Orientations for DT Invariants on Quasi-Projective Calabi-Yau 4-Folds

Orientations for DT invariants on quasi-projective Calabi–Yau 4-folds

Arkadij Bojko∗

Abstract For a Calabi–Yau 4-fold (X, ω), where X is quasi-projective and ω is a nowhere vanishing section of its canonical bundle KX , the (derived) moduli stack of compactly supported perfect complexes MX is −2-shifted symplectic and thus has an orientation ω bundle O → MX in the sense of Borisov–Joyce [11] necessary for defining Donaldson– Thomas type invariants of X. We extend first the orientability result of Cao–Gross– Joyce [15] to projective spin 4-folds. Then for any smooth projective compactification Y , such that D = Y \X is strictly normal crossing, we define orientation bundles on

the stack MY ×MD MY and express these as pullbacks of Z2-bundles in gauge theory, constructed using positive Dirac operators on the double of X. As a result, we relate the ω orientation bundle O → MX to a gauge-theoretic orientation on the classifying space of compactly supported K-theory. Using orientability of the latter, we obtain orientability of MX . We also prove orientability of moduli spaces of stable pairs and Hilbert schemes of proper subschemes. Finally, we consider the compatibility of orientations under direct sums.

1 Introduction

1.1 Background and results Donaldson-Thomas type theory for Calabi–Yau 4-folds has been developed by Borisov– Joyce [11], Cao–Leung [22] and Oh–Thomas [73] . The construction relies on having Serre duality Ext2(E•,E•) =∼ Ext2(E•,E•)∗ . for a perfect complex E•. This gives a real structure on Ext2(E•,E•), and one can take 2 • • its real subspace Ext (E ,E )R. Borisov–Joyce [11, Definition 3.25] define orientations as a square root of the natural isomorphism arXiv:2008.08441v3 [math.AG] 1 Jul 2021 2 ∼ det(LX ) = OX ∗Address: Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, U.K., E-mail: [email protected]

1 for any −2-shifted symplectic derived scheme X. On moduli stacks of sheaves, this is equiv- 2 • • alent to the orientation of Ext (E ,E )R or in the language of [73] to a choice of a class of positive isotropic subspaces.

For the Borisov–Joyce class or Oh–Thomas class to be defined in singular homology, re- −1 spectively Chow homology with Z[2 ] coefficients, one needs to show that these choices can be made continuously. This was proved for compact Calabi–Yau 4-folds in Cao–Gross–Joyce [15]. However, many conjectural formulas have been written down in [17, 18, 19, 20, 21, 23, 25] and [26] for DT4 invariants of non-compact Calabi–Yau 4-folds, where additionally the moduli spaces can be non-compact and one has to use localization (see Oh–Thomas [73, §6] and Cao–Leung [22, §8]). The construction of virtual structure sheaves of Oh–Thomas [73] for non-compact Calabi–Yau 4-folds and its localization also depends on existence of global orientation. Moreover, in string theory the DT4-invariants over Hilbert (respectively Quot- ) schemes of points were studied by Nekrasov [69] and Nekrasov–Piazzalunga [70]. The dependence on signs (orientations) is explained in [70, §2.4.2]. The main goal of this work is to justify these results geometrically by proving orientability of the moduli stack MX of compactly supported perfect complexes for any quasi- projective Calabi–Yau 4-fold extending the result of Cao–Gross–Joyce [15]. We also prove orientability of moduli spaces of stable pairs (as in [24, 21]) and of Hilbert schemes of proper subschemes. Our result is also used by Oh–Thomas [73] to extend their theory to quasi- projective Calabi–Yau 4-folds and allows for the construction of global virtual structure sheaves even over quasi-projective moduli spaces when X is also quasi-projective. Of particular importance are general toric Calabi–Yau 4-folds and due to work of 4 Nekrasov and his collaborators especially C which was considered from a mathematical point of view in Cao–Kool [17] and Cao–Kool–Monavari [20]. Recently. Kool–Rennemo 4 [56] described explicitly the orientations on Quot-schemes of points on C by entirely different means giving evidence for our more general result. A Calabi–Yau n-fold (X, Ω) here is a smooth quasi-projective variety X of dimension n over C, such that Ω is a nowhere vanishing (algebraic) section of its canonical bundle KX . An important feature of our results is that we are able to express our orientations on MX in terms of differential geometric ones. This allows us to consider compatibility of ω choices of trivializations of the orientation bundle O → MX under direct sums, which is related to constructing vertex algebras and Lie algebras on the homology of MX as done by Joyce in [49] and used for expressing wall-crossing formulae by Gross–Joyce–Tanaka in [43]. As another consequence we obtain that if Mα is a moduli scheme or stack of sheaves 0 with given compactly supported K-theory class α ∈ Kcs (X), then there is a natural choice of orientations on it up to a global sign coming from differential geometry independent of how many connected components Mα has after fixing a choice of compactification Y of X by a strictly normal crossing divisor in the sense of Definition 2.21.

2 1.2 Content of sections and main theorems In §2 we introduce briefly the language of derived algebraic geometry and recall the results of Pantev–Toën–Vaquié–Vezzosi [74] and Brav–Dyckerhoff [13] about existence of shifted symplectic structures. For any smooth (quasi-)projective variety X, we introduce twisted virtual canonical bundles ΛL for any coherent sheaf L on the moduli stack of (compactly supported) perfect complexes MX . If (X, Θ) is a spin manifold as in Definition 2.10, then the virtual canonical bundle is defined by KMX = ΛΘ, where Θ is the corresponding S theta characteristic. In Definition 2.11, we define the orientation bundle O → MX over the moduli stack of perfect complexes on a projective spin 4-fold X and prove that it is trivializable:

Theorem 1.1 (see Theorem 2.14). Let (X, Θ) be smooth, projective and spin of complex dimension 4, and S O → MX (1.1) top the orientation bundle from Definition 2.11. Let ΓX :(MX ) → CX be as in Definition 2.13 and apply the topological realization functor (−)top from Blanc [8, Definition 3.1] to 1.1 S top top to obtain a Z2-bundle (O ) → (MX ) . There is a canonical isomorphism of Z2-bundles

/ S top ∼ ∗ D+ (O ) = ΓX (OC ) ,

D/ + where OC → CX is the Z2-bundle from Joyce–Tanaka–Upmeier [51, Deﬁnition 2.22] applied to the positive Dirac operator D/ + : S+ → S− as in Cao–Gross–Joyce [15, Theorem S 1.11]. In particular, O → MX is trivializable. We describe examples where this can be used to construct orientations on non-compact Calabi–Yau 4-folds using Corollary 2.16. These include

4 a) C .

b) Tot(KV → V ) for every smooth projective variety V , where KV is the canonical line bundle and dimC(V ) = 3.

c) Tot(L1 ⊕ L2 → S) for any smooth projective surface S, where L1,L2 are line bundles ∼ such that L1 ⊗ L2 = KS. d) Tot(E → C) for any smooth projective curve C, where E is a rank 3 vector bundle, 3 such that Λ E = KC . This is a simple generalization of the work of Cao–Gross–Joyce [15], . Instead of trying to answer the question of existence of “spin compactifications” of Calabi–Yau 4-folds, we define in Definition 2.24 for any non-compact Calabi–Yau 4-fold X and its smooth projective compactification Y the moduli stack MY ×MD MY of perfect complexes identified at the divisor D = Y \X which we require to be strictly normal crossing ./ (see Definition 2.21). We define an orientation bundle O → MY ×MD MY depending on

3 extension data ./, which contains the information about a choice of order ord of the smooth SN irreducible divisors in D = i=1 Di. Under the inclusion into the ﬁrst component

ζ : MX → MY ×MD MY

./ ω the bundle O pulls back to O → MX . This construction mimics the excision techniques from gauge theory as in Donaldson–Kronheimer [30], Donaldson [29] and Upmeier [95] and is more natural. The analog in topology would be considering vector bundles E,F → X+ identiﬁed on +, where (X+, +) is the one point compactiﬁcation of X by the point +. 0 This is known to generate Kcs (X) which is the natural topological analog of compactly supported perfect complexes on X. We collect here the orientability results stated in §2 (see Theorem 2.26 and Theorem 2.28):

Theorem 1.2 (Theorem 2.26). Let (X, Ω) be a smooth Calabi–Yau 4-fold and Y its smooth projective compactiﬁcation by a strictly normal crossing divisor D. Choose ord and the extension data ./ as in Deﬁnition 2.23, then the Z2-bundle

./ O → MY ×MD MY is trivializable. Let CD, CY be the topological spaces of maps from Deﬁnition 2.13. Let C DO → CY ×CD CY be the trivializable Z2-bundle from (4.13). If

top Γ:(MY,D) → CY ×CD CY is the natural map from Deﬁnition 2.25, then there exists a canonical isomorphism

./ ∗ C ∼ ./ top I :Γ (DO) = (O ) .

As a consequence of this we prove:

Theorem 1.3 (Theorem 2.28). Let (X, Ω) be a Calabi–Yau 4-fold, then the Z2-bundle

ω O → MX (1.2) from Definition 2.9 is trivializable. Moreover, fix a smooth projective compactification Y , such that D = Y \X is a strictly ω top top normal crossing divisor. Let (O ) → (MX ) be the Z2-bundle obtained by applying the topological realization functor (−)top to (1.2), then there is a canonical isomorphism

cs ∗ cs ∼ ω top IY : (ΓX ) (O ) = (O ) ,

cs cs cs + where O → CX is the trivializable Z2-bundle from Deﬁnition 2.27, CX = MapC0 (X , +), (BU× 0 cs top cs Z, 0) is the topological space classifying Kcs (X) and Γ :(MX ) → CX is the natural map from Deﬁnition 2.27.

4 We argue in Remark 2.29, that with some extra care, the isomorphism IY could be shown to be independent of Y , but leave proof of this for once applications arise. If X is quasi-projective and M is a moduli stack of pairs of the form OX → F , where F is compactly supported, we need a modification, because the complex is not compactly supported. Let E → M be the universal family, then for fixed compactification X ⊂ Y there is a natural isomorphism from Definition 2.30:

∼ ∗ det HomM(E, E) 0 = det HomM(E, E) 0 , where (−)0 denotes the trace-less part and we use the notation HomZ (E,F ) for two perfect ∨ complexes E,F on X × Z to denote π2 ∗(E ⊗ F ), where π2 : X × Z → Z is the projection. 0 To this, there is an associated Z2-bundle O → M. We can now state the result. Theorem 1.4 (Thm. 2.31). Let i : X → Y be a compactiﬁcation with Y \X = D strictly ¯ ¯ normal crossing. Let η : M → MX ×MD MY be given by [E] →, [E, OY ], where E is the extension by a structure sheaf to the divisor, then there is a canonical isomorphism

η∗(O./) =∼ O0 .

In particular, O0 is trivializable.

Consequentially, if all stable pairs parameterized by M are of a fixed class OX + α, 0 J K where α ∈ Kcs(X), this determines unique orientations up to a global sign for a fixed compactification Y of X. In §3, we recall some background from Cao–Gross–Joyce [15] and their orientability result [15, Theorem 1.11]. We then develop some technical tools for transport along complex determinant line bundles of complex pseudo-differential operators generalizing the work of work of Upmeier [95], Donaldson [29], [30] and Atiyah–Singer [4]. We require such generalizations, because in allowing Y to be any compactification of X, we lose the ability to work with real structures and real line bundles, as there is no analog for the real Dirac operator D/ + on Y . We can still do excisions natural up to contractible choices of isotopies and use these to restrict everything back into X where the operator D/ + exists. In §4.1 we construct the doubled spin manifold Y˜ from X, and its moduli space (topo- ˜ ˜ logical stack) BY,˜ T˜ of connections on pairs of principal bundle P,Q → Y identified on T . This space has a product orientation as defined in Definition 4.2, which corresponds to the product of the orientation Z2-torsors at each pair of connections. The resulting orientation bundle is denoted DO(Y˜ ) and its pullback to pairs of vector bundles on Y identified on D is denoted by DO. ./ The bundles DO and O restricted to holomorphic vector bundles generated by global sections are related in §4.3 as strong H-principal Z2-bundles (see Definition 3.4) by constructing isotopies (natural up to contractible choices) between the real structures used to obtain them. This tells us that O./ is trivializable and allows us to express compatibility under direct sums in §5. As the theory developed for identifying the Z2-bundles relies on constructing natural isotopies of algebraic and gauge theoretic isomorphisms of complex

5 determinant line bundles, and this part of it works in great generality in any dimension, the author hopes to use it to construct orientation data as in Joyce–Upmeier [52] for any non-compact Calabi–Yau 3-fold. Finally, in §5, we discuss the relations of orientations under direct sums in K-theory. For given extension data ./, we obtain the result in Proposition 5.3, which expresses choices ./ of trivialization of O → MY ×MD MY in terms of orientations on the K-theoretic space 0 CY ×CD CY with the property π0(CY ×CD CY ) = K (Y ∪D Y ). The signs comparing the orientations are more complicated, but restrict to the expected result under the inclusions cs cs ζ : MX → MY ×MD MY and κ : CX → CY ×CD CY : cs cs Theorem 1.5 (Theorem 5.4). Let Cα denote the connected component of CX corresponding 0 cs cs cs ω cs to α ∈ Kcs(X) = π0(CX ) and Oα = O |Cα . There is a canonical isomorphism φ : ω ω ∗ ω cs cs O O → µ (O ) such that for ﬁxed choices of trivialization oα of Oα , we have ω cs ∗ cs cs ∗ cs ∼ cs ∗ cs φ I (ΓX ) oα I (ΓX ) oβ = α,βI (ΓX ) oα+β .

χ¯(α,α)¯χ(β,β)+¯χ(α,β) where the α,β ∈ {±1} satisfy β,α = (−1) α,β and α,βα+β,γ = β,γα,β+γ 0 for all α, β, γ ∈ Kcs(X). Using this, we formulate a version of [51, Theorem 2.27] for the compactly supported cs orientation group ΩX adapted to the compactly supported case. As a result, we obtain that for all three stacks MZ , MY ×MD MY and MX for (Z, Θ) projective spin 4-fold, (X, Ω) a non-compact Calabi–Yau 4-fold and Y a smooth compacti- ﬁcation of X, one can construct Joyce’s vertex algebras on their homology as in [49] using the signs from Proposition 5.3 and Remark 2.15. For MX the signs α,β from Theorem 1.5 are used in constructing the vertex algebra, which could be used to express wall-crossing formulas for (localized) DT4 invariants using the framework of Gross–Joyce–Tanaka [43].

Acknowledgments

The author wishes to express his gratitude to Dominic Joyce for his supervision and many helpful suggestions and ideas. We are indebted to Markus Upmeier for patient explanation of his work. We also thank Chris Brav, Yalong Cao, Simon Donaldson, Mathieu Florence, Michel van Garrel, Jacob Gross, Maxim Kontsevich, Martijn Kool, Edwin Kutas, Sven Meinhardt, Hector Papoulias, Yuuji Tanaka, Richard Thomas, Bertrand To¨enand Andrzej Weber. Finally, we would like to thank the referee for invaluable remarks. The author was supported by a Clarendon Fund Scholarship at the University of Oxford.

Notation

dim If E → X is a complex vector bundle, we denote det(E) = Λ C E its complex determinant ∗ dim line bundle. We write det (E) to denote its dual. If E is real we write detR(E) = Λ R E.

6 If L1,L2 → Y are two line bundles we will sometimes use the notation L1L2 = L1 ⊗ L2. ∗ If P : H0 → H1 is an operator between Hilbert spaces, we write P to denote its adjoint.

If D ⊂ X is a divisor, we write E(D) = E ⊗OX OD. If s : V → W is a map of vector bundles V,W → X and Q → X another vector bundle, then we will write s = s ⊗ idQ : V ⊗ Q → W ⊗ Q. an We often do not distinguish between a scheme X over C and its analytiﬁcation X .

2 Orientation on non-compact Calabi–Yau 4-folds

In this section, we review the definition of orientation on the moduli stack of perfect complexes of a Calabi–Yau 4-fold which uses the existence of −2-shifted symplectic structure as defined by Pantev–Toën–Vaquié–Vezzosi in [74]. We introduce some new notions of orientation in more general cases which we use together with compactifications of non-compact Calabi–Yau 4-folds to prove that the moduli stack of compactly supported perfect complexes is orientable.

2.1 Moduli stacks of perfect complexes and shifted symplectic structures Here we recall the construction of moduli stacks of perfect complexes that we will be working with and the shifted symplectic structures on them.

Let X be a smooth algebraic variety. Its category DbCoh(X) of complexes of sheaves with coherent cohomologies does not have a moduli stack of its objects in the setting of standard algebraic stacks. Instead, one needs to rely on the methods provided by the theory of higher stacks and derived stacks. For a thorough discussion of these two terms, one can look at Toënand Vezzosi [94, 91, 93] in the setting of model categories (see Hovey [48] and Hirschhorn [46] ) and Lurie [62] in the setting of ∞-categories (see Lurie [64] or [63]). Both higher stacks and derived stacks over C form the ∞-categories HStaC and DStaC. There exist an inclusion ∞-functor i : HStaC → DStaC and its adjoint truncation ∞-functor t0 : DStaC → HStaC, which relate the two categories. Note that the C-points of the stacks are left invariant under these functors. Let T be a dg-category (for background on dg-categories see Keller [54] and Toën[90]). Toën[90] introduces homotopy theory of dg-categories which is then used in Toënand Vaquié [92] to define for T its associated moduli stack as a derived stack MT which classifies the pseudo-perfect dg-modules of T op. This defines a functor from the homotopy category op of dg-categories to the homotopy category of derived stacks M(−) : Ho(dg - Cat) → Ho(DSta). One can compose this with the truncation functor t0 : Ho(DSta) → Ho(HSta) mapping to the homotopy category of higher (infinity) stacks. We denote the composition by M(−) = t0 ◦ M(−) .

7 Deﬁnition 2.1. Let X be quasi-projective variety over C, then we use the notation MX ,

MX for MLpe(X), respectively MLpe(X), where Lpe(X) is the dg-category of perfect complexes on X. When X is smooth and projective, it is shown by Toën–Vaquiéin [92, Corollary 3.29] that MX is a locally geometric derived stack which is locally of finite type. A locally geometric derived stack corresponds to a union of open geometric sub-stacks, and thus it has a well defined cotangent complex LMX in the dg-category of perfect modules on MX denoted by Lpe(MX ).

For X smooth and projective, both MX and MX can be expressed as mapping stacks in the homotopy categories of the ∞-categories DStaC and HStaC. Let PerfC be the derived stack of perfect dg-modules/complexes over C as deﬁned in [91, Deﬁnition 1.3.7.5] when applied to complexes of vector spaces over C as the chosen homotopy algebraic geometric context. Taking its truncation PerfC = t0(PerfC), we have

MX = Map(X, PerfC) MX = Map(X, PerfC) . (2.1) Deﬁnition 2.2. If X is just a quasi-projective variety that is not smooth or is not projective, then we will denote the mapping stacks using a superscript:

X X M = Map(X, PerfC) M = Map(X, PerfC) . (2.2) X Remark 2.3. One can show that if X is not smooth, then MX will not be equal to M in general, as the former can classify objects of QCoh(X) which are not perfect. When X is a quasi-projective smooth variety over C, then Lpe(X) is no longer a finite type dg-category and its moduli stack MX behaves differently. For any affine scheme Spec(A) ∈ AffC, the Spec(A)-points of MX are families of perfect complexes on X × Spec(A) with proper support in X.

