H p + q ( p, p + r ) ψ 1 / / δ 1   H p + q ( p − r + 1 , p + 1) δ 2   H p + q +1 ( p + r , p + 2 r ) ψ 2 / / H p + q +1 ( p + 1 , p + r + 1) (20) the follow ing iden tities ho ld: • Im( ψ 1 ) = E p, q r and Im( ψ 2 ) = E p + r, q − r +1 r ; • d p, q r = δ 2   Im( ψ 1 ) : E p, q r → E p + r, q − r +1 r . The limit of the sequence L p F p, q H /F p +1 , q − 1 H can also b e defined as ([5], eq. (3) p. 3 16): E p, q 0 H := E p, q ∞ = Im  H p + q ( p, + ∞ ) ψ p + q − → H p + q (0 , p + 1)  (21) i.e. E p, q 0 H ≃ F p, q H /F p +1 , q − 1 H canonically . 8 The map δ is calle d in the same wa y in [5]. Instead, w e introduce the name ψ since the analog o us ma p in [5] has no na me. 9 The maps ψ 1 , ψ 2 , δ 1 , δ 2 of the diagram are maps of the family (18); here and in the following w e use this notation in order not to write to o ma ny indices . 15 Giv en a top ological space X with a finite filtration: ∅ = X − 1 ⊂ X 0 ⊂ · · · ⊂ X m = X w e can consider a cohomology theory h • [12] and define (for p ≤ p ′ ≤ p ′′ ; a, b ≥ 0; p + a ≤ p ′ + b ): • H n ( p, p ′ ) = h n ( X p ′ − 1 , X p − 1 ); • ψ n : H n ( p + a, p ′ + b ) → H n ( p, p ′ ) is induced (thanks to t he axioms o f cohomology) b y the map of couples i : ( X p ′ − 1 , X p − 1 ) → ( X p ′ + b − 1 , X p + a − 1 ); • δ n : H n ( p, p ′ ) → H n ( p ′ , p ′′ ) is the comp osition of the map π ∗ : h n ( X p ′ − 1 , X p − 1 ) → h n ( X p ′ − 1 ) induced by the map of couple s π : ( X p ′ − 1 , ∅ ) → ( X p ′ − 1 , X p − 1 ), and t he Bo c kstein map β n : h n ( X p ′ − 1 ) → h n +1 ( X p ′′ − 1 , X p ′ − 1 ). Remark: the shift by − 1 in the definition of H n ( p, p ′ ) is necessary in order to ha v e the equalit y H n (0 , + ∞ ) = h n ( X ). It would not b e necessary if we decleared X 0 = ∅ , but this is not coheren t with the case of simplicial complexes , since, in that case, X 0 denotes the 0-sk eleton. Since K-theory is a cohomology theory , it is natural to consider the sp ectral sequence asso ciated to it for a giv en filtratio n ∅ = X − 1 ⊂ X 0 ⊂ · · · ⊂ X m = X : suc h a sequence is called A tiy ah-Hirzebruc h sp ectral sequence (AHSS). In particular, groups and maps are defined in the following wa y (for p ≤ p ′ ≤ p ′′ ; a, b ≥ 0; p + a ≤ p ′ + b ): • H n ( p, p ′ ) = K n ( X p ′ − 1 , X p − 1 ); • ψ n : K n ( X p ′ + b − 1 , X p + a − 1 ) → K n ( X p ′ − 1 , X p − 1 ) is the pull-back via the ma p i : X p ′ − 1 /X p − 1 → X p ′ + b − 1 /X p + a − 1 (w e recall that, for spaces having the homotop y type of a finite simplicial complex, K n ( X , Y ) = ˜ K n ( X/ Y ) by definition [2]); • δ n : K n ( X p ′ − 1 , X p − 1 ) − → K n +1 ( X p ′′ − 1 , X p ′ − 1 ) is the comp osition of the map π ∗ : K n ( X p ′ − 1 , X p − 1 ) − → K n ( X p ′ − 1 ) induced b y π : X p ′ − 1 → X p ′ − 1 /X p − 1 , a nd the K- theory Bo ck stein map δ n : K n ( X p ′ − 1 ) − → K n +1 ( X p ′′ − 1 , X p ′ − 1 ). 4.2 K-theory and simplicial cohomolo gy In the pro of of the following lemma w e will need the definition of r e duc e d and unr e duc e d susp ension of a top olo g ical space X . W e recall that the unreduced susp ension is defined a s ˆ S 1 X = ( X × [ − 1 , 1]) / ( X × {− 1 } , X × { 1 } ), i.e. as the do uble cone built o n X . Instead, fixing a mar ked p oint x 0 ∈ X , the reduced susp ension is defined as S 1 X = ˆ S 1 X/ ( { x 0 } × [ − 1 , 1]). The group K 1 ( X ) is defined as K ( S 1 X ), but, since S 1 X is obtained f r om ˆ S 1 X quotienting out b y a con tractible subspace, it follo ws that K ( S 1 X ) ≃ K ( ˆ S 1 X ) [2]. Lemma 4.1 F or k ∈ N and 0 ≤ i ≤ k , let: X = ˙ [ i =0 ,...,k X i 16 b e the one-p o i n t union of k top olo gic al s p ac es. Then: ˜ K n ( X ) ≃ k M i =0 ˜ K n ( X i ) . Pro of: F or n = 0, let us construct the isomorphism ϕ : ˜ K ( X ) → L ˜ K ( X i ): it is simply giv en by ϕ ( α ) i = α | X i , where α | X i is the pull- bac k via the immersion X i → X . T o build ϕ − 1 , let us consider { [ E i ] − [ n i ] } ∈ L ˜ K ( X i ), where [ n i ] is the K -theory class represen ted b y the trivial bundle of rank n i . Since the sum is finite, by adding and subtracting a trivial bundle w e can suppose n i = n j for ev ery i, j , so that we consider { [ E i ] − [ n ] } . Since the in tersection of the X i is a p oin t and the bundles E i ha v e the same ra nk, w e can g lue them to a bundle E on X ( see [2] pp. 20-21): then w e declare ϕ − 1 ( { [ E i ] − [ n ] } ) = ([ E ] − [ n ]). F or n = 1, w e first note that ˜ K ( ˆ S 1 ( X 1 ˙ ∪ X 2 )) = ˜ K ( ˆ S 1 X 1 ˙ ∪ ˆ S 1 X 2 ), since quotien t ing by a con tractible space (the linking betw een v ertices of the cones and the joining p oin t) w e obtain the same space. Hence ˜ K 1 ( X 1 ˙ ∪ X 2 ) ≃ ˜ K 1 ( X 1 ) ⊕ ˜ K 1 ( X 2 ). Then, by induction, the thesis extends to finite families. Hence w e ha v e pro v en the result f or ˜ K n with n = 0 and n = 1: b y Bott p erio dicit y [2] the result holds fo r an y n .  