The age grading and the Chen-Ruan cup product

THE A GE GRADING AND THE CH EN-RU AN CUP PRODUCT RICHARD HEPW OR TH Abstract. W e prov e that the obstruction bundle used to define the cup- product in Chen-Ruan c ohomology is determined b y the so-called age g r ading or de gr e e- shifting numb ers . Indeed, the obstruction bundle can b e directly computed using the age grading. W e obtain a K¨ unneth Theorem for C hen- Ruan cohomology as a dir ect consequence of an elemen tary property of t he age grading, and explain how several other r esults – including asso ciativity of the cup-product – can be prov ed in a si milar w ay . Introduction In [CR04] Chen and Ruan defined the orbifold or Chen-Ruan cohomolo gy H ∗ CR ( X ) of an almost-complex orbifold X . As a group H ∗ CR ( X ) is simply the cohomology H ∗ (Λ X ) of the inertia orbifold Λ X , with degrees shifted b y a quan tity known as the age gr adi ng or de gr e e-shifting numb er . The striking feature of Chen and Ruan’s work is that this group is a n asso cia tive gra ded r ing under the Chen-Ruan cup pr o duct . This cup pro duct has prov ed difficult to compute b eca use its definition inv olves a certain obstruction bund le that Chen and Ruan defined using orbifold Riemann surfaces. Since the work of Chen and Ruan several interesting results rega r ding the ob- struction bundle hav e a pp e ared. Chen a nd Hu’s de Rham descriptio n of the Chen- Ruan cohomolo gy of ab elian or bifolds [CH06] in volved a c omputation of the ob- struction bundle for ab elia n orbifolds. F antec hi and G¨ ottsche [FG03] refined Chen and Ruan’s co nstruction in the case of globa l quotient o rbifolds. Jarvis, Kauf- mann and Kimura [JKK07] gav e a n explicit fo r mula for the r ational equiv ariant K-theory class of the F antec hi-G¨ ottsche obstruction bundle, so deter mining it up to iso morphism. The purp os e of this note is to show that the obstruction bundle can b e computed directly in terms of the ag e g rading. As a res ult we obtain a K ¨ unneth Theorem for Chen-Ruan cohomolo gy . W e also explain how several known theor ems – including asso ciativity of the Chen-Rua n pro duct – can b e pr ov ed as direct consequence s o f elementary pr op erties of the a ge gr ading. Let us recall from [CR04] that the obstruction bundle is a vector-bundle E → Λ 2 X over the 2-sectors of X , and is defined us ing the local description of Λ 2 X in terms o f or bifold-charts on X . W e will show that ea ch comp onent X ( g 1 ,g 2 ) of Λ 2 X is na tur ally equipp ed with a twisting gr oup h g 1 , g 2 i and a fibrewise -linear a ction of this group on the pullback ε ∗ V → X ( g 1 ,g 2 ) of any vector-bundle V → X . W e obtain the following descr iption o f the obstruc tio n bundle. Theorem 1. Over a c omp onent X ( g 1 ,g 2 ) of Λ 2 X t he obstruction bund le E is given by E ( g 1 ,g 2 ) = ( ε ∗ T X ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i wher e Σ is a Riemann surfac e with action of h g 1 , g 2 i su ch that Σ / h g 1 , g 2 i is an orbifold Riemann-spher e with s ingular p oints of or der o ( g 1 ) , o ( g 2 ) and o ( g 1 g 2 ) . The author is supported by E.P .S.R.C. Postd o ctoral Researc h F ello wship EP/D066980. 1 2 RICHARD HEPWOR TH Chen and Rua n ga ve a formula for the dimension of th e obstruction bundle in terms o f the age gr ading [CR04]. Equipp ed with Theorem 1, the study of the obstruction bundle becomes a matter of understanding the assignmen t V 7→ ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i . By observing that this assignment is deter mined b y the dimension of its v alues on the irreducible repr esentations of h g 1 , g 2 i we ar e able to determine the obstruction bundle directly using Chen and Ruan’s for mu la. Let ι V ( g ) denote the age of g in V and let V g 1 ,...,g k denote the elements of