If X is proper, we can use the description of MX as a mapping stack to construct a universal complex on X × MX : If

u : X × MX → PerfC (2.3) is the canonical morphisms for MX as a mapping stack, and U0 is the universal complex on PerfC used by Pantev–Toën–Vaquié–Vezzosi in [74], then one defines the universal complex ∗ UX = u (U0) on X ×MX . When X is quasi-projective and not necessarily proper, we need a different construction of the universal complex. Definition 2.4. Let iX : X,→ Y be a smooth compactification of X and UY ∈ Lpe Y × MY the universal complex. Let

ξ = M ∗ : M → M (2.4) Y (iX ) X Y ∗ be the image of the pullback iX : Lpe(Y ) → Lpe(Y ), then it acts on Spec(A)-points by ∗ the right adjoint of iX × idSpec(A) as follows from its construction in To¨en–Vaqui´e[92,

8 §3.1] and therefore by the pushforward (iX × idSpec(A))∗ of families of compactly supported perfect complexes on X. We deﬁne UX → X × MX by

∗ UX = ξY (UY ) . It is independent of the choice of a compactiﬁcation∗.

Q Q 0 When Z = i∈I Zi, we will use πI0 : Z → i∈I0 Zi for I ⊂ I to denote the projection to I0 components of the product. We use this also for general ﬁber products. Let Y now be any projective smooth four-fold and L a coherent sheaf on Y and UY ∈ Lpe(Y × MY ) its universal complex. We deﬁne

Ext = π (π∗ U ∨ ⊗ π∗ U ⊗ π∗L) , P = ∆∗ Ext . (2.5) L 2,3 ∗ 1,2 Y 1,3 Y 1 L MY L

As pushforward along π2,3 : Y × MY × MY → MY × MY maps (compactly supported) ! perfect complexes in Y to perfect complexes, it has a right adjoint π2,3 by Lurie’s adjoint functor theorem Gaitsgory–Rozenblyum [39, Thm. 2.5.4, §1.1.2], Lurie [64, Cor. 5.5.2.9]. ! ∗ Moreover, π2,3 = π2,3(−) ⊗ KY [4], which gives us the usual Serre duality in families ∼ ∗ ∨ Ext = σ (Ext ∨ )[−4] , (2.6) L (KY ⊗L ) where σ : MY × MY → MY × MY is the map interchanging the factors. Deﬁnition 2.5. Let Y be smooth and and L a coherent sheaf on Y , then as (2.5) are perfect, we construct the L-twisted virtual canonical bundle

ΣL = det(ExtL) , ΛL = det(PL) .

Moreover, ΣL, ΛL are Z2-graded with degree given by a map deg(ΛL): MX → Z2, such that

deg(ΣL)|Mα×Mβ ≡ χ(α, β · L) (mod 2) , deg(ΛL)|Mα ≡ χ(α, α · L) (mod 2) .

0 where α, β ∈ K (X) and Mα is the stack of complexes with class E = α. See Deﬁnition 3.12 and (5.5) for more details. J K

From the duality (2.6), we obtain the isomorphisms

∼ ∗ ∗ ∨ ∗ Σ = σ (Σ ∨ ) , θ : −→ ∨ [−4] , i :Λ −→ Λ ∨ . (2.7) L KX ⊗L L PL P(KX ⊗L ) L L (KX ⊗L )

It follows from [13, Proposition 3.3] that the tangent complex of MX can be expressed as T (MX ) = POX [1] (2.8) We use the term Calabi–Yau manifold in the following sense.

∗Simply choose a common compactiﬁcation Y ← Y 00 → Y 0 and compare the resulting universal complexes.

9 Deﬁnition 2.6. A Calabi–Yau n-fold is a pair (X, Ω), where X is a smooth quasi-projective variety of dimension n over C and Ω is a global non-vanishing algebraic section of the canonical bundle KX of X. In this case, we use the notation KMX = ΛOX . Then applying the above to L = OX , we obtain the isomorphisms ω θ : T(MX ) → L(MX )[−2] , (2.9) iω : K → K∗ . (2.10) MX MX Brav and Dyckerhoﬀ prove in [13, Proposition 5.3] and [13, Theorem 5.5 (1)] that the ω isomorphism θ comes from a −2-shifted symplectic form ω on MX .

This extra condition to have a −2-shifted symplectic stack (MX , ω) is necessary for constructing Borisov–Joyce fundamental classes as in [11] and Oh–Thomas classes from [73].

2.2 Orientation bundles on moduli stacks of perfect complexes In this subsection, we review the known results of orientation on compact Calabi–Yau 4- folds, and we deﬁne new orientation bundles which we will use in later sections to prove the orientability for the non-compact case.

There is a diﬀerent but equivalent approach to constructing a Z2-bundle other than taking a real line bundle (or a complex line bundle with a real structure) and its associated orientation bundle. Deﬁnition 2.7. Let L → X be a complex line bundle over some variety, stack or a topological space. Additionally, let us assume, that there is an isomorphism I : L → L∗. Then consider the adjoint isomorphism J which is the composition of

idL⊗I ∗ L ⊗ L −−−−→ L ⊗ L → C , where the second isomorphism is the canonical one. Then one can define the square root I Z2-bundle associated with I denoted by O . This bundle is given by the sheaf of its sections (which we denote the same way): I ∼ O (U) = {o : L|U −→ CU : o ⊗ o = J} . As this will be important for the proof of Theorem 2.26, we mention here how this construction is related to the one using real structures. Remark 2.8. Let I : L → L∗ be an isomorphism and h−, −i a metric on L. There is the induced isomorphism µ0 : L → L¯ such that h−, µ0(l)i = I(l) for all l ∈ L. As an anti- linear endomorphism, we take its second power (µ0)2 which is given by a multiplication by a 0 √µ strictly positive real function s on X. Then µ = s is a well defined real structure on L. Let Lµ denote the real line bundle of fixed points and or(Lµ) its associated orientation bundle, I then it is easy to see that or(Lµ) is canonically isomorphic to O . For a nice exposition of orientations on O(n, C) bundles see Oh-Thomas [73].

10 We recall the deﬁnition of orientations on −2-shifted derived stacks (see Borisov–Joyce [11, Deﬁnition 2.12]), which can now be applied also to non-compact Calabi–Yau 4-folds.

ω Deﬁnition 2.9. Let (S, ω) be a −2-shifted symplectic derived stack. Let θ : T(S) → L(S)[−2] be the isomorphism associated to the −2 shifted symplectic form. Taking the ω ∗ determinants and inverting, one obtains the isomorphism i : det(LS) → det(LS) . The ω ω orientation bundle O → S is the square root Z2 bundle associated to i .

Let (X, ω) be a Calabi–Yau 4-fold as in Deﬁnition 2.6, then we have KMX = det(LMX ). The isomorphisms θω and iω from (2.9) and (2.10) are associated to the −2-shifted symplectic derived stack from Deﬁnitions 2.1 and 2.6. We denote in this case the orientation ω bundle by O → MX .

From now now on we will restrict ourselves to working only in higher stacks HStaC. By this we mean that we take truncations S = t0(S) everywhere and restrict bundles and their isomorphism constructed on S to S by the canonical inclusion S,→ S. When X is a compact Calabi–Yau 4-fold, Cao–Gross–Joyce [15, Theorem 1.15] prove ω that O → MX is trivializable. One could generalize their result by replacing the require- ment of X being Calabi–Yau by a weaker one.

Deﬁnition 2.10. Let X be a smooth projective variety and KX its canonical divisor class. A divisor class Θ, such that 2Θ = KX is called a theta characteristic. We say that (X, Θ) for a given choice of a theta characteristic Θ is spin.

Using this, we construct orientation for the case when X is smooth projective and spin.

Deﬁnition 2.11. When X is spin and a choice of Θ is made, we will also use the notation

KMX = ΛΘ. Remark 2.12. A choice of Θ is equivalent to a choice of spin structure on Xan (see Atiyah [5, Proposition 3.2]).

Before we state the generalization of Cao–Gross–Joyce [15, Theorem 1.15] to projective spin varieties, we recall some terminology used to formulate it. Blanc [8] and Simpson [83], define the topological realization ∞-functor: (−)top : an HStaC → Top as the simplically enriched left Kan extension of the functor (−) : AffC → Top , which maps every finite type affine scheme over C to its analytification.

Z Deﬁnition 2.13. Let Z be a projective variety over C. Let M , be the mapping stack Z top from (2.2). Let uZ : Z × M → PerfC be the canonical map. Applying (−) and using Blanc [8, 4.2], we obtain (u )top : Zan ×(MZ )top → BU × , where BU = lim BU(n). § Z Z −→n→∞ This gives us Z top an ΓZ :(M ) → MapC0 (Z ,BU × Z) , by the universal property of MapC0 (−, −), where MapC0 (−, −) denotes the mapping space bifunctor in Top. For any topological space T we use the notation CT = MapC0 (T,BU ×Z) .

11 S Theorem 2.14. Let (X, Θ) be spin with the orientation bundle O → MX . Let ΓX : top top S top (MX ) → CX be as in Deﬁnition 2.13 and apply (−) to obtain a Z2-bundle (O ) → top (MX ) . There is a canonical isomorphism of Z2-bundles / S top ∼ ∗ D+ (O ) = ΓX (OC ) ,

D/ + where OC → CX is the Z2-bundle from Joyce–Tanaka–Upmeier [51, Definition 2.22] applied to the positive Dirac operator D/ + : S+ → S− as in Cao–Gross–Joyce [15, Theorem S 1.11]. In particular, O → MX is trivializable by the aforementioned theorem. Proof. This is a simple generalization of the proof of [15, Theorem 1.15] relying on the fact that Theorem 3.10 requires Xan to be a spin manifold to trivialize the orientation bundle on BX . We mention it, as we hope that it will be useful in the future for generalization of DT4 invariants. We only discuss the corresponding real structure on the differential geometric side replacing [15, Definition 3.24]. We have the pairing ∧S :(A0,k ⊗ Θ) ⊗ (A0,4−k ⊗ Θ) → A4,4 , and the corresponding spin Hodge star ?S : A0,q ⊗ Θ → A0,n−q ⊗ Θ

S S 0,k ∞ (β ⊗ t) ∧ ?k (α ⊗ s) = hβ ⊗ s, α ⊗ tiΩ α, β ∈ A , s, t ∈ Γ (Ω) , 4,4 S 0,even where Ω ∈ A is the volume form. As a result, we have the real structures: #1 : A ⊗ 0,even S 0,odd 0,odd S q S Θ → A ⊗ Θ and #2 : A ⊗ Θ → A ⊗ Θ, where again #1 |A0,2q⊗Θ = (−1) ? S q+1 S S and #2 |A0,2q+1⊗Θ = (−1) ? . The Dolbeault operator commutes with these DΘ ◦ #1 = S #2 ◦DΘ and its real part is the positive Dirac operator D/ : S+ → S− by Friedrich [35, §3.4]. As twisting by connections only corresponds to tensoring symbols of operators by identity, this extends also to real structures on det(D∇End(E) ).

Remark 2.15. Note that one can also state the equivalent of Cao–Gross–Joyce [15, Theo- rem 1.15 (c)], expressing the comparison of orientations under direct sums on MX in terms of the comparison on CX . Suppose, that X is Calabi–Yau and that there exists Y smooth with an open embedding iY : X,→ Y , where Y is spin. Recall that we have the map ξY : MX → MY from Definition 2.4. We say that Y is a spin compactification of X. We now state the weaker result about orientability for non-compact Calabi–Yau 4-folds. Corollary 2.16. Let X be a Calabi–Yau 4-fold, and let Y be a spin compactification of ∼ X with a choice of Θ and an isomorphism φ : OX −→ Θ|X , then there exists an induced isomorphism of Z2 bundles on MX : ω ∼ ∗ Θ O = ξY (O ) .

In particular, MX is orientable.

Proof. Let E be a perfect complex on X with a proper support, then the Z2 torsors over a Spec(A)-point [E] of both of the above Z2-bundles are given by ∼ ω {oE : det Hom(E,E) −−→ C s.t. oE ⊗ oE = ad(i )|[E]}

12 where iω is the Serre duality, and we used the isomorphism E ⊗ Θ =∼ E induced by φ. Thus we have a natural identiﬁcation of both Z2-bundles. The last statement follows from Theorem 2.14.

Remark 2.17. If Y is a spin compactiﬁcation of X and Y \X = D is a divisor. Let N D = ∪i=1Di be its decomposition into irreducible components. If we can write the canonical PN divisor of Y as KY = i=1 aiDi, where ai ≡ 0(mod 2), then one can take

N X ai Θ = D 2 i i=1

1 as the square root. After choosing a meromorphic section Ω¯ 2 of Θ with poles and zeros on ∼ D, one obtains an isomorphism φ : OX −→ Θ|X . Then the condition of Corollary 2.16 is satisﬁed.

4 4 Example 2.18. The simplest example is C . While its natural compactiﬁcation P is not 1 3 1 ×4 spin, one can choose to compactify it as P × P or (P ) which are both spin, both of which satisfy the property in Remark 2.17.

Example 2.19. Let S be a smooth projective variety 0 ≤ dimC(S) = k ≤ 4 and let E → S be a vector bundle, s.t. det(E) = KS. Then X = Tot(E → S) is Calabi–Yau. Taking its smooth compactification Y = P(E⊕OS) with the divisor at infinity D = P(E) ⊂ P(E⊕OS), one can show that KY = −(rk(E) + 1)D. If rk(E) ∈ 2Z + 1, we see that we can choose (rk(E)+1) Θ = OY (− 2 D) which satisfies the property of Remark 2.17. Then if rk(E) + k = 4, this is an example of Corollary 2.16, when rk(E) = 1, 3. If X = Tot(L1 ⊕ L2 → S) for a smooth projective surface S and its line bundles L1,L2, s.t. L1L2 = KS, then the spin compactification can be obtained as P(L1⊕OS)×SP(L2⊕OS). Example 2.20. Suppose we have a toric variety X (see Fulton [36], Cox [28]) given by a n n fan in the lattice Z ⊂ R . Suppose it is smooth and it contains the natural cone spanned n by (ei)i=1. Define the hyperplanes

n n X Hi = {(x1, . . . , xn) ∈ R : xj = i} . j=1

Then X is Calabi–Yau if and only if all the primitive vectors of rays of the fan lie in H1 and all the cones are spanned by a basis. A simple generalization of this well known statement shows that X is spin if and only if all the primitive vectors lie in H = S H . odd i∈2Z+1 i Starting from a toric Calabi–Yau X, one can compactify X to a projective smooth toric variety Y by adding divisors corresponding to primitive vectors. In general, we will not 2 have spin compactiﬁcations: Consider the fan in R with more than 3 primitive vectors in H1, then any compactiﬁcation will be consecutive blow ups of a Hirzebruch surface at points, with at least one blow up.

13 A common way of constructing Calabi–Yau manifolds is by removing anti-canonical divisors from a Fano manifold. To further illustrate the scarceness of spin-compactiﬁcations in even dimension, we study the classiﬁcation of toric projective Fano fourfolds by Batyrev [6]. Using the condition described above, we can show that there exist only 4 smooth toric 1 3 1 Fano fourfolds with a spin structure. These are PP3 (O ⊕ O(2)), P × P , P × PP2 (O ⊕ O(1)) 1 and 4P corresponding to the polytopes B2,B4,D12 and L8 respectively. Note that there are 123 smooth projective toric Fano fourfolds in total.

To avoid having to answer the question of existence of spin-compactiﬁcations, we develop a diﬀerent general approach in the next section.

2.3 Orientations for a non-compact Calabi–Yau via algebraic excision principle Let (X, Ω) be a Calabi–Yau fourfold, then we ﬁx a compactiﬁcation Y with D = Y \X a strictly normal crossing divisor, where we are using the following convention.

Deﬁnition 2.21. A strictly normal crossing divisor is a normal crossing divisor, which is a union of smooth divisors with transversal intersections (any k-fold intersection is in particular smooth).

Let (X, Ω) be a Calabi–Yau fourfold, then we fix a compactification Y with Y \D a strictly normal crossing divisor. By Hironaka [45, Main Theorem 1], Bierstone–Milman [7] there exists such a compactification by embedding into a projective space and taking ∼ resolutions. Consider triples complexes (E, F, φ), where E,F ∈ Lpe(Y ) and φ : E|D −→ F |D . We will take the difference of the determinants det Hom(E,E) and det Hom((F,F ) and cancel the contributions which live purely on the divisor. One could think of this as an algebraic version of the excision principle defined for complex operators in §3.3. Let us now make the described method more rigorous. Let X, Y and D be as in the paragraph above, then we can write D as the union

N [ D = Di (2.11) i=1 where each Di is a smooth divisor. We require Ω to be algebraic, then there exists a unique meromorphic section Ω¯ of KY , s.t. Ω¯|X = Ω. The poles and zeroes of Ω¯ express KY uniquely PN in the following form KY = i=1 aiDi, where ai ∈ Z. We may write for the canonical line bundle: N N O O ⊗|ki| KY = O(kiDi) = O(sgn(ki)Di) . (2.12) i=1 i=1

Let ND be the free lattice spanned by the divisors Di which we from now on denote by the N elements ei ∈ ND. For a line bundle L = ⊗i=1O(aiDi) we write La, where a = (a1, . . . , aN ).

14 We will also use the notation Lk = KY . Then for a non-zero global section si of O(Di) one has the usual exact sequence

·si 0 La La+ei La+ei ⊗OY ODi 0 .

As all the operations used to deﬁne Exta = ExtLa and Pa = PLa in Deﬁnition 2.5 are derived, we obtain distinguished triangles

[1] Exta Exta+e Ext Exta[1] , i La+ei ⊗OY ODi (2.13) [1] a a+e a[1] . P P i PLa+ei ⊗OY ODi P By (2.2) both MY and MD can be expressed as mapping stacks Map Y, Perf and i C Map Di, PerfC , respectively. Let incDi : Di → Y be the inclusion, then we denote by ∗ ρi : MY → MDi the morphisms induced by the pullback (incDi ) : Lpe(Y ) → Lpe(Di), where we are using that both stacks are mapping stacks. For each divisor Di we set

La|Di = La,i and

∗ ∨ ∗ ∗ ∗ Exta,i = π2,3 ∗(π1,2 UDi ⊗ π1,3 UDi ⊗ π1La,i) , Pa,i = ∆ Exta,i , Lemma 2.22. We have the isomorphism

∼ ∗ Ext = (ρi × ρi) (Exta+e ,i) , La+ei ⊗OY ODi i ∼ ∗ = ρ ( a+e ,i) . PLa+ei ⊗OY ODi i P i where we use the same notation for the complexes P on MY and MDi .

Proof. For universal complexes UY , UDi on Y × MY , Di × MDi , we have UY |Di×MY = ∗ (idDi × ρi) UDi as follows from the commutative diagram

Y × MY PerfC

incDi ×idMY

idDi ×ρi Di × MY Di × MDi .