Remark: w e stress the fact that the previous lemma holds only for the one-p oin t union of a finite n um b er of spaces. In the following theorem w e supp ose that the group of simplicial co c hains C p ( X , Z ) of a finite simplicial complex coincides with the gr o up o f c hains C p ( X , Z ): that’s b ecause, b eing the dimension finite, w e can define the cob oundary o p erator δ p directly on c hains, asking that t he cob oundary of a simplicial p -simplex σ p is the alternated sum of the ( p + 1)-simplices whose b oundary con tains σ p (while the b oundary o p erator ∂ p giv es the alternated sum o f the ( p − 1)-simplices con tained in the b oundary of σ p ). W e can use this definition since the group of p -co c hiains as usually defined, i.e. Hom( C p ( X , Z ) , Z ), is canonically isomorphic to C p ( X , Z ) in the case o f finite simplicial complexes, and t he usual cob oundary op erat o r corresp onds to the one we defined ab ov e under suc h a n isomorphism. Theorem 4.2 L et X b e a n -dim e nsional finite simplicial c omplex , X p b e the p -ske l e ton of X for 0 ≤ p ≤ n a nd C p ( X , Z ) b e the gr oup of simplicia l p -c o chain s . T h en, for any p such that 0 ≤ 2 p ≤ n or 0 ≤ 2 p + 1 ≤ n , ther e ar e isomorphis m s: Φ 2 p : C 2 p ( X , Z ) ≃ − → K ( X 2 p , X 2 p − 1 ) Φ 2 p +1 : C 2 p +1 ( X , Z ) ≃ − → K 1 ( X 2 p +1 , X 2 p ) which c an b e summarize d by: Ψ p : C p ( X , Z ) ≃ − → K p ( X p , X p − 1 ) . Mor e over: K 1 ( X 2 p , X 2 p − 1 ) = K ( X 2 p +1 , X 2 p ) = 0 . 17 Pro of: W e denote the simplicial structure of X b y ∆ = { ∆ m i } , where m is the dimension of the simplex and i enume rates the m -simplices, so that X 2 p = k [ i =0 ∆ 2 p i . Then the quotien t b y X 2 p − 1 is homeomorphic to k spheres of dimension 2 p attac hed to a p oint: X 2 p /X 2 p − 1 = ˙ [ i S 2 p i . By lemma 4.1 w e obtain ˜ K ( X 2 p /X 2 p − 1 ) ≃ L i ˜ K ( S 2 p ), and, by Bott p erio dicit y , ˜ K ( S 2 p ) = ˜ K ( S 0 ) = Z . Hence: K ( X 2 p , X 2 p − 1 ) ≃ M i Z = C 2 p ( X , Z ) . F or the o dd case, let X 2 p +1 = h [ j =0 ∆ 2 p +1 j . W e ha v e b y lemma 4.1: K 1 ( X 2 p +1 , X 2 p ) = ˜ K 1  ˙ [ j S 2 p +1 j  = M j ˜ K 1  S 2 p +1 j  = M j ˜ K ( S 2 p +2 j ) = M j Z = C 2 p +1 ( X , Z ) . In the same wa y , K 1 ( X 2 p , X 2 p − 1 ) = L j ˜ K 1 ( S 2 p j ) = L j ˜ K ( S 2 p +1 j ) = 0, and similarly for K ( X 2 p +1 , X 2 p ).  F or η the dual of the tautolog ical line bundle on C P 1 , whose sheaf of sections is usually denoted as O CP 1 (1), if we iden tify S 2 with CP 1 top ologically , the explicit isomorphisms Φ 2 p and Φ 2 p +1 of theorem 4.2 are: Φ 2 p  ∆ 2 p i  =    ( − 1) p ( η − 1) ⊠ p ∈ ˜ K  S 2 p i  0 ∈ ˜ K  S 2 p j  for j 6 = i and: Φ 2 p +1  ∆ 2 p +1 i  =    ( − 1) p +1 ( η − 1) ⊠ ( p +1) ∈ ˜ K 1  S 2 p +1 i  0 ∈ ˜ K 1  S 2 p +1 j  for j 6 = i where w e put the ov erall factor s ( − 1) p and ( − 1) p +1 for coherence with (14). Remark: suc h isomorphisms are canonical, since ev ery simplex is supposed to b e orien ted and η − 1 is distinguishable from 1 − η also up t o automorphisms of X (in the first case the trivial bundle has negative co efficien t, in the second case the non-trivial one, so that , for example, they hav e opp osite first Chern class). 18 4.3 The sp ectral sequence W e no w recall ho w to build the sp ectral sequence. The assigned groups are: H n ( p, p ′ ) = K n ( X p ′ − 1 , X p − 1 ) . 4.3.1 The first st ep The first step, from (1 9), is: E p, q 1 = H p + q ( p, p + 1) = K p + q ( X p , X p − 1 ) . By theorem 4.2 we hav e the isomorphisms: E 2 p, 0 1 ≃ C 2 p ( X , Z ) E 2 p, 1 1 = 0 E 2 p +1 , 0 1 ≃ C 2 p +1 ( X , Z ) E 2 p +1 , 1 1 = 0 . Since, for a p oint x 0 , K ( { x 0 } ) = Z and K 1 ( { x 0 } ) = 0, w e can write these isomorphisms in a compact form: E p, q 1 ≃ C p ( X , K q ( x 0 )) . (22) An ywa y , since E p, 1 1 = 0 fo r ev ery p , and since only t he parity o f q is meaningful, the only in teresting case is q = 0. Therefore, from no w on w e deal o nly with the groups E p, 0 r . F or the cob oundaries, since d p, q r : E p, q r → E p + r, q − r +1 r , in part icular d p, 0 r : E p, 0 r → E p + r, − r +1 r , if r is ev en the cob oundary is surely 0 , th us only the o dd cob oundaries are in teresting. Therefore, from no w on w e deal only with the cob oundaries d p, 0 r with r o dd. F or r = 1, in the diagram (20) one ha s ψ 1 = ψ 2 = id, hence d p, 0 1 = δ 2 , i.e. d p, 0 1 = ( δ p ) p,p +1 ,p +2 . In particular: d p, 0 1 : K p ( X p , X p − 1 ) − → K p +1 ( X p +1 , X p ) is the comp osition: ˜ K p ( X p /X p − 1 ) d p, 0 1 / / ( π p,p − 1 ) ∗ ' ' O O O O O O O O O O O ˜ K p +1 ( X p +1 /X p ) ˜ K p ( X p ) . δ p 7 7 n n n n n n n n n n n n for π p,p − 1 : X p → X p /X p − 1 the natural pro j ection and δ p is the Bo c kstein map. Another w ay to describ e d p, 0 1 can b e obtained considering the exact sequence induced b y X p /X p − 1 − → X p +1 /X p − 1 − → X p +1 /X p : then d p, 0 1 is the corresp onding Bo c kstein map: d p, 0 1 : ˜ K p ( X p /X p − 1 ) − → ˜ K p +1 ( X p +1 /X p ) . (23) 19 4.3.2 The second step W