INVERSES OF STRUCTURED VECTOR BUNDLES 5 INDRANILBISWASANDVAMSIP.PINGALI 1 0 2 Abstract. StructuredvectorbundleswereintroducedbyJ.SimonsandD.Sullivanin[SS-2010]. We provethatallstructuredvectorbundleswhoseholonomieslieinGL(N,C), SO(N,C),orSp(2N,C)have r p structuredinverses. ThisgeneralizesatheoremofSimonsandSullivanprovedin[SS-2010]. A 2 2 1. Introduction ] G Differential K-theory is an enhanced version of topological K-theory constructed by incorporat- D ing connections and differential forms. It was developed in order to refine the families of Atiyah- . Singer index theorem and also to classify the Ramond-Ramond field strengths in string theory h t [BS-2012], [Fr-2000]. One model of the first group in differential K-theory K0(X) for a manifold X a ff m is a Grothendieck group of vector bundles equipped with connections and odd di erential forms b [ [Ka-1987]. In[SS-2010], Simons and Sullivan constructedanothermodelofK0(X)usingjustvector ff bundlesandconnections. Theygotridofthedi erentialformatthecostofintroducinganequiva- 2 b v lencerelationamongtheconnectionsonthevectorbundles. Moreprecisely,theyconstructedK0(X) 1 as the Grothendieck group of “structured” vector bundles; the definition of a structured vector 7 b bundleisrecalledinSection2. 0 0 In addition to constructing a model of K0(X), Simons and Sullivan in [SS-2010] proved an inter- 0 estingresult about the existenceof stable inversesof hermitian structuredbundles. (It is Theorem . b 2 1.15 in [SS-2010] which is essentially Theorem 1.1 below for the unitary group.) However, their 0 proofworks only for the unitary group (and also to the more general case of compact Lie groups) 5 becausetheyusedtheexistenceofuniversal connectionsa´ laNarasimhan-Ramanan [NR-1961]. In 1 ff : [PT-2014] a slightly di erent proof of the theorem of Simons and Sullivan was given that did not v involve universalconnections. In fact, theproofin [PT-2014] is valid evenfor connectionsthat are i X notcompatiblewiththemetric. r Alltheflatconnectionsconsideredherewillhavetrivialmonodromyrepresentation. Notethatif a is a flat connectionwith trivial monodromy on a vector bundle V over X, and x is a point ofX, 0 ∇ thenthereisauniqueisomorphism f ofV withthetrivial vectorbundleX V X, equipped × x0 −→ withthetrivialconnection,suchthat f isconnectionpreservingandcoincideswiththeidentitymap overx (hereV denotesthefiberofV overx ). 0 x0 0 Inthispaperweprovethefollowingtheorem. Theorem1.1. LetGbeoneofthegroupsGL(N,C),SO(N,C),Sp(2N,C). Givenastructuredvectorbundle = [V, ] on a smooth manifold X such that the holonomy of some equivalent connection is in G, V {∇} ∇ there exists astructured inverse = [W, ]with theproperty that theholonomy of isin the sameG, W {∇} ∇ satisfying e e = [X CM, ] F V⊕W × {∇ } where isaflatconnectionwithtrivialmonodromyonthetrivialvectorbundleX CM overX. F ∇ × AsanimmediateconsequenceofTheorem1.1wehavethefollowingcorollary. 2000MathematicsSubjectClassification. 