Relative Seiberg-Witten invariants and a sum formula
aa r X i v : . [ m a t h . DG ] S e p Relative Seiberg-Witten invariantsand a sum formula
Mohammad Farajzadeh-Tehrani ∗ and Pedram SafariSeptember 22, 2020 Abstract
We study relative Seiberg-Witten moduli spaces and define relative invariants for a pair( X, Σ) consisting of a smooth, closed, oriented 4-manifold X and a smooth, closed, oriented2-dimensional submanifold Σ ⊂ X with positive genus. These relative Seiberg-Witten invariantsare meant to be the counterparts of relative Gromov-Witten invariants. We also obtain a sumformula (aka a product formula) that relates the SW invariants of a sum X of two closed oriented4-manifolds X and X along a common oriented surface Σ with dual self-intersections to therelative SW invariants of ( X , Σ) and ( X , Σ). Our formula generalizes Morgan-Szab´o-Taubes’product formula.
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22 A tour of Seiberg-Witten theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 SW equations in dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 SW equations in dimension 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 SW equations on a circle bundle over a Riemann surface . . . . . . . . . . . . . . . . 142.5 SW equations on a four-manifold with a cylindrical end . . . . . . . . . . . . . . . . 212.6 Gluing monopoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . . c structures . . . . . . . . . . . . . . . 253.2 The main component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Proof of Proposition 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Tunneling spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Compactification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.7 Conclusion; proofs of Theorems A–C . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ∗ Supported by NSF grant DMS-2003340 Introduction
Relative moduli spaces associated to a pair (
X, D ) of a symplectic manifold/complex projectivevariety X and a (symplectic) divisor D have many applications in both symplectic and complexalgebraic geometry. In particular, we use moduli spaces of stable relative maps to define relativeGromov-Witten (or GW) invariants. We also use them to derive a GW sum formula when a smoothone-parameter family of symplectic manifolds/varieties { Y λ } λ ∈ C degenerates to a simple normalcrossing variety Y = S i ∈I X i .In real dimension four, Seiberg-Witten (or SW) moduli spaces have been an excellent tool forexploring smooth closed orientable 4-manifolds. When X is symplectic, by a celebrated work ofTaubes, we know that the SW invariants of X are equal to certain (possibly disconnected) GWinvariants [T1]. This correspondence has been a great tool for classifying symplectic 4-manifolds;e.g. see [L]. In light of SW-GW correspondence, it is natural to seek a construction of relative SWinvariants for every pair ( X, Σ) of a closed oriented 4-manifold X and a closed, oriented, possiblynon-connected, 2-dimensional submanifold Σ ⊂ X . It is also natural to expect a sum formula thatexpresses the SW invariants of a sum X of two oriented 4-manifolds X and X along a commonoriented surface Σ with dual self-intersections in terms of the relative SW invariants of ( X , Σ) and( X , Σ). It is expected that in the symplectic case and under Taubes’ GW-SW correspondence,such an SW sum formula should correspond to the well-known GW sum formula.Seiberg-Witten invariants o o Taubes’ correspondence / / O O ? SW