V fixed b y g 1 , . . . , g k . Theorem 2. Write V 1 , . . . , V n for the irr e ducibl e r epr esentations of h g 1 , g 2 i and let T i → X ( g 1 ,g 2 ) b e ve ctor bund les for which ε ∗ T X = L V i ⊗ T i as bun d les of h g 1 , g 2 i r epr esent ations. Then E ( g 1 ,g 2 ) = M h i T i wher e h i = ι V i ( g 1 ) + ι V i ( g 2 ) − ι V i ( g 1 g 2 ) + dim V i g 1 ,g 2 − dim V i g 1 g 2 . (1) Example 3. Suppose that h g , h i = {± 1 , ± g , ± h, ± g h } is the quaternion gr oup of order 8 . Then E ( g,h ) is the bundle Hom h g,h i ( Q, ε ∗ T X ), where Q is the 2-dimensional irreducible representation of h g , h i . Similar metho ds to those us e d to prove Theorem 2 will b e use d to r ecov er thre e existing results. These are asso ciativity of the Chen-Ruan cup- pro duct [CR04], Chen and Hu’s description of the o bstruction bundle for ab elia n orbifolds [CH0 6 ], and Gonz´ alez et al.’s computation of the Chen-Ruan co ho mology of c o tangent or b- ifolds [GLS + 07]. W e als o obtain the following K ¨ unneth Theorem: Theorem 4. L et X , Y b e almost-c omplex orbifolds with SL singularities, so that H ∗ CR ( X ) and H ∗ CR ( Y ) ar e c onc ent ra te d in inte gr al de gr e es and we c an form t he gr ade d ring H ∗ CR ( X ) ⊗ H ∗ CR ( Y ) . Then ther e is a gr ade d ring iso morphism H ∗ CR ( X ) ⊗ H ∗ CR ( Y ) ∼ = H ∗ CR ( X × Y ) . Remark 5. In de Rham cohomolo gy the cup pro duct ca n be rega rded as the comp osite H ∗ ( X ) ⊗ H ∗ ( X ) ∼ = H ∗ ( X × X ) ∆ ∗ − − → H ∗ ( X ) of the K¨ unneth Iso morphism with the map induced by the diagonal ∆ : X → X × X . Asso ciativity of the cup-pr o duct is then equiv alen t to ∆ ∗ (Id × ∆) ∗ = ∆ ∗ (∆ × Id) ∗ , which is just the usual functoria lit y o f induced maps. Using Theo rem 4 we ca n therefore regard Chen a nd Ruan’s definitio n of the cup-pro duct as a definition of ∆ ∗ , and their pro of o f as so ciativity a s a pro o f that ∆ ∗ (Id × ∆) ∗ = ∆ ∗ (∆ × Id ) ∗ . Question 6. Is it possible to define the induced ma p f ∗ : H ∗ CR ( Y ) → H ∗ CR ( X ) asso ciated to a g e neral ma p of o rbifolds f : X → Y ? Do es functorialit y g ∗ f ∗ = ( f g ) ∗ hold? Remark 7. One might wis h to take Theorems 1 a nd 2 as the definition of the obstruction bundle, so r emoving the need for o rbifold Riemann surfa ces. This is no t po ssible since the pro of of Theor e m 2 r e quires an a pplication of Chen and Ruan’s formula [CR04] for the dimension of the obstructio n bundle in o r der to show that the right-hand-side of (1 ) is non-negative. Nevertheless, after this single app eal to the theor y of orbifold Riema nn sur faces, whose conclusion is simply an inequality regar ding the age gra ding, one can rega rd the re s ults her e a s a n elementary wa y to define the Chen-Ruan cup pro duct and to pr ov e its ass o ciativity . A p ositive answer to either part of the following question would remov e the need for orbifold Riemann surfaces entirely . THE AGE GRADING AND THE CHEN-RUAN CUP PR ODUCT 3 Question 8. Is there an elementary pr o of o f the inequality ι V ( g 1 ) + ι V ( g 2 ) − ι V ( g 1 g 2 ) + dim V g 1 ,g 2 − dim V g 1 g 2 > 0? Is ther e a n elementary description of the assig nment V 7→ ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i ? As men tioned earlie r Ja rvis, Kaufmann and Kimura [JKK07] hav e determined the obstructio n bundle o f a global quotient b y giving an explicit des cription o f the rational K -theory class of the F antec hi-G¨ ottsche obs truction bundle. Their metho ds, combined with Theo r em 1, co uld b e used to give an analog o us r esult for gene r al orbifolds. The main step – co rresp onding to [JKK07, Lemma 8.5 ] – in the pro of of suc h a r e sult w ould rely on the sa me key observ ation that is used in the pro of of Theorem 2: that Chen a nd Ruan’s formula for the dimensio n of the