58A10,53B15. Keywordsandphrases. Structuredvectorbundle,connection,symplecticbundle,orthogonalbundle. 1 2 I.BISWASANDV.P.PINGALI Corollary1.2. LetK0 (X)betheGrothendieck groupofstructured vectorbundles = [V, ]satisfying G V {∇} the condition that the connection has holonomy in G (both G and X are as in Theorem 1.1). Let d denote b ∇ thetrivialflatconnectiononatrivialbundleX Ck. Thenthefollowingtwohold: × (1) EveryelementofK0 (X)isoftheform [k]where[k] = [X Ck, d ](sok = NifG = GL(N,C) G orSO(N,C)andk = 2N ifG = Sp(2VN,−C)),and × { } b (2) = inK0 (X)ifandonlyif [ ] = [ ]asstructuredbundlesforsomeflatbundle G V W V⊕ N W⊕ N [ ]withtrivialmonodromy. N b In [SS-2010], Simons and Sullivan proved what they called the Venice lemma which essentially saysthateveryexactformarisesoutofCherncharacterformsoftrivialbundles(seealso[PT-2014]). Using the same ideas as in the proof of Theorem 1.1 we give a proof of the following holonomy versionoftheVenicelemma. Proposition 1.3. Fix G to be one the groups GL(N,C), SO(N,C) and Sp(2N,C). If η is any odd smooth form onX,then there exists atrivial bundleT = X Ck (k isasinCorollary 1.2)andaconnection onit × ∇ whoseholonomyisinGsuchthat ch(T, ) ch(T,d) = dη. (1.1) ∇ − 2. Preliminaries As mentioned in the introduction, in order to define structured vector bundles we need to define an equivalence relation between connections on vector bundles on a smooth manifold. To do so, we recall the definition of the Chern-Simons forms. Throughout, V and W are smooth complex vector bundles on a smooth manifold X. For a connection on a vector bundle V, let ∇ F C (X, End(V) 2T X)bethecurvatureof ,andlet ∞ ∗ ∇ ∈ ⊗ ∇ V √ 1 ch( ) = Trexp − F ∇ 2π ∇! bethecorrespondingCherncharacterformonX. Definition 2.1. If and are smooth connections on V, then the Chern-Simons form between 1 2 them is defined as∇a sum∇of odd differential forms CS( , ) modulo exact forms satisfying the 1 2 ∇ ∇ followingtwoconditions: (1) (Transgression) dCS( , ) = ch( ) ch( ). 1 2 1 2 ∇ ∇ ∇ − ∇ (2) (Functoriality) If f : Y X is a smooth map between smooth manifolds Y and X, then CS(f , f ) = f CS(−→, )moduloexactforms. ∗ 1 ∗ 2 ∗ 1 2 ∇ ∇ ∇ ∇ In[SS-2010]anequivalencerelationbetweenconnectionswasdefined,whichwenowrecall. Definition2.2. If and aretwosmoothconnectionsonavectorbundleV onX,then 1 2 ∇ ∇ ∼ 1 2 ∇ ∇ ifCS( , ) = 0moduloexactformsonX. Theequivalenceclassof isdenotedby . 1 2 ∇ ∇ ∇ {∇} An isomorphism class = [V, ] as in Definition 2.2 is called a structured vector bundle. The directsumof = [V, V]and {∇=} [W, ]isdefinedas V W V {∇ } W {∇ } = [V W, ]. V W V⊕W ⊕ {∇ ⊕∇ } Asymplectic (respectively,orthogonal) bundleonXisapair(E,ϕ),whereEisaC vectorbundle ∞ onXandϕisasmoothsectionofE E ,suchthat ∗ ∗ ⊗ (1) ThebilinearformonEdefinedbyϕisanti-symmetric(respectively,symmetric),and INVERSESOFSTRUCTUREDVECTORBUNDLES 3 (2) thehomomorphism E E (2.1) ∗ −→ defined by contraction of ϕ is an isomorphism, equivalently, the form ϕ is fiber-wise non- degenerate. A connection on a vector bundle E induces a connection on E E . A symplectic (respectively, ∗ ∗ ⊗ orthogonal)connectiononasymplectic(respectively,orthogonal)bundle(E,ϕ)isaC connection ∞ ∇ on the vector bundle E such that the section ϕ is parallel with respect to the connection on E E ∗ ∗ ⊗ inducedby . ∇ If(E,ϕ)isasymplectic(respectively,orthogonal)bundle,thentheinverseoftheisomorphismin (2.1) produces a symplectic (respectively, orthogonal)structure ϕ on E , because (E ) = E. Let ′ ∗ ∗ ∗ ∇ be a symplectic (respectively, orthogonal) connection on the symplectic (respectively, orthogonal) bundle(E,ϕ). Thentheconnection onE inducedby isasymplectic(respectively,orthogonal) ′ ∗ ∇ ∇ connectionon(E ,ϕ ),whereϕ isdefinedabove. Wenotethattheisomorphismin(2.1)takesϕand ∗ ′ ′ toϕ and respectively. ′ ′ ∇ ∇ ff Wedefinetheholonomyversionofdi erentialK-theorynext. Definition2.3. LetVbeavectorbundlewithaconnection whoseholonomyisinagroupG. Let V ∇ denotetheequivalenceclassofallsuchconnectionsanddenotethecorrespondingstructured V G {∇ } bundlesby . Also,letT denotethefreegroupofsuchstructuredbundles. Thefollowinggroup G istheholonoVmyversionofdifferentialK-theoryonasmoothmanifoldX: T K0 (X) = G (2.2) G + V W−V⊕W b Remark 2.4. Notice that we require all the equivalent connections in to have holonomy in V G {∇ } G as a part of the definition of equivalence. In particular, equivalent connections in the sense of [SS-2010]donotnecessarilyhavetheirholonomyinthesamegroup. FromnowonwardswedropthesubscriptGwheneveritisclearfromthecontext. 3. ProofofTheorem1.1 WedividetheproofofTheorem1.1intotwocases. 3.1. ThecaseofG = GL(N,C). ff This case has already been covered in [PT-2014]. Howeverhere we provide a di erent proof. Our approachreliesonLemma3.1provedbelow. WebelievethatLemma3.1maybeofinterestinitsown right. ThegeometricalcontentoftheabovementionedlemmaisthatonRneverytrivialbundlewith a connection is a subbundle, equipped with the induced connection, of a trivial bundle equipped withaflatconnection. Lemma 3.1. Let V be a trivial complex vector bundle of rank r on Rn, and let A be a connection on V. Then there exists an invertible, smooth (2n +2)r (2n +2)r complex matrix valued function g such that A = [dgg 1] ,where1 i,j r. × ij − ij ≤ ≤ n Proof. Notice that A = A dxk, where A are smoothr r complex matrix valued functions and k k × Xk=1 xk arecoordinatesonRn. WemaywriteA as k A = 2I+A A +A (2I+A A ). k †k k k− †k k 4 I.BISWASANDV.P.PINGALI Using this it can be deduced that A is a difference of two smooth functions with values in r r k positive definite matrices (an r r matrix B is called positive definite if v (B+B )v > 0 v ,×0). † † × ∀ Indeed,wehave I+A A + Ak+A†k = I+ Ak † I+ Ak + 3A A 0. †k k 2 (cid:18) 2 (cid:19) (cid:18) 2 (cid:19) 4 †k k ≥ Also,dxk = e xkd(exk)and dxk = exkd(e xk). Hence − − − 2n A = f dh k k Xk=1 wheretheh arepositivesmoothfunctionsandthe f arer rpositive-definitesmoothmatrix-valued i i × functions. Wemayattempttofind gbyforcingthefirstr (2n+2)rsub-matrixofdgtobe × dh Id ... dh Id 0 0 1 r r 2n r r × × h i andthefirst(2n+2)r rsub-matrixof g 1 tobe − × f 1 f  2  Ifwemanagetofindsucha g,thenA = [dg−gIP1df]2...rh×nkr.fk . ij − ij Indeed,weclaimthatthematrix gdefinedby h Id h Id ... h Id Id Id 1 r r 2 r r 2n r r r r r r × × × × 1 ×  −  does the job. (HeregI=d Iisd000r...t×hre r Idr000r...