sum formula (cid:15) (cid:15) Gromov-Witten invariants O O GW sum formula (cid:15) (cid:15) ? Relative SW invariants o o ? / / Relative GW invariantsIf Σ has genus g > X and X , then Morgan-Szab´o-Taubes’ productformula [MST, Thm. 3.1], when the restriction of the characteristic line bundle to Σ has degree2 g − , is a particular case of this picture. Applications of their formula include the proof of Thomconjecture [MST] — also proved independently in [KM1] using a vanishing argument — as well asconstruction of non-symplectic 4-manifolds with non-trivial SW invariants [FS]. Also, the vanishingresult of [OS, Thm. 2.1] is another particular case of the SW sum formula envisioned above.In this paper, we introduce a setup, in two formats, for constructing relative SW moduli spaces andinvariants without any major restriction on the topology of ( X, Σ). In one format, as formulated inTheorem B, we obtain relative SW invariants g SW X, Σ and a sum formula that generalizes Morgan-Szab´o-Taubes’ product formula, but its relation to the GW side is not clear to us. In the otherformat, formulated in Theorem A, we obtain relative SW invariants SW X, Σ and a sum formula,subject to the regularity of the tunneling spaces, which should be equivalent to the GW side ofthe diagram above under Taubes’ GW-SW correspondence. We believe, moreover, that for certainchoice of Ω in (1.5), the relative invariants provided by these two approaches are equivalent.Given a smooth, closed, oriented 4-manifold X and a transverse union Σ = S i ∈I Σ i of positivelyintersecting, closed, oriented, 2-dimensional sub-manifolds Σ i ⊂ X , we can define a notion oflogarithmic tangent bundle T X ( − log Σ) that coincides with the corresponding notions in algebraicgeometry/symplectic topology whenever ( X, Σ) is complex/symplectic; see [FMZ]. There is a vector2undle homomorphism ι : T X ( − log Σ) −→ T X, covering id X , which is an isomorphism away from Σ. In this paper, for simplicity, we assume thatΣ is smooth. We will then have T X ( − log Σ) | Σ ∼ = T Σ ⊕ C Σ , where C Σ is the trivial complex line bundle Σ × C . In order to define relative SW invariants, ouridea is to use the logarithmic tangent bundle T X ( − log Σ) instead of the classical tangent bundle T X, as is traditionally used in SW theory. The advantage of this approach is two-fold. First ofall, it is amenable to working with cylindrical-end manifolds, for which a good deal of literatureis available at our disposal [MMR, T, MST, MOY, N, N1, S, KM]. We will review the essentialmaterial in Section 2 for the sake of coherence of the presentation and establishing the notation. Inthis context, we will work over the cylindrical-end manifold X − Σ and simply use the logarithmictangent bundle to distinguish different topological components of particular SW moduli spaces indimensions 3 and 4.From another perspective, using the logarithmic tangent bundle enables us to directly work overthe closed manifold X and generalize the results to arbitrary normal crossing case, where it isnot feasible to study X − Σ as a manifold with one cylindrical end. This approach is outlinedin Section 4, where we use a so-called logarithmic connection on