obstruction bundle in fact determines the obstruction bundle entirely . The same observ ation was also used in the pro of o f [CH06, Pro po sition 3.4]. Here is an outline of the pap er. In Section 1 we recall the definition of the k - sectors Λ k X of X . W e characterize Λ k X in a w ay tha t allows us to intro duce, for each comp onent X ( g 1 ,...,g k ) , a twisting gr oup h g 1 , . . . , g k i and its twisting action on the pullback to X ( g 1 ,...,g k ) of any vector-bundle V → X . W e then prov e Theorem 1 . In Section 2 we recall the age grading a nd list some o f its prop erties . W e then prov e Theorem 2. Section 3 sho ws how to perform the calculation s tated in Example 3. In Section 4 we prove Theorem 4 and outline how similar metho ds may b e used in the pro o f of the results of Chen and Ruan [CR04], Chen-Hu [CH06], and Gonz´ alez et. al [GLS + 07] men tioned earlier . Ac kno wledgments. The a uthor is supp orted by an E.P .S.R.C. Postdo ctoral Re- search F e llowship, grant num ber EP/ D06698 0. 1. Twisted sectors and the twisting group T o an orbifold X one can asso ciate the k -se ctors or twiste d k -se ctors of X , whic h are orbifolds Λ k X for k > 0. The 0-sectors Λ 0 X is just X itself. The 1-s ectors Λ X := Λ 1 X is called the inertia o rbifold . There is an evaluation map ε : Λ k X → X , and mo re gener al ev aluation maps ε I 1 ...I j : Λ k X → Λ j X for any seq ue nce I 1 , . . . , I j of o r dered tuples in { 1 , . . . , k } . See [ARZ0 6, § 2], o r [ALR07, § 4 .1]. The following prop o s ition gives a new characterization of the twisted sectors, in terms of whic h we can immediately define the t wisting gro up a nd the t wisting action. The section ends with the pr o of of Theo rem 1. In what follows we w ill not distinguish betw een the orbifold X and the group oid G repr e senting it; our constructions are Mo rita-inv a riant. Prop ositi o n 9 . L et G b e a pr op er ´ etale Lie gr oup oid and H a Lie gr oup oid. (1) Morphisms H → Λ k G c orr esp ond pr e cisely to diagr a ms of t he form H f % % f 9 9 G . φ 1 ,...,φ k   The morphism c orr esp onding to the diagr am ab ove wil l b e written as ( f , φ i ) . 4 RICHARD HEPWOR TH (2) 2 -morphisms H ( f ,φ i ) # # ( g,γ i ) ; ; Λ k G ψ   c orr esp ond pr e cisely t o 2 -morphisms ψ : f ⇒ g for which ψ φ i = γ i ψ . (3) Id : Λ k G → Λ k G c orr esp onds to Λ k G ε % % ε 9 9 G E 1 ,...,E k   wher e ε is the usual evaluation map and t he E i ar e c anonic al ly-determine d 2 -automorphisms of ε . (4) ε I 1 ...I j : Λ k G → Λ j G c orr esp onds to Λ k G ε % % ε 9 9 G E I 1 ,...,E I j   wher e E I l = E l 1 ◦ · · · ◦ E l m for I l = ( l 1 , . . . , l m ) . Definition 10. W e will denote the c o mpo nents o f Λ k X by X ( g 1 ,...,g k ) , X ( h 1 ,...,h k ) , etcetera; the g i and the h i are simply lab els fo r the comp onents. (1) Let X ( g 1 ,...,g k ) be a comp onent o f Λ k X . The twisting gr oup h g 1 , . . . , g k i of X ( g 1 ,...,g k ) is the group on ge nerators g 1 , . . . , g k , isomorphic under g i 7→ E i to h E 1 , . . . , E k i ⊂ Aut( ε | X ( g 1 ,...,g k ) : X ( g 1 ,...,g k ) → X ). (2) Let V → X b e a vector bundle. The tautolo gi c al action of h g 1 , . . . , g k i on the bundle ε ∗ V → X ( g 1 ,...,g k ) is given by the fibrewise-linea r automor phisms g i : ε ∗ V → ε ∗ V induced b y the 2-automo rphisms E i of ε . Finiteness of the twisting gr o up is guaranteed by Lemma 1 2 below. F or the second pa r t of the definition rec a ll that, g iven f : A → C and g : B → C , 2- automorphisms of f induce automorphisms o f A × f ,g B , and that when g : B → C is a vector-bundle these automorphisms are fibre wise-linear maps. The twisting group a nd twisting action a ls o satisfy certain naturality pr op erties with resp ect to maps of X , maps of V , and ev aluation maps, all of whic