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Spin structure on moduli space of sheaves on Calabi-Yau threefold 3 1 0 2 Zheng Hua ∗† n January 10, 2013 a J 9 ] Abstract G A Kontsevich and Soibelman defined a notion of orientation data on Calabi-Yaucategory. Itcanbeviewedasaconsistentchoiceofspinstruc- . h tureonmodulispaceofobjectsinthegivenCYcategory. Theorientation t data plays an important role in Donaldson-Thomas theory. Let X be a a m compactCY3-foldsatisfyingappropriatetopologicalcondition. Weprove the existence and uniqueness of orientation data on the derived category [ of coherent sheavesDb(X). 2 v 1 Introduction 0 9 7 The goal of this paper is to construct C∞ orientation data (See Definition 7.2) 3 on moduli spaces of coherent sheaves on Calabi-Yau threefolds. Orientation . 2 data is introduced by Kontsevich and Soibelman in [12]. Roughly speaking, 1 an orientation data on a Calabi-Yau category (defined in section 3.3 of [12]) 2 C is a choice of square root of the determinant line bundle on the moduli space 1 of objects, satisfying an additional compatibility condition. Depending on the : v structure we put on the determinant line bundle, one can consider different i X types of square roots. In the original definition of Kontsevich and Soibelman, they consider the determine bundle as a (ind-)constructible super line bundle r a overthemodulispaceofallobjectsin . Wewillnotstudyinsuchagenerality. C Let be Db(coh(X)), the derived category of coherent sheaves on a smooth C projective CY 3-fold X. Moreover, we will restrict to moduli stack of coherent sheaves, denoted by ( for short), instead of moduli of arbitrary com- X M M plexes. One can stratify such that over each strata there is a graded vector bundlewithfiberExt• (EM,E)atpointE . Byabusingnotations,wedenote this(ind-)constructibXlegradedvectorbun∈dlMebyExt• (E,E). Inparticular,this X means rank of Exti (E,E) is constant for all i over each strata. We define the X (ind-)constructible determinant line bundle L to be sdet(Ext• (E,E)) where X ∗DepartmentofMathematics, KansasStateUniversity †Institute of Mathematical Science, Chinese University of Hong Kong Email:[email protected] 1 sdet is the super determinant. Kontsevich-Soibelman’s orientation data is a choice of (ind-)constructible square root of L that is compatible under Hall multiplication. The orientation data is a critical ingredient in the definition of motivic Donaldson-Thomas invariant (See section 6 [12]). The (ind-)constructible structure is the weakest structure that one can put ondeterminantlinebundle. However,itissufficientforthepurposeof[12]since motivicDTinvariantsaredefinedbyfirststratifythemodulistackandthenadd the different strata with appropriate motivic weights. By the work of Quillen and Knudsen-Mumford, the determinant line bundle can be equipped with a holomorphic or algebraic structure, without stratifying the moduli space. We will review these constructions in Section 2. Given the holomorphic determinant line bundle L over the underlying analytic stack associated to , a C∞ orientation data is a choice of square root of L that is compatible uMnder Hall multiplication(SeeDefinition7.2). Fromthepointofview ofderivedgeometry, the determinant line bundle L plays the role of canonical bundle of moduli space. We call a square root of L a spin structure on . The existence and classification of C∞ orientationMdata on moduli space is a purely topological question, essentially determined by the homotopy type of X. Inthispaper,weconsideraspecialclassofCY3-foldscalledadmissible CY 3-folds (Definition 6.8). The main theorem of the paper is: Theorem1.1. (Theorem 7.6)IfX isanadmissibleCY3-fold, thenthereexists a C∞ orientation data on Db(X). Admissible condition is a condition on torsion part of homology of X. Any simply connected torsion free CY threefold is admissible (Theorem 6.10). The proof of Theorem 7.6 is a combination of gauge theory and surgery theory. Thesametechniquehasbeenusedwidelyinthestudy offourmanifold. A very good reference is chapter 5 of [6]. In the first step, we reduce the question about moduli space of coherent sheaves to