T X ( − log Σ) and logarithmicconnections on the related spinor bundles to derive a logarithmic version of the SW equations. Wewill treat the details of this direct approach and the general normal crossing case in a separatepaper.Here is an outline of the rest of the paper. In Section 3.1, we define the notion of a relativelycanonical spin c structure for the logarithmic tangent bundle. Every other spin c structure will thenbe obtained by tensoring a relatively canonical spin c structure with a hermitian line bundle E on X . Let s be a spin c structure on T X ( − log Σ) with spinor bundle S = S + ⊕ S − and characteristicline bundle L = L s = det( S + ) = det( S − ) . Define the degree of s along Σ to be the integer d ( s ) .. = ( g − − | m ( s ) | , m ( s ) = 12 deg( L Σ ) , (1.1)where g is the genus of Σ and L Σ .. = L s | Σ . The motivation for this definition comes from complexgeometry: if X is a symplectic 4-manifold, then T X ( − log Σ) can be equipped with a complexstructure. In this case, if s can is the canonical spin c structure on T X ( − log Σ) and s E = s can ⊗ E is the canonical spin c structure twisted by a complex line bundle E , then d ( s E ) in (1.1) is simplythe degree of E Σ .. = E | Σ or of the Serre dual K Σ ⊗ E ∗ Σ , depending on the sign of deg( L Σ ) being non-positive or positive. If we set ß = PD( c ( E )) ∈ H ( X, Z ) , then the spin c structure s E is determinedby ß and d ( s E ) is equal to the product of homology classes Σ · ß (or Σ · ( K Σ − ß)).We focus next on X − Σ. The restriction of a suitable metric on
T X ( − log Σ) to X − Σ gives X − Σthe structure of a manifold with cylindrical end [0 , ∞ ) × Y, where Y is a circle bundle over Σ . We will then proceed in two steps, which we call gliding and descent , as described below. Let For simplicity, we are assuming that Σ is connected. If Σ is not connected, some of the numbers and structuresin the following description, such as the integer d ( s ) , should be defined component-wise. ( X − Σ , s X − Σ ) denote the moduli space of monopoles on X − Σ with finite energy on the end withrespect to the induced spin c structure s X − Σ .. = s | X − Σ on X − Σ . Different spin c structures s and s ′ on T X ( − log Σ) may have the same restriction on X − Σ. Therefore, for such s and s ′ , M ( X − Σ , s X − Σ )will be the same as M ( X − Σ , s ′ X − Σ ). In the following, by looking at the limits of the monopolesover the cylindrical end, we will choose a component of M ( X − Σ , s X − Σ ) corresponding to s .In temporal gauge, a monopole on X − Σ with finite energy on the end is the gradient flow line ofthe Chern-Simons-Dirac function CSD on the cylindrical end (see Section 2.5), where a stationarysolution on the end corresponds to a monopole on Y . Therefore, we have a limiting map ∂ : M ( X − Σ , s X − Σ ) −→ M ( Y, s Y ) , (1.2)where s Y is the induced spin c structure on Y . Meanwhile, as we will see in Section 2.4, an irreduciblemonopole on Y in this setup actually descends to a monopole on Σ , i.e., it will be the pullbackof a solution of SW on Σ via the bundle projection Y −→ Σ . Because of this, the moduli space M ( Y, s Y ) decomposes into a disjoint union of components M ( Y, s Y ) = J ∪ [ m M ( Y, s Y ) m , where J ∼ = Map(Σ , S ) ∼ = T g is the subspace of reducible solutions and the second union is overthe set of integers m = m ( s ) for all spin c structures s on T X ( − log Σ) that restrict to s X − Σ on X − Σ. The integer m ( s ) and the restriction s X − Σ uniquely determine s ; therefore, for each s , wedefine M ( X − Σ , s ) .. = ∂ − (cid:0) M ( Y, s Y ) m ( s ) (cid:1) ⊂ M ( X − Σ , s X − Σ ) . By definition, all the monopoles in M ( X − Σ , s ) are automatically irreducible. We also define M ( X − Σ , s X − Σ , J ) = ∂ − (cid:0) J (cid:1) . Thus, we have a decomposition M ( X − Σ , s X − Σ ) = M ( X − Σ , s X − Σ , J ) ∪ [ s M ( X − Σ , s ) , where the second union runs over all spin c structures s on T X ( − log Σ) that restrict to the fixedspin c structure s X − Σ on X − Σ . A monopole in M ( Y, s Y ) m ( s ) can be identified with an effective divisor of degree d ( s ) > , ora single point if d ( s ) = 0 . Depending on whether m ( s ) < m ( s ) >