e ha v e sho wn that E p, 0 1 ≃ C p ( X , Z ); we also hav e that E p, 0 2 ≃ H p ( X , Z ) (see [3]), i.e. d p, 0 1 is the simplicial cob o undary o p erator under the isomorphism (22). By the first formula of (19) w e hav e E p, 0 2 = Im  H p ( p, p + 2) ψ p − → H p ( p − 1 , p + 1)  , i.e.: E p, 0 2 = Im  K p ( X p +1 , X p − 1 ) ψ p − → K p ( X p , X p − 2 )  . (24) Th us there is a canonical isomorphism: Ξ p : H p ( X , Z ) − → Im ψ p ⊂ K p ( X p , X p − 2 ) . (25) Co cycles and cob oundaries W e now consider the maps: ˜ j p : X p /X p − 1 − → X p +1 /X p − 1 ˜ π p : X p /X p − 2 − → X p /X p − 1 = X p /X p − 2 X p − 1 /X p − 2 ˜ f p : X p /X p − 2 − → X p +1 /X p − 1 These maps induce a commutativ e diagram: E p, 0 1 = ˜ K p ( X p /X p − 1 ) ( ˜ π p ) ∗ ) ) S S S S S S S S S S S S S S S ˜ K p ( X p +1 /X p − 1 ) ( ˜ j p ) ∗ O O ( ˜ f p ) ∗ / / ˜ K p ( X p /X p − 2 ) (26) where ( ˜ f p ) ∗ , ( ˜ j p ) ∗ , ( ˜ π p ) ∗ are maps of the ψ -type. W e ha v e that E p, 0 2 = Im( ˜ f p ) ∗ b y (24 ). W e no w pro v e that: 1. Ker d p, 0 1 = Im( ˜ j p ) ∗ ; 2. Im d p − 1 , 0 1 = Ker( ˜ π p ) ∗ . The first statemen t follows directly from (23) using the exact sequence: · · · − → ˜ K p ( X p +1 /X p − 1 ) ( ˜ j p ) ∗ − → ˜ K p ( X p /X p − 1 ) d p, 0 1 − → ˜ K p +1 ( X p +1 /X p ) − → · · · and the second by the exact sequenc e: · · · − → ˜ K p − 1 ( X p − 1 /X p − 2 ) d p − 1 , 0 1 − → ˜ K p ( X p /X p − 1 ) ( ˜ π p ) ∗ − → ˜ K p ( X p /X p − 2 ) − → · · · . Since Im( ˜ f p ) ∗ ≃ H p ( X , Z ) and d p, 0 1 corresp onds to the simplicial cob oundary under this isomorphism, it follows that : • co cycles in C p ( X , Z ) corresp ond to classes in Im( ˜ j p ) ∗ , i.e. to classes in ˜ K p ( X p /X p − 1 ) that are restriction o f classes in ˜ K p ( X p +1 /X p − 1 ); 20 • cob oundaries in C p ( X , Z ) corresp o nds t o classes in Ker( ˜ π p ) ∗ , i.e. to classes in ˜ K p ( X p /X p − 1 ) that are 0 when lifted to ˜ K p ( X p /X p − 2 ); • Im π ∗ corresp onds to co c hains (not only co cycles) up to cob oundaries and its subset Im( ˜ f p ) ∗ corresp onds to cohomolog y classes; • giv en α ∈ Im ( ˜ f p ) ∗ , we can lift it to a class in ˜ K p ( X p /X p − 1 ) choo sing differen t triv- ializations on X p − 1 /X p − 2 , a nd the differen t homotopy classes of such trivializatio ns determine the differen t respresen ta tiv e co cycles of t he class. 4.3.3 The last step W e recall equation (2 1): E p, q ∞ = Im  H p + q ( p, + ∞ ) ψ p + q − → H p + q (0 , p + 1)  whic h, in our case, b ecomes: E p, 0 ∞ = Im  K p ( X , X p − 1 ) ψ p − → K p ( X p )  (27) where ψ is obtained by the pull-bac k of f p : X p → X/X p − 1 . Since, for i p : X p → X the natural immersion and π p : X → X/X p the natural pro jection, f p = π p − 1 ◦ i p holds, the follo wing diagram comm utes: ˜ K p ( X/X p − 1 ) ( π p − 1 ) ∗ & & N N N N N N N N N N N ( f p ) ∗ / / ˜ K p ( X p ) ˜ K p ( X ) . ( i p ) ∗ 9 9 s s s s s s s s s (28) Remark: in the previous triangle we cannot say that ( i p ) ∗ ◦ ( π p − 1 ) ∗ = 0 by exactness, since b y exactness ( i p ) ∗ ◦ ( π p ) ∗ = 0 at the same lev el p , as follow s from X p → X → X/X p . The sequenc e K p ( X , X p − 1 ) ( π p − 1 ) ∗ − → K p ( X ) ( i p − 1 ) ∗ − → K p ( X p − 1 ) is exact, i.e.: Im( π p − 1 ) ∗ = Ker( i p − 1 ) ∗ . Since trivially Ker( i p ) ∗ ⊂ K er( i p − 1 ) ∗ , w e obtain that Ker( i p ) ∗ ⊂ Im( π p − 1 ) ∗ . Moreov er: Im ψ = Im  ( i p ) ∗ ◦ ( π p − 1 ) ∗  = Im  ( i p ) ∗   Im( π p − 1 ) ∗  ≃ Im( π p − 1 ) ∗ Ker( i p ) ∗ = Ker ( i p − 1 ) ∗ Ker( i p ) ∗ hence, finally: E p, 0 ∞ ≃ Ker  K p ( X ) − → K p ( X p − 1 )  Ker  K p ( X ) − → K p ( X p )  (29) i.e. E p, 0 ∞ is made, up to canonical isomorphism, by p -classes on X whic h are 0 on X p − 1 , up to classes whic h are 0 on X p . 21 4.3.4 F rom t he first to the last st ep W e no w see ho w to link the first a nd the last step of the sequence. In general we hav e: E p, q 1 = H p + q ( p, p + 1 ) E p, q ∞ = Im  H p + q ( p, + ∞ ) ψ 1 − → H p + q (0 , p + 1)  . for ψ 1 = ( ψ p + q ) p, + ∞ 0 ,p +1 . W e also consider the map: ψ 2 : H p + q ( p, + ∞ ) − → H p + q ( p, p + 1 ) where ψ 2 = ( ψ p + q ) p, + ∞ p,p +1 . An elemen t α ∈ E p, q 1 surviv es until the last step if and only if α ∈ Im ψ 2 and its class in E p, q ∞ is ψ 1 ◦ ( ψ − 1 2 )( α ), whic h is we ll-defined since Ker ψ 2 ⊂ Ker ψ 1 . F or: ψ 3 : H p + q ( p, p + 1 ) − → H p + q (0 , p + 1) i.e. ψ 3 = ( ψ p + q ) p,p +1 0 ,p +1 , it holds that ψ 1 = ψ 3 ◦ ψ 2 , so that ψ 1 ◦ ( ψ − 1 2 ) = ψ 3 . F or α ∈ Im ψ 2 ⊂ E p, q 1 , w e call { α } E p, q ∞ the class it reac hes in E p, q ∞ . Then we ha ve: { α } E p, q ∞ = ψ 3 ( α ) . F or AHSS this b ecomes: E p, 0 1 = K p ( X p , X p − 1 ) E p, 0 ∞ = Im  K p ( X , X p − 1 ) ψ 1 − → K p ( X p )  and: ψ 2 : K p ( X , X p − 1 ) − → K p ( X p , X p − 1 ) . In this case, ψ 2 = ( i p,p − 1 ) ∗ for i p,p − 1 : X p /X p − 1 → X/X p − 1 . Th us, the classes in E p, 0 1 surviving until the last s tep are the ones whic h are restrictions of a class defin