For the dual, we also have U ∨| = (id × ρ )∗U ∨ . Thus we have the following Y Di×MY Di i Di

15 equivalences

Ext La+ei ⊗OY ODi ∗ ∨ ∗ ∗ =π2,3 ∗ π1,2(U ) ⊗ π1,3(U) ⊗ π1(La+ei ⊗OY ODi ) ∼ ∨ ∗ =π2,3 ∗ (incDi × idMY ×MY )∗ U ⊗ U ⊗ π1La+ei |Di×MY ×MY ∼ ∗ ∗ ∨ ∗ ∗ =π2,3 ∗ (incDi × idMY ×MY )∗ ◦ (idDi × ρi × ρi) (π1,2(UDi ) ⊗ π1,3(UDi ) ⊗ π1La+ei,i) ∼ ∗ ∗ ∨ ∗ ∗ =π2,3 ∗ (idDi × ρi × ρi) (π1,2(UDi ) ⊗ π1,3(UDi )) ⊗ π1(La+ei,i) ∼ ∗ =(ρi × ρi) (Exta+ei,i) , the ﬁrst isomorphism is the projection formula [38, Lem. 3.2.4] and the last step follows ∗ ∼ ∗ from the base change isomorphism π2,3 ∗ ◦ (idDi × ρi) = (ρi × ρi) ◦ π2,3 ∗ using Gaitsgory [38, Prop. 2.2.2]† and that the diagram

D × MX × MX D × MD × MD D

MX × MX MD × MD ∗ consists of Cartesian diagrams by the pasting law in ∞-categories. The second formula follows using (ρ × ρ ) ◦ ∆ = ∆ ◦ ρ : M → M × M . i i MY MDi i X Di Di After taking determinants of (2.13), we obtain the isomorphisms

∼ ∗ ∼ ∗ Σa+ei = Σa ⊗ ρi Σa+ei,i , Λa+ei = Λa ⊗ ρi Λa+ei,i) , (2.14)

where we omit writing L. We have the maps iDi : Di → Y , iD : D → Y inducing

D ρi :MY −→ MDi , ρD : MY −→ M , N N sp Y Y sp MD := MDi and ρ = ρi : MY −→ MD i=1 i=1

D Note that we have the obvious map M → MDi induced by the inclusion Di ,→ D. This gives sp sp : MY × D MY = MY,D −→ MY × sp MY = M . (2.15) M MD Y,D

Definition 2.23. For given X, Y as above let Ω¯ be a meromorphic section of KY restricting to Ω. Let ord denote the decomposition of D into irreducible components as in (2.11), which also specifies their order, such that there exist 0 ≤ N1 ≤ N2 ≤ N, such that ai = 0 for 0 < i ≤ N1, ai > 0 for N1 < i ≤ N2 and ai < 0 for N2 < i ≤ N, where ai are the coefficients

†These references are stated for derived stacks. So we should work with derived stacks until we construct the isomorphisms in Deﬁnition 2.24, which we can then restrict to its truncation.

16 from (2.12). For the construction, we may assume N1 = 0. We deﬁne extension data as the following ordered collection of sections ./= (si,k)i∈{1,...N2}, (tj,l)j∈{N2+1,...N} , si,k : OY → OY (Di) , tj,l : OY → OY (Dj) . 1≤k≤ai 1≤l≤−aj

Q Q −1 such that i∈{1,...N2} si,k j∈{N2+1,...,N}(tj,l) = Ω and si,k, tj,k are holomorphic with zeros 1≤k≤ai 1≤l≤−aj only on Di, resp. Dj.

This leads to a deﬁnition of a new Z2-bundle: sp Deﬁnition 2.24. On MY,D we have the line bundle

∗ ∗ ∗ LY,D = π1Λ0 ⊗ (π2Λ0) , (2.16)

π1 sp π2 where MY ←−MY,D −→MY are the natural projections. For a ﬁxed choice ./, there is a natural isomorphism

sp ∼ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ϑ./ : LY,D = π1Λ0 ⊗ π1 ◦ ρ (ΛD) ⊗ π1 ◦ ρ (ΛD) ⊗ (π2Λ0) ∼ ∗ ∗ ∗ ∼ ∗ ∗ ∗ ∼ ∗ = π1Λk ⊗ π2(Λk) = π1(Λ0) ⊗ π2(Λ0) = LY,D , (2.17)

sp Here ΛD → MD are line bundles, and we used the commutativity of

sp π2 MY,D MY ρ π1 ρ sp MY MD

∗ in the ﬁrst step. The bundles ΛD = ΛD,− ⊗ ΛD,+ appear as the result of using chosen si,k to construct isomorphism (2.14) for the ﬁrst N2 divisors, then ΛD,− is obtained from using −1 tj,k and (2.14). Thus we will have the expressions:

ΛD,+ =Λ PN2−1 ... ⊗ Λ PN2−1 ⊗ ... k− i=1 aiei+(aN2 −1)eN2 ,N2 (k− i=1 aiei),N2

⊗ Λk−(a−1)e1,1 ⊗ ... ⊗ Λk,1

ΛD,− =Λ PN2 ⊗ ...... ⊗ Λ−eN ,N . k− i=1 aiei,N2+1

The second to last step uses (2.7). We deﬁne the Z2-bundles by using Deﬁnition 2.7:

sp ∗ ./ sp ϑ./ : LY,D →(LY,D) ,Osp → MY,D , ./ ∗ ./ O = sp Osp (2.18)

./ sp where Osp associated to ϑ./ .

17 ./ The important property of the Z2-bundle O is that it is going to allow us to use index theoretic excision on the side of gauge theory to prove its triviality. One should think of the triples [E, F, φ] which are the points in MY,D as similar objects to the relative pairs in [95, Definition 2.5] with identification given in some neighborhood of the divisor D. The ./ Z2-bundle O only cares about the behavior of the complexes in X. top Definition 2.25. Recall that from Definition 2.13 we have the maps ΓY :(MY ) → CY D top and ΓD :(M ) → CD, We define Γ as the composition

top top h top (MY,D) −→ (MY ) ×(MD)top (MY ) −→ C ×h C 'C × C = C . (2.19) Y CD Y Y CD Y Y,D The ﬁrst map is induced by the homotopy commutative diagram obtained from applying (−)top to the Cartesian diagram

MY,D MY .

D MY M

The second map uses homotopy commutativity of

top D top top (MY ) (M ) (MY ) . ΓY ΓD ΓY

CY CD CY

an an an The final homotopy equivalence is the result of the map (incD) :(D) → Y being a cofibration for the standard model structure on Top. The map CY → CD is a fibration so the homotopy fiber-product is given by the strict fiber-product up to homotopy equivalences.

We now state the theorem which follows from Proposition 4.15 below and is the main tool in proving orientability of MX . Theorem 2.26. For X,Y and D ﬁx ord and the extension data ./ as in Deﬁnition 2.24, then the Z2-bundle ./ O → MY,D (2.20) C is trivializable. Let DO → CY,D be the trivializable Z2-bundle from (4.1), then there exists a canonical isomorphism ./ ∗ C ∼ ./ top I :Γ (DO) = (O ) . (2.21) We now reinterpret this result to apply it to the orientation bundle of interest Oω → MX .

18 Deﬁnition 2.27. Let ζ : MX → MY,D induced by the map ξY × [0] : MX × SpecC → MY,D. We have the commutative diagram

top ζtop MX MY,D

Γcs Γ (2.22) cs ics CX CY,D , where cs cs κ : CX = CY ×CD {0} ,→ CY,D, + cs + and CY ×CD {0} = MapC0 (X , +), (BU × Z, 0) . The space C = MapC0 (X , +), (BU × X Z, 0) is the classifying space of compactly supported K-theory on X (see Spanier [85], cs 0 Ranicki–Roe [78, §2], May [67, Chapter 21]): π0(CX ) := Kcs(X) . We define cs cs ∗ C O := (κ ) (DO) . (2.23) The following is the first important consequence of Theorem 2.26 and leads to the construction of virtual fundamental classes using Borisov–Joyce [11] or Oh–Thomas [73] using the −2-shifted symplectic structures. It also gives preferred choices of orientations at fixed points up to a global sign when defining/computing invariants using localization as in [73, 22] for moduli spaces of compactly supported sheaves Mα with a fixed K-theory class 0 α ∈ Kcs and for a given compactification Y . ω Theorem 2.28. Let (X, Ω) be a Calabi–Yau fourfold then the Z2-bundle O → MX is trivializable. Moreover, for a fixed choice of embedding X → Y , with D = Y \X strictly normal crossing, there exists a canonical isomorphism cs ∗ cs ∼ ω top I : (ΓX ) (O ) = (O ) . Proof. We prove this in 3 steps: ∗ ./ ∼ ω 1. We have a natural isomorphism ζ (O ) = O : Consider a Spec(A)-point [E] ∈ MX , then at the corresponding point ([E˜], 0) ∈ MY,D, the isomorphism ϑ./ is given by ˜ ˜ ∼ ∗ ˜ ˜ ∼ ∗ ˜ ˜ det(Hom(E, E)) ⊗ C = det (Hom(E, E ⊗ KY )) = det (Hom(E, E)) , ∼ where we use that ΛL|[0] = C, the first isomorphism is Serre duality, and the second one si,k tj,l is the composition of isomorphisms induced by E −−→ E(Di) and E(−Dj) −−→ E. As Ω E is compactly supported in X, these isomorphisms compose into E −→ E ⊗ KY by the ω assumption on ./. Therefore ϑ./|([E˜],0) coincides with i |[E] and their associated Z2-bundles are identified. 2. By Lemma 4.8, we know Ocs, Oω are independent of the choice of ./. We define a family of I(./) for fixed ord:

(2.22) 1. cs ∗ cs ∼ top ∗ ∗ C ∼ top ∗ ./ top ∼ ω top I(./) : (ΓY ) (O ) = (ζ ) ◦ Γ (DO) = (ζ ) ◦ (O ) = (O ) .

19 ∗ Any two choices of si,k diﬀer by C (and same holds for tj,l). Therefore the set of ./ ∗ PN |a |−1 corresponds to (C ) i i which is connected and I(./) does not depend on ./. 3. For simplicity, let us assume we only have two diﬀerent divisors D1,D2. We then have the isomorphism obtained from applying (2.14) twice

∗ ∼ ∗ ∗ ∗ π1 Λe1+e2 =π1(Λ0)(ρ2 ◦ π1) Λe2,2(ρ1 ◦ π1) Λe1+e2,1 , (2.24) ∗ ∼ ∗ ∗ ∗ π1 Λe1+e2 =π1(Λ0)(ρ1 ◦ π1) Λe1,1(ρ2 ◦ π1) Λe1+e2,2 , (2.25) ∗ ∼ ∗ ∗ ∗ π2 Λe1+e2 =π2(Λ0)(ρ2 ◦ π2) Λe2,2(ρ1 ◦ π2) Λe1+e2,1 , (2.26) ∗ ∼ ∗ ∗ ∗ π2 Λe1+e2 =π2(Λ0)(ρ1 ◦ π2) Λe1,1(ρ2 ◦ π2) Λe1+e2,2 . (2.27)

To show that there is no diﬀerence between the chosen two orders, we use the commutative diagram, where all rows and columns ﬁt into distinguished triangles:

∗ P Pe1 ρ1 Pe1,1

∗ (2.28) Pe2 Pe1+e2 ρ1 Pe1+e2,1

∗ ∗ ∗ ρ2 Pe2,2 ρ2(Pe1+e2,2) ρ1,2 Pe1+e2,1,2

We used in the bottom right corner term the restriction ρ1,2 : MY → MD1∩D2 and e +e ,1,2 = . Taking determinants of all four corner terms of the diagram P 1 2 PO(D1+D2)|D1∩D2 (see Knudsen–Mumford [55, Prop. 1]) and pulling back by π1, we get both (2.24) and (2.25) ∗ ∗ deg (ρD ◦π ) ΛO(D )| deg (ρ1◦π1) ΛO(D )| where the latter comes with (−1) 2 1 1 D1 2 D2 . This holds also for (2.26), (2.27). By commutativity of the diagram, we see that choosing the steps (2.24), (2.26) or (2.25), (2.27) we obtain the same as the signs cancel. As this permutes any two divisors, we obtain independence in the general case. From Lemma 4.8, Proposition 4.7 and C Proposition 4.15, DO and I are independent of the order. Note that this should be all con- sp sidered under pull-back by sp : MY,D → MY,D to have natural isomorphism independent of choices on the smooth intersection D1 ∩ D2: ∗ ∗ ∗ ∼ ∗ ∗ ∗ sp π ◦ ρ1,2(Pe1+e2,1,2) = sp π2 ◦ ρ1,2(Pe1+e2,1,2) .

Remark 2.29. If the need arises to show independence of compactiﬁcation, one can relate two compactiﬁcations Y1 ← Y˜ → Y2 by a common one obtained as a blow up of the closure of X,→ Y1 × Y2. Then one could use [79, Thm. 1.2] comparing Hodge cohomologies for locally free sheaves under blow up in hopes of showing independence of the isomorphism I.

Let us discuss another straight-forward consequence of the framework used in Theo- rem 2.26. For (X, Ω) a quasi-projective Calabi–Yau fourfold, let M be a moduli scheme of

20 stable pairs OX → F where F is compactly supported (see [24, 21, 26, 50, 89]) or ideal sheaves of proper subvarieties. To make sense out of Serre duality, generalizing the approach in Kool–Thomas [57, §3] and Maulik–Pandharipande–Thomas [65, §3.2], we choose a compactiﬁcation Y as in Theorem 2.26.

Definition 2.30. Let E = (OX×M → F) → X × M be the universal perfect complex. Using the inclusion iX : X → Y we obtain the universal complex E¯ = OY ×M → (iX × idM )∗(F) → Y ×M. We have the following isomorphism, where (−)0 denotes the trace-less part: ω ∼ ¯ ¯ ∼ ∗ ¯ ¯ iM : det HomM (E, E)0 = det HomM (E, E)0 = det HomM (E, E ⊗ Lk)0 κ ∼ ∗ ¯ ¯ ∼ ∗ = det HomM (E, E)0 = det HomM (E, E)0 , where κ is constructed using the isomorphisms ¯ ¯ ∼ ¯ ¯ HomM (E, E ⊗ La)0 = HomM (E, E ⊗ La−ei )0 (2.29) ω in each step determined by ./ as in Definition 2.24. The orientation bundle OM → M is ω defined as the square root Z2-bundle associated with iM . Let M¯ be a moduli stack of stable pairs or ideal sheaves on Y of subvarieties with ¯ proper support in X with the projection πGm : M → M which is a [∗/Gm] torsor. We have an inclusion η : M¯ → MY,D given on Spec(A)-points by mapping [E] 7→ ([E,¯ OY ]), where E¯ = iX ∗(E). The following result leads to the construction of virtual fundamental classes when M is compact (assuming one believes the existence of shifted symplectic structures on pairs as in Preygel [76] or Bussi [14]) and preferred choices of orientations up to a global sign at fixed 0 points when using localization for a fixed K-theory class OX + α for α ∈ Kcs(X) and a choice of compactification Y . J K Theorem 2.31. Let (X, Ω) be a quasi-projective Calabi–Yau fourfold and let Y be its com- ω pactification as in Theorem 2.26. Let OM → M be the orientation bundle from Definition 2.30 for M a moduli scheme of stable pairs or ideal sheaves of proper subschemes of X. There is a canonical isomorphism of Z2-bundles π∗ (Oω ) ∼ η∗(O./) . Gm M = ω In particular, OM → M is trivializable. ¯ ∗ ¯ Proof. The universal perfect complex EM¯ on M is given by (idY × πGm ) (E). We have: ∼ ∗ ∼ ∗ γ : det HomM¯ (EM¯ , EM¯ )0 = det HomM¯ (EM¯ , EM¯ ) det HomM¯ (O, O) = η (LY,D) , such that the following diagram of isomorphism commutes:

∗ ∼ ∗ ∗ η (LY,D) η (LY,D) η∗(ϑ./) ∼ γ γ−∗ ∼

∼ ∗ ∗ det Hom ¯ (E ¯ , E ¯ )0 det Hom ¯ (E ¯ , E ¯ )0 M M M π∗ (iω ) M M M Gm M

21 which follows from the commutativity of

tr HomM¯ EM¯ , EM¯ ⊗ La−ei 0 HomM¯ EM¯ , EM¯ ⊗ La−ei HomM¯ O, O ⊗ La−ei

tr HomM¯ EM¯ , EM¯ ⊗ La 0 HomM¯ EM¯ , EM¯ ⊗ La HomM¯ O, O ⊗ La

tr 0 HomM¯ EM¯ , EM¯ ⊗ La,i HomM¯ O, O ⊗ La,i in each step (2.29). As a result, the Z2-bundles associated to these are canonically isomorphic and we apply Theorem 2.26.

The reader interested in computations is welcome to skip the next two sections. We discuss sign comparisons under direct sums of perfect complexes in §5.

3 Background and some new methods

In this section, we review and develop further the necessary language for working with orientations. This includes developing an excision principle for complex determinant line bundles generalizing the work of Upmeier [95], Donaldson [29], [30] and Atiyah–Singer [4].

3.1 Topological stacks The definition of a topological stack follows at first the standard definition of stacks over the standard site of topological spaces. It can be found together with all basic results in Noohi [71] and Metzler [68], the homotopy theory of topological stacks is developed by Noohi in [72]. For each groupoid of topological spaces [G ⇒ X], one defines a prestack bX/Gc, such that the objects of the groupoid bX/Gc(W ) correspond to the continuous maps W → X for any W ∈ Ob(Top). The morphisms between α : W → X and β : W → X correspond to λ : W → G which are mapped respectively to α and β under the two maps G ⇒ X. One also defines [X/G] as the stack associated to this prestack. The following result makes working with topological stacks much easier. Proposition 3.1 (Noohi [71, p.26]). Every topological stack X has the form of an associated stack [X/G] for some topological groupoid [G ⇒ X]. The canonical map X → [X/G] gives a chart of X . Conversely [X/G] associated to any groupoid is a topological stack. Remark 3.2. The definition of a topological stack given in [71] is more complicated and depends on the choice of a class of morphisms called local fibrations (LF). Instead, we are using Noohi’s definition of topological stacks from [72] which corresponds to pretopological stacks in [71]. In [72], Noohi proposes a homotopy theory for a class of topological stacks called hoparacompact. Let tSthp denote the 2-category of hoparacompact topological stacks. A clas- cla sifying space of X in tSthp is a topological space X = X with a representable map

22 πcla : X → X such that for any T → X , where T is a topological space, its base change T ×X X → T is a weak homotopy equivalence. In [72, §8.1], Noohi provides a functorial construction of the classifying space X cla for every hoparacompact topological stack X , such that the resulting space is paracompact. In fact, [72, Corollary 8.9] states that the functor

cla (−) : Ho(tSthp) −→ Ho(pTop) is an equivalence of categories, where pTop denotes the category of paracompact topological spaces. Note that as, we are interested in comparing Z2-bundles, it is enough to restrict to ﬁnite CW complexes and weak homotopy equivalences are replaced by usual ones avoiding the question of ghost maps.

3.2 Moduli stack of connections and their Z2-graded H-principal Z2- bundles Let X be a smooth connected manifold of dimension n and π : P → X be a principal G bundle for a connected Lie group G with the Lie algebra g. We have the action of the gauge group GP on the space of connections AP . This action will be continuous and the spaces are paracompact because they are infinity CW-complexes, so we get a hoparacompact stack BP = [AP /GP ]. If X is a compact spin Kählerfourfold, let S+,S− denote the positive and negative spinor bundles and D/ : S+ → S− the positive Dirac operator, then for each connection ∇P ∈ AP one can define the twisted Dirac operator

∇ad(P ) ∞ ∞ D/ :Γ (ad(P ) ⊗ S+) → Γ (ad(P ) ⊗ S−) . (3.1)

This induces an AP family of real elliptic operators and therefore gives by §3.3 a real D/ line bundle detP → AP . Because GP maps the kernel of (3.1) to the kernel and same for the cokernels, this R-bundle is Gp equivariant and descends to an R-bundle on BP . D/ The orientation bundle of which we denote by OP → BP . One takes the unions over all isomorphism classes of U(n)-bundles for all n:

G D/ G D/ BX = BP ,O = OP . (3.2) [P ] [P ]

These are the orientation bundles of Joyce–Tanaka–Upmeier [51] and Cao–Gross–Joyce [15]. For the proof of Theorem 2.26, we will rely on the properties of special principal Z2-bundles under homotopy-theoretic group completion of H-spaces. For background on H-spaces, see Hatcher [44, §3.C], May–Ponto [67, §9.2] and Cao-Gross–Joyce [15, §3.1]. Recall that an (admissible) H-space is a triple (X, eX , µX ) of a topological space X, its point ex ∈ X and a continuous map µX : X × X → X is called an H-space, if it induces a commutative monoid in Ho(Top). An admissible H-space X is group-like if π0(X) is a group. Note that there are many ways how to include higher homotopies into the theory of H-spaces. For An-spaces

23 see Stasheff [86] and [87]. For E∞-spaces see May [66], for Γ-spaces see Segal [82]. While E∞-spaces and Γ-spaces are roughly the same, A∞ spaces do not require commutativity. All our spaces fit into these frameworks which by [51, Example 2.19] give us additional control over the Z2-bundles on them. One also defines H-maps as the obvious maps in the category of H-spaces. We use the notion of homotopy-theoretic group completions from May [66, §1]. One has the following universality result for homotopy theoretic group completion, that we will use throughout.