0, this divisor is the zero setof a non-zero holomorphic section of E Σ or K Σ ⊗ E ∗ Σ , respectively. For d > , letDiv d (Σ) ∼ = Sym d (Σ)denote the space of effective divisors of degree d on Σ; if d = 0 , this space is taken to be a singlepoint. By the argument above, we obtain a landing map ♭ : M ( Y, s Y ) m ( s ) −→ Div d ( s ) (Σ) This component is empty if Σ · Σ = 0 and deg( L Σ ) = 2 − g, and is equal to T g +1 if Σ · Σ = 0 and deg( L Σ ) = 2 − g .Since the case Σ · Σ = 0 is treated extensively in [MST], throughout the paper, we often implicitly or explicitly assumethat Σ · Σ = 0. For m [1 − g, g − M ( Y, s Y ) m is empty, and for m = 0 , we get the reducible component J instead. Different m ’s among non-trivial components differ by a multiple of Σ · Σ. Y the corresponding effective divisor on Σ of degree d ( s ) onwhich the non-zero spinor lands; this will be the “empty” divisor if d ( s ) = 0. The landing map is adiffeomorphism. Combining the two steps, and using the nomenclature of Gromov-Witten theory,we obtain an evaluation mapev .. = ♭ ◦ ∂ : M ( X − Σ , s ) −→ Div d ( s ) (Σ) . (1.3)In order to make sure that the moduli space is cut transversely and contains no reducible solutions,we need to perturb the SW equations on X − Σ in a way that (1.3) is still defined. This can beachieved by using either a compact perturbation or an adapted perturbation ; see Section 2.5. In thefirst case, the perturbation term is a compactly supported self-adjoint 2-form η o . In the secondcase, restricted to the neck [0 , ∞ ) × Y, the perturbation term is additionally equal to the self-adjointpart of the pullback of a non-trivial 2-form η on Y . Furthermore, following [MST], we considera special type of adapted perturbations η ν , where η is obtained from a holomorphic 1-form ν onΣ; see Lemma 2.4. We denote the resulting moduli spaces by M η o ( X − Σ , s ) and M η ν ( X − Σ , s ) , respectively. The discussion leading to (1.3) readily generalizes to the compactly-perturbed modulispaces M η o ( X − Σ , s ). In the second case, CSD is circle-valued, but (1.3) is still defined and we geta major restriction on its image. Given a non-trivial holomorphic 1-form ν , vanishing onDiv( ν ) = { p , . . . , p g − } ⊂ Σ(counted with multiplicities), let S d ( ν ) ( ∼ = S g − − d ( ν )) denote the set of subsets of Div( ν ) of size d .In the case of M η ν ( X − Σ , s ), J will be empty and (1.3) takes values in S d ( s ) ( ν ). Therefore, M η ν ( X − Σ , s X − Σ ) = [ s M η ν ( X − Σ , s ) . Remark 1.1.
We have assumed g >
0; otherwise, there is no such ν . Furthermore, if g = 0, wehave M η o ( X − Σ , s X − Σ ) = M η o ( X − Σ , s X − Σ , J ). Theorem A.
Let X be a smooth, closed, oriented -manifold and Σ ⊂ M a smooth, closed, orientedsurface with positive genus. If b + X − Σ > , for a generic compact perturbation η o , M η o ( X − Σ , s ) isa smooth orientable (but not necessarily compact) manifold of real dimension d = ( c ( L ) + Σ) − χ ( X ) − σ ( X )4 . (1.4) Furthermore, the evaluation map (1.3) is a smooth submersion.