h follow from Pr op osition 9. Remark 11. Pro p osition 9 suggests a definition o f the twisted sector s Λ k X of a differentiable Deligne-Mumford stack X , where one defines morphis ms U → Λ k X using a n a nalogue o f Pr op osition 9 , par t 1, and one defines 2 -morphisms a nd ev al- uation maps using analogues of par ts 2, 3 , and 4. This is indeed pos s ible, a nd o ne finds that if G is a group o id representing X , then Λ k G is precisely the gr oup oid representing X that one obtains from G by pulling ba ck under ε : Λ k X → X . When k = 1 the resulting iner tia stack Λ X is equiv alen t to, but leaner tha n, the more usual Λ X := X × X × X X . THE AGE GRADING AND THE CHEN-RUAN CUP PR ODUCT 5 Lemma 12. Supp ose given a diagr am of t he form a / / A m ! ! m = = B φ   wher e a is a p oint in a co nnected Lie gr oup oid A , and B i s a pr o p er ´ etale Lie gr oup oid. Then φ is trivial if and only if φ | a : m | a ⇒ m | a is trivial. Pr o of. This is immediate in the cas e B = U ⋊ G for G a finite group acting on U , and follows in g eneral b eca use B lo ca lly ha s this for m.  Pr o of of Pr op osition 9. Recall from [ARZ0 6] that Λ k G is the gr o up oid with o b jects and a rrows Λ k G 0 = { ( a 1 , . . . , a k ) ∈ G k 1 | s ( a i ) , t ( a j ) all coinc ide } , Λ k G 1 = { ( u, a 1 , . . . , a k ) ∈ G k +1 1 | s ( a i ) , t ( a j ) , s ( u ) all coincide } , with source and tar get s ( u, a 1 , . . . , a k ) = ( a 1 , . . . , a k ) , t ( u, a 1 , . . . , a k ) = ( u a 1 u − 1 , . . . , ua k u − 1 ) , and str ucture-maps e ( a 1 , . . . , a k ) = ( e ( α ) , a 1 , . . . , a k ) , α = s ( a i ) = t ( e j ) , i ( u, a 1 , . . . , a k ) = ( u − 1 , ua 1 u − 1 , . . . , ua k u − 1 ) , m (( v , ua 1 u − 1 , . . . , ua k u − 1 ) , ( u, a 1 , . . . , a k )) = ( v u, a 1 , . . . , a k ) . Suppo se g iven f : H → G and φ i : f ⇒ f , i = 1 , . . . , k . W e obtain F 0 : H 0 → G k 1 , F 1 : H 1 → G k +1 1 , given by F 0 ( h ) = ( φ 1 ( h ) , . . . , φ k ( h )) , F 1 ( H ) = ( f 1 ( H ) , φ 1 ( sH ) , . . . , φ k ( sH )) . W e claim that these maps tog ether g ive a g roup oid morphism H → Λ k G . That F 0 , F 1 are maps in to Λ k G 0 , Λ k G 1 resp ectively follows from s ◦ φ i = f 0 = t ◦ φ i . That F 0 , F 1 resp ect the source and tar get maps follo ws from the sa me fact for f , together with the natura lity of the φ i . That F 1 commutes with comp osition follows from the same fact for f 1 together with the naturalit y of the φ i . T hus we have co nstructed a g roup oid mor phism F : H → Λ k G as requir ed. This reaso ning c a n b e reversed to pro duce f : H → G , φ i : f ⇒ f from a groupo id- morphism F : H → Λ k G , and par t 1 is pr ov ed. The pro of of par t 2 is similar. The ev aluation map ε : Λ k G → G is the map obtained by sending the a i to their common source and ta rget. F ro m the pro of of part 1 it is therefore immediate that Λ k G → Λ k G corresp onds to Λ k G ε % % ε 9 9 G E 1 ,...,E k   6 RICHARD HEPWOR TH where E i : Λ k G 0 → G 1 is ( a 1 , . . . , a k ) 7→ a i . The diag ram Λ k G ε % % ε 9 9 G E I 1 ,...,E I j   induces Λ k G → Λ j G , ( a 1 , . . . , a k ) 7→ ( a I 1 , . . . , a I j ) on ob jects, and similarly for arrows, where a I l = a l 1 · · · a l m , I l = ( l 1 , . . . , l m ); this is just the ev aluation map ε I 1 ··· I j . P arts 3 and 4 are prov ed.  