Proposition 3.3 (Caruso–Cohen–May–Taylor [27, Proposition 1.2]). Let f : X → Y be a homotopy-theoretic group-completion. If π0(X) contains a countable coﬁnal sequence, then for each weak H-map g : X → Z, where Z is group-like, there exists a weak H-map g0 : Y → Z unique up to weak homotopy equivalence, such that g0 ◦ f is weakly homotopy equivalent to g.

Note that weak H-maps correspond to relaxing the commutativity to hold only up to weak homotopy equivalences. We will again not diﬀerentiate between the two. Let us now merge the deﬁnition of Z2-graded commutativity with the notion of H-principal Z2-bundles of Cao–Gross–Joyce [15].

Deﬁnition 3.4. Let (X, eX , µX ) be an H-space. A Z2-bundle O → X together with a continuous map deg(O): X → Z2 satisfying

deg(O) ◦ µ(x, y) = deg(O)(x) + deg(O)(y) is a Z2-graded Z2-bundle. If O1,O2 are Z2-graded then the isomorphism ∼ O1 ⊗Z2 O2 = O2 ⊗Z2 O1

deg(O1)deg(O2) diﬀers by the sign (−1) from the naive one. Moreover, O1 ⊗Z2 O2 has grading deg(O1) + deg(O2). A pullback of a Z2-graded Z2-bundle, is naturally Z2-graded. An isomorphism of Z2-graded Z2-bundles has to preserve the grading. A weak H-principal Z2-graded Z2-bundle on X is a Z2-graded Z2-bundle P → X, such that there exists an isomorphism p of Z2-graded Z2-bundles on X × X

∗ p : P Z2 P → µX (P ) .

A Z2-graded strong H-principal Z2-bundle on X is a pair (Q, q) of a trivializable Z2 graded Z2-bundle Q → X and an isomorphism of Z2-graded Z2 bundles on X × X

∗ q : Q Z2 Q → µX (Q) , such that under the homotopy h : µX ◦ (idX × µX ) ' µX ◦ (µX × idX ) the following two isomorphisms of the Z2-bundles on X × X × X are identiﬁed

∗ ∗ (idX × µX ) (q) ◦ (id × q): Q Z2 Q Z2 Q → µX ◦ (idX × µX ) Q

24 and ∗ ∗ (µX × idX ) (q) ◦ (q × id) : Q Z2 Q Z2 Q → µX ◦ (µX × idX ) Q. The isomorphism i :(P, p) → (Q, q) of Z2-graded strong H-principal Z2-bundles has to solve ∗ µX i ◦ p = q ◦ (i i) .

Including the Z2-gradedness, we obtain a minor modiﬁcation of Cao–Gross–Joyce [15, Proposition 3.5]. Proposition 3.5. Let f : X → Y be a homotopy-theoretic group completion of H-spaces, then for

(i)a Z2-graded weak H-principal Z2-bundle P → X, there exists a unique Z2-graded weak 0 ∗ 0 H-principal Z2-bundle P → Y such that f (P ) is isomorphic to P .

(ii)a Z2-graded strong H-principal Z2-bundle (Q, q) on X, there exists a unique Z2-graded 0 0 strong H-principal Z2-bundle (Q , q ) on Y unique up to a canonical isomorphism, such that (f ∗Q0, (f × f)∗q0) is isomorphic to (Q, q).

Proof. Without the Z2-graded condition the result is stated in Cao–Gross–Joyce [15, Propo- sition 3.5]. Then as deg(P ) respectively deg(Q) can be viewed as additive maps π0(X) → Z2 and π0(Y ) is a group-completion, there exist unique extensions of the grading.

We often suppress the maps µX and eX for an H-space X, we also write Q instead of (Q, q) for a strong H-principal Z2-bundle when q is understood.

Lemma 3.6. Let O1,O2 → X be Z2-graded strong (resp. weak) H-principal Z2-bundles. Then O1 ⊗Z2 O2 is a Z2-graded strong (resp. weak) H-principal Z2-bundle. ∗ Proof. Let qi : Oi Oi → µX (Oi) be the isomorphisms from Deﬁnition 3.4. Then we deﬁne Def∼ 3.4 q :(O1 ⊗Z2 O2) Z2 (O1 ⊗Z2 O2) = (O1 Z2 O1) ⊗Z2 (O2 Z2 O2) p ⊗p 1∼ 2 ∗ ∗ ∼ ∗ = µX (O1) ⊗Z2 µX (O2) = µ (O1 ⊗Z2 O2) . deg(π∗(O ))deg(π∗(O )) Notice that we get an extra sign (−1) 1 2 2 1 . To check associativity 3.4, we need commutativity of

∗ ∗ (−1)deg(π2 (O2))deg(π3 (O1)) ∗ (O1 ⊗ O2) (O1 ⊗ O2) (O1 ⊗ O2) (O1 ⊗ O2) µX (O1 ⊗ O2)

deg µ∗ (π∗(O ) π ∗(O ) deg π∗(O ) ∗ ∗ (−1) X 2 1 3 1 1 2 (−1)deg(π1 (O2)deg(π2 (O1))

∗ ∗ deg µX (π1 (O2) ∗ ∗ ⊗π2 (O2) deg π2 (O1) (−1) ∗ ∗ ∼ ∗ (µX ×idX ) ◦µX (O1⊗O2)= µ (O1 ⊗ O2) 2 (O1 ⊗ O2) ∗ ∗ X Z ◦(idX ×µX ) ◦µX (O1⊗O2) .

25 Without the extra signs, it would be commutative because Oi are strong H-principal. To check the signs note that going down and right, resp. right and down we get

deg(π∗(O )deg(π∗)(O )+(degπ∗(O )deg+(π∗)(O ))degπ∗(O ) (−1) 2 2 3 1 2 1 3 1 1 1

deg(deg(π∗(O )deg(π∗)(O )+(degπ∗(O )deg+(π∗)(O ))degπ∗(O ) = (−1) 2 2 3 1 1 2 2 1 1 2 .

With the Z2-grading we need to distinguish between duals of strong H-principal Z2- bundles. ∗ ∗ Deﬁnition 3.7. Let (O, p) be a strong H-principal Z2-graded Z2-bundle. Its dual (O , p ) will be deﬁned to be a strong H-principal Z2-graded Z2-bundle, such that as Z2-bundles O∗ = O and the isomorphism

∗ ∗ ∗ ∼ ∗ ∗ p : O Z2 O −→ µX (O ) , ∗ deg(π∗(O))deg(π∗(O)) is given by p = (−1) 1 2 p, where π1, π2 are the projections X × X → X. cla Example 3.8. An example of an H-space is the topological space (BX ) , where the multiplication µBX : BX ×BX → BX is given by mapping ([∇P ], [∇Q] 7→ [∇P ⊕∇Q]), and we take cla cla cla cla (µBX ) :(BX ) × (BX ) → (BX ) . It is Z2-graded (see [95, 97] for the corresponding grading of real determinant line bundles) in the following sense: Let [∇P ] ∈ BX , then

D/ D/ deg(O + )([∇P ]) = χ (E,E) , (3.3)

n D/ where E is the C vector bundle associated to the U(n)-bundle P and χ (E,E) = indD/ ∇End(E) is the complex index from Deﬁnition 3.12.

Example 3.9. Let X be a quasi-projective Calabi–Yau 4-fold. Let µMX : MX × MX → MX be the map corresponding to taking sums of perfect complexes. Then a natural isomorphism φω : Oω Oω → µ∗ (Oω) Z2 MX was constructed in [15, Deﬁnition 3.12] for projective X. This also works when X is quasi- ∼ ∗ projective by restricting the isomorphism Λ0 −→ Λk for any compactiﬁcation Y along ξY ω ω top top from (2.4). Therefore if O is trivializable, then the pair (O ) → (MX ) together with ω top (φ ) gives a strong H-principal Z2-bundle. We also construct in Proposition 4.7 the isomorphism φϑ./ : O./ O./ → µ∗ (O./), Z2 MY,D where

µMY,D : MY,D × MY,D → MY,D corresponds to summation on both components of MY,D. We will need the following formulation of [15, Thm. 1.11]: Theorem 3.10 (Cao–Gross–Joyce [15, Thm 1.11]). Let X be a compact spin manifold of D/ dimension 8, then the Z2-bundle O + → BX are Z2-graded strong H-principal Z2-bundles.

26 3.3 Complex excision n Pseudo-differential operators over R are explained in Hörmander[47]. For background on pseudo-differential operators on manifolds, we recommend Lawson–Michelson [60, §3.3], Atiyah–Singer [4, §5], Donaldson–Kronheimer [30, p. 7.1.1], and Upmeier [95, Appendix A]. We will not review the definition due to its highly analytic nature, as we do not use it explicitly. The excision principle for differential operators was initiated by Seeley [81] and used by Atiyah–Singer [4]. Its refinement to excision for Z2-bundles was applied by Donaldson [29], Donaldson–Kronheimer [30] and categorified by Upmeier [95]. We use these ideas and extend them to complex determinant line bundles. From now on, we will be assuming that all real bundles come with a choice of a metric and all complex vector bundles with a choice of a hermitian metric. Note that the spaces of metrics are convex and therefore contractible. When we use convex, we automatically mean non-empty. ∞ ∞ Let X be a manifold, E,F → X complex vector bundles, P :Γcs (E) → Γ (F ) pseudo-differential operator of degree m, then its symbol σ(P ): π∗(E) → π∗(F ), where π : T ∗X → X is the projection map, is a homogeneous of degree m on each fiber of T ∗X linear homomorphism. One says that P is elliptic, when its symbol σ(P ) is an isomorphism outside of the zero section X ⊂ T ∗X. We will be working with continuous families of symbols and pseudo-differential operators as defined in [4, p. 491] or as in Upmeier [95, Appendix]. For a topological space M, we denote the corresponding set of elliptic pseudo-differential M-families by Ψm(E,F ; M) and the elliptic symbol M-families by Sm(E,F ; M) with the map

σ :Ψm(E,F ; M) → Sm(E,F ; M) . (3.4)

It is compatible with respect to addition, scalar multiplication, composition and taking duals (see [4, §5] for details). It is standard to restrict to degree 0 operators and symbols using P σ σ(P )

Ψ0 S0 .

∗ − 1 σ ∗ − 1 (1 + PP ) 2 P (σ(P )σ(P ) ) 2 σ(P )

If X is compact then each P ∈ Ψm(E,F ; M) gives an M-family of Fredholm operators ∞ ∞ between Hilbert spaces containing Γcs (E) and Γ (E) such that ker(P ) and coker(P ) lie in ∞ ∞ Γ (E0) and Γ (E1) respectively. Let P be a continuous Y -family of Fredholm operators Py : H0 → H1 for each y, where Hi are Hilbert spaces. Determinant line bundle det(P ) → Y of P is defined in Zinger [97] using stabilization (in this case one only needs the Hi to be Banach spaces) and in Upmeier [95, Definition 3.4], Freed [32] or Quillen [77]. We will use the conventions from [95, Definition 3.4].

27 Definition 3.11 (Phillips [75]). Let P : H → H be a self adjoint Fredholm operator on the Hilbert space H. The essential spectrum specess(P ) is the set λ ∈ R, such that P − λId is not Fredholm. We denote by spec(P ) the spectrum of P . For each µ > 0, such that ±µ∈ / spec(P ) and (−µ, µ) ∩ specess(P ) = ∅, one defines V(−µ,µ)(P ) ⊂ H as the subspace of eigenspaces of P for eigenvalues −µ < λ < µ. If P is positive semi-definite, we will also write V[0,µ)(P ) . If P is skew adjoint, we will also denote the set of its eigenvalues by spec(P ) (note that spec(P ) = ispec(−iP )).

For a Y family of self adjoint Fredholm operators, one can choose U ⊂ Y sufficiently small and µ from Definition 3.11, such that V(−µ,µ)(P ) becomes a vector bundle on U. This can be used to define topology on the union of determinant lines

∗ ∗ det(Py) = det(Py) ⊗ det(Py ) , y ∈ Y as in [95, Deﬁnition 3.4].

Deﬁnition 3.12. The bundle det(P ) is Z2-graded with degree ind(P ), where ind(P ) = ∗ dim(Ker(Py)) − dim(Ker(Py )) = ind(Py). If we have two Y -families P1 and P2, then the isomorphism ∼ det(P1) ⊗ det(P2) = det(P2) ⊗ det(P1) (3.5) diﬀers from the naive one by the sign (−1)ind(P1)ind(P2).

We have the “inverse” of (3.4)

−1 S0(E,F,M) 3 p 7−→ P ∈ S0 E,F,M × σ (p)

Which we use to abuse the notation

det(p) det(P0) det(P )

iP ×id Y ∗ × Y 0 σ−1(p) × Y

∗ −1 Here det(P0) = (iP0 × id) det(P ). Note that as σ (p) is convex ([95, Theorem 4.6]), for −1 ∼ two diﬀerent choices P0,P1 ∈ σ (P ) we have natural isomorphisms det(P0) = det(P1). Therefore

Lemma 3.13. The complex line bundle det(p) is well-deﬁned up to natural choices of isomorphisms.

The following lemma is meant for book-keeping purposes.

Lemma 3.14. Let pi ∈ Sm(Ei,Fi; M) for i = 1, 2 and q ∈ Sm(E,F,Y × I).

28 (i) (Functoriality.) If µE : E1 → E2, µF : F1 → F2 are isomorphisms such that

∗ p1 ∗ π (E1) π (F1)

∗ ∗ π (µE ) π (µF ) (3.6) ∗ ∗ p2 ∗ π (E2)π π (F2)

commutes, then there is a natural isomorphism det(p1) → det(p2) . (ii) (Direct sums.) There is a natural isomorphism

det(p1 ⊕ p2) −→ det(p1)det(p2) . (3.7)

(iii) (Adjoints.) There is a natural isomorphism

∗ ∗ det((p1) ) −→ det (p1) . (3.8)

∗ (iv) (Triviality.) If p1 = π (µ) for some isomorphism µ : E1 → F1, then there is a natural isomorphism + det(p ) −→ C . (3.9) ∼ (v) (Transport.) There is a natural isomorphism det(q)|Y ×{0} = det(q)|Y ×{1}. such that for qi ∈ Sm(Ei,Fi,Y × I) we have the commutative diagram

(v) det(q1 ⊕ q2)|Y ×{0} det(q1 ⊕ q2)|Y ×{1}

(ii) (ii) (v)⊗(v) det(q1)|Y ×{0} ⊗ det(q2)|Y ×{0} det(q1)|Y ×{1} ⊗ det(q2)|Y ×{1}

Proof. For

−1 −1 (i) make a natural choice of a pair (P1,P2) ∈ σ (p1) × σ (p2) commuting with µE, µF and apply [95, Proposition 3.5 (i)].

−1 −1 (ii) make a natural choice of any P1 × P2 ∈ σ (p1) × σ (p2) in loc cit.

−1 (iii) make a natural choice P1 ∈ σ (p1) in loc cit.

(iv) make the choice P1 = µ in loc cit.

−1 (v) make a contractible choice of Q0 ∈ σ (q), then we have natural isomorphism τ such ∼ that τt : det(Q0)|Y ×{0} = det(Q0)|Y ×{t} for all t ∈ I and τ0 = id, then consider the one for t = 1. The commutativity of the diagram follows immediately from the deﬁnition.

29 The following deﬁnition is the main reason, why we introduced the above concepts.

Deﬁnition 3.15. Let Ei, Fi be vector bundles on compact manifold X and pi ∈ S0(Ei,Fi; Y ). Let U, V ⊂ X be open, U ∪V = X and µE : E1|U → E2|U , µF : F1|U → F2|U isomorphisms, such that ∗ p1|T ∗U ∗ π (E1|U ) π (F1|U )

∗ ∗ π (µE ) π (µF ) (3.10) ∗ p2|T ∗U ∗ π (E2|U ) π (F2|U )

∞ commutes. Choose a function χ ∈ Ccpt(V, [0, 1]) with χ|X\U = 1. Then we obtain that:

∗ ∗ χ (1 − t + tχ)p1 t(1 − χ)π µF t ∈ I 7−→ (p1, p2, µE, µF )t = ∗ ∗ (3.11) t(1 − χ)π µE −(1 − t + tχ)(p2) is elliptic. The following result might appear deceptively obvious, but the usual I2-family argument does not go through.

Lemma 3.16. Let X be compact, Ei,Fi, complex vector bundles on X and pi ∈ S0(Ei,Fi; Y ) with isomorphism µE : E1 → E2, µF : F1 → F2 , satisfying (3.10) on X then we have the commutativity up to contractible choices

∗ Prop. 3.14 (ii) 0 det(p1)det(p2) det (p1, p2, µE, µF )0

Prop. 3.14 (iii) + (i) Prop. 3.14 (v)

0 det (p1, p2, µE, µF ) C Prop. 3.14 (iv) 1

−1 −1 Proof. Choose (P1,P2) ∈ σ (p1) × σ (p2) commuting with µE, µF and construct

∗ (1 − t)P1 tµF −1 0 Ψt = ∗ ∈ σ (p1, p2, µE, µF )t . tµE −(1 − t)P2

−1 0 µE By composing Ψt with ∗ −1 we obtain (µF ) 0

t id −(1 − t)P ∗ Ψ˜ = : E ⊕ F −→ E ⊕ F . t (1 − t)P t id 1 2 1 2

>0 Let ν ∈ R and U ⊂ Y be chosen suﬃciently small as in Upmeier [95, Deﬁnition 3.4], such ˜ ∗ ˜ that V[0,ν)(Ψ0Ψ0) is a vector bundle. ˜ ∗ ˜ ˜ ˜ ∗ 2 Notice that Ψt Ψt = ΨtΨt . Moreover, by spectral theorem each non-zero eigenvalue λ ∈ ˜ ∗ ˜ ˜ (0, ν) of Ψ0Ψ0 has multiplicity 2k for some positive integer k and then Ψ0 has eigenvalues

30 ˜ ˜ ∗ ˜ iλ, −iλ each of multiplicity k in its set of eigenvalues spec(Ψ0). The eigenvectors of Ψt Ψt remain the same, but corresponding eigenvalues are λ2(1 − t)2 + t2. We therefore deﬁne ν(t) = ν(1 − t)2 + t2 and we have a natural isomorphism

˜ ∗ ˜ ∼ ˜ ∗ V[0,ν)(Ψ0Ψ0) = V[0,ν(t))(Ψt Ψt) (3.12) given by the identity for all t ∈ I (here one extends to t = 1 by considering the same ﬁnite set of eigenvectors which now have eigenvalue 1), which gives a continuous isomorphism of vector bundles on U×I and restricts to identity for t = 0. The isomorphisms of determinant line bundles is then given by

[95, Def. 3.4] ∼ ∗ ∗ ∗ αν(t) : det(Ψ0) = det(V[0,ν)(Ψ0Ψ0))det (V[0,ν)(Ψ0Ψ0)) (3.12) [95, Def. 3.4] ∼ ∗ ∗ ∗ ∼ = det(V[0,ν(t))(Ψt Ψt))det (V[0,ν(t))(ΨtΨt )) = det(Ψt) .