Unlike the classic case in SW theory, the moduli spaces M η o ( X − Σ , s ) are not necessarily compact.Because of the tunneling phenomenon (see [KM, Sec. 16] or [N, Sec. 4.4.2]), a sequence of monopolesin M η o ( X − Σ , s ) will, after passing to a sub-sequence, “converge” to a finite ordered set of monopoles(called a broken trajectory in [KM]), where the first one is a monopole on X − Σ and the rest arenon-trivial (i.e., non-stationary) monopoles on the cylinder R × Y. Furthermore,(1) the limit at + ∞ of the monopole on X − Σ defined in (1.2) coincides with the limit at −∞ ofthe monopole on the first copy of R × Y , and(2) the limit at + ∞ of the monopole on the i -th copy of R × Y coincides with the limit at −∞ ofthe monopole on the ( i + 1)-th copy of R × Y .5n order to obtain a compact moduli space without boundary, we will consider the monopoleson the cylinder R × Y up to the natural C ∗ -action generated by translation in the R -direction androtation in the Y -direction. We denote the space containing such limits by M η o ( X − Σ , s ). With theexception of taking quotients by C ∗ = R × S instead of just by R , this is the same compactificationconsidered in [KM, OS]. The limiting map (1.2) on M η o ( X − Σ , s ) ⊂ M η o ( X − Σ , s X − Σ )is taken at + ∞ of the last monopole. This compactification is the direct analogue of the relativecompactification of the moduli space of pseudo-holomorphic curves relative to Σ (with the contactorders all equal to one).While we show that the complement M η o ( X − Σ , s ) − M η o ( X − Σ , s )is a finite union of strata of expected codimension at least 2, it is not clear to us if the tunnelingspaces are always manifolds of the expected dimension; see Section 3.5. Therefore, without furtherrestrictions (e.g. as in [OS]), it may not always be the case that, for generic η o , M η o ( X − Σ , s ) is a C -manifold of the expected dimension (1.4). Whenever the latter happens, the relative Seiberg-Witten invariants of ( X, Σ) in the class of a logarithmic spin c structure s are defined by integrationon M η o ( X − Σ , s ) in the following way.Let c ∈ H ( M η o ( X − Σ , s ) , Z )denote the first Chern class of the natural circle bundle on the moduli space as in the classical case(see [M, Sec. 6.7] or [Sal, p. 249, (7.24)]). For Ω ∈ H ∗ (Div d ( s ) (Σ)) satisfyingdeg(Ω) + 2 r = d , define SW X, Σ η o ( s ; Ω) .. = Z M ηo ( X − Σ , s ) c r ∧ ev ∗ Ω = Z M ηo ( X − Σ , s ) c r ∧ ev ∗ Ω . (1.5)Under the assumption that the tunneling spaces have their expected dimensions, the inclusion M η o ( X − Σ , s ) ⊂ M η o ( X − Σ , s ) defines a pseudo-cycle in the sense of [Z]; this justifies the secondequality in (1.5). In this situation, unlike in the definition of the classical SW invariants, thecondition b + X − Σ > X, Σ η o ( s ; − ) is independent of the choice of generic η o and the cylindrical metric, since all the monopoles in M η o ( X − Σ , s ) are automatically irreducibleby definition. We will elaborate more on this point in Section 3.6. Therefore, we drop η o fromthe notation and denote the invariants by SW X, Σ ( s ; − ) . When X is symplectic, we believe that aspecial case of (1.5), described below, is equal to certain relative Gromov-Witten invariant in thesense of Taubes’ correspondence.Given a spin c structure s on T X ( − log Σ) such that d ( s ) ≥ t = { t , . . . , t k } be a (possibly empty) partition of d ( s ) into k = k ( t ) positive integers, i.e., d ( s ) = t + · · · + t k . (1.6)Let Div t (Σ) = n k X i =1 t i p i : p i ∈ Σ ∀ i = 1 , . . . , k o ⊂ Div d ( s ) (Σ)6enote the subspace of effective divisors that can be written as a sum of k (not necessarily distinct)points { p i } ki =1 with multiplicities { t i } ki =1 . If we sort the numbers in (1.6) so that t = · · · = t i < t i +1 = · · · = t i + i < · · · < t i + ··· + i n − +1 = · · · = t i + ··· + i n , k = i + · · · + i n , then Div t (Σ) ∼ = Sym i (Σ) × · · · × Sym i n (Σ) . For Ω = PD(Div t (Σ)) in (1.5), we define a particular type of relative SW invariants bySW X, Σ ( s ; t ) .. = SW X, Σ (cid:0) s ; PD(Div t (Σ)) (cid:1) . If X is symplectic, as mentioned earlier, we can identify the set of spin c structures s on T X ( − log Σ)with the set of homology classes ß ∈ H ( X, Z ). Under this identification, (1.4) simplifies to d = ß · ß − K X · ßand t is a partition of d ( s ) = Σ · ß (or Σ · ( K Σ − ß)) into a sum t + · · · + t k of positive integers.Then we believe that, similar to Taubes’ GW-SW correspondence theorem, SW X, Σ (ß; t ) is equalto certain relative Gromov-Witten invariant GW X, Σ (ß; t ); it is a count of (possibly disconnected) J -holomorphic curves of degree ß that intersect Σ in k points with tangency orders t = { t , . . . , t k } and pass through r generic points in X .In order to resolve the issues above regarding regularity of tunneling spaces and non-compactness,we may consider adapted perturbations η ν corresponding to holomorphic 1-forms ν on Σ. In thiscase, we show that the tunneling spaces are empty. Theorem B.