ed on a ll X/X p − 1 . Moreov er, ψ 1 = ( f p ) ∗ for f p : X p → X/X p − 1 , and f p = i p,p − 1 ◦ π p,p − 1 for π p,p − 1 : X p → X p /X p − 1 . Hence ψ 1 = ( π p,p − 1 ) ∗ ◦ ψ 2 , and, in fact, ψ 3 = ( π p,p − 1 ) ∗ . This implies that, for α ∈ Im ψ 2 ⊂ E p, 0 1 : { α } E p, 0 ∞ = ( π p,p − 1 ) ∗ ( α ) . (30) 4.4 Rational K-theory and cohomology W e no w consider the A tiyah-Hirzebruc h sp ectral sequence in the rational case [3]. In partic- ular, w e consider the groups: H n ( p, p ′ ) = K n Q ( X p ′ − 1 , X p − 1 ) where K n Q ( X , Y ) := K n ( X , Y ) ⊗ Z Q . In this case the sequence is made by t he groups Q p, q r = E p, q r ⊗ Z Q . In pa r ticular: Q p, 0 1 ≃ C p ( X , Q ) Q p, 1 1 = 0 Q p, 0 2 ≃ H p ( X , Q ) Q p, 1 2 = 0 Q p, 0 ∞ ≃ Ker  K p Q ( X ) − → K p Q ( X p − 1 )  Ker  K p Q ( X ) − → K p Q ( X p )  Q p, 1 ∞ = 0 . (31) Suc h a sequence collapses at the second step [3], hence Q p, 0 ∞ ≃ Q p, 0 2 . Since: 22 • L p Q p, 0 ∞ is the graded group asso ciated to the c hosen filtration of K ( X ) ⊕ K 1 ( X ); • in pa r t icular, by (31), L 2 p Q 2 p, 0 ∞ is the gra ded group of K Q ( X ) and L 2 p +1 Q 2 p +1 , 0 ∞ is the graded group of K 1 Q ( X ); • Q p, 0 ∞ ≃ H p ( X , Q ), thus it has no torsion; it follows that: K Q ( X ) = M 2 p Q 2 p, 0 ∞ K 1 Q ( X ) = M 2 p +1 Q 2 p +1 , 0 ∞ hence: K Q ( X ) ≃ H ev ( X , Q ) K 1 Q ( X ) ≃ H odd ( X , Q ) . In particular, the isomorphisms of the last equation ar e g iven b y the Chern c haracter: c h : K Q ( X ) − → H ev ( X , Q ) c h : K 1 Q ( X ) − → H ev ( S 1 X , Q ) ≃ H odd ( X , Q ) and they are also isomorphism of rings. 5 Gysin map and AHSS W e are no w r eady t o describ e the explicit link b etw een the Gysin map and the Atiy ah- Hirzebruc h sp ectral sequence . W e start with the case of an em b edded sumb anifold of ev en co dimension, corresponding, from a phys ical p oint of view, to a D- brane world-v olume in t yp e I IB sup erstring theory , then w e repro duce the same result in t he case o f o dd co dimension, corresp onding to a D -brane w o rld-v olume in type I IA superstring theory . 5.1 Ev en case W e call X a compact smo oth n -dimensional manifold and Y a compact em b edded p -dimensional submanifold. W e c ho ose a finite triangulation of X whic h restricts to a triangula t ion of Y [22]. W e use the follow ing notation: • w e denote the triang ulation of X by ∆ = { ∆ m i } , where m is the dimension of the simplex and i en umerates the m -simplices; • w e denote by X p ∆ the p -sk eleton of X with resp ect to ∆. The same no t ation is used f o r other tr ia ngulations or simplicial decompositions of X a nd Y . In the follo wing theorem w e need the definition of “dual cell decomp osition” with respect to a triangulation: w e refer to [11] pp. 5 3 -54. Theorem 5.1 L et X b e an n -dimensional c om p act man ifold and Y ⊂ X a p -di m ensional emb e dde d c omp act submanif old. L et: • ∆ = { ∆ m i } b e a triangulation of X which r estricts to a triangulation ∆ ′ = { ∆ m i ′ } of Y ; 23 • D = { D n − m i } b e the dual de c omp o sition of X with r e s p e ct to ∆ ; • ˜ D ⊂ D b e subse t of D made by the duals o f the simplic es in ∆ ′ . Then, c al ling | ˜ D | the supp ort of ˜ D : • the interior of | ˜ D | is a tubular neighb orho o d of Y in X ; • the interior of | ˜ D | do es not interse ct X n − p − 1 D , i.e.: | ˜ D | ∩ X n − p − 1 D ⊂ ∂ | ˜ D | . Pro of: The n -simplices of ˜ D are the duals of the v ertices of ∆ ′ . Let τ = { τ m j } b e the first baricen tr ic sub division of ∆ [11, 12]. F o r eac h v ertex ∆ 0 i ′ in Y (thought of as an elemen t of ∆), its dual is: ˜ D n i ′ = [ ∆ 0 i ′ ∈ τ n j τ n j . (32) Moreo v er, if τ ′ = { τ ′ m j ′ } is the fir st baricen tric sub division of ∆ ′ (of course τ ′ ⊂ τ ) and D ′ = { D ′ m i ′ } is the dual of ∆ ′ in Y , t hen ( reminding that p is t he dimension of Y ): D ′ p i ′ = [ ∆ 0 i ′ ∈ τ ′ p j ′ τ ′ p j ′ (33) and: ˜ D n i ′ ∩ Y = D ′ p i ′ . Moreo v er, let us consider the ( n − p )-simplices in ˜ D con tained in ∂ ˜ D n i ′ (for a fixed i ′ in form ula (32)), i.e. X n − p ˜ D ∩ ˜ D n i ′ : they in tersect Y transv ersally in the baricen ters of eac h p -simplex of ∆ ′ con taining ∆ 0 i ′ : w e call suc h baricente rs { b 1 , . . . , b k } and the in tersecting ( n − p )-cells { ˜ D n − p l } l =1 ,...,k . Since (f o r a fixed i ′ ) ˜ D n i ′ retracts o n ∆ 0 i ′ , w e can consider a lo cal c ha r t ( U i ′ , ϕ i ′ ), with U i ′ ⊂ R n neigh b orho o d of 0, suc h that: • ϕ − 1 i ′ ( U i ′ ) is a neigh b orho o d of ˜ D n i ′ ; • ϕ i ′ ( D ′ p i ′ ) ⊂ U i ′ ∩ ( { 0 } × R p ), for 0 ∈ R n − p (see eq. (3 3)); • ϕ i ′ ( ˜ D n − p l ) ⊂ U i ′ ∩  R n − p × π p ( ϕ i ′ ( b l ))  , for π p : R n → { 0 } × R p the pro jection. W e now consider the natural foliation of U i ′ giv en by the in tersection with the hyperplanes R n − p × { x } and its image via ϕ − 1 i ′ : in this wa y , we obain a foliation of ˜ D n i ′ transv ersal to Y . If w e do this for any i ′ , by construction the v arious fo lia tions glue on the in t ersections, since such inters ections are giv en b y the ( n − p )-cells { ˜ D n − p l } l =1 ,...,k , and the inte rior giv es a C 0 -tubular neigh b orho o d of Y . Moreo v er, a ( n − p − r )-cell of ˜ D , f o r r > 0, cannot interse ct Y since it is contained in the b oundary of a ( n − p )-cell, and suc h cells inters ect Y , whic h is done b y p -cells, only in their inte rior p oin ts b j . Being the simplicial decomp osition finite, it follo ws that the in terior of | ˜ D | do es not inters ect X n − p − 1 D .  