moduli of vector bundles. This is essentially consequence of a theorem of Joyce and Song (Theorem 7.3). Using gauge theory we realize the moduli space of holomorphic vector bundles as analytic subspace of a complex Banach manifold modulo automorphisms. A simple but important observation is that the determinant line bundle L extends to the ambient Banach manifold. Instead of constructing square root of L on the analytic subspace,we do it onthe ambientBanachmanifold. This is equivalent with proving c (L) is divisible by two. Over rational number, 1 it follows from Grothendieck-Riemann-Roch theorem and Atiyah-Singer index theorem(Theorem3.1,Theorem5.1). Thetorsionpartofc (L)isnotcaptured 1 by GRR or AS. We get around it by proving the even torsion cannot occur if X is admissible. The odd torsion never matters anway. To be more specific, we consider those CY 3-folds that can be uniformized to connected sum of S3 S3 by a sequence of conifoldtransitions and study how the torsionpartof × the secondcohomologygroupofthe Banachmanifoldmentionedabovechanges under the conifold transition. Moduli space of sheaves on CY 3-fold is locally an intersection of two holo- morphicLagrangiansubvarietiesinsideholomorphicsymplecticmanifold. Donaldson- 2 Thomas invariant is the (weighted) sum of Lagrangian intersection numbers. There are two generalizations of DT invariants. The first one is to replace the DTinvariants,whicharenumbers,byelementsinappropriateGrothendieckring ofstacks. Suchgeneralization,due to Joyce-SongandKontsevich-Soibelman,is calledthe motivic DT invariants([11][12]). Thesecondwayisto replaceitby a perversesheaf on . Such perversesheafis knownto existlocally by the work M of Joyce and Song [11]. However, the gluing problem of these locally defined perversesheavesis nontrivial. We refer to [3] and [13] for some recent progress. In both of these two generalized DT theories, the orientation data, i.e. spin structure on plays an essential role. M This paper is organized as follows. In section 2, we recall two definitions of determinant line bundle. In section 3 and 5, we prove the rational c (L) is 1 divisible by two. In section4, we recallsome results in topology of loopspaces. In section 6, we define admissible CY 3-fold study the torsion in cohomology of space of principal bundles. The main theorem 7.6 is proved in Section 7. In the lastsection,weexplainalinkbetweenorientationdataandvolumeformon Lagrangiandistributionwhichis usedto categorifyDT invariants. This section is independent from other sections. The readers might consider reading it first to get some motivations. Acknowledgement Theexistenceoforientationdataisprovedinthecaseof representations of quiver with potential by Davison [5]. Theorem 3.1 is known to Kontsevich. Thanks to Dominic Joyce for pointing out that. I would like to thank Conan Leung, Dominic Joyce for some very helpful discussion. Thanks to Prof. Simon Donaldson for his comments about η-invariants. Thanks to Young-HoonKiem for pointing out a mistake in the earlier version. Partof the work was done during my visit to Oxford. I am grateful to Balazs Szendroi for his hospitality. 2 Determinant line bundle We recall two definitions of determinant line bundle, one in algebraic geometry and one in differential geometry. Both definitions will be used later. Let ( for short) be the moduli space of sheaves on X. This is an X M M Artin stack locally of finite type. One can write as disjoint union indexed M by classes in topological K-theory: = . β M M β∈KG0(X) 2.1 Algebraic definition Let be the universal sheaf over X. Denote the projection X E M× M× →M by π. 3 Definition 2.1. The (algebraic)determinant line bundle L over is defined M to be sdet(π RHom( , )[1]) ∗ E E where RHom is the sheaf derived endomorphism. Because isflatover andX issmooth,RHom( , )isaperfectcomplex. E M E E The determinant is well defined. Let E be a sheaf over X. Its equivalence class [E] represents a point in the moduli stack . The tangent complex of at [E] is defined to be the graded M M vector space T = Exti (E,E). [E]M X Mi IthasanobviousgradedLiealgebrastructurebyanti-commutingtheassociative product. The degree zero piece is the subalgebra of endomorphisms whose corresponding group is Aut(E). The fiber of the determinant line bundle L at [E] is top top ( Exteven(E,E))−1 Extodd(E,E). ⊗ ^ ^ It inherits an action of Aut(E). AsheafE iscalledsimple ifExt0(E,E)=C. Wedenotethemodulispaceof ∼ simple sheavesby si. It is a C∗ gerb overa scheme (Deligne-Mumford stack) M locally of finite type. By restricting to sheaves with fixed determinant we can removethis C∗ gerbconsistently and obtain a moduli scheme (DM stack). The determinant line bundle descends. 