Let X be a smooth, closed, oriented 4-manifold and Σ ⊂ M a smooth, closed, orientedsurface with positive genus. If b + X − Σ > , for every ν = 0 and generic adapted perturbation η ν , M η ν ( X − Σ , s ) is a smooth closed manifold of real dimension e d = ( c ( L ) + Σ) − χ ( X ) − σ ( X )4 − d ( s ) . (1.7) For different such η ν and η ν ′ , M η ν ( X − Σ , s ) and M η ν ′ ( X − Σ , s ) are smoothly cobordant. In this case, for each point q ∈ S d ( s ) ( ν ), if e d is even, we define g SW X, Σ ν ( s ; q ) .. = Z ev − ( q ) ⊂M ην ( X − Σ , s ) c e d / (1.8)By the second statement of Theorem B, the right-hand side (1.8) is independent of the choice ofthe compactly-supported part of η ν , which justifies the notation on the left-hand side. Recall that,by definition, the elements of M η ν ( X − Σ , s ) are automatically irreducible. Furthermore, the finitesum g SW X, Σ ( s ) = X q ∈ S d ( s ) ( ν ) g SW X, Σ ν ( s ; q )is independent of the choice of ν. The dimension formula (1.7) differs from (1.4) by a term d( s ). Particularly, if X is symplectic,identifying the set of spin c structures s on T X ( − log Σ) with the set of homology classes ß ∈ H ( X, Z ) again, (1.7) simplifies to e d = ß · ß − ( K X + Σ) · ß . K X + Σ is the logarithmic canonical line bundle of ( X, Σ). The integer e d is even iff d ( s ) = Σ · ß is even. It is not clear to us if g SW is equal to certain count of J -holomorphic curves.Next, we prove that there is a SW sum formula that relates the SW invariant of a sum X = X Σ X of two closed oriented 4-manifolds X and X along a common oriented surface Σ with dual self-intersections to the relative SW invariants of ( X , Σ) and ( X , Σ) . Here, we only need to considerspin c structures on X that are of “pullback type” on the separating hypersurface Y of this summanifold. Recall that the SW invariants of the connected sum X of two closed oriented 4-manifolds X and X with b +2 ( X i ) > X = X Σ X and s , s ′ are two spin c structures on X . We say s is equivalent to s ′ if bothrestrict to the same spin c structures on X − Σ and X − Σ. We denote the equivalence class of s by [ s ]. Theorem C.
Suppose X = X Σ X and s is a spin c structure on X that is of pullback type onthe separating hypersurface Y . For each holomorphic -form ν = 0 we have the sum formula X s ′ ∈ [ s ] SW X ( s ′ ) = X [ s ]= s s X q ∈ S d ( s ) ( ν ) ε s , s ,q g SW X , Σ ν ( s ; q ) · g SW X , Σ ν ( s ; q ) . (1.9)In (1.9), ε s , s ,q ∈ {± } depend on the choice of orientations. Whenever the relative invariants (1.5)are defined, we get another sum formula X s ′ ∈ [ s ] SW X ( s ′ ) = X [ s ]= s s ε s , s X Ω SW X , Σ ( s ; Ω) · SW X , Σ ( s ; Ω ∗ ) (1.10)where Ω runs over the terms in the K¨unneth decomposition P Ω ⊗ Ω ∗ of the diagonal in Div(Σ) × Div(Σ). Under the GW-SW correspondence, the SW sum formula (1.10) should correspond to thewell-known GW sum formula, with X being the symplectic sum of X and X along Σ. Remark 1.2. If s is a spin c structure on X that is not of pullback type on the separating hyper-surface Y , then it follows from Fact 1 in [N1, p. 94] and the same convergence argument used inthe proof of Theorem C that all the terms in (1.10) are zero. Remark 1.3.