Pr o of of The or em 1. Consider an o r bifold gr oup oid of the for m U ⋊ G , wher e G is a finite gr oup acting on a manifold U . The n Λ k ( U ⋊ G ) = F U h 1 ,...,h k ⋊ G , where the union is taken over all k -tuples in G . The e v aluation map ε : Λ k ( U ⋊ G ) → U ⋊ G is induced by the compo nent wise inclusion F U h 1 ,...,h k → U , a nd E i : F U h 1 ,...,h k → U × G sends u ∈ U h 1 ,...,h k to ( u, h i ). A co mp o nent of Λ k ( U ⋊ G ) is then a compo ne nt of U ⋊ G ( g 1 ,...,g k ) = G U h 1 ,...,h k ⋊ G ≃ U h 1 ,...,h k ⋊ C G ( h 1 , . . . , h k ) where ( g 1 , . . . , g k ) is a diagona l conjuga c y cla ss in G ; on the fir s t line the union runs over ( h 1 , . . . , h k ) ∈ ( g 1 , . . . , g k ) and on the s econd line a single c hoice of such an ( h 1 , . . . , h k ) has bee n made . Then the twisting gr oup h g 1 , . . . , g k i is isomorphic under g i 7→ h i to h h 1 , . . . , h k i ⊂ G . Now let V → U ⋊ G be a vector-bundle, which is to say that V is a G -equiv ariant bundle ov er U . Then at an y point u ∈ U h 1 ,...,h k , h h 1 , . . . , h k i acts on th e fibre of V a t U . Under g i 7→ h i this is precisely the t wisting action of h g 1 , . . . , g k i on the fibres of ε ∗ V → U ⋊ G ( g 1 ,...,g k ) . Recall from [ALR07, § 4.3 ] the construction o f the o bstruction-bundle E ( g 1 ,g 2 ) → G ( g 1 ,g 2 ) ov er a co mpo nent of Λ 2 G . Let ( y , h 1 , h 2 ) be a po int of G ( g 1 ,g 2 ) . T ake an orbifold-chart U y ⋊ G y around y in G , so that F ( h ′ 1 ,h ′ 2 ) ∼ ( h 1 ,h 2 ) U h ′ 1 ,h ′ 2 y ⋊ G y ≃ U h 1 ,h 2 y ⋊ C G ( h 1 , h 2 ) is an or bifold-chart around ( y , h 1 , h 2 ) in G ( g 1 ,g 2 ) . Let N y = h h 1 , h 2 i ⊂ G y . Consider the pullback tangen t-bundle ε ∗ T G → G ( g 1 ,g 2 ) and the vector-space H 0 , 1 ¯ ∂ (Σ y ), where Σ y is the Riemann surfa ce with N y -action such that Σ y / N y is the or bifold Riemann s pher e with mar ked p o int s of or der o ( h 1 ), o ( h 2 ), o ( h − 1 2 h − 1 1 ), resp ectively . Then N y acts on both H 0 , 1 ¯ ∂ (Σ y ) and ε ∗ T G , and over the chosen chart for G ( g 1 ,g 2 ) the obstruction bundle is defined to be ( H 0 , 1 ¯ ∂ (Σ y ) ⊗ ε ∗ T G ) N y . Allowing y to v ary , one obta ins E ( g 1 ,g 2 ) → G ( g 1 ,g 2 ) . Now let Σ b e the Riemann surface with h g 1 , g 2 i -action re quired for the theorem. Note, using the first parag r aph, that g i 7→ h i ident ifies the twisting gro up h g 1 , g 2 i with N y , so tha t we may tak e Σ y = Σ. Note also tha t under N y ∼ = h g 1 , g 2 i , the action o f N y on ε ∗ T G is just the tauto logical a ction. Th us, ov er the ch osen orbifold- chart for ( y , h 1 , h 2 ), the la st par agra ph states that the obstruction-bundle is ( H 0 , 1 ¯ ∂ (Σ) ⊗ ε ∗ T G ) h g 1 ,g 2 i . By a llowing y to v ary , the theorem is prov ed.  2. The a ge grading. In this sectio n we recall the age-gr adi ng or de gr e e- shifting n umb ers and we list some prop erties. W e then discuss additive functors and ad ditive functions b efore proving Theorem 2. THE AGE GRADING AND THE CHEN-RUAN CUP PR ODUCT 7 Definition 13. Let G b e a finite gro up, g an e le ment of G , a nd V a complex representation of G . The age of g , deno ted b y ι V ( g ), is ι V ( g ) = X λ i , where g ha s the form    e 2 π iλ 1 . . . e 2 π iλ n    , 0 6 λ i < 1 with respect to an appropriate bas is of V . The age g rading appea rs in [IR96] for G ⊂ SL( n, C ) and V = C n . Chen and Ruan [CR04] gave the slig htly mor e g eneral definition ab ov e under the name de gr e e-shif ting numb er . Lemma 14. (1) ι V ( g ) dep ends o nly on t he isomorphism cla ss of V and t he c onjugacy class of g . (2) ι V ⊕ W ( g ) = ι V ( g ) + ι W ( g ) . (3) exp(2 π iι V ( g )) = det( g : V → V ) . (4) ι V ( g 1 ) + ι V ( g 2 ) − ι V ( g 1 g 2 ) + dim V g 1 ,g 2 − dim V g 1 g 2 > 0 . Pr o of. It is trivial to v erify the first three prop erties. The last is due to Chen and Ruan: the statement in [CR04, Theo rem 4.1 .5 ], that the orbifold cup-pro duct pr e- serves the gr ading of H ∗ CR ( X ), when applied to the orbifold X = V /G , is precisely the statement that the left-hand-side is the dimension of the o bstruction bundle, and so is a non-negative integer. This can also b e deduced from the Riemann- Ro ch formula as in [FG03, § 1], and from the Eichler tra ce formula as explaine d in [JKK07, § 8].  