2.2 Analytic definition WegiveaseconddefinitionofL formodulispaceofholomorphicvectorbundles. Our main reference is [7] and [15] LetE beacomplexvectorbundleoverX. Weidentifyitwiththeframebun- dleofaprincipalGc bundleforGc =GL(n,C)wherenistherankofE. Denote A for the infinite dimensional linear manifold A0,1(ad E) of (0,1) forms with value inadjointbundle. An infinite dimensionalgaugegroupGc :=C∞(X,Gc) acts on A. The orbit space A/Gc parameterizes equivalence classes of (0,1)- connections. Typically, we complete A to a complex Banach space under appropriate Sobolev norm. However, the results in this paper are insensitive to particular choice of Sobolev norm. So we will ignore the completion. A (0,1) connection :A0,0(E) A0,1(E) is called integrable if its curva- ture F = 2 vanishes∇. Strictly spe→aking, A is identified with the underlying ∇ affine space∇of A0,1(ad E). Given two (0,1)-connections A and A′, their differ- ence A A′ lies in A0,1(ad E). − If we fix an integrable reference connection , a connection := +A A for A A0,1(ad E) is integrable if and only if∇it satisfies the M∇aurer-C∇artan ∈ equation d A+A A=0 ∇ ∧ 4 where d := [ , ]. The space of integrable connections, denoted by A(1,1), ∇ is an analytic s∇ub−space of A. Because integrable (0,1) connections are in one to one correspondence with holomorphic structures on E, the Banach analytic stackA(1,1)/Gcparameterizesequivalenceclassesofholomorphicvectorbundles with underlying complex vector bundle being E. If we pick a hermitian metric on E, then the gauge group Gc is reduced to the unitary group G = U(n). Similar to the fixing determinant trick in algebraicgeometry,wecanconsiderthegaugegrouptobespecialunitarygroup. By abusing the notations, we use E to denote the underlying principal SU(n) bundle. Given a connection A, there is a first order elliptic operator D acting on A the Dolbeault complex L:=A0,•(adE), defined as D = + ∗ :Leven Lodd A ∇A ∇A → where ∗ istheadjointwithrespecttotheL2 metricinducedbythehermitian ∇A metric on E. The adjoint D∗ maps Lodd to Leven. A When is integrable, there are isomorphisms A ∇ Ker DA ∼=Exteven(E∇A,E∇A),Ker DA∗ ∼=Extodd(E∇A,E∇A) where E is the corresponding holomorphic vector bundle. When is non- ∇A ∇A integrable, D is invertible. A Definition 2.2. For the family ofelliptic operatorsD with A A, we define A the fiber of its determinant line bundle L at A to be ∈ top top ( Ker D )−1 Ker D∗. A ⊗ A ^ ^ One can turn L into a holomorphic line bundle over A. Define the ∂¯- Laplacian ∆ to be the second order elliptic operator D D∗. For A A and A A A ∈ a positiverealnumber l thatis notinthe spectrumof∆ ,there exists anopen A neighborhood U of A such that the direct sum of eigenspaces of ∆ A A H+ = H+ <l λ Mλ<l and H− = H− <l λ Mλ<l form holomorphic vector bundles over U . The determinant line bundle L is A defined locally to be top top ( H+)−1 H−. <l ⊗ <l ^ ^ Because D is an isomorphism from H+ to H− for λ positive, the above defi- A λ λ nition is independent with choice of l. 5 The elliptic operator D restricts to a linear transform from H+ to H−. A <l <l Its determinant det(D ) defines a section of L over U that vanishes exactly A A when Ker(DA) is nonzero. Onthe overlapof two charts UA andUA′, there are two numbers 0<l <l′ that arenot in the spectrum ofLaplacians suchthat L are super determinants defined above. The section det(D ) of (detH+ )−1 A (l,l′) ⊗ detH− isinvertible. Bymultiplyingsuchsection,thedeterminantlinebundle (l,l′) glues on the overlap. It is clear that the above construction is compatible with action of Gc, i.e. L is a line bundle over the stack A/Gc. When both algebraic and analytic definitions of the determine line bundle apply, they coincide. Remark 2.3. 