We see that this is a representative of the transport Prop. 3.14 (v) because it restricts to identity at t = 1. To see that this isomorphism is independent of ν, we can restrict 0 to a single point y ∈ Y . Let ν > ν > 0, then for Ψ0(y) choose its diagonalization when ∗ restricted to V[0,ν0)(Ψ0Ψ0). From looking at [95, Deﬁnition 3.4] it is then easy to see that

Y (1 − t) + µ−1t 0 αν (t) = 1 αν(t) . [(1 − t)2 + |µ|−2t2] 2 µ∈spec(Ψ0) ν<|µ|2<ν0

As each µ = iλ comes with its conjugate of the same multiplicity, the factor is equal to one. 0 ∼ ∗ ∼ Let α : det(Ψ0) = det(P )det(P ) = C be isomorphism combining (3.6),(3.7) and (3.8), then it can be checked in the same way that

Y |µ|2 α (1) = α0 , ν µ µ∈Spec(Ψ0) 0<|µ|<ν where the factor again becomes one. By covering Y by such sets Ui and choosing appropriate νi, we can glue the isomorphisms on Ui×I, because they coincide on the overlaps (Ui∩Uj)×I. 0 µ−1 ˜ ˜ E Composing α(t) : det(Ψ0) → det(Ψt) with ∗ −1 , we obtain Prop. 3.14 (v) and the (µF ) 0 commutativity of the diagram.

Remark 3.17. Note that when pi ∈ S0(Ei,Fi; Y ) have a real structure and µE, µF pre- χ χ serve it, then there exists a natural Z2-bundle or(p1, p2, µE, µF ) ⊂ det(p1, p2, µE, µF ) as in Donaldson–Kronheimer [30, §7.1.1] or Upmeier [95] The transport isomorphism of Propo- sition 3.14 (v) for the Y × I family along or(p1, p2, µE, µF ) is canonical, because it is the standard transport along ﬁbers Z2.

31 Our main object of study are going to be twisted Dirac operators and Dolbeault operators. Let X be a manifold, P a U(n)-principal bundle, Vn a representation of U(n) and E the associated vector bundle, then for a given connection ∇P on P and its associated ∇ connection ∇E, the twisted operator D E has the degree 0 symbol

S σ(D) ⊗ id =: σ (D) . 0 π∗E E

If Φ : V → W is an isomorphism of vector bundles, we will also write Φ = id ⊗ Φ: E ⊗ V → E ⊗ W .

Let us now formulate the excision isomorphism for complex operators in the form we will need in 4.3. This generalizes [95, Thm. 2.10] to complex determinant line bundles. Moreover, for real operators it is slightly more general then [95, Thm. 2.13] in that, we do not require a framing of bundles, but isomorphisms in [95, Thm. 2.13(b)]. This would already follow from [95, Thm. 2.10], but we obtain it as a consequence of Remark 3.17. Note that we also do not require the isomorphisms below to be unitary, as this is not necessary for the operators in (3.11) to be elliptic.

Deﬁnition 3.18. Let Xi be compact, Ei,Fi, vector bundles on Xi for i = 1, 2 and Di : ∞ ∞ Γ (Ei) → Γ (Fi) complex/real elliptic diﬀerential operators. Moreover, let Si,Ti ⊂ Xi open, such that Si ∪ Ti = Xi and IS : S1 → S2 an isomorphism. We then denote by

Φ1 σV1 (D1) σW1 (D1)

ξV ξW (3.13)

Φ2 σV2 (D2) σW2 (D2)

Lemma 3.19. For the data given by (3.13) and a compact subsets Ki, s.t. Xi\Ki ⊂ Ti are identiﬁed by IS,we have natural isomorphisms in families

∗ ∼ ∗ Ξ(Di, ξV/W , Φi) : det σV1 (D1) det σW1 (D1) −→ det σV2 (D2) det σW2 (D2) , such that for another set of data

0 Φ1 σ 0 (D ) σ 0 (D ) V1 1 W1 1

ξV0 ξW 0

σ 0 (D ) σ 0 (D ) V2 2 0 W2 2 Φ2

32 for the same Si,Ti,K the diagram is commutative up to natural isotopies

Ξ(D ,ξ ,Φ⊕Φ0) i V ⊕V 0/W ⊕W 0 ∗ ∗ det σ 0 (D ) det σ 0 (D ) det σ 0 (D ) det σ 0 (D ) V1⊕V1 1 W1⊕W1 1 V2⊕V2 2 W2⊕W2 2

(3.7) (3.7) (3.8) (3.8)

Ξ(Di,ξV /W ,Φ) 0 det σV (D1) det σ 0 (D1) Ξ(Di,ξ 0 0 ,Φ ) det σV (D2) det σ 0 (D2) 1 V1 V /W 2 V2 ∗ ∗ ∗ ∗ det σ 0 (D1) det σW (D1) det σ 0 (D2) det σW (D2) W1 1 W2 2

−1 Moreover, if Si = Xi, then Ξ(Di, ξV/W , Φi) = ((3.6)) ◦ ((3.6)). Proof. The following is standard, and we simply lift it to complex determinant line bundles. ∞ Making a contractible choice of χi ∈ Ccs (Si), χi|Ki = 1 identified under IV , the composition of the following isomorphisms gives Ξ(Di, ξV/W , Φi): 3.14(ii),(i) ∗ ∼ χ1 det σV1 ) ⊗ det(σW1 = det (σV1 (D1), σW1 (D1), Φ1, Φ1)0 3.14(v) ∼ χ1 = det (σV1 (D1), σW1 (D1), Φ1, Φ1)1 ∗ ∼ χ2 = det (σV2 (D2), σW2 (D2), Φ2, Φ2)1 3.14(v) ∼ χ2 = det (σV2 (D2), σW2 (D2), Φ2, Φ2)0 ∼ ∗ = det σV2 (D2))det (σW2(D1) , where for the step ∗, we are making a contractible choice of Pi ∈ −1 χ σ (σVi (Di), σWi (Di), Φi, Φi)1 supported representatives in Si of the two symbols on both sides as in Upmeier [95, Thm. A.6] identified by the isomorphism ξV 0 , ξW 0 and using that ∞ ker(Pi) ∈ Γcs Si, (Ei ⊗ Vi) ⊕ (Fi ⊗ Wi) , ∞ coker(Pi) ∈ Γcs Si, (Fi ⊗ Vi) ⊕ (Ei ⊗ Wi) . The second statement follows from the compatibility under direct sums in 3.14(v). The final statement is just Lemma 3.16.

4 Proof of Theorem 2.26

We construct here a double Y˜ for our manifold X, such that the “compactly supported” orientation on X can be identified with the one on Y˜ . We use homotopy theoretic group completion to reduce the problem to trivializing the orientation Z2-bundles on the moduli space of pairs of vector bundles generated by global sections identified on the normal crossing divisor. Then we express the isomorphism ϑ./ from Definition 2.24 using purely vector bundles in §4.3. We then construct the isotopy between the two different real structures to obtain an isomorphism of Z2-bundles by hand. The final result of this section is contained in Proposition 4.11 and Proposition 4.15.

33 ∂U −U U

T˜ K

Figure 4.1: Spin manifold Y˜ and the subset T˜ ⊂ Y˜ .

4.1 Relative framing on the double Here we construct the double of a non-compact X, such that it can be used in §4.2 to deﬁne orientations back on moduli spaces over X.

Definition 4.1. Let X be a non-compact spin manifold dimR(X) = n. Let K ⊂ X be a compact subset. Choose a smooth exhaustion function d : X → [0, ∞) Then by Sard’s theorem for a generic c > max{d(x): x ∈ K} the set U = {x ∈ X | d(x) ≤ c} is a manifold with the boundary ∂U = {U ∈ X | d(g) ≤ c} . Normalizing the gradient grad(dg) restricted to ∂U, we obtain a normal vector field ν to ∂U. Let V be the tubular neighborhood of ∂U in X, then it is diffeomorphic to (−1, 1) × ∂U and is a collar. We define Y˜ := U ∪∂U (−U), where −U denotes a copy of U with negative orientation. Then Y˜ admits a natural spin- structure which restricts to the original one on U (see for example [40, p. 193]). Since we do not need it here explicitly, we do not give its description.

Let T = X\K, where K is compact, then define T˜ = (T¯ ∩ U) ∪ (−U) (see Figure 4.1). ˜ ∼ Let P,Q → Y , be two U(n), bundles, such that there exists an isomorphism P |T˜ = Q|T˜. We define now the moduli stack of pairs of connections on principal bundles identified on T˜.

Deﬁnition 4.2. Consider the space AP × AQ × GP,Q,T˜, where GP,Q,T˜ is the set of smooth ˜ isomorphisms φ : P |T˜ → Q|T˜. Let GP × GQ be the product of gauge groups. We have a natural action

(GP × GQ) × (AP × AQ × GP,Q,T˜) → AP × AQ × GP,Q,T˜ ˜ ˜ −1 (γP , γQ, ∇P , ∇Q, φ) 7→ (γP (∇P ), γQ(∇Q), γQ ◦ φ ◦ (γP ) ) .

We denote the quotient stack by BP,Q,T˜ = [AP × AQ × GP,Q,T˜/GP × GQ] . Let us deﬁne the union [ BY,˜ T˜ = BP,Q,T˜ , [P ],[Q]: [P |T˜]=[Q|T˜] where we chose representatives P,Q for the isomorphism classes.

34 p1 p2 There exist natural maps BY˜ ←−BY,˜ T˜ −→BY˜ induced by AP × AQ × GP,Q,T˜ → AQ and D/ + AP × AQ × GP,Q,T˜ → AP . Let O → BY˜ be the Z2-graded H-principal Z2-bundles from (3.2), then we deﬁne ˜ ∗ D/ + ∗ D/ + ∗ DO(Y ) = p1(O ) Z2 p2((O ) ) , (4.1) / where (OD+ )∗ is from Deﬁnition 3.8. Let us now construct an explicit representative cla (BY,˜ T˜) .

Deﬁnition 4.3. Let P and Q be U(n)-bundles on Y˜ isomorphic on T˜. Consider the following two quotient stacks

PQ = [AP × AQ × GP,Q,T˜ × P/GP × GQ] ,

QP = [AP × AQ × GP,Q,T˜ × Q/GP × GQ] , which are U(n)-bundles on Y˜ ×B ˜. We have a natural isomorphism τP,Q : PQ| ¯ → P,Q,T T ×BP,Q,T˜ ˜ ˜ ˜ QP | ¯ given by [∇P , ∇Q, φ, p] 7→ [∇P , ∇Q, φ, φ(p)]. After taking appropriate unions, T ×BP,Q,T˜ ∼ we obtain bundles P1, P2 → Y˜ ×B ˜ ˜ with an isomorphism P1| ˜ = P2| ˜ . Pulling Y,T T ×BY,˜ T˜ T ×BY,˜ T˜ ˜ cla cla Pi back to Y × (BY,˜ T˜) , we obtain Pi ﬁber bundles, which are U(n)-bundles on each connected components for some n ≥ 0. Together with the isomorphism τ cla, these induce two maps ˜ G p1, p2 : Y × BY,˜ T˜ −→ BU(n) , n≥0 ˜ F with a unique (up to contractible choices) homotopy Hp : T × BY,˜ T˜ × I → n≥0 BU(n) ˜ between p1 and p2 restricted to T × BY,˜ T˜. We obtain the following homotopy commutative diagram cla F (B × B ) Map 0 (Y,˜ BU(n)) Y˜ BT˜ Y˜ C n≥0 (4.2) ˜ F ˜ F MapC0 (Y, n≥0 BU(n)) MapC0 (T, n≥0 BU(n)) .

h This induces a map B ˜ ˜ → V ˜ × V ˜ , where we use the notation Y,T Y VT˜ Y G VZ = MapC0 (Z, BU(n)) n≥0 for each topological space Z. If T,¯ → X is a neighborhood deformation retract pair then so is T,˜ → Y . It is then a coﬁbration in Top and the left vertical and lower horizontal arrow of (4.2) are ﬁbrations in Top. This implies that the natural map

h V ˜ ˜ = V ˜ ×V V ˜ −→ V ˜ × V ˜ Y,T Y T˜ Y Y VT˜ Y

35 cla is a homotopy equivalence. By homotopy inverting, we construct R :(BY,˜ T˜) → VY,˜ T˜ . It can be shown by following the arguments of Atiyah–Jones [3], Singer [84], Donaldson [30, Prop 5.1.4] and Atiyah—Bott [2] that this is a weak homotopy equivalence. We therefore ˜ cla have the natural Z2-bundle (DO(Y )) → VY,˜ T˜. We summarize some obvious statements about the above constructions. There is a natural map un : BU(n) → BU × Z, such that π2 ◦ un = n. This induces maps VZ → CZ which are homotopy theoretic group completions for any Z. In particular we have a natural map Ω:˜ V → C := C × C . Y,˜ T˜ Y,˜ T˜ Y˜ CT˜ Y˜ cla cla cla Lemma 4.4. The spaces (BY,˜ T˜) , VY,˜ T˜ are H-spaces. The maps (p1) , (p2) , R are ˜ cla cla H-maps. In particular, (DO(Y )) → (BY,˜ T˜) 'VY,˜ T˜ is a Z2-graded strong H-principal C ˜ Z2-bundle and there exists a unique Z2-graded strong H-principal Z2-bundle DO(Y ) → CY,˜ T˜ up to canonical isomorphisms, such that there is a canonical isomorphism ∗ C ˜ ∼ ˜ q (DO(Y )) = DO(Y ) , where q : VY,˜ T˜ → CY,˜ T˜ is the homotopy theoretic group completion. Proof. The last statement follows using Proposition 3.5 (ii).

4.2 Moduli space of vector bundles generated by global sections We use the definitions of moduli spaces of vector bundles generated by global section of Friedlander–Walker [33] used by Cao-Gross–Joyce [15, Definition 3.18]. For definition of Ind-schemes see for example Gaitsgory–Rozenblyum [39], for general treatment of indization of categories see Kashiwara-Shapira [53, §6]. For Z a scheme over C, this moduli space is defined as the mapping Ind-scheme :

∞ TZ = MapIndSch(Z, Gr(C )) , ∞ where IndSch is the category of Ind-schemes over C and we view Gr(C ) as an object in this category.

vb Deﬁnition 4.5. Induced by the embedding of schemes D,→ Y , we obtain a map ρD : TY → TD. We can construct the ﬁber-product in Ind-schemes TY,D . There is a natural map ∞ ΩY : TY → MY given by composing with the natural Gr(C ) → PerfC. Together with the ag D map ΩD : TD → M constructed in the same way, we obtain a homotopy commutative diagram of higher stacks:

ρvb T D T T Y D vb Y ρD ag Ωag ag ΩY D ΩY ρ M D MD M , Y ρD Y

ag which induces Ω : TY,D = TY ×TD TY → MY,D .

36 As (−)top commutes with homotopy colimits by Blanc [8, Prop. 3.7] for an Ind-scheme S considered as a higher stack represented by the sequence of closed embeddings of finite top type schemes S0 → S1 → S2 → ..., its topological realization (S) is the co-limit in Top an an an of the sequence S0 → S1 → S2 → ..., because the maps are closed embeddings of CW- complexes and thus cofibrations. Using that filtered co-limits commute with finite limits, p p p p we can express TY,D as the filtered co-limit of T × p T , where T = Map (Z, Gr( )) Y TD Y Z Sch C for any scheme Z. From this, it also follows that

top p an p an an an (TY,D) = lim (T ) × p an (T ) = T ×T an T . −→ Y (TD) Y Y D Y p→∞ We have therefore constructed a map

top an an top Ω : T × an T → (M × D M ) . (4.3) Y TD Y Y M Y The following is a non-trivial modiﬁcation of [15, Prop. 3.22], [42, Prop. 4.5] to the case of MY,D. We use in the proof the language of spectra (see Strickland [88] and Lewis–May [41]). We only use that the inﬁnite loop space functor Ω∞ : Sp → Top preserves homotopy equivalences, where Sp is the category of topological spectra.

top an an top Proposition 4.6. The map Ω : T × an T → (M ) is a homotopy theoretic Y TD Y Y,D group completion of H-spaces. Proof. Let us recall that in a symmetric closed monoidal category C with the internal hom functor MapC(−, −) the contravariant functor C 7→ MapC(C,D) maps co-limits to limits. Thus push-outs are mapped to pullbacks because the homotopy category of higher stacks is symmetric closed monoidal as shown by To¨en–Vezzosi in [94, Theorem 1.0.4]. The following diagram i D D Y

iD (4.4) Y has a push-out Y ∪D Y in the category of schemes over C using that iD is a closed embedding and Schwede [80, Corollary 3.7]. Moreover, the result of Ferrand [31, §6.3] tells us that Y ∪D Y is projective. We conclude that there are natural isomorphisms

∼ Y ∪DY MY,D = MapHSt(Y ∪D Y, PerfC) = M , ∼ ∞ TY,D = MapIndSch(Y ∪D Y, Gr (C)) = TY ∪DY . In fact, under these isomorphisms, the map Ω from Definition 4.5 corresponds to the natural Y ∪DY map ΩY ∪DY : TY ∪DY → M . For a quasi-projective variety Z over C, Friedlander–Walker define in [33, Definition semi ∞ an an 2.9] the space K (Z) as the infinity loop space Ω TZ , where they use that T is an an semi E∞-space. Therefore there is a map TZ → K (Z), which is a homotopy theoretic group

37 completion by [61, p. 6.4] and [59, §2]. For a dg-category D over C, Blanc [8, Deﬁnition 4.1] st deﬁnes the connective semi-topological K-theory K˜ (D) in the category Sp. Moreover, in st [8, Theorem 4.21], he constructs an equivalence between the K˜ (D) and the spectrum of the topological realization of the higher moduli stack of perfect modules of D‡. This induces st a homotopy equivalence Ω∞K˜ (Perf(Z)) → (MZ )top of H-spaces. In [1, Theorem 2.3], Antieau–Heller prove existence of a natural homotopy equivalence between the H-spaces st Ω∞K˜ (Perf(Z)) and Ksemi(Z). The composition

an semi ∞ ˜ st Z top TZ −→ K (Z) −→ Ω K (Perf(Z)) −→ (M ) for Z = Y ∪ Y is homotopy equivalent to Ωtop . We have thus shown that Ωtop is a D Y ∪DY homotopy theoretic group-completion.

./ top top We now make (O ) → (MY,D) into a weak H-principal Z2-bundle with respect to the binary operation

top top top top (µMY,D ) :(MY,D) × (MY,D) → (MY,D) , (4.5) which is determined by

µMY MY × MY MY π1,3 ρD×ρD ρD µ D D MD D MY,D × MY,D M × M M (4.6)

ρD×ρD ρD π2,4 µMY MY × MY MY ,

It can be checked to be commutative and associative in Ho(HSta) . In fact, as M is a C Y,D homotopy fiber product of Γ-objects, it is itself one in HStaC (see Bousfield–Friedlander [12, §3] for definition of Γ-objects in model categories and Blanc [8, p. 45] for the construction in this case). Let us set some notation. For any a, b, we have the isomorphisms

∗ ∗ ∗ ∼ ∗ ∗ ∗ π1,3(Σa) ⊗ π2,4(Σa) = π1,3(Σb) ⊗ π2,4(Σb) by similar construction as in (2.17). In particular, for ﬁxed ./ we obtain the isomorphism

∗ ∗ ∗ ∼ ∗ ∗ ∗ σ./ : π1,3(Σk) ⊗ π2,4(Σk) −→ π1,3(Σ0) ⊗ π2,4(Σ0) . (4.7) ‡This moduli stack is denoted in Blanc [8] by MD. Unlike the moduli stacks in Toën–Vaquié,it classifies only perfect dg-modules over D and not the pseudo-perfect ones. For the case D = Lpe(Y ) it therefore coincides with the mapping stack MY . When Y is projective and smooth, we already know that MY and MY are equivalent.