24 W e no w consider quin tuples ( X , Y , ∆ , D , ˜ D ) satisfying the followin g condition: (#) X is an n -dimensional compact manifo ld and Y ⊂ X a p - dimensional em b edded com- pact submanifold, suc h that n − p is ev en and the normal bundle N ( Y ) is spin c . Moreo v er, ∆, D and ˜ D are defined as in theorem 5.1. Lemma 5.2 L et ( X , Y , ∆ , D , ˜ D ) b e a quintuple satisfying (#) , U = In t | ˜ D | and α ∈ K ( Y ) . Then: • ther e exists a neighb orho o d V of X \ U such that i ! ( α ) | V = 0 ; • in p articular, i ! ( α ) | X n − p − 1 D = 0 . Pro of: By equation (17): i ! ( α ) = ψ ∗ β , β = ( ϕ + U ) ∗ ◦ T ( α ) ∈ ˜ K ( U + ) . Let β = [ E ] − [ n ], and let V ∞ ⊂ U + b e a neigh b orho o d of ∞ whic h trivializes E . Then ( ψ ∗ E )   ψ − 1 ( V ∞ ) is trivial. Hence, for V = ψ − 1 ( V ∞ ): ( ψ ∗ β )   V =  ( ψ ∗ E )   V  − [ n ] = [ n ] − [ n ] = 0 . By theorem 5.1, X n − p − 1 D do es not interse ct the tubular neighborho o d In t | ˜ D | of Y , hence X n − p − 1 D ⊂ ψ − 1 ( V ∞ ) = V , so that ( ψ ∗ β )   X n − p − 1 D = 0.  5.1.1 T rivial bundle W e start considering t he case o f a trivial bundle. Theorem 5.3 L et ( X , Y , ∆ , D , ˜ D ) b e a quintuple satisfying (#) and Φ n − p D : C n − p ( X , Z ) → K ( X n − p D , X n − p − 1 D ) b e the isomorphism state d in the or em 4.2. L et: π n − p, n − p − 1 : X n − p D − → X n − p D /X n − p − 1 D b e the pr oje ction and PD ∆ Y b e the r epr esentative of PD X [ Y ] ( f or [ Y ] the homolo gy class of Y ) given by the sum of the c el l s dual to the p -c el ls o f ∆ c overing Y . T hen: i ! ( Y × C ) | X n − p D = ( π n − p, n − p − 1 ) ∗ (Φ n − p D (PD ∆ Y )) . Pro of: W e define: ( U + ) n − p D = X n − p D | U X n − p − 1 D | ∂ U so that t here is a natural immersion ( U + ) n − p D ⊂ U + defined sending the denominator t o ∞ (the n umerator is exactly X n − p ˜ D of theorem 5.1). W e also define, considering the map ψ of equation (17): ψ n − p = ψ   X n − p D : X n − p D − → ( U + ) n − p D . 25 The lat t er is w ell-defined since the ( n − p )-simplices outside U and all the ( n − p − 1)-simplices are sen t to ∞ by ψ . Calling I the set of indices of the ( n − p )-simplices in D , calling S k the k -dimensional sphere and denoting b y ˙ ∪ the one-p oint union of top olog ical spaces , there are the follow ing canonical homeomorphisms: ξ n − p X : π n − p ( X n − p D ) ≃ − → ˙ [ i ∈ I S n − p i ξ n − p U + : ψ n − p ( X n − p D ) ≃ − → ˙ [ j ∈ J S n − p j where { S n − p j } j ∈ J , with J ⊂ I , is the set o f ( n − p )- spheres corresp onding to the ( n − p )- simplices with interior contanine d in U , i.e. corresp onding to π n − p  X n − p D   U  . The homeo- morphism ξ n − p U + is due to the fact that the b o undar y of the ( n − p )-cells in tersecting U is con tained in ∂ U , hence it is sen t to ∞ b y ψ n − p , while all the ( n − p )-cells outside U are sen t to ∞ : hence, t he imag e of ψ n − p is homeomorphic to ˙ S j ∈ J S n − p j sending ∞ to the attac hment p oin t. W e define: ρ : ˙ [ i ∈ I S n − p i − → ˙ [ j ∈ J S n − p j as the natural pro jection, i.e. ρ is the iden tity of S n − p j for ev ery j ∈ J and sends all the spheres in { S n − p i } i ∈ I \ J to the atta chmen t p oin t. W e hav e that : ξ n − p U + ◦ ψ n − p = ρ ◦ ξ n − p X ◦ π n − p, n − p − 1 hence: ( ψ n − p ) ∗ ◦ ( ξ n − p U + ) ∗ = ( π n − p, n − p − 1 ) ∗ ◦ ( ξ n − p X ) ∗ ◦ ρ ∗ . (34) W e put N = N ( Y ) and ˜ λ N = ( ϕ + U ) ∗ ( λ N ), where λ N is the Thom class of the normal bundle defined in equation (15). By lemma 3.1 and equation (17) w e hav e i ! ( Y × C ) = ψ ∗ ◦ ( ϕ + U ) ∗ ( λ N ). Then: i ! ( Y × C )   X n − p D = ψ ∗ ( ˜ λ N )   X n − p D = ( ψ n − p ) ∗  ˜ λ N   ( U + ) n − p D  and ( ξ n − p X ) ∗ ◦ ρ ∗ ◦ (( ξ n − p U + ) − 1 ) ∗  ˜ λ N   ( U + ) n − p D  = Φ n − p D (PD ∆ Y ) since: • PD ∆ Y is t he sum of the ( n − p )-cells in tersecting U ; • hence (( ξ n − p X ) − 1 ) ∗ ◦ Φ n − p D (PD ∆ Y ) gives a ( − 1) n − p 2 ( η − 1) ⊠ n − p 2 factor to eac h sphere S n − p j for j ∈ J and 0 ot herwise; • but this is exactly ρ ∗ ◦ (( ξ n − p U + ) − 1 ) ∗  ˜ λ N   ( U + ) n − p D  since b y equation (16) w e ha v e, for y ∈ Y : ( λ N )   N + y = λ R n − p ≃ ( − 1) n − p 2 ( η − 1) ⊠ n − p 2 26 and for the spheres outside U , that ρ sends to ∞ , w e ha v e that: ρ ∗  ˜ λ N   ( U + ) n − p D     ˙ S i ∈ I \ J S n − p i = ρ ∗  ˜ λ N   ρ ( ˙ S i ∈ I \ J S n − p i )  = ρ ∗  ˜ λ N   {∞}  = ρ ∗ (0) = 0 . Hence, from equation (3 4): i ! ( Y × C )   X n − p D = ( ψ n − p ) ∗  ˜ λ N   ( U + ) n − p D  = ( π n − p, n − p − 1 ) ∗ ◦ ( ξ n − p X ) ∗ ◦ ρ ∗ ◦ (( ξ n − p U + ) − 1 ) ∗  ˜ λ N   ( U + ) n − p D  = ( π n − p, n − p − 1 ) ∗ Φ n − p D (PD ∆ Y ) .  