T o prov e Theo rem 2 we must consider the as signment V 7→ ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i , which w e r egard a s a functor from repr esentations of h g 1 , g 2 i to vector spac e s. Mo re generally we shall consider additive functor s V G → V . Here V is the catego ry of finite-dimensional complex v ector- spaces, V G is the category of finite-dimensio nal complex representations of a finite gr oup G , and a dditive means that the functor preserves dir ect sums. W e shall also consider additive functions | V G | → N ∪ { 0 } , where | V G | denotes the isomorphism cla sses in V G and additiv e means that the function sends direct sums to sums. E a ch additive functor H yields a n a dditive function dim H by taking the dimension, a nd any additive function f a rises in this way by s e tting H f ( − ) = L i f ( V i ) hom G ( V i , − ), where the V i are the distinct irreducible r e presentations of G . By basic represe ntation theory this establis hes a 1 − 1 corr esp ondence b etw een the additive functions and natural isomor phism classes o f additive functor s. Our decision to c o nsider additive functor s, rather tha n the representations that afford them, will be justified in the next section, wher e we will hav e to cons ide r functors suc h as V 7→ V g . These functors are easy to write down, but the re pre- sentations that a fford them a r e no t. Pr o of of The or em 2. Chen and Ruan’s re s ult [CR04] that the orbifold cup-pro duct preserves the grading of H ∗ CR ( X ), when applied to the or bifold V / h g 1 , g 2 i , states precisely that ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i has dimension ι V ( g 1 ) + ι V ( g 2 ) − ι V ( g 1 g 2 ) + dim V g 1 ,g 2 − dim V g 1 g 2 . (2) The assig nment V 7→ ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i is an additiv e functor V h g 1 ,g 2 i → V and the co rresp onding a dditiv e function sends [ V ] to the expr ession (2). But the as- signment V 7→ L h i Hom h g 1 ,g 2 i ( V i , V ) is a se cond additive functor that cor resp onds 8 RICHARD HEPWOR TH to this additiv e f unction, b y the second par t of Lemma 14. Consequently the tw o functors are naturally isomo rphic, a nd so E ( g 1 ,g 2 ) = ( H 0 , 1 ¯ ∂ (Σ) ⊗ ε ∗ T X ) h g 1 ,g 2 i ∼ = M h i Hom( V i , ε ∗ T X ) = M h i T i . This co mpletes the pro of.  3. An example Suppo se that w e wish to compute the obstruction bundle ov er a 2-sector X ( g,h ) of an or bifold X , and that the twisting g roup h g , h i is the quaternion group of o r der 8. Thus h g , h i = {± 1 , ± g , ± h, ± g h } where − 1 is central, − k denotes − 1 · k , and g 2 = h 2 = − 1 and g h = − hg . T o begin we must find the irreducible representations of h g , h i . These are 1 , G, H , GH , Q, where 1 is the tr ivial r epresentation, G is the linear repres ent ation o n which g = − 1 and h = 1, H is the linear r epresentation on whic h g = 1 and h = − 1, GH is the linear repr esentation on which g = h = − 1, a nd Q is the 2-dimensio nal representation on whic h g =  i 0 0 − i  and h =  0 1 − 1 0  . Now we must co mpute the quantities h V = ι V ( g ) + ι V ( h ) − ι V ( g h ) + dim V g,h − dim V gh for each ir reducible represenation V . W e find that h 1 = 0 + 0 − 0 + 0 − 0 h G = 1 2 + 0 − 1 2 + 0 − 0 h H = 0 + 1 2 − 1 2 + 0 − 0 h GH = 1 2 + 1 2 − 0 + 0 − 1 h Q = 1 + 1 − 1 + 0 − 0 so tha t the h V all v anish ex cept for h Q , which is equal to 1 . Now we can apply The- orem 2 a nd compute E ( g,h ) . Note that in the theorem T i = Ho m h g 1 ,g 2 i ( V i , ε ∗ T X ), and so immediately we obtain E ( g,h ) = Hom h g,h i ( Q, ε ∗ T X ) as claimed in Exa mple 3. 