38 ./ Proposition 4.7. Let O → MY,D be the Z2-bundle from Deﬁnition 2.24, then there ./ ./ ./ ∗ ./ exists an isomorphism φ : O Z2 O → µM(O ) depending on ./ but independent of ord. Moreover, we have

∗ ./ ./ ∗ ./ ./ (idMY,D × µM) (φ )(id × φ ) = (µM × idMY,D ) (φ ) ◦ (φ × id) . Proof. First note, that we have the isomorphism

µ∗ (Λ ) ∼ π∗Λ ⊗ π∗ Σ ⊗ π∗ ◦ σ∗Σ ⊗ π∗Λ . MY 0 = 1 0 1,2 0 1,2 0 2 0 Using this together with (4.6) we obtain the following commutative diagram

∗ ∗ ∗ ∗ ∗ ∗ π1Λ0 ⊗ π3Λ0 ⊗ π4Λ0 ⊗ π2Λ0

∗ ∗ ∗ ∗ ∗ π1 Λ0⊗π1,3Σ0⊗π1,3Σk⊗π3 Λ0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ⊗π4 Λ0⊗π2,4Σ0⊗π2,4Σk⊗π2 Λ0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π1 Λ0⊗π1,3Σ0⊗π1,3◦σ Σ0⊗π3 (Λ0) µ (π Λ0 ⊗ π Λ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 1 2 0 ⊗π4 Λ0⊗π2,4◦σ Σ0⊗π2,4Σ0⊗π2 Λ0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π1 Λk⊗π1,3◦σ Σk⊗π1,3Σk⊗π3 (Λk) ∗ ∗ ∗ ∗ ∗ ∗ ⊗π4 Λk⊗π2,4Σk⊗π2,4◦σ Σk⊗π2 Λk

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ π1 Λ0⊗π1,3Σ0⊗π1,3◦σ Σ0⊗π3 Λ0 µ (π Λ ⊗ π Λ0) ∗ ∗ ∗ ∗ ∗ 1 0 2 ⊗π4 Λ0⊗π2,4Σ0⊗π2,4◦σ Σ0⊗π2 Λ0

∗ ∗ ∗ ∗ ∗ ∗ ∗ π1 Λ0⊗π1,3Σ0⊗π1,3Σk⊗π3 Λ0 ∗ ∗ ∗ ∗ ∗ ⊗π4 Λ0⊗π2,4Σ0⊗π2,4Σk⊗⊗π2Λ0

∗ ∗ ∗ ∗ ∗ ∗ π1Λ0 ⊗ π3Λ0 ⊗ π4Λ0 ⊗ π2Λ0

∗ Where the left vertical arrow is µ (ϑ./) and the composition of all arrows on the right is ∗ ∗ π1,2(ϑ./) ⊗ π3,4(ϑ./) by generalization of the arguments in Cao–Gross–Joyce [15, p. 43]. To construct arrows on the right, we use multiple times Serre duality and (4.7). This is what ./ ∗ ∗ ∗ ∗ induces the isomorphism φ . Note that we need to permute π2Λ0 through π3Λ0 and π4Λ0 on both ends, giving the extra sign

deg(π∗Λ )deg(π∗Λ )+deg(π∗Λ ) (−1) 2 0 3 0 4 0 (4.8) for the isomorphism of Z2-bundles. Checking the associativity of the isomorphism combines the ideas of the proof of associativity in Lemma 3.6 and the ones used in the diagram above.

39 The independence of ord follows by the same arguments as used in 3. of the proof of Theorem 2.26.

Using the notation from Deﬁnition 4.3, we have an obvious map

an an Λ: T × an T −→ V (4.9) Y TD Y Y,D which corresponds to the inclusion of holomorphic maps into the continuous maps to ∞ an ¯ Gr(C ) . This map is continuous (see Friedlander–Walker [34]). Let Ti ⊃ Di be closed tubular neighborhoods for i = 1,...,N. One can construct homotopy retracts Hi of T¯ to D which can be extended to H˜ : I × Y → Y , such that H˜ | = H and i i i i I×T¯i i H˜i(t, −)| = id , where (1 + i)Ti denotes some tubular neighborhood con- Y \(1+i)Ti Y \(1+i)Ti ¯ ˜ ˜ taining Ti. We concatenate them to get H, H = H|T¯×I . Using that D has locally analyti- 4−k k ˜ cally the form C × {(z1, . . . , zk) ∈ C : z1 · ... · zk = 0}, one can assume that H(t, T ) ⊂ T and H˜ (t, D) = D§. The pullback along H˜ (1, −) and H(1, −) induces homotopy equivalences

Υ: VY,D −→ VY,T¯ , ΥC : CY,D −→ CY,T¯ , (4.10) which we use from now on to identify the spaces. As X ⊂ Y is Calabi–Yau, choosing K = X\T , where T is the interior of T¯, we construct spin Y˜ as in Deﬁnition 4.1. Deﬁne

G : V −→ V ,GC : C −→ C , (4.11) Y˜ Y,T¯ Y,˜ T˜ Y˜ Y,T¯ Y,˜ T˜ by G (m , m ) = (m ˜ , m˜ ) for each (m , m ) ∈ V × V such that Y˜ 1 2 1 2 1 2 Y VT¯ Y

m˜ 1|U = m1|U , m˜ 1|−U = m1|U , m˜ 2|U = m2|U , m˜ 2|−U = m1|U . (4.12)

Which gives us Z2-bundles:

D := Υ∗ ◦ G∗ (D (Y˜ )) ,DC −→ C . (4.13) O Y˜ O O Y,D

Lemma 4.8. Let E, F, φ be smooth vector bundles and φ : E|D → F |D be an isomorphism, ¯ smooth on each Di. Then there exists a contractible choice of isomorphism Φ: E|T¯ → F |T¯, Φ : E| → F | such that Φ | = φ| and Φ can be deformed into Φ¯ along isomorphism. i T¯i T¯i i Di Di i Moreover, the map (4.11) corresponds to

[E, F, φ] 7−→ [E,F, Φ]¯ .

C The Z2-graded strong H-principal Z2-bundles DO and DO are independent of the choices made.

§ One can construct this by taking a splitting of 0 → TDi → TY → NDi → 0, taking geodesic ﬂow in the normal direction for all Di. Then around each intersection projecting the ﬂow to be parallel to each of the other divisors.

40 ∗ ∼ Proof. The isomorphism H (E|D) = E|T¯ can be constructed by parallel transport along a contractible choice of partial connections in the I direction (see e.g. Lang [58, §IV.1]) which ∼ ∗ ∼ ∗ ∼ are piece-wise smooth. Doing the same for F gives us Φi : F |T¯ = H (F |D) = H (E|D) = N ET¯. As, we can re-parameterize the order using an I -family of homotopies, it will be independent of it. Moreover, each Φ is defined by F | ∼ H∗(F | ) ∼ H∗(E| ) ∼ E| , i T¯i = i Di = i Di = T¯i which can be deformed to Φ¯ along the transport. The choices of splittings 0 → TDi → TY → NDi → 0,where NDi is the normal bundle are contractible and so is the choice of metric for geodesic flow. Different choices of sizes of these neighborhoods correspond to a choice of some small i > 0. For each choice of the data above, the Z2-bundle DO → VY,D is independent of the choices made during the construction of Y˜ in Definition 4.1. For this let (Y˜1, T˜1), (Y˜2, T˜2) be two pairs constructed using Definition 4.1. Recall that this corresponds to fixing two different sets U1,2 ⊃ K with a boundary. A family Z → VY,D gives Z −→z1 V ,Z −→z2 V Which can be interpreted as the following diagram of Y˜1,T˜1 Y˜2,T˜2 (families) of vector bundles:

Φ˜ 1 on T˜1 E˜1 F˜1

id on U1∩U2 id on U1∩U2 ,

E˜2 F˜2 φ˜2 on T˜2

∗ ˜ ∼ Induced by [95, Thm. 2.10], we have the isomorphism z1 DO(Y1) = ∇ ∇ ∗ ∇ ∇ ∗ ad(P˜ ) ad(Q˜ ) ∼ ad(P˜ ) ad(Q˜ ) ∼ ∗ ˜ ˜ ˜ or D/ 1 or D/ 1 = or D/ 2 or D/ 2 = z2 DO(Y1) , where Pi, Qi are the unitary frame bundle for E˜i, F˜i. The fact that it is an isomorphism of strong H-principal Z2-bundles follows from compatibility under sums in Theorem [95, Thm.2.10 (iii)] and the ∇(P˜ × Q˜ ) ¯n⊗ m ∇(P˜ × Q˜ ) ¯n⊗ m natural orientations of or(D/ 1 Y˜ 1 U(n)×U(m)C C ) and or(D/ 2 Y˜ 2 U(n)×U(m)C C ) as in [51, Ex. 2.11] compatible under excision as they are determined by the complex structures which are identiﬁed.

From now on, we will not distinguish between Ind-schemes and their analytiﬁcations. Note that by Proposition 3.3, we have the commutative diagram

Λ Υ GY˜ TY,D VY,D VY,T¯ VY,˜ T˜

ag top (Ω ) Ω Ω˜ . (4.14) GC top Γ ΥC Y˜ (MY,D) CY,D CY,T¯ CY,˜ T˜

The map Γ was expressed explicitly in Deﬁnition 2.25 and all the vertical arrows are homotopy theoretic group completions.

41 4.3 Comparing excisions The results of this section have been also obtained in the author’s work [10] by different but equivalent means. We begin by defining a new set of differential geometric line bundles ¯ ¯∗ ∞ 0,even ∞ 0,odd on TY,D × TY,D. Let D = ∂ + ∂ :Γ (A ) → Γ (A ), then we define

ˆ dg Σa,P,Q −→ AP × AQ

∇Hom(E,F ⊗L ) given by a complex line det(D a ) at each point (∇P , ∇Q), where E,F are the associated complex vector bundles to P,Q and ∇Hom(E,F ) is the induced connection on ∗ ˆ dg E ⊗ F . This descends to a line bundle on Σa,P,Q → BP × BQ. Taking the union over isomorphism classes [P ], [Q], we obtain a line bundle

ˆ dg Σa −→ BY × BY .

cla cla Using the natural map Λ × Λ: TY × TY → VY × VY ' (BY ) × (BY ) , we can pull back these bundles to obtain

dg dg ∗ dg Σa −→ TY × TY and Λa = ∆ (Σa ). (4.15)

Lemma 4.9. After a choice of ./ there exist natural isomorphisms

dg ∗ dg ∗ dg ∗ ∼ ∗ dg ∗ dg ∗ σ./ :π1,3(Σk ) ⊗ π2,4(Σk ) −→ π1,3(Σ0 ) ⊗ π2,4(Σ0 ) , dg ∗ dg ∗ dg ∗ ∼ ∗ dg ∗ dg ∗ τ./ :π1(Λk ) ⊗ π2(Λk ) −→ π1(Λ0 ) ⊗ π2(Λ0 ) . Moreover, we have the isomorphisms

dg ∼ ∗ dg ∗ dg ∼ dg ∗ # dg :Σ −−→ σ (Σ ) , #a :Λ −−→ (Λ ) . (4.16) Σa a k−a a k−a

Proof. The following construction works in families due to the work done in §3.3, so we restrict ourselves to a point (p, q) = ([E1,F1, φ1], [E2,F2, φ2]), where φ1/2,i : E1/2|Di →

F1/2|Di are isomorphism. We also set the notation

Va = End(E1,F1 ⊗ La) and Wa = End(E2,F2 ⊗ La) . (4.17)

Using Φ¯ a to denote the isomorphism Va|T → Wa|T constructed in Lemma 4.8. We then have the data iΦk σVk σWk

Ω−1 Ω−1

σV0 σW0 iΦ0 using that Ω is invertible outside of D. We therefore obtain the isomorphism σ./ from Lemma 3.19. Then we have the standard deﬁnition of the Hodge star ? :Γ∞(Ap,q) →

42 Γ∞(A4−p,4−q) , given by α ∧ ∗β = hα, βiΩ ∧ Ω¯, where h−, −i is the hermitian metric on forms. We deﬁne then the anti-linear maps

0,even 0,even 0,odd 0,odd #1 : A → A ⊗ KX , #2 : A → A ⊗ KX op 0,even 0,even op 0,odd 0,odd #1 : A ⊗ KX → A , #2 : A ⊗ KX → A , (4.18)

q q+1 op q op q+1 by #1|A0,2q = (−1) ?, #2|A0,2q+1 = (−1) ?, #1 |A4,2q = (−1) ?, #2 |A4,2q+1 = (−1) ?. op op These solve #i ◦ #i = id and #i ◦ #i = id. We have the commutativity relations op op ∼ DKX ◦ #1 = #2 ◦ D and #2 ◦ DKX = D ◦ #1 and obtain the isomorphisms det(σVa ) = ∼ ∗ det(σVk−a ) = det(σVk−a ) , where the second isomorphism on both lines uses the hermitian metrics on forms which descend to a hermitian metric on the determinant.

vb Deﬁnition 4.10. Let U → Y × TY be the universal vector bundle generated by global sections. We deﬁne

vb ∗ ∗ ∗ ∗ vb ∗ vb Exta = π1,2 ∗(π1,3 Uvb ⊗ π1,3 Uvb ⊗ π1(La)) , Pa = ∆ Exta vb vb vb ∗ vb Σa = det(Exta ) , Λa = ∆ det(Σa )

We then have the isomorphisms

vb ag ∗ ∗ vb ∗ vb ∗ ∼ ∗ vb ∗ vb ∗ σ./ = (Ω ) (σ./): π1,3(Σk ) ⊗ π2,4(Σk ) −→ π1,3(Σ0 ) ⊗ π2,4(Σ0 ) , vb ∗ vb ∗ vb ∗ vb ∗ ∼ ∗ vb ∗ vb ∗ τ./ = ∆ (σ./ ): π1(Λk ) ⊗ π2(Λk ) −→ π1(Λ0 ) ⊗ π2(Λ0 ) .

./ ./ ∼ ag ∗ ./ In particular, we have the Z2-bundle Ovb → TY,D, such that naturally Ovb = (Ω ) O . The following proposition is the result of trying to develop a more general framework of relating compactly supported coherent sheaves to compactly supported pseudo-diﬀerential operators using cohesive modules of Block [9] and Yu [96].

a,d vb ∼ dg Proposition 4.11. There are natural isomorphisms κa :Σa = Σa such that the the following diagram commutes up to natural isotopies:

vb ∗ vb ∗ vb ∗ σ./ ∗ vb ∗ vb ∗ π1,3(Σk ) ⊗ π2,4(Σk ) π1,3(Σ0 ) ⊗ π2,4(Σ0 )

∗ a,d ∗ a,d −∗ ∗ a,d ∗ a,d −∗ π1,3(κk )⊗π2,4(κk ) π1,3(κ0 )⊗π2,4(κ0 ) dg ∗ dg ∗ dg ∗ σ./ ∗ dg ∗ dg ∗ π1,3(Σk ) ⊗ π2,4(Σk ) π1,3(Σ0 ) ⊗ π2,4(Σ0 ) , (4.19)

vb ∗ vb ∗ vb ∗ τ./ ∗ vb ∗ vb ∗ π1(Λk ) ⊗ π2(Λk ) π1(Λ0 ) ⊗ π2(Λ0 )

dg ∗ dg ∗ dg ∗ τ./ ∗ dg ∗ dg ∗ π1(Λk ) ⊗ π2(Λk ) π1(Λ0 ) ⊗ π2(Λ0 ) .

43 Proof. We examine up close the deﬁnitions of each object involved and show that up to natural isotopies in families the diagram commutes. We begin therefore with the deﬁnition a,d of τa . We restrict again to a point (p, q) = ([E1,F1, φ1], [E2,F2, φ2]) as it can be shown by using the arguments of §3.3, [15, Prop. 3.25] that our methods work in families. We also use

Let E → Y be a holomorphic vector bundle, then RΓ•(E) = Γ(E ⊗ A0,•). where the ¯ ¯∇E diﬀerential is given by ∂E = ∂ . Here ∇E is the corresponding Chern connection. Let ¯ ¯∗ 0,even 0,odd DE = ∂E + ∂E : Γ(E ⊗ A ) → Γ(E ⊗ A ) , then Hodge theory gives us the natural ∼ • isomorphisms det(DE) = det RΓ (E) after making a contractible choice of metric on E. Continuing to use the notation from Lemma 4.9, we obtain the isomorphisms

a,d vb ∼ • ∼ ∼ ∇Va ∼ ag κa |p :Σa |p = det RΓ (Va) = det DVa = det(D ) = Σa |p generalizing those of Cao–Gross–Joyce [15, Prop. 3.25], Cao–Leung [16, Thm. 2.2], Joyce– ∗ ∗ ∗ ∼ Upmeier [52, p. 38]. Recall now that at (p, q) the isomorphism π1,3(Σa) ⊗ π2,4(Σa) = ∗ ∗ ∗ π1,3(Σa−ei ) ⊗ π2,4(Σa−ei ) is given by

• • ∗ ∼ • • det RΓ (Va) ⊗ det RΓ (Wa) = det RΓ (Va−ei ) ⊗ det RΓ (Va,i) • ∗ • ∗ ∼ • • ∗ ⊗ det RΓ (Wa,i) ⊗ det RΓ (Wa−ei ) = det RΓ (Va−ei ) ⊗ det RΓ (Wa−ei ) , (4.20) where we are using the short exact sequences 0 → Va−e → Va → V → 0 and 0 → i a|Di Wa−e → Wa → W → 0 . We have the exact complex i a|Di

! pVa (fVa , fWa ) (rVa , −φa,i ◦ rWa ) pWa Va−ei ⊕ Wa−ei Ka,i Va ⊕ Wa Va,i → 0 , (4.21)

Here ρVa/Wa are restrictions, fVa/Wa the factors of inclusion and pVa/Wa ◦fVa/Wa = si for the si section si : O −→O(Di). Moreover, Ka,i := ker rVa −φa,i ◦rWa is locally free, because Va|Di has homological dimension 1. This holds also for the corresponding family on Y ×TY,D ×TY,D by the same argument. The following is a simple consequence of the construction. Lemma 4.12. We have the quasi-isomorphisms

(fVa ,fWa ) (fVa ,fWa ) Va ⊕ Wa Ka,i Va−ei ⊕ Wa−ei Ka,i

πV pV , πW pW . a−ei a a−ei a

si si Va−ei Va Wa−ei Wa

44 Using Ca to denote both upper cones and CVa , CWa to denote the lower cones, this gives a

Ca CVa commutative diagram of quasi-isomorphisms.

CWa Wa,i Therefore (4.20) becomes

• ∗ • • ∗ • det RΓ (Va) ⊗ det RΓ (Va−ei ) ⊗ det(RΓ (Wa−ei )) ⊗ det (RΓ (Wa) ∼ • ∗ • ∼ = det RΓ Ca ⊗ det RΓ Ca = C .