1. The algebraic definition applies to any perfect complexes, in particular coherent sheaves, over X. The analytic definition only applies for vector bundles. 2. The analytic definition applies for family (not necessarily bounded) of elliptic operators on space of sections of vector bundle. When the elliptic operator is ∆ for an integrable connection A, it coincides with the al- A gebraic definition fiberwise. However, the determinant still makes sense when F is nonzero while the algebraic definition stops to work. A The key observation of the paper is that the analytic definition of determi- nantbundle isabetter definitiontodiscussKontsevich-Soibelman’sorientation data. Thisisbecausebyworkingwithnon-integrableconnectionsmosttechnical difficulties coming from singularities of moduli space are gone. 3 Rational c (L) I 1 Let andL be definedasabove. If is a smoothmanifoldthen firstChern classMc (L) lies in H2( ,Z). L hasMa topological square root if and only if 1 c (L) is divisible by twMo. 1 Overrational,c (L)canbecomputedbyGrothendieck-Riemann-Rochthe- 1 orem in algebraic context or Atiyah-Singer index theorem in analytic context. It suffices to check the case when is a compact Riemann surface. In this M section, we compute the rational c using GRR theorem,under the assumption 1 that is a smooth proper curve. Proof of the general case, which requires M Atiyah-Singer index theorem, will be given in section 5. Let X be a simply connected CY 3-fold and C be a compact Riemman surface. Let be the universalsheaf over X C, π be the projectionto C and E × p be the projection to X. Theorem 3.1. Modulo torsion, the first Chern class of π RHom( , ) is di- ∗ E E visible by 2. 6 Proof. For simplicity we denote F for RHom( , ). GRR theorem says E E ch(πF) td =π (ch(F) td ). ! C ∗ X×C · · By adjunction, we can rewrite it as ch(πF)=π (ch(F) p∗td ). ! ∗ X · The Todd class of X is c (X) 2 1+ . 12 The Chern character of is E c2 2c c3 3c c +3c c4 4c2c +2c2+4c c 4c ch( )=r+c + 1− 2 + 1− 1 2 3 + 1− 1 2 2 1 3− 4. 1 E 2 6 24 The Chern character of F is (r 1)c4 4rc2c +2(r+6)c2+4(r 3)c c 4rc ch(F)=r2+((r 1)c2 2rc )+ − 1− 1 2 2 − 1 3− 4. − 1− 2 12 Apply GRR, we obtain (r 1)(c4+c2c (X)) rc c (X) rc2c (r+6)c2 (r 3)c c rc c (πF)=[ − 1 1 2 2 2 1 2+ 2+ − 1 3 4][X]. 1 ! 12 − 6 − 3 6 3 − 3 · Separate terms depending on rank and terms independent of rank. c2c (X)+c4 c c (X) c2 c2c c c c c2c (X)+c4 c (πF)=r[ 1 2 1 2 2 + 2 1 2 + 1 3 4] [X] [ 1 2 1 c2+c c ] [X] 1 ! 12 − 6 6 − 3 3 − 3 · − 12 − 2 1 3 · c2c (X)+c4 =2rc (π ) [ 1 2 1 c2+c c ] [X] 1 !E − 12 − 2 1 3 · (3.1) The rank depending term is even since c (π ) belongs to H2(C,Z). We 1 ! E need to show the rank independent term is even. Lemma 3.2. Let A be a class in H2(X,Z). Then 2A3+A c (X) 0 mod 12. 2 ∪ ≡ Proof. Let A be c (D) for some divisor D on X. By GRR, 1 A3 A c (X) 2 χ( (D))= + ∪ . X O 6 12 Since this must be an integer, the lemma follows. By Kunneth formula, c can be written as p∗A+π∗B where A H2(X,Z) 1 and B H2(C,Z). By previous lemma, the term ∈ ∈ c2c (X)+c4 2ABc (X)+4A3B 1 2 1 [X]= 2 [X]. 12 · 12 · 7 is even. Supposerisodd,thereexistsadivisorDsuchthatc ( (D))iseven. There- 1 E fore, we can assume c c to be even. Finally, c2 is even again by Kunneth 1 3 2 formula. If is a line bundle, the theorem holds trivially. Let be a sheaf of rank E E r =2k >0, we consider a short exact sequence 0 //L // //Q //0 E where L is a line bundle and Q is a sheaf of rank 2k 1. − ch( ∨ )=ch(L∨ L)+ch(Q∨ Q)+ch(L∨ Q)+ch(L Q∨). E ⊗E ⊗ ⊗ ⊗ ⊗ The sum of the last two terms is even because the odd terms get canceled and the even terms get doubled. Then the even rank case is proved. 