4. Applica tion s In this section w e prove Theorem 4 using a n elemen tary prop erty of the age grading. W e then expla in how a s imilar metho d can b e used in pro ofs of Chen and Rua n’s result on the as s o ciativity of the cup-pro duct [CR04], Chen and Hu’s computation o f the obstruction bundle of a b elian o rbifolds [C H0 6], and Gonz´ a lez et al.’s computation o f the Chen- Rua n co ho mology of cotang ent or bifolds [GLS + 07]. Pr o of of The or em 4. T o b egin with w e note tha t, given complex r epresentations V of G and W o f H , a nd elements g ∈ G , h ∈ H , we have ι V ⊕ W ( g × h ) = ι V ( g ) + ι W ( h ) . (3) Prop ositio n 9 g ives an iso morphism Λ k ( X × Y ) ∼ = Λ k X × Λ k Y under which ε : Λ k ( X × Y ) → X × Y and its 2-automorphisms E i corres p o nd to ε × ε and E i × E i resp ectively . W e c a n therefore write ( X × Y ) ( g 1 × h 1 ,...,g k × h k ) for the compo nent THE AGE GRADING AND THE CHEN-RUAN CUP PR ODUCT 9 X ( g 1 ,...,g k ) × Y ( h 1 ,...,h k ) , where h g 1 × h 1 , . . . , g k × h k i is indeed identified with the subgroup of h g 1 , . . . , g k i × h h 1 , . . . , h k i genera ted by the g i × h i . F urthermore, ε ∗ T ( X × Y ) → ( X × Y ) ( g 1 × h 1 ,...,g k × h k ) is equiv ariantly identified with ε ∗ T X ⊕ ε ∗ T Y . Using the last par a graph w e hav e the usual K ¨ unneth Isomorphism H ∗ (Λ X ) ⊗ H ∗ (Λ Y ) ∼ = H ∗ (Λ( X × Y )), and by (3) and the last para graph the grading shifts resp ect this isomor phism, so that we have an isomorphism o f gr aded vector-spa ces H ∗ CR ( X ) ⊗ H ∗ CR ( Y ) ∼ = H ∗ CR ( X × Y ). W e must prove that this is a ring-homo morphism. Let τ 1 , τ 2 ∈ H ∗ CR ( X ), σ 1 , σ 2 ∈ H ∗ CR ( Y ). Then ( τ 1 ∪ CR τ 2 ) × ( σ 1 ∪ CR σ 2 ) = ε 12 ∗ ( ε ∗ 1 τ 1 ∪ ε ∗ 2 τ 2 ∪ e ( E )) × ε 12 ∗ ( ε ∗ 1 σ 1 ∪ ε ∗ 2 σ 2 ∪ e ( E )) = ε 12 ∗ (( ε ∗ 1 τ 1 ∪ ε ∗ 2 τ 2 ∪ e ( E )) × ( ε ∗ 1 σ 1 ∪ ε ∗ 2 σ 2 ∪ e ( E ))) = ( − 1) d ε 12 ∗ ( ε ∗ 1 ( τ 1 × σ 1 ) ∪ ε ∗ 2 ( τ 2 × σ 2 ) ∪ ( e ( E ) × e ( E ))) = ( − 1) d ( τ 1 × σ 1 ) ∪ CR ( τ 2 × σ 2 ) , where d = deg ( τ 2 ) · deg ( σ 1 ), as require d. The third line ho lds b ecause the E uler classes have even degree s and – since X and Y have SL singularities – the honest degrees (as element s of H ∗ (Λ X ), H ∗ (Λ Y )) of τ 2 and σ 1 agree with their shifted degrees mo dulo 2. The la st line relies on an isomorphism π ∗ 1 E ⊕ π ∗ 2 E ∼ = E (4) of o bstruction bundles ov er ( X × Y ) ( g 1 × h 1 ,g 2 × h 2 ) ∼ = X ( g 1 ,g 2 ) × Y ( h 1 ,h 2 ) . Let us write H ( g 1 ,g 2 ) for the functor V 7→ ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i . Then the required isomorphism (4) will follow, using Theor em 1, Theo rem 2 and the comments at the star t of the pro of, from a natural isomor phism H ( g 1 × h 1 ,g 2 × h 2 ) ( V ⊕ W ) ∼ = H ( g 1 ,g 2 ) ( V ) ⊕ H ( h 1 ,h 2 ) ( W ) of functor s V G × V H → V . Since the functors ar e additive, this will follow from natural isomorphisms H ( g 1 × h 1 ,g 2 × h 2 ) ( V ⊕ 0) ∼ = H ( g 1 ,g 2 ) ( V ) , H ( g 1 × h 1 ,g 2 × h 2 ) (0 ⊕ W ) ∼ = H ( h 1 ,h 2 ) ( W ) . But by (3) the dimensions of the left ha nd sides are equal to the dimens ions o f the right ha nd sides. The isomorphisms now follow from the discus s ion in Section 2.  Now we sha ll sketc h the pro ofs o f three other well-known results, explaining in each cas e how o ne can simplify the pr o of using the tec hniques pres ented in this pap er. Theorem 15 (Chen-Ruan [CR04]) . The C hen-Ruan cup pr o duct is asso ciative. Sketch Pr o of. Chen and Ruan’s proo f of this result com bines a cohomologic a l ar- gument with [CR04, Le mma 4.3.2 ], the essential consequence of which is that there is a n isomorphism o f v ector- bundles: ǫ ∗ 12 , 3 E ⊕ ǫ ∗ 1 , 2 E ⊕ Exc 12 ∼ = ǫ ∗ 1 , 23 E ⊕ ǫ ∗ 2 , 3 E ⊕ Exc 23 . (5) Here ǫ 12 , 3 , ǫ 1 , 2 , ǫ 1 , 23 , ǫ 2 , 3 are ev aluation-maps Λ 3 X → Λ 2 X , and E x c 12 , E xc 23 are the ‘excess bundles’ obtained by applying the functor s V 7→ V g 1 g 2 / ( V g 1 ,g 2 + V g 1 g 2 ,g 3 ), V 7→ V g 2 