The follo wing theorem enco des the link b et w een the Gysin map and t he AHSS. Theorem 5.4 L et ( X , Y , ∆ , D , ˜ D ) b e a quintuple satisfying (#) and Φ n − p D : C n − p ( X , Z ) → K ( X n − p D , X n − p − 1 D ) b e the isomo rphism s tate d in the or em 4.2. L et us supp ose that PD ∆ Y is c ontaine d in the kernel of al l the b ound a ries d n − p, 0 r for r ≥ 1 . Then it defines a class: { Φ n − p D (PD ∆ Y ) } E n − p, 0 ∞ ∈ E n − p, 0 ∞ ≃ Ker( K ( X ) − → K ( X n − p − 1 )) Ker( K ( X ) − → K ( X n − p )) . The fol lowing e quality holds: { Φ n − p D (PD ∆ Y ) } E n − p, 0 ∞ = [ i ! ( Y × C )] . Pro of: By equations (27) and (2 8) w e ha ve the follow ing comm utativ e diag r am: E n − p, 0 ∞ = Im  ˜ K ( X/X n − p − 1 D ) ( π n − p − 1 ) ∗ ) ) S S S S S S S S S S S S S S S S ( f n − p ) ∗ / / ˜ K ( X n − p D )  ˜ K ( X ) ( i n − p ) ∗ 9 9 s s s s s s s s s s (35) and, giv en a represen tat ive α ∈ Im( π n − p − 1 ) ∗ = Ker( ˜ K ( X ) → ˜ K ( X n − p − 1 D )), w e ha v e that { α } E n − p, 0 ∞ = ( i n − p ) ∗ ( α ) = α | X n − p D . Moreov er: • the class { Φ n − p D (PD ∆ Y ) } E n − p, 0 ∞ , b y f orm ula (30), corresp o nds to the elemen t of ˜ K ( X n − p D ) defined b y ( π n − p, n − p − 1 ) ∗ (Φ n − p D (PD ∆ Y )) , for π n − p, n − p − 1 : X n − p D → X n − p D /X n − p − 1 D ; • b y lemma 5.2 w e ha v e i ! ( Y × C ) ∈ Ker( K ( X ) − → K ( X n − p − 1 D )), hence [ i ! ( Y × C )] is w ell-defined as an elemen t of E n − p, 0 ∞ and, b y exactness, i ! ( Y × C ) ∈ Im( π n − p − 1 ) ∗ ; • b y theorem 5.3 we ha ve ( i n − p ) ∗ ( i ! ( Y × C )) = ( π n − p, n − p − 1 ) ∗ (Φ n − p D (PD( Y ))); • hence { Φ n − p D (PD ∆ Y ) } E n − p, 0 ∞ = [ i ! ( Y × C )].  27 Let us consider a trivial ve ctor bundle of generic ra nk Y × C r . W e denote b y [ r ] it s K-theory class o n Y . By lemma 3.1 w e hav e that [ r ] · λ N = λ ⊕ r N , hence theorem 5.3 b ecomes: i ! ( Y × C r )   X n − p D = ( π n − p, n − p − 1 ) ∗  Φ n − p D (PD ∆ ( r · Y ))  and theorem 5.4 b ecomes: { Φ n − p D (PD ∆ ( r · Y )) } E n − p, 0 ∞ = [ i ! ( Y × C r )] . 5.1.2 Generic bundle If w e consider a generic bundle E o v er Y of rank r , w e can prov e that i ! ( E ) and i ! ( Y × C r ) ha v e the same restriction to X n − p D : in fact, the Thom isomorphism giv es T ( E ) = E · λ N and, if w e restrict E · λ N to a finite family of fib ers, whic h are transv ersal to Y , the contribution of E b ecomes trivial, so it has the same effect of the tr ivial bundle Y × C r . W e no w giv e a precise pro of of this statemen t. Lemma 5.5 L et ( X, Y , ∆ , D , ˜ D ) b e a quintuple satisfying (#) and π : E → Y a ve ctor bund le of r ank r . Th en: i ! ( E )   X n − p D = i ! ( Y × C r )   X n − p D . Pro of: referring to the not a tions in the pro of of lemma 3.1 , w e ha v e that: E · λ N = i ∗ ( ˜ π ∗ ) − 1 ( E ⊠ λ N ) = i ∗ ( ˜ π ∗ ) − 1 ( π ∗ 1 E ⊗ π ∗ 2 λ N ) . Since X n − p D in tersects the tubular neigh b orho o d in a finite n um b er of cells, corresp onding under ϕ + U to a finite num b er of fib ers of N , it is sufficien t to prov e that, for any y ∈ Y , ( E · λ N )   N + y = λ ⊕ r N   N + y . F irst of all: • i ( N + y ) = ( { y } × N y ) + ⊂ ( { y } × N ) + ; • E · λ N   N + y = ( i | N + y ) ∗  ( ˜ π ∗ ) − 1 ( π ∗ 1 E ⊗ π ∗ 2 λ N )    i ( N + y )  . T o obtain the bundle  ( ˜ π ∗ ) − 1 ( π ∗ 1 E ⊗ π ∗ 2 λ N )    i ( N + y ) , w e can restrict ˜ π to: A = ˜ π − 1 [ i ( N + y )] = ˜ π − 1  ( { y } × N y ) +  =  { y } × N + y  ∪  Y × {∞}  ∪  {∞} × N +  and consider ( ˜ π | A ∗ ) − 1  ( π ∗ 1 E ⊗ π ∗ 2 λ N )   A  . Mor eov er: • ( π ∗ 1 E ⊗ π ∗ 2 λ N )   { y }×N + y = ( C r ⊗ π ∗ 2 λ N )   { y }×N + y ≃ λ ⊕ r N   N + y ; • ( π ∗ 1 E ⊗ π ∗ 2 λ N )   Y ×{∞} = ( π ∗ 1 E ⊗ 0)   Y ×{∞} = 0; • ( π ∗ 1 E ⊗ π ∗ 2 λ N )   {∞}×N + = (0 ⊗ π ∗ 2 λ N )   {∞}×N + = 0. Hence, since the three comp onen t s o f A in tersect eac h other at most at one p o int, b y lemma 4.1 w e obtain: ( π ∗ 1 E ⊗ π ∗ 2 λ N )   A =  π ∗ 1 ( Y × C r ) ⊗ π ∗ 2 λ N    A .  28 Remark: In the statement of theorem 5 .4 (and of its generalization to a n y v ector bundle) it w as necessary to explicitly introduce a triangulation ∆ o n X , since the first step of the sp ectral seque nce consists of simplicial co c hains, whic h b y definition dep end on the simplicial structure c hosen. An yw ay , the groups E p, 0 r for r ≥ 2 and the filtra tion Ker( K ( X ) → K ( X n − p )) of K ( X ) do not dep end on the particular simplicial structure chos en [3], th us, if w e start fro m the cohomology class PD X [ Y ] a t t he second step of the sp ectral sequence (whic h is the D-br a ne c harge densit y with resp ect to the cohomological classification) w e can drop the dep endence on ∆, D and ˜ D . Therefore the c ho ice of the triangulation has no effect on the ph ysical classification of D-brane c harges. 