Using compatibility with respect to diﬀerent ﬁltrations discussed in [55, p. 22] and that a dual of an evaluation is a coevaluation in the monoidal category of line bundles together with checking the correct signs one can show that this is expressed as

• ∗ • • ∗ • det(RΓ (Va)) ⊗ det (RΓ (Va−ei )) ⊗ det(RΓ (Wa−ei ) ⊗ det RΓ Wa detRΓ•V ⊗det∗RΓ•(V )⊗det∗RΓ•(K )⊗detRΓ•(V ⊕W ) ∼ a a−ei a,i a−ei a−ei ∼ , (4.22) = ∗ • • • ∗ • = C det RΓ (Va−ei ⊕Wa−ei ) ⊗det RΓ (Ka,i) ⊗det RΓ (Wa−ei ) ⊗det RΓ (Wa) where the last isomorphism is the consequence of the following short exact sequences:

0 −→ Va−ei ⊕ Wa−ei −→ Ka,i ⊕ Va−ei −→ Va −→ 0 ,

0 −→ Va−ei ⊕ Wa−ei −→ Ka,i ⊕ Wa−ei −→ Wa −→ 0 . (4.23)

Let ∂¯1, ∂¯2, ∂¯3 be the holomorphic structures on each of the three terms of the ﬁrst sequence, ¯ ¯ ¯t ∂1 t∂(1,1) then choosing its splitting we can deﬁne ∂2 = and deforming to t = 0 the 0 ∂¯3 sequence splits. This gives us the following diagram commuting up to isotopy:

[55, Cor. 2] • • • det RΓ (Va−ei ⊕ Wa−ei ) det(RΓ (Va)) det RΓ (Ka,i ⊕ Va−ei )

detD ∗ detD , Va−ei ⊕Wa−ei ⊕Va Ka,i⊕Va−ei

(and a similar one for the second sequence) where ∗ corresponds to the isomorphism (3.6) for f f p∗ Va Wa Va ∗ : Va−ei ⊕ Wa−ei ⊕ Va −→ Ka,i ⊕ Va−ei . (4.24) 0 id si

In Lemma 4.14 below, we show that there exists a natural isomorphism ΦKa,i : Ka,i|Ti →

Ka,i|Ti , such that all the diagrams below satisfy the conditions in Deﬁnition 3.18. We

45 therefore get

 0 iΦ−1 0  a−ei,i  iΦa−ei,i 0 0  0 0 iΦ σ a,i σ Va−ei ⊕Wa−ei ⊕Va Va−ei ⊕Wa−ei ⊕Wa

f f p∗ f f p∗ Va Wa Va Va Wa Va ∗ ∗ 0 id si 0 id si

σ σ Ka,i⊕Va−ei Ka,i⊕Wa−ei iΦKa,i 0 0 iΦa−ei

We may restrict (4.24) to X\(1 − )T¯i because we already cover Ti by the other isomorphisms in the diagram. We choose the compact set Ki = (1 − /2)Ti. Then deform f f tp∗ Va Wa Va t 7→ ∗ as these are now isomorphisms in X\(1 − )Ti. Moreover, rotating 0 tid si

 sin(t) id icos(t)Φ−1 0  a−ei,i icos(t)ΦKa,i + sin(t) id 0 t 7→ icos(t)Φa−ei,i sin(t) id 0  , 0 Φa−ei 0 0 Φa,i we obtain the separate two diagrams

id 0 Φa−e ,i σ 0 id σ σ i σ Va−ei ⊕Wa−ei Va−ei ⊕Wa−ei Va−ei Wa−ei

, ∗ ∗ ( fVa fWa ) ( fVa fWa ) si si

σKa,i σKa,i σVa σWa id Φa,i

In the left diagram we can extend the identities to all of Y , so by Lemma 3.19, we showed that (4.22) coincides with the adjoint of Ξ from Lemma 3.19 for the right diagram. Using P this for each step k = i kiei, deforming the isomorphisms Φa,i on Ti into Φa on T and TN Q ∗ Q −∗ ∗ taking K = i=1 Ki, we obtain (4.19) for the data ./ because (si,k) (tj,k) = Ω . The second diagram in (4.19) is obtained by pulling back along ∆ : TY,D → TY,D × TY,D. Remark 4.13. We could replace in (4.19) the labels k, 0 by arbitrary a, b.

Lemma 4.14. There exists a natural isomorphism ΦKa,i : Ka,i|Ti → Ka,i|Ti , such that icos(t)ΦKa,i +sin(t)idKa,i are invertible for all t and such that all diagrams used in the proof of Proposition 4.11 satisfy the condition of Deﬁnition 3.18.

Proof. Using the octahedral axiom, one can show that there are short exact sequences fVa pWa fWa pVa 0 → Va−ei −−→ Ka,i −−→ Wa → 0 and 0 → Wa−ei −−→ Ka,i −−→ Va → 0. We obtain the

46 following commutative diagram

0 Va−ei |Di Ka,i|Di Wa|Di 0

−φa−ei,i id φa,i

0 Wa−ei |Di Ka,i|Di Va|Di 0 as can be seen from the following diagram:

0 Va−ei,i Ka,i|Di Va,i ⊕ Wa,i Va,i 0 ,

−φa−ei,i id id −φa,i

0 Wa−ei,i Ka,i|Di Va,i ⊕ Wa,i Wa,i 0 induced by restricting

0 Ka,i|Di Va,i ⊕ Wa,i Va,i 0

id id −φa,i

0 Ka,i|Di Va,i ⊕ Wa,i Wa,i 0

to the divisor. Choosing a splitting of the ﬁrst exact sequence in Ti. we obtain (fVa , fWa ) = id −Φ0 a−ei,i 0 0 , where Φa−e ,i|Di = φa−ei,i, so we can take Φa−ei,i = Φa−e ,i. This induces 0 si i i also the splitting of the second sequence in Ti and we can deﬁne the isomorphism by

In this splitting, we then have the invertible isomorphisms icos(t)ΦKa,i + sin(t)id = it ie Φa−e 0 −1 i −1 −Φi 0 −it where we used idKs = (fVa , fWa )◦(fVa , fWa ) = and one can si ie Φa i si Φi check directly, these satisfy the necessary commutativity for all steps needed in Proposition 4.11.

∗ ∼ ag ∗ ./ Proposition 4.15. There are natural isomorphisms Λ (DO) = (Ω ) (O ) of strong H- principal Z2-bundles independent of ord. ./ Proof. Let Odg be the Z2-bundle associated to

dg ∗ ∗ −1 dg −1 ∗ dg ∗ dg ∗ ∗ dg ∗ ∗ dg ϑ./ = π1(#k) ⊗ π2(#k) ◦ (τ./ ) : π1(Λ0 ) ⊗ π2(Λ0 ) → π1(Λ0 ) ⊗ π2(Λ0 ) , then it is a strong H-principal Z2-bundle by similar arguments to 4.7 using Lemma 4.9 and part 2 of Lemma 3.17, where we obtain similarly as in (4.8) additional signs deg(π∗Λdg)deg(π∗Λdg)+deg(π∗Λdg) (−1) 2 0 3 0 4 0 .

47 We have the isomorphisms of strong H-principal Z2-bundles ∗ ./ ∼ ./ ∼ Λ (O ) = Odg = DO , where the ﬁrst isomorphism is constructed by applying Proposition 4.11 and Lemma 3.19 and is clearly strong H-principal. For the second one, let us again restrict ourselves to the point (p, q) as in the proof of Proposition 4.19. Note that for all t we have isomorphisms:

¯ ¯ t Prop. 3.14(v) ¯ ¯ 1 det(σV0 , σW0 , Φ, Φ)χ det(σV0 , σW0 , Φ, Φ)χ

∗ ∗ −∗ π1 (Ω)⊗π2 (Ω ) ¯ ¯ t Prop. 3.14(v) ¯ ¯ 1 det(σV0 , σW0 , Φ, Φ)χ det(σVk , σWk , Φ, Φ)χ

(4.18) (4.18) ∗ ¯ ¯ t Prop. 3.14(v) ∗ ¯ ¯ 1 det (σV0 , σW0 , Φ, Φ)χ det (σV0 , σW0 , Φ, Φ)χ

dg The composition of the vertical arrows on the left for t = 0 is precisely ϑ./ , while the associated Z2-torsor to the real structure corresponding to vertical arrows on the right is ¯ by excision as in Lemma 4.8 isomorphic to DO|(p,q) using that Φ are unitary and (4.18) restrict outside of T to [43, Def. 3.24]. As the diagram is commutative for all t, we obtain ∼ ./ the required isomorphisms DO|(p,q) = Odg. It is an isomorphisms of strong H-principal Z2-bundles by the compatibility under direct sums of Lemma 3.17 and using the arguments of [15, Prop. 3.25] (see also proof of Lemma 4.8) together with paying attention to the signs above. The independence of any choice of ord can then be shown because diagram (4.19) vb dg commutes and both σ./ and σ./ are independent (see proof of Theorem 2.28). We only sketch the idea, as the precise formulation is comparably more tedious than the proof of Proposition 4.11: one uses excision on the diagram (2.28) using the common resolutions of

Lemma 4.12, where the top right corner of (2.28) has resolution V0 ⊕ W0 → Ke2,2, bottom left V0 ⊕ W0 → Ke1,1 and the bottom right one ∼ (V0 ⊕ W0 → Ke1,1 ⊕ Ve2 ⊕ We2 → Ke1+e2,1) = (V0 ⊕ W0 → Ke2,2 ⊕ Ve1 ⊕ We1 → Ke1+e2,2) .

The automorphism on Ke1,1|T1 , Ke2,2|T2 , Ke1+e2,1|T1 , Ke1+e2,2|Ti are then the ones constructed in Lemma 4.14 and one follows the arguments of Proposition 4.11 to remove contributions of all K’s.

Theorem 2.26 now follows from the above corollary together with applying Proposition 3.5 (i) and then (ii).

5 Orientation groups for non-compact Calabi–Yau fourfolds

In this ﬁnal section, we describe the behavior of orientations under direct sums. We recall the notion of orientation group from [51] and formulate the equivalent version of [51, Theorem

48 2.27] for the non-compact setting, where we replace K-theory with compactly supported K-theory. For background on compactly supported cohomology theories, see Spanier [85], Ranicki–Roe [78, §2]. From the algebraic point of view, see the discussion in Joyce–Song [50, §6.7] and Fulton [37, §18.1].

5.1 Orientation on compactly supported K-theory C In (4.13), we define the strong H-principal Z2-bundle DO → CY,D. We first describe its commutativity rules as in [51, Definition 2.22]. cs cs cs Definition 5.1. Let µC : CY,D × CY,D → CY,D, µcs : C × C → C be the binary maps and C C ∗ C cs cs cs ∗ cs τ : DO DO −→ µC(DO) , τ : O O −→ µcs(O ) (5.1) C be the isomorphisms of Z2-bundles on CY,D × CY,D, which make (DO, τ) into a strong H- cs principal Z2-bundle and τ a restriction of τ. We recall the notion of Euler-form as defined in Joyce–Tanaka–Upmeier [51, Definition 2.20] for real elliptic differential operators.

Definition 5.2. Let X be a smooth compact manifold, E0,E1 vector bundles on X and P : E0 → E1 a real or complex elliptic differential operator. Let E,F → X be complex 0 0 vector bundles, the Euler form χP : K (X) × K (X) → Z is defined by

χP ( E , F ) = indC(σ(P ) ⊗ idπ∗(Hom(E,F ))) J K J K together with bi-additivity of χP . We used the notation from §3.3 for symbols of operators. R If X is spin and P = D/ +, we write χX := χD/ . Similarly, if X is a complex manifold and ¯ ¯∗ 0,even 0,odd D = ∂ + ∂ : A → A is the Dolbeault operator, then we use χX := χD. 0 Recall that we have a comparison map c : G0(X) → K (X), where G0(X) is the b ag Grothendieck group associated to D Coh(X) . Let χX : K0(X) × K0(X) → Z be deﬁned ag P i i by χX (E,F ) = i∈ (−1) dimC(Ext (E,F )), then when X is smooth , we have χX ◦(c×c) = ag Z χX .

Proposition 5.3. Let i1, i2 : Y → Y ∪D Y be the inclusions of the two copies of Y and 0 ∗ ∗ δ(α, β) ∈ K (Y ∪DY ) denote a K-theory class, such that i1(δ(α, β)) = α and i2(δ(α, β)) = β. 0 We have the bijection π0(CY,D) = K (Y ∪D Y ). Let Cδ(α,β) be the components corresponding C C to δ(α, β) and DO|δ(α,β) the restriction of DO to it. Suppose a choice of trivialization oδ(α,β) : C 0 Z2 → DO|δ(α,β) is given for each δ(α, β) ∈ K (Y ∪D Y ), then deﬁne δ(α1,β1),δ(α2,β2) ∈ {−1, 1} by τ oδ(α1,β1) Z2 oδ(α2,β2) = δ(α1,β1),δ(α2,β2)oδ(α1,β1)+δ(α2,β2) These signs satisfy: χY (α1,α1)−χY (β1,β1) χY (α2,α2)−χY (β2,β2) +χY (α1,α2)−χY (β1,β2) δ(α2,β2),δ(α1,β1) =(−1)

δ(α1,β1),δ(α2,β2) .

δ(α1,β1),δ(α2,β2)δ(α1+α2,β1+β2),δ(α3,β3) = δ(α2,β2),δ(α3,β3)δ(α2+α3,β2+β3),δ(α1,β1) . (5.2)

49 top −1 ./ top ./ top ag Let (M ) = Γ (C ), (O ) = (O ) | top and o = γ(α,β) γ(α,β) γ(α,β) (Mγ(α,β)) γ(α,β) ./ ∗ ./ top ./ ./ I Γ (oγ(α,β)) the trivializations of (Oγ(α,β)) obtained using I from (2.21). Let φ be from Proposition 4.7, then it satisﬁes

φ./(oag oag = oag . δ(α1,β1) Z2 δ(α2,β2) δ(α1,β1),δ(α2,β2) δ(α1,β1)+δ(α2,β2)

˜ ˜ C ˜ Proof. Recall the definition of Y , T from Definition 4.1. One can express DO(Y ) as a / / ∗ DY˜ ∗ DY˜ ∗ product of Z2-graded Z2-bundles p1(OC ) ⊗ p2(OC ) obtained by Proposition 3.5 from / p1 p2 D/ ˜ DY˜ Y O in Example 3.8, where CY˜ ←−CY˜ ×CD CY˜ −→CY˜ are the projections and deg OC |Cα˜ = χR (˜α, α˜) . Using Definition 3.8 and Lemma 3.6 together with Joyce–Tanaka–Upmeier [51, Y˜ p. 2.26], a simple computation shows that for each δ˜i(˜αi, β˜i), which under inclusion ˜ι1,2 : ˜ ˜ ˜ ˜ Y → Y ∪T˜ Y restrict toα ˜i, βi respectively, we have the formula χR (˜α1,α˜1)−χR (β˜1,β˜1) χR (˜α2,α˜2)−χR (β˜2,β˜2) +χR (˜α1,α˜2)−χR (β˜1,β˜2) τ˜ = (−1) Y˜ Y˜ Y˜ Y˜ Y˜ Y˜ δ˜2(˜α2,β2),δ˜1(˜α1,β˜1) τ˜ (5.3) δ˜1(˜α1,β˜1),δ˜2(˜α2,β˜2)

Two points [E±,F ±, φ±] of V × V map to [E˜±, F˜±, φ˜±] ∈ V , as described in (4.12). Y VT¯ Y Y,˜ T˜ Using excision on index and Deﬁnition 5.2, it follows that

R ˜+ ˜− R ˜+ ˜− + − + − χ ˜ ( E , E ) − χ ˜ ( F , F ) = χY ( E , E ) − χY ( F , F ) . Y J K J K Y J K J K J K J K J K J K Using (4.14) and biadditivity of χ, we obtain

χR (˜α , α˜ ) − χR (β˜ , β˜ ) = χ (α , α ) − χ (β , β ) , Y˜ 1 2 Y˜ 1 2 Y 1 2 Y 1 2 whereα ˜i, β˜i are K-theory classes glued from αi, βi as in (4.12), from which we obtain χY (α1,α1)−χY (β1,β1) χY (α2,α2)−χY (β2,β2) +χY (α1,α2)−χY (β1,β2) τδ(α2,β2),δ(α1,β1) =(−1)

τδ(α1,β1),δ(α2,β2) .

C This leads to (5.2) by using that DO is strong H-principal. To conclude the ﬁnal statement of the proposition, one applies Proposition 4.15.

cs top cs Recall from Deﬁnition 2.27 that we have the map Γ :(MX ) → CX . There exists a compactly supported Chern character which is an isomorphism

∗ ∗ chcs : Kcs(X) ⊗Z Q −→ Hcs(X, Q) (5.4)

even of Z2-graded rings. We also have the Euler form on Hcs (X, Q):

even even χ¯ : Hcs (X, Q) × Hcs (X, Q) −→ Q ∨ χ¯(a, b) = deg(a · b · td(TX))4 . (5.5)

50 0 0 Combining (5.4) and (5.5), one getsχ ¯ : Kcs (X) × Kcs (X) → Z . Note that, we have ¯ 0 0 χ¯(α, β) = χY (¯α, β), where for a class α ∈ Kcs (X),α ¯ denotes the class of K (Y ) extended trivially. ω ω ω ∗ ω Recall from Example 3.9 that we have the isomorphisms φ : O Z2 O → µ (O ). We can now state the main result of comparison of signs under sums in non-compact Calabi–Yau fourfolds.

cs cs Theorem 5.4. Let Cα denote the connected component of CX corresponding to α ∈ 0 cs cs cs top cs −1 cs cs Kcs(X) = π0(CX ) and Oα = O |Cα . Let (Mα) = (Γ ) (Cα ). After ﬁxing choices of cs cs trivializations oα of Oα , we deﬁne α,β

ω ∗ cs ∗ cs ∼ ∗ cs φ I Γ oα I Γ oβ = α,βI Γ oα+β .

cs If one moreover ﬁxes the preferred choice of o0 , such that

cs cs cs cs τ (o0 o0 ) = o0 , (5.6) then :(α, β) 7→ α,β ∈ {±1} is up to equivalences the unique group 2-cocycle satisfying

χ¯(α,α)¯χ(β,β)+¯χ(α,β) α,β = (−1) β,α (5.7)

Proof. Recall from the proof of Theorem 2.26 that we have the isomorphism ζ∗O./ =∼ Oω which is by construction in Proposition 4.7 strong H-principal as can be checked directly by comparing Z2-torsors at each Spec(A)-point [i∗(E), 0]. By Proposition 5.3 after setting β1 and β2 equal to 0 and α1 =α ¯, α2 = β¯ it then follows that, we have (5.7) together with

α,0 = 0,α = 1 , α,βα+β,γ = β,γα,β+γ (5.8) which it precisely the cocycle condition. The ﬁrst condition follows from (5.6).

We now discuss the orientation group from Joyce–Tanaka–Upmeier [51, Deﬁnition 2.26] 0 0 applied to Kcs (X) instead of K (X) in our non-compact setting. The compactly supported orientation group is deﬁned as

cs 0 cs cs Ωcs(X) = {(α, oα ): α ∈ Kcs(X), oα orientation on Cα } .

cs cs cs cs cs The multiplication is given by (α, oα ) ? (β, oβ ) = α + β, τα,β(oα Z2 oβ ) . The resulting 0 group is the unique group extension 0 → Z2 → Ωcs(X) → Kcs(X) → 0 for the group 2-cycle 0 of Theorem 5.4. Choices of orientations induce a splitting Kcs → Ωcs(X) as sets. This 0 0 ﬁxed : Kcs(X) × Kcs(X) → Z2 in Theorem 5.4. Let us describe the method used in [51, 0 Thm. 2.27] for extending orientations. Choosing generators of Kcs(X), one obtains p q Y Y 0 ∼ r p Kcs (X) = Z × Zmk × Z2 j , (5.9) k=1 j=1

51 where mk > 2 odd and pj > 0. Fixing a choice of isomorphism (5.9), choose orientation cs cs on each Cαi , αi = (0,..., 0, 1,, 0 ..., 0), where 1 is in position i. Use τ to obtain ori- 0 entations for all α ∈ Kcs(X) by adding generators going from left to right in the form q p cs cs cs cs (a1, . . . , ap, (bj)j=1, (ck)k=1) and using in each step oα0+g = τ (oα0 og ), where g is a generator. As a result, one obtains the splitting:

q p Y Y ∼ 0 ∼ r p Or(o):Ωcs (X) = Kcs (X) × {−1, 1} = Z × Zmk × Z2 j × {−1, 1} , (5.10) k=1 j=1

cs where o is the set of orientation on Cα for the chosen generators α. Letχ ¯ij :=χ ¯(αi, αj). The next result replaces K-theory by compactly supported K-theory in Joyce–Tanaka– cs Upmeier [51, Thm. 2.27] and considers the Z2-bundle O we constructed. Theorem 5.5 ([51, Thm. 2.27]). Let Or(o) be the isomorphism (5.10) for a given choice of orientations o on generators corresponding to the isomorphism (5.9). Let T2 be the 2-torsion 0 subgroup of Kcs (X). Then:

(i) Deﬁne the map ξ : T2 → Z2 as Ξ(γ) = γ,γ, Then it is a group homomorphism. p q cs r Q Q p (ii) Using Or(o) from (5.10) to identify Ω (X) with Z × k=1 Zmk × j=1 Z2 j ×{−1, 1} the induced group structure on the latter becomes

p q 0 0 0 p 0 q 0 a1, . . . , ar, (bj)j=1, (ck)k=1, o ? a1, . . . , ar, (bj)j=1, (ck)k=1, o 0 0 0 p 0 q = a1 + a1, . . . , ar + ar, (bj − bj)j=1, (ck + ck)k=1, P (¯χ +¯χ χ¯ )a0 a 0 (−1) 1≤h

q q where γ = (0,..., 0, (0)k=1, (˜cj)j=1) and 0 c¯j +c ¯j c˜ = 2pj −1 j 0 cj + cj

0 0 pj for the unique representatives 0 ≤ c¯j, c¯j, cj + cj < 2 .