g 3 / ( V g 2 ,g 3 + V g 1 ,g 2 g 3 ) to ǫ ∗ T X . Equation (5) was prov ed in [CR04] by manipulating or bifold Riema nn- surfaces. Here we s ha ll show how it follows by considering the age grading. Over a fixed comp onent X ( g 1 ,g 2 ,g 3 ) of Λ 3 X each side o f (5) is obtained b y applying an additive 10 RICHARD HEPWOR TH functor to ε ∗ T X . The tw o corr esp onding a dditive functions send a representation V to the quantit ies ι V ( g 1 g 2 ) + ι V ( g 3 ) − ι V ( g 1 g 2 g 3 ) − dim V g 1 g 2 g 3 + dim V g 1 g 2 ,g 3 + ι V ( g 1 ) + ι V ( g 2 ) − ι V ( g 1 g 2 ) − dim V g 1 g 2 + dim V g 1 ,g 2 + dim V g 1 g 2 − dim V g 1 ,g 2 − dim V g 1 g 2 ,g 3 + dim V g 1 ,g 2 ,g 3 and ι V ( g 1 ) + ι V ( g 2 g 3 ) − ι V ( g 1 g 2 g 3 ) − dim V g 1 g 2 g 3 + dim V g 1 ,g 2 g 3 + ι V ( g 2 ) + ι V ( g 3 ) − ι V ( g 2 g 3 ) − dim V g 2 g 3 + dim V g 2 ,g 3 + dim V g 2 g 3 − dim V g 2 ,g 3 − dim V g 1 ,g 2 g 3 + dim V g 1 ,g 2 ,g 3 resp ectively . B ut these are easily seen to b e equal, and the isomorphism (5) no w follows from the dis cussion in Section 2.  Theorem 16 (Chen-Hu [CH06 ]) . L et X b e an almost-c omplex ab elian orbifo ld. Then E ( g 1 ,g 2 ) → X ( g 1 ,g 2 ) is the su m mand of ε ∗ T X sp anne d by t he h g 1 , g 2 i -invariant lines L on which ι L ( g 1 ) + ι L ( g 2 ) > 1 . Sketch Pr o of. By Theorem 1 it suffice s to show that the functor V 7→ ( V ⊗ H 0 , 1 ¯ ∂ (Σ)) h g 1 ,g 2 i is isomo r phic to the functor that sends V to the span of those linear subr epresen- tations L 6 V for which ι L ( g 1 ) + ι L ( g 2 ) > 1. These functors are additive and so to prov e this we m ust show that the corr esp onding a dditive functions obtained by computing the dimensions are equal, and to do this it suffices to chec k the cla im for linear representations. But it is trivia l to verify tha t if L is a linear representation of a finite gr oup G and g 1 , g 2 ∈ G , then ι L ( g 1 ) + ι L ( g 2 ) − ι L ( g 1 g 2 ) − dim L g 1 g 2 + dim L g 1 ,g 2 is equal to 0 if ι L ( g 1 ) + ι L ( g 2 ) 6 1 , and is equal to 1 otherwise. This completes the pro of.  Theorem 17 (Gonz´ alez et al. [GLS + 07]) . F or an almost-c omplex orbifold X we have a ring-isomorphism H ∗ CR ( T ∗ X ) ∼ = H ∗ virt (Λ X ) , wher e H ∗ virt (Λ X ) is t he ‘virtual c oho molo gy’ of Λ X . Sketch Pr o of. In [GLS + 07] the pro of w as r educed using a coho mological ar gument to the claim that over a comp onent ( T ∗ X ) ( g 1 ,g 2 ) of Λ 2 ( T ∗ X ) we have E ⊕ π ∗ ǫ ∗ T X g 1 g 2 ǫ ∗ T X g 1 ,g 2 ∼ = π ∗ ǫ ∗ T X ǫ ∗ T X g 1 + ǫ ∗ T X g 2 , (6) where π deno tes the pro jection T ∗ X → X and the map it induces on 2-sector s. Each side of (6) is obtained b y applying an additive functor to ε ∗ T X , s o to prov e (6) it suffices to show that the dimension o f these functors when applied to a r epresentation V ar e alwa ys eq ual. But the dimensions are ι V ⊕ ¯ V ( g 1 ) + ι V ⊕ ¯ V ( g 2 ) − ι V ⊕ ¯ V ( g 1 g 2 ) − dim( V ⊕ ¯ V ) g 1 g 2 + dim( V ⊕ ¯ V ) g 1 ,g 2 + dim V g 1 g 2 − dim V g 1 ,g 2 and dim V − dim V g 1 − dim V g 2 . W e hav e used the fact that T ( T ∗ X ) ∼ = π ∗ T X ⊕ π ∗ T X . By no ting that ι V ⊕ ¯ V ( g ) = ι V ( g ) + ι ¯ V ( g ) = dim V − dim V g one verifies that the dimensions are e q ual, and this prov es the theor em.  THE AGE GRADING AND THE CHEN-RUAN CUP PR ODUCT 11 References [ALR07] Alejandro Adem, Johann Leida, and Y ong bin Ruan. Orbifolds and stringy top olo gy , v olume 171 of Cambridge T r acts in Mathematics . Cam bridge Universit y Press, Cam- bridge, 2007. [ARZ06] Alejandro Adem, Y ong bin Ruan, and Bi n Zhang. 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Stringy K -theory and the Chern ch aracter. Invent. Math. , 168(1):23 –81, 2007. Dep ar tment of Pure Ma thema tics, University of S heffield, Shef field, S3 7RH E-mail addr ess : r.hepworth@shef field.ac.uk