5.2 Odd case W e now consider the case of n − p o dd (for n the dimension of X and p the dimension of Y ), corresp onding by a ph ysical p oint of view to ty p e I IA sup erstring theory . In this case the Gysin map ta kes v alue in K 1 ( X ), whic h is isomorphic to K ( ˆ S 1 X ), fo r ˆ S 1 X the unreduced suspension of X defined as: ˆ S 1 X = ( X × [ − 1 , 1]) / ( X × {− 1 } , X × { 1 } ) i.e. as a double cone built on X . W e thus consider the natural em b edding i 1 : Y → ˆ S 1 X and the corresp onding G ysin map: ( i 1 ) ! : K ( Y ) → K ( ˆ S 1 X ) ≃ K 1 ( X ) . Let U b e a tubular neighborho o d of Y in X , a nd let U 1 ⊂ ˆ S 1 X b e the tubular neigh b or- ho o d o f Y in ˆ S 1 X defined remo ving the v ertices of the double cone to ˆ S 1 U . W e hav e that ˆ S 1 ( X n − p D | U ) ⊂ U 1 and ˆ S 1 ( X n − p − 1 D | ∂ U ) ⊂ ∂ U 1 , where ∂ U 1 con taines also the v ertices of the double cone. In this w ay w e can riform ulate the previous results in the o dd case, considering ˆ S 1 ( X n − p D ) and ˆ S 1 ( X n − p − 1 D ) rather than X n − p D and X n − p − 1 D . W e consider quin tuples ( X , Y , ∆ , D , ˜ D ) safisfying the follow ing condition: (# 1 ) X is an n -dimensional compact manifo ld and Y ⊂ X a p - dimensional em b edded com- pact submanifold, such that n − p is o dd and N ( Y ) is spin c . Moreov er, ∆, D and ˜ D are defined as in theorem 5.1. W e now reform ulate the same theorems stated for the ev en case, whic h can b e prov ed in the same w ay . W e remark that N ˆ S 1 X Y is spin c if and o nly N X Y is, since N ˆ S 1 X Y = N X Y ⊕ 1 so that, b y the axioms of characteristic classes [19], W 3 is the same. Lemma 5.6 L et ( X , Y , ∆ , D , ˜ D ) b e a quintuple satisfying (# 1 ) and α ∈ K ( Y ) . Then: • ther e exists a neighb orho o d V of ˆ S 1 X \ U 1 such that i 1 ! ( α )   V = 0 ; • in p articular, i 1 ! ( α )   ˆ S 1 ( X n − p − 1 D ) = 0 .  29 Theorem 5.7 L et ( X , Y , ∆ , D , ˜ D ) b e a quintuple satisfying (# 1 ) and Φ n − p D : C n − p ( X , Z ) → K 1 ( X n − p D , X n − p − 1 D ) ≃ K ( ˆ S 1 ( X n − p D ) , ˆ S 1 ( X n − p − 1 D )) b e the isomo rphism state d in the or em 4.2. L et: π n − p, n − p − 1 : ˆ S 1 ( X n − p D ) − → ˆ S 1 ( X n − p D ) / ˆ S 1 ( X n − p − 1 D ) b e the pr oje ction and PD ∆ Y b e the r epr esentative of PD X [ Y ] ( f or [ Y ] the homolo gy class of Y ) given by the sum of the c el l s dual to the p -c el ls o f ∆ c overing Y . T hen: i 1 ! ( Y × C )   ˆ S 1 ( X n − p D ) = ( π n − p, n − p − 1 ) ∗ (Φ n − p D (PD ∆ Y )) .  Theorem 5.8 L et ( X , Y , ∆ , D , ˜ D ) b e a quintuple satisfying (# 1 ) and Φ n − p D : C n − p ( X , Z ) → K 1 ( X n − p D , X n − p − 1 D ) b e the i s omorphism state d in the or e m 4.2. L et us supp ose that PD ∆ Y is c ontaine d in the kernel of al l the b ound a ries d n − p, 0 r for r ≥ 1 . Then it defines a class: { Φ n − p D (PD ∆ Y ) } E n − p, 0 ∞ ∈ E n − p, 0 ∞ ≃ Ker( K 1 ( X ) − → K 1 ( X n − p − 1 )) Ker( K 1 ( X ) − → K 1 ( X n − p )) . The fol lowing e quality holds: { Φ n − p D (PD ∆ Y ) } E n − p, 0 ∞ = [( i 1 ) ! ( Y × C )] .  5.3 The rational case 5.3.1 Even case W e no w analyze the case of rationa l co efficien ts. W e define: K Q ( X ) := K ( X ) ⊗ Z Q . W e can th us classify the D-br a ne c harge densit y at ra t io nal lev el as i ! ( E ) ⊗ Z Q . The Chern c ha r a cter provide s an isomorphism c h : K Q ( X ) → H ev ( X , Q ). Since the square ro ot of ˆ A ( T X ) is a po lyform starting with 1, it a lso defines an isomorphism, so t ha t the comp osition: b c h : K Q ( X ) − → H ev ( X , Q ) b c h( α ) = c h ( α ) ∧ q ˆ A ( T X ) remains an isomorphism. Thus , the classifications with rational K- theory and rational co- homology are completely equiv alen t. W e can also define the r a tional Atiy ah-Hirzebruc h spectral sequence Q 2 k, 0 r ( X ) := E 2 k, 0 r ( X ) ⊗ Z Q . Such a seque nce collapses at the second step [3], i.e. at the lev el of cohomolog y: thus Q 2 k, 0 ∞ ( X ) ≃ Q 2 k, 0 2 ( X ). An explicit isomorphism is giv en by the appropriate comp onent of the Chern c haracter: c h n − p 2 : Ker  K Q ( X ) − → K Q ( X n − p − 1 )  Ker  K Q ( X ) − → K Q ( X n − p )  − → H n − p ( X , Q ) . 30 This map is w ell-defined since, for a bundle whic h is trivial on the ( n − p )-sk eleton, the Chern c haracters of degree less or equal to n − p 2 are zero [3] (in particular c h n − p 2 = b c h n − p 2 for a bundle whic h is trivial on the ( n − p − 1)-sk eleton). Moreo v er, since Q 2 k, 0 ∞ has no torsion: K Q ( X ) = M 2 k Q 2 k, 0 ∞ and an isomorphism can b e obtained splitting α ∈ K Q ( X ) as α = P 2 k α 2 k where c h( α 2 k ) = c h k ( α ). 5.3.2 Odd case In this case, w e hav e the isomorphism c h : K 1 Q ( X ) → H odd ( X , Q ). Moreo v er, H odd ( X , Q ) ≃ H ev ( ˆ S 1 X , Q ). Hence w e ha v e the corresp ondence among: • i 1 ! ( E ) ∈ K 1 Q ( X ); • b c h ( i 1 ! E ) ∈ H ev ( ˆ S 1 X , Q ) ≃ H odd ( X , Q ); • ⊕ 2 k  ( i 1 k ) ! ( Y k × C q k )  Q 2 k +1 , 0 ∞ . 