4 Some results in topology and gauge theory We recall some basics on gauge theory. Our main reference is Chapter 5 of [6]. Let X be a compact complex manifold and G be a compact Lie group. Unless we specify, G will be taken to be SU(n). Let P be a principal G bundle over X. Let A = A be the space of X,P connections on P and G be the gauge group. The main theme of this section is the topology of the orbit space B = A/G. Sometime, when there is doubt about which manifold is involved, we write B instead. X Because the action of G is not free, it is much easier to work with framed connections. If x is a base point on X, a framed connection is a pair (A,φ) 0 whereAisaconnectionandφisanisomorphismofG-spacesφ:G P . The gauge group G acts naturally on space of framed connections and→we wx0rite B˜ for the space of equivalence classes B˜ =(A Hom(G,P ))/G. × x0 Thereis anaturalactionofthefinite dimensionalgaugegroupGonB˜ such that the quotient stack is B. One way to think of this quotient is to regard a framing φ as fixed and define G G to be its stabilizer, that is 0 ⊂ G = g G g(x )=1 . 0 0 { ∈ | } Then B˜ may be described as A/G and the projection B˜ B is simply the 0 → quotient map for the remainder of the gauge group, G/G =G. 0 ∼ We recall a standard theorem in algebraic topology. 8 Theorem 4.1. There is a weak homotopy equivalence B˜ Map∗(X,BG) X,P P ≃ where Map∗ denotes base-point-preserving maps and Map∗(X,BG) denotes P the homotopy class corresponding to the bundle P X. → Proof. See proposition 5.1.4 of [6]. Proposition 4.2. When G equals SU(n) for n 0, H2(B˜ ,Z) is torsion S6 ≫ free. Proof. Because a principal G bundle over S6 is determined by its transition function over the equator, we have a homotopy equivalence B˜ Map∗(S5,SU(n))=Ω5SU(n). S6 ≃ Lemma 4.3. Assume k<2n. There is an isomorphism HN(ΩkSU(n))=HN(ΩkSU(n+1)) ∼ for N <2n k. − Proof. We first compute the cohomologyof iterated loop spaces of odd spheres since they are the building blocks of loop spaces of SU(n). When k < 2n+1, we claim Z j =0,2n+1 k Hj(ΩkS2n+1,Z)= − (4.1) (cid:26) 0 0<j <2n+1 k − Thiscanbeprovedbyinductiononk. Whenk =0,thisisclearlyright. Assume k is even. Consider the fibration ΩkS2n+1 PΩk−1S2n+1 Ωk−1S2n+1. → → The condition k < 2n + 1 guarantees the base to be connected and simply connected. By the induction assumption, the bottom row of E2 page of the Serre spectral sequence looks like Z ...0... Z ...0... where the second copy of Z appear in degree 2n+2 k. Since the path space − is contractible, the claim follows. The other half of the claim can be check similarly. Consider the spectral sequence for the fibration ΩkSU(n) ΩkSU(n+1) ΩkS2n+1. → → The j-th row of E2 page is zero when j = 2,3,...,2n k. The convergence − of spectral sequence gives HN(ΩkSU(n + 1)) = HN(ΩkSU(n)) when N = ∼ 0,1,...,2n 1 k. − − 9 Remark 4.4. The integral cohomology of ΩkS2n+1 is in general not torsion free. However,for big n the lower degree component will be torsion free. Recall the stable special unitary group SU is defined to be the direct limit SU =limSU(n). −→ In general, the cohomology functor doesn’t commute with direct limit. The discrepancy is measured by the derived functor lim1. However, by Lemma 4.3 lim1 vanishes. Therefore, for fixed N and k −→ −→ HN(ΩkSU)=HN(ΩkSU(n)) ∼ when n 0. ≫ The Bott periodicity theorem says there is a homotopy equivalence ΩSU BSU. ≃ We are interested in the first and the second cohomology group of Ω5SU(n), i.e. N = 1,2. By Bott periodicity, HN(Ω5SU(n),Z) = HN(Ω5SU,Z) = ∼ ∼ HN(ΩSU,Z) when n 0. By Bott’s theorem [4], ΩSU is torsion free and generated by universal≫Chern classes. As a consequence, H1(B˜ ,Z) = 0 and S6 H2(B˜ ,Z)=Z for G=SU(n) with n 0. S6 ≫ Because B˜ Map∗(S2,BSU(n)) ΩSU(n), its integral cohomology is S2 ≃ ≃ torsion free and generated by the universal Chern classes c ,...,c . This is a 1 n special case of Proposition 2.20 of [1]. Similarly, B˜ is homotopic to Ω2SU(n). By Lemma 4.3, when n 0 S3 ≫ Hi(Ω2SU(n)) = Hi(SU) = 0 for i = 1,2 and H3(Ω2SU(n)) = H3(SU) = π (S3)=Z. 3 5 Rational c (L) II 1 In this section, we prove theorem 3.1 for not necessarily smooth scheme. M Let X be a simply connected CY 3-fold. As we see in section 2, there is a family of elliptic operators D over the space of framed connections B˜ . The A X family has determinant line bundle L. Denote the index bundle of the elliptic complex DA Leven nn -- Lodd DA∗ by ind(D,E). We are interested in the case of index bundle of the universal principal bundle over B˜ X. X P × 10