References

[1] B. Antieau and J. Heller. “Some remarks on topological K-theory of DG categories”. Proceedings of the American Mathematical Society 146.10 (2018), pp. 4211–4219. arXiv: 1709.01587 (cit. on p. 38). [2] M. F. Atiyah and R. Bott. “The Yang-Mills Equations over Riemann Surfaces”. Philo- sophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 308.1505 (1983), pp. 523–615 (cit. on p. 36).

52 [3] M. F. Atiyah and J. D. S. Jones. “Topological aspects of Yang-Mills theory”. Comm. Math. Phys. 61.2 (1978), pp. 97–118 (cit. on p. 36). [4] M. F. Atiyah and I. M. Singer. “The Index of Elliptic Operators: I”. Annals of Math- ematics 87.3 (1968), pp. 484–530 (cit. on pp. 5, 22, 27). [5] M. F. Atiyah. “Riemann surfaces and spin structures”. Annales scientifiques de l’Ecole´ Normale Supérieure Ser. 4, 4.1 (1971), pp. 47–62 (cit. on p. 11). [6] V. V. Batyrev. “On the classification of toric Fano 4-folds”. Journal of Mathematical Sciences 94.1 (1999), pp. 1021–1050. arXiv: math/9801107 (cit. on p. 14). [7] E. Bierstone and P. D. Milman. “Functoriality in resolution of singularities”. Publ. Res. Inst. Math. Sci. 44 (2008), pp. 609–639. arXiv: math/0702375 (cit. on p. 14). [8] A. Blanc. “Topological K-theory of complex noncommutative spaces”. Compositio Mathematica 152 (2016), pp. 489–555. arXiv: 1211.7360 (cit. on pp. 3, 11, 37, 38). [9] J. Block. “Duality and equivalence of module categories in noncommutative geometry”. A celebration of the mathematical legacy of Raoul Bott. Vol. 50. Amer. Math. Soc., Providence, RI, 2010, pp. 311–339. arXiv: math/0509284 (cit. on p. 43). [10] A. Bojko. Orientations for DT invariants on quasi-projective Calabi-Yau 4-folds. 2020. arXiv: 2008.08441 (cit. on p. 42). [11] D. Borisov and D. Joyce. “Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds”. Geom. Topol. 21.6 (2017), pp. 3231–3311. arXiv: 1504. 00690v2 (cit. on pp. 1, 10, 11, 19). [12] A. K. Bousfield and E. M. Friedlander. “Homotopy theory of Γ-spaces, spectra, and bisimplicial sets”. Geometric Applications of Homotopy Theory II. Springer Berlin, 1978, pp. 80–130 (cit. on p. 38). [13] C. Brav and T. Dyckerhoff. Relative Calabi-Yau structures II: Shifted Lagrangians in the moduli of objects. 2018. arXiv: 1812.11913 (cit. on pp. 3, 9, 10). [14] V. Bussi. Derived symplectic structures in generalized Donaldson–Thomas theory and categorification [Oxford PhD thesis]. 2014. url: https://ora.ox.ac.uk/objects/ uuid:54896cc4-b3fa-4d93-9fa9-2a842ad5e4df (cit. on p. 21). [15] Y. Cao, J. Gross, and D. Joyce. “Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds”. Advances in Mathematics 368 (2020). arXiv: arXiv:1811.09658 (cit. on pp. 1–3, 5, 11, 12, 23–26, 36, 37, 39, 44, 48). [16] Y. Cao, J. Gross, and D. Joyce. “Orientability of moduli spaces of Spin(7)-instantons and coherent sheaves on Calabi–Yau 4-folds”. Advances in Mathematics 368 (2020). arXiv: arXiv:1502.01141 (cit. on p. 44). [17] Y. Cao and M. Kool. “Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds”. Adv. Math. 338 (2018), pp. 601–648. arXiv: 1712.07347 (cit. on p. 2). [18] Y. Cao and M. Kool. “Counting zero-dimensional subschemes in higher dimensions”. J. Geom. Phys. 136 (2019), pp. 119–137. arXiv: 1805.04746 (cit. on p. 2).

53 [19] Y. Cao and M. Kool. Curve counting and DT/PT correspondence for Calabi-Yau 4-folds. 2019. arXiv: 1903.12171 (cit. on p. 2). [20] Y. Cao, M. Kool, and S. Monavari. K-theoretic DT/PT correspondence for toric Calabi-Yau 4-folds. 2020. arXiv: 1906.07856 (cit. on p. 2). [21] Y. Cao, M. Kool, and S. Monavari. “Stable pair invariants of local Calabi-Yau 4-folds” (2020). arXiv: 2004.09355 (cit. on pp. 2, 21). [22] Y. Cao and N. C. Leung. Donaldson-Thomas theory for Calabi-Yau 4-folds. 2015. arXiv: 1407.7659 (cit. on pp. 1, 2, 19). [23] Y. Cao, D. Maulik, and Y. Toda. “Genus zero Gopakumar–Vafa type invariants for Calabi–Yau 4-folds”. Adv. Math. 338 (2018), pp. 41–92. arXiv: 1801.02513 (cit. on p. 2). [24] Y. Cao, D. Maulik, and Y. Toda. “Stable pairs and Gopakumar-Vafa type invariants for Calabi-Yau 4-folds” (2019). arXiv: 1902.00003 (cit. on pp. 2, 21). [25] Y. Cao and Y. Toda. Curve counting via stable objects in derived categories of Calabi- Yau 4-folds. 2019. arXiv: 1909.04897 (cit. on p. 2). [26] Y. Cao and Y. Toda. Gopakumar-Vafa type invariants on Calabi-Yau 4-folds via de- scendent insertions. 2020. arXiv: 2003.00787 (cit. on pp. 2, 21). [27] J. Caruso, F. R. Cohen, J. P. May, and L. R. Taylor. “James Maps, Segal Maps, and the Kahn-Priddy Theorem”. Transactions of the American Mathematical Society 281.1 (1984), pp. 243–283 (cit. on p. 24). [28] D. Cox, J. Little, and H. Schenck. Toric Varieties. Graduate studies in mathematics. American Mathematical Soc., 2011 (cit. on p. 13). [29] S. K. Donaldson. “The orientation of Yang-Mills moduli spaces and 4-manifold topology”. J. Diﬀerential Geom. 26.3 (1987), pp. 397–428 (cit. on pp. 4, 5, 22, 27). [30] S. Donaldson and P. Kronheimer. The Geometry of Four-manifolds. Oxford mathematical monographs. Clarendon Press, 1990 (cit. on pp. 4, 5, 22, 27, 31, 36). [31] D. Ferrand. “Conducteur, descente et pincement”. Bull. Soc. Math. 131.4 (2003), pp. 553–585 (cit. on p. 37). [32] D. Freed. “On determinant line bundles”. Mathematical Aspects of String Theory. World Sci. Publishing, 1987, pp. 189–238 (cit. on p. 27). [33] E. M. Friedlander and M. E. Walker. “Semi-topological K-theory using function complexes”. Topology 41.3 (2002), pp. 591–644 (cit. on pp. 36, 37). [34] E. M. Friedlander and M. E. Walker. “Function Spaces and Continuous Algebraic Pairings for Varieties”. Compositio Mathematica 125.1 (2001), pp. 69–110 (cit. on p. 40). [35] T. Friedrich. Dirac-Operatoren in der Riemannschen Geometrie. Friedr. Vieweg & Sohn, Braunschweig, 1997 (cit. on p. 12).

54 [36] W. Fulton. Introduction to toric varieties. 131. Princeton University Press, 1993 (cit. on p. 13). [37] W. Fulton. Intersection theory. Vol. 2. Springer Science & Business Media, 2013 (cit. on p. 49). [38] D. Gaitsgory and N. Rozenblyum. A study in derived algebraic geometry Quasi- coherent sheaves on prestacks. url: %7Bhttp : / / people . math . harvard . edu / ~gaitsgde/GL/%7D (cit. on p. 16). [39] D. Gaitsgory and N. Rozenblyum. “DG indschemes”. Perspectives in representation theory 610 (2014), pp. 139–251. arXiv: 1108.1738 (cit. on pp. 9, 36). [40] R. E. Gompf and A. I. Stipsicz. 4-manifolds and Kirby calculus. Vol. 20. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1999 (cit. on p. 34). [41] J. P. C. Greenlees and J. P. May. “Equivariant stable homotopy theory”. Handbook of algebraic topology. North-Holland, Amsterdam, 1995, pp. 277–323 (cit. on p. 37). [42] J. Gross. The homology of moduli stacks of complexes. 2019. arXiv: 1907.03269 (cit. on p. 37). [43] J. Gross, D. Joyce, and Y. Tanaka. Universal structures in C-linear enumerative invariant theories. I. 2020. arXiv: 2005.05637 (cit. on pp. 2, 6, 48). [44] A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002. url: http://pi.math.cornell.edu/~hatcher/AT/AT.pdf (cit. on p. 23). [45] H. Hironaka. “Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I”. Annals of Mathematics 79.1 (1964), pp. 109–203 (cit. on p. 14). [46] P. S. Hirschhorn. Model categories and their localizations. 99. American Mathematical Soc., 2009 (cit. on p. 7). [47] L. H¨ormander.“Pseudo-diﬀerential operators and hypoelliptic equations”. Singular integrals (Proc. Sympos. Pure Math., Vol. X, Chicago, Ill., 1966). 1967, pp. 138–183 (cit. on p. 27). [48] M. Hovey. Model categories. 63. American Mathematical Soc., 2007 (cit. on p. 7). [49] D. Joyce. Ringel–Hall style Lie algebra structures on the homology of moduli spaces. preprint. 2019. url: https://people.maths.ox.ac.uk/joyce/hall.pdf (cit. on pp. 2, 6). [50] D. Joyce and Y. Song. “A theory of generalized Donaldson-Thomas invariants”. Mem- oirs of the American Mathematical Society 217.1020 (2012). arXiv: 0810.5645v6 (cit. on pp. 21, 49). [51] D. Joyce, Y. Tanaka, and M. Upmeier. “On orientations for gauge-theoretic moduli spaces”. Advances in Mathematics 362 (2020). arXiv: 1811.01096 (cit. on pp. 3, 6, 12, 23, 24, 41, 48–52).

55 [52] D. Joyce and M. Upmeier. Orientation data for moduli spaces of coherent sheaves over Calabi-Yau 3-folds. 2020. arXiv: 2001.00113 (cit. on pp. 6, 44). [53] M. Kashiwara and P. Schapira. Categories and Sheaves. Grundlehren der mathema- tischen Wissenschaften. Springer Berlin, 2005 (cit. on p. 36). [54] B. Keller. “On diﬀerential graded categories”. Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006 (2006), pp. 151–190. arXiv: arXiv:math/0601185 (cit. on p. 7). [55] F. Knudsen and D. Mumford. “The Projectivity of the Moduli Space of Stable Curves I: Preliminaries on “det” and “Div””. Mathematica Scandinavica 39.1 (1977), pp. 19– 55 (cit. on pp. 20, 45). [56] M. Kool and J. Rennemo. In preparation (cit. on p. 2). [57] M. Kool and R. P. Thomas. “Stable pairs with descendents on local surfaces I: the vertical component”. Pure and Applied Mathematics Quarterly 13.4 (2017), pp. 581– 638. arXiv: 1605.02576 (cit. on p. 21). [58] S. Lang. Introduction to diﬀerentiable manifolds. Second. Universitext. Springer- Verlag, New York, 2002 (cit. on p. 41). [59] H. B. Lawson Jr., P. Lima-Filho, and M.-L. Michelsohn. “Algebraic cycles and equivariant cohomology theories”. Proceedings of the London Mathematical Society. Third Series 73.3 (1996), pp. 679–720 (cit. on p. 38). [60] H. B. Lawson Jr and M.-L. Michelsohn. Spin geometry. Princeton University Press, 1989 (cit. on p. 27).

[61] P. Lima-Filho. “Algebraic Families of E∞-spectra”. Topology and its Applications 98.1-3 (1999), pp. 217–233 (cit. on p. 38). [62] J. Lurie. Derived Algebraic Geometry. Ph.D. thesis, Massachusetts Institute of Tech- nology, Dept. of Mathematics. 2004 (cit. on p. 7). [63] J. Lurie. On Inﬁnity Topoi. 2003. arXiv: math/0306109 (cit. on p. 7). [64] J. Lurie. Higher topos theory. Vol. 170. Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009, pp. xviii+925 (cit. on pp. 7, 9). [65] D. Maulik, R. Pandharipande, and R. P. Thomas. “Curves on K3 surfaces and modular forms”. Journal of Topology 3 (2010), pp. 937–996. arXiv: arXiv:1001.2719 (cit. on p. 21).

[66] J. P. May. “E∞-spaces, group completions and permutative categories”. New De- velopments in Topology, London Math. Soc. Lecture Note Series 11 (1974) (cit. on p. 24). [67] J. P. May and K. Ponto. More concise algebraic topology: localization, completion, and model categories. University of Chicago Press, 2011 (cit. on pp. 19, 23). [68] D. Metzler. Topological and Smooth Stacks. 2003. arXiv: math/0306176 (cit. on p. 22).

56 [69] N. Nekrasov. “Magnificent four”. Ann. Inst. Henri PoincaréD 7.4 (2020), pp. 505– 534. arXiv: 1712.08128 (cit. on p. 2). [70] N. Nekrasov and N. Piazzalunga. “Magnificent four with colors”. Communications in Mathematical Physics 372.2 (2019), pp. 573–597 (cit. on p. 2). [71] B. Noohi. Foundations of Topological Stacks I. 2005. arXiv: math/0503247 (cit. on p. 22). [72] B. Noohi. “Homotopy types of topological stacks”. Advances in Mathematics 230.4-6 (2012), pp. 2014–2047. arXiv: 0808.3799v2 (cit. on pp. 22, 23). [73] J. Oh and R. P. Thomas. Counting sheaves on Calabi–Yau 4-folds, I. 2020. arXiv: 2009.05542 (cit. on pp. 1, 2, 10, 19). [74] T. Pantev, B. Toën,M. Vaquié,and G. Vezzosi. “Shifted symplectic structures”. Publications mathématiquesde l’IHES´ 117.1 (2013), pp. 271–328 (cit. on pp. 3, 7, 8). [75] J. Phillips. “Self-adjoint Fredholm operators and spectral flow”. Canadian Mathemat- ical Bulletin 39.4 (1996), pp. 460–467 (cit. on p. 28). [76] A. Preygel. “Some remarks on shifted symplectic structures on non-compact mapping spaces” (2013). url: http://www.tolypreygel.com/papers/note_shifted_sympl_ open.pdf (cit. on p. 21). [77] D. Quillen. “Determinants of Cauchy-Riemann operators over a Riemann surface”. Functional Analysis and Its Applications 19.1 (1985), pp. 31–34 (cit. on p. 27). [78] A. Ranicki and J. Roe. Surgery for Amateurs. 2017. url: https://sites.psu.edu/ surgeryforamateurs/%5C%5Cfiles/2017/12/surgerybook2017-2gfid7m.pdf (cit. on pp. 19, 49). [79] S. Rao, S. Yang, X. Yang, and X. Yu. Hodge cohomology on blow-ups along subvarieties. 2019. arXiv: 1907.13281 (cit. on p. 20). [80] K. Schwede. “Gluing schemes and a scheme without closed points”. Contemporary Mathematics 386 (2005), p. 157 (cit. on p. 37). [81] R. T. Seeley. “Integro-Differential Operators on Vector Bundles”. Transactions of the American Mathematical Society (1965), pp. 167–204 (cit. on p. 27). [82] G. Segal. “Categories and cohomology theories”. Topology 13.3 (1974), pp. 293–312 (cit. on p. 24). [83] C. Simpson. The topological realization of a simplicial presheaf. 1996. arXiv: q-alg/ 9609004 (cit. on p. 11). [84] I. M. Singer. “Some remarks on the Gribov ambiguity”. Communications in Mathe- matical Physics 60.1 (1978), pp. 7–12 (cit. on p. 36). [85] E. Spanier. “Cohomology with supports”. Pacific Journal of Mathematics 123.2 (1986), pp. 447–464 (cit. on pp. 19, 49).

57 [86] J. D. Stasheff. “Homotopy associativity of H-spaces. I”. Transactions of the American Mathematical Society 108.2 (1963), pp. 275–292 (cit. on p. 24). [87] J. D. Stasheff. “Homotopy associativity of H-spaces. II”. Transactions of the American Mathematical Society 108.2 (1963), pp. 293–312 (cit. on p. 24). [88] N. Strickland. An introduction to the category of spectra. 2020. arXiv: 2001.08196 (cit. on p. 37). [89] Y. Toda. “Curve counting theories via stable objects I. DT/PT correspondence”. Journal of the American Mathematical Society 23.4 (2010), pp. 1119–1157. arXiv: 0902.4371 (cit. on p. 21). [90] B. Toën.“The homotopy theory of dg-categories and derived Morita theory”. Inven- tiones mathematicae 167.3 (2007), pp. 615–667 (cit. on p. 7). [91] B. Toën,B. Toën,and G. Vezzosi. Homotopical Algebraic Geometry II: Geometric Stacks and Applications: Geometric Stacks and Applications. Vol. 2. American Math- ematical Soc., 2008. arXiv: arXiv:math/0404373 (cit. on pp. 7, 8). [92] B. Toënand M. Vaquié.“Moduli of objects in dg-categories”. Annales scientifiques de l’Ecole normale supérieure. Vol. 40. 3. 2007, pp. 387–444. arXiv: math/0503269v5 (cit. on pp. 7, 8). [93] B. Toënand G. Vezzosi. “From HAG to DAG: Derived Moduli Stacks”. Axiomatic, Enriched and Motivic Homotopy Theory (2004), pp. 173–216. arXiv: math/0210407 (cit. on p. 7). [94] B. Toënand G. Vezzosi. “Homotopical algebraic geometry I: Topos theory”. Advances in mathematics 193.2 (2005), pp. 257–372. arXiv: math/0207028v4 (cit. on pp. 7, 37). [95] M. Upmeier. A categorified excision principle for elliptic symbol families. 2019. arXiv: 1901.10818 (cit. on pp. 4, 5, 18, 22, 26–33, 41). [96] S. Yu. “The Dolbeault dga of a formal neighborhood”. Trans. Amer. Math. Soc. 368.11 (2016), pp. 7809–7843. arXiv: 1206.5155 (cit. on p. 43). [97] A. Zinger. “The Determinant Line Bundle for Fredholm Operators: Construction, Properties, and Classification”. Mathematica Scandinavia 118.2 (2016), p. 203. arXiv: 1304.6368 (cit. on pp. 26, 27).