6 Conclus ions and future p ers p ectiv es T o summarize, we hav e considered the classifications of D-brane c harges in a compact eu- clidean space-time S sh o wn in table 2. W e can no w explain the relations b etw een them. In t eger Rational Cohomology PD S [ q · W Y p ] ∈ H 9 − p ( S, Z ) i #  c h ( E ) ∧ G ( W Y p )  ∈ H ev ( S, Q ) K-theory (Gysin map) i ! ( E ) ∈ K ( S ) i ! ( E ) ∈ K Q ( S ) K-theory (AHSS) { PD S [ q · W Y p ] } ∈ E 9 − p, 0 ∞ ( S ) { i # (c h ( E ) ∧ G ( W Y p )) } ∈ Q ev , 0 ∞ ( S ) T able 2: Classifications W e already sa w the complete equiv alence of the three rational classifications, due to the isomorphisms H ∗ ( S, Q ) ≃ K ∗ Q ( S ) ≃ L k Q k , 0 ∞ , whic h split into ev en and o dd parts. F o r the in tegral classifications, the three approa ches are not equiv alen t, and o ur aim for this pap er w a s to clarify their relationships. The cohomological and AHSS appro ac hes hav e a clear link as one can see in the table, but they do not t ak e into accoun t the gauge and gravitational 31 couplings. Since we ha v e seen the link b et w een Gysin ma p and A tiy ah- Hirzebruc h sp ectral sequence , w e can also link the t w o corresp onding approache s. W e hav e pro v ed that, for a w o r ld- v o lume W Y p with gauge bundle E of rank q , i ! ( E ) ∈ Ker( K 9 − p ( S ) → K 9 − p ( S 8 − p )) and that: { PD S [ q · W Y p ] } E 9 − p, 0 ∞ = [ i ! ( E )] . Th us, w e can use AHSS to detect p ossible a nomalies, then we can use the Gysin map to get the charge of a non-anomalous brane: suc h a c harg e b elongs to the equiv alence class reac hed by AHSS, so that the Gysin map give s ric her information. Some commen ts are in order. One could ask why the additiona l inf o rmation prov ided b y the Gysin map has to b e considered: in fact, we ha v e prov en that it concerns the c hoice of a represen tativ e of the class, while, discussing AHSS in c ha pt er 2, w e ha v e seen tha t one o f its adv antages is that it quotien ts out unstable configur a tions. It seems that suc h additional information k eeps in to accoun t only instabilities. Actually , this is not the case. L et us consider a couple ( W Y p , i ! ( E )) made b y a D-brane w o r ld-v olume and its c harge with resp ect to the Gysin map approac h. The c harge do es no t provide complete information ab out the w orld- v o lume, since i ! E is a class in the whole space-time, exactly as the c harge q of a par t icle do es not pro vide infor ma t io n ab out its tra jectory . This is true a lso for the cohomo lo gical a nd AHSS classifications: t wo ho mologous w orld-v olumes are not the same tra jectory . If we consider t w o couples ( W Y p , i ! ( E )) and ( W Y p , i ! ( F )), we know that [ i ! ( E ) − i ! ( F )] E 9 − p, 0 ∞ = 0, which means that i ! ( E ) − i ! ( F ) lies in the image of some b oundaries of AHSS. Let us supp o se that it lies in the image of d 3 . This means t hat there exists an unstable w orld-v olume W U p with a gauge bundle, e.g. the trivial one, suc h that i ! ( W U p × C ) = i ! ( E ) − i ! ( F ), but the tw o terms of the latter equalit y concern different w orld-v olumes with the same zero c harge: in fact, W U p has c harge 0 b ecause it lies in the image of d 3 , while i ! ( E − F ) has c harge 0 since, b eing rk( E − F ) = 0, it is a represen tativ e of the class reached star t ing from 0 · W Y p . An yw ay , the w olrd-v olume W Y p is not anomalo us in general and the fact that the gauge bundle on it is E or F is a meaningful informatio n. Actually the informat io n contained in i ! ( E − F ) is partially contained in the c harges of the sub-bra nes of W Y p . Th us, w e can apply AHSS to the w or ld- v o lume of the D- brane, then, if it corresp onds to the trivial class w e consider it as an unstable one, otherwise w e can consider each represen tativ e o f the class as an additional meaningful information. P o ssibile generalizations of this work are the follow ing: • admitting the presence of the B -field compatibly with F reed-Witten anomaly , consid- ering also twis ted K-t heory and the correspo nding twis ted AHSS; • considering the case of non-compact space-time a nd w orld-v olumes, using the appro- priate form of AHSS; • studying branes with singularities, using the a ppropriate form of the Gysin map. Ac knowledgemen ts W e w ould lik e to thank Loria no Bono ra f or the helpfulness he alw ays show ed since w e started to w ork with him. W e a r e also really gr ateful to Jarah Evslin for man y suggestions and for 32 ha ving p oin ted out man y subtleties. W e a lso thank Giulio Bonelli and Ugo Bruzzo for useful discussions . References [1] A. Adams, J. Polc hinski, E. Silvers tein, D on ’t Panic! Close d String T achyons in ALE Sp ac etimes , JHEP 0110 (2001) 02 9 a rXiv: hep-th/01080 75. [2] M. Atiy ah, K-the ory , Addison-W esley Publishing Compan y , Inc., 1978. [3] M. 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