×irden....t.......it....y mIadt000r...r×irx.) Tfff2h21nXisXXkXkkckha...knhhhkkfkkbfffkkek−1−−v11erifiIde000r...d×rby a straightforward r r × × computation. Notethat h f isinvertiblebecauseh > 0and f + f > 0. (cid:3) k k k k k† X k Since Rn is simply connected, any flat bundle on it has trivial monodromy. Lemma 3.1 implies thefollowinggeneralization: Proposition 3.2. Let (V, = d + A) be a complex rank r vector bundle equipped with a connection ∇ on a smooth manifold X of dimension n. Then there exists a trivial complex vector bundle T of rank (4n+8r+2)(n+2r)onX,andasmoothflatconnectionwithtrivialmonodromy = d+AonT,suchthat ∇ e e INVERSESOFSTRUCTUREDVECTORBUNDLES 5 V W = TforsomesmoothcomplexvectorbundleW,and • ⊕ theconnectionAisinducedfromA. • Proof. Using the Whitney embeddingetheorem, there is an embedding of the total space of V in R2n+4r. The zero section of V is diffeomorphic to X (and hence X also sits in R2n+4r). The tangent bundleTV ofV isasubbundleofTR2n+4r . V By endowing V with the metric induce|d from the Euclidean metric on R2n+4r, we may find the orthogonalcomplementTV ofTV. Itsatisfies ⊥ TV TV = TR2(n+2r) . X ⊥ X | ⊕ | The vector bundle V itself may be identified with a subbundle of TV. Using the induced metric we may find the orthogonal complement V of V in TV. Therefore, there exists a vector bundle ⊥ U = V TV onXsuchthatV U = X Cn+2r = Q. ⊥ ⊥ Wem⊕ayendowUwithsomea⊕rbitraryc×onnection . Thisinducestheconnection = U Q U ∇ ∇ ∇⊕∇ on the bundle Q. Using a tubular neighborhood and a partition of unity we may extend from Q X to a connection on the trivial vector bundle of rank (n +2r) defined on all of R2n+4∇r. Now ∇Q we may use Lemma 3.1 to come up with a vector bundle T of rank (4n+8r+2)(n+2r) on R2n+4r, e equippedwith aflat connection , such that is inducedfromit. RestrictingourattentiontoX ∇T ∇Q e weseethatthevectorbundleT = T equippedwiththeconnection satisfiestheconditionsin e |X e ∇T|X (cid:3) theproposition. e e Proposition3.2maybeviewedasavectorbundleversionoftheNashembeddingtheorembecause itstatesthateveryconnectionarisesoutofaflatconnectionwithtrivialmonodromy. Wenowstate ausefullemma[SS-2010,Lemma1.16]. Lemma3.3 (Simons-Sullivan). LetV andW besmooth vector bundleson asmooth manifold X. Let be ∇ a smooth connection on the direct sum V W with curvature R. Let and be the connections on V V W ⊕ ∇ ∇ and W respectively constructed from using the decomposition of V W. Suppose that R (V) V and r,s ∇ ⊕ ⊆ R (W) W foralltangentvectorsr,satanypointofX. Then r,s ⊆ CS( , ) = 0 modulo exact forms on X. V W ∇ ⊕∇ ∇ Lemma3.3inconjunctionwithlemma3.2impliesTheorem1.1inthecaseG = GL(N,C). Indeed, givenastructuredbundle = [V, , lemma3.2furnishesaflat bundle = [T, ]suchthat V T V {∇ } T {∇ } VisasubbundleofTwiththeconnection beinginducedfrom . UsingthenaturalmetriconT V T wemayfindanorthogonalcomplementW∇toVsothatV W = T∇. EndowingW withtheinduced ⊕ connection from (which is flat), it is straight-forward tocheck that theconditions oflemma W T ∇ ∇ 3.3aresatisfied. Let = [W, ]. W W {∇ } Usinglemma3.3weseethat [T, ] = [V W, ] = [V W, ] = . T T V W {∇ } ⊕ {∇ } ⊕ {∇ ⊕∇ } V⊕W 3.2. G = Sp(2N,C)orG = SO(N,C). From Section 3.1 we knowthat there is a structuredinverse = [W, ] of(E, ). We clarify W {∇} ∇ thatW doesnotnecessarilyhaveaG–structure. Let denotetheconnectiononW inducedby . ∇′ e ∗ ∇ Usingthenaturalpairing ofW with W ,thevectorbundleW W hasacanonical G–structureϕ . ∗ ∗ 0 e ⊕ e Wenotethat isaG–connectionon(W W ,ϕ ). ′ ∗ 0 ∇⊕∇ ⊕ e e 6 I.BISWASANDV.P.PINGALI Clearly,(W , )isastructuredinverseof(E , ). Therefore, ∗ ′ ∗ ′ ∇ ∇ e (E∗ W W∗, ′ ′) ⊕ ⊕ ∇ ⊕∇⊕∇ isastructuredinverseof(E, ). Theconnection ′ e ′pereservestheG–structureϕ′ ϕ0onthe ∇ ∇ ⊕∇⊕∇ ⊕ vectorbundleE W W . ∗ ∗ ⊕ ⊕ e e 4. Applications InthissectionweproveCorollary1.2andProposition1.3. 4.1. ProofofCorollary1.2. (1) AnyelementofK0 (X)isoftheform[ ] [ ]bydefinition. Sincethereexistsaninverse G to such that = [k], whVere−[kW] is flat with trivial monodromy, we see that Q[ ] W[ ] = [ b Q]⊕ W[k]. V − W V⊕Q − (2) If [ ] = [ ] in K0 (X), then = for some structuredbundle . Let be an G invVerse ofW. Adding to bothVs⊕idPes weWsee⊕tPhat [N] = [N] for soPme flatEvector P b E V⊕ W ⊕ bundleN withtrivialmonodromy. 4.2. ProofofProposition1.3. UsingtheVenicelemmain[PT-2014]weseethatthereexistsatrivial bundleTwithaconnection = d+Asuchthat ∇T dη e e = ch(T, ) ch(T,d). 2 ∇T − Itisnotnecessarilythecasethattheholonomyeofe liesienG. However,weknowthat ∇T dη = ch(T , e) ch(T ,d) ∗ ∗ 2 ∇T∗ − becausedηisanevenform. Therefore, e e e dη = ch(T T, ) ch(T T,d). ∗⊕ ∇T∗ ⊕∇T − ∗⊕ UsingthesamereasoningasinSection3.2weobtainthedesiredresult. e e e e e e Acknowledgements Wearegratefultotherefereefordetailedcommentstoimprovetheexposition. Thefirst–named authoracknowledgesthesupportofaJ.C.BoseFellowship. References [BS-2012] U.BunkeandT.Schick.DifferentialK-theory: asurvey.Globaldifferentialgeometry.SpringerBerlinHeidelberg, 2012.303–357. [Fr-2000] D.Freed.Diracchargequantizationandgeneralizeddifferentialcohomology.Surv.Diff.Geom.7(2000),129–194. [Ka-1987] M.Karoubi.HomologiecycliqueetK-the´orie.Asterisque(149): 147,(1987). [NR-1961] M.S.NarasimhanandS.Ramanan.Existenceofuniversalconnections.Am.Jour.Math.83(1961),563–572. [PT-2014] V.PingaliandL.Takhtajan.OnBottChernformsandtheirapplications.Math.Ann.360(2014),519–546. [SS-2010] J.SimonsandD.Sullivan.StructuredvectorbundlesdefinedifferentialK-theory.Quantaofmaths,ClayMath.Proc., Vol.11,Amer.Math.Soc.,Providence,RI,579–599(2010). SchoolofMathematics,TataInstituteoffundamentalresearch,HomiBhabharoad,Mumbai400005,India E-mailaddress: [email protected] DepartmentofMathematics,412KriegerHall,JohnsHopkinsUniversity,Baltimore,MD21218,USA E-mailaddress: [email protected]