The issue that the sum s s of two spin c structures on X and X is only well-defined up to the equivalence relation defined before Theorem C also appears in the GW sumformula: two homology classes A ∈ H ( X , Z ) and A ∈ H ( X , Z ) that have the same intersectionnumber with Σ can be glued together to produce a homology class A ∈ H ( X, Z ); however, A isonly well-defined up to addition with homology classes associated to certain elements of H (Σ , Z ),known as rim tori. A natural question is whether we can refine the sum formulas (1.9) and (1.10)to express each individual SW X ( s ′ ) in terms of some “refined relative invariants” of X and X . Inthe context of GW sum formula, this problem is extensively studied in [FZ1, FZ2]. c structures Given an oriented riemannian rank 2 n (real) vector bundle V −→ X over a smooth manifold X, aspin c structure s = ( S , ρ ) on V consists of a rank 2 n (complex) hermitian vector bundle S −→ X, called the spinor bundle, and a Clifford multiplication, which is a linear map of vector bundles ρ : V −→ End( S ) , V x ∋ v −→ ρ ( v ) ∈ End( S | x ) , ρ ( v ) ∗ + ρ ( v ) = 0 , ρ ( v ) ∗ ρ ( v ) = | v | id , ∀ v ∈ V. (2.1)We sometimes denote the action of ρ ( v ) on ψ ∈ S by v · ψ when there is no chance of confusion.Every such S admits a canonical splitting into rank 2 n − hermitian vector bundles, S = S + ⊕ S − ,satisfying ρ ( v ) S ± | x = S ∓ | x , ∀ x ∈ X, v ∈ V x . In other words, we have ρ ( v ) = (cid:20) γ ( v ) − γ ( v ) ∗ (cid:21) , γ : V −→ Hom( S − , S + ) , where γ ( v ) ∗ γ ( v ) = | v | id . A spin c isomorphism from ( S , ρ ) to ( S , ρ ) is a unitary bundle isomorphism f : S −→ S whichinduces a bundle isomorphism End( f ) : End( S ) −→ End( S ) such that End( f ) ◦ ρ = ρ . Denoteby S c ( V ) the set of isomorphism classes of spin c structures on V . Given a riemannian 2 n -manifold X, let S c ( X ) be the set of isomorphism classes of spin c structures on T X.
This is in fact the set ofprincipal spin c bundles lifting the principal tangent bundle of X up to bundle isomorphism.If ( S , ρ ) is a spin c structure on V and n > , thendet( S + ) = det( S − ) = L ⊗ n − s for a unique complex line bundle L s , called the characteristic line bundle of the spin c structure,for which c ( L s ) is an integral lift of w ( V ) . In fact, L s is the determinant line bundle of theprincipal Spin c (2 n )-bundle which lifts the principal SO(2 n )-bundle associated to V −→ X. Wedefine c ( s ) .. = c ( L s ) . Similarly, we say that the spin c structure s is torsion if c ( L s ) is torsion.When there is no chance of confusion, we will drop the subscript s and simply write L. Given a spin c structure s = ( S , ρ ), every other spin c structure s = ( S , ρ ) ∈ S c ( V ) has the form S ∼ = S ⊗ E, ρ = ρ ⊗ id , where E is a hermitian line bundle. We express the above relation between s and s by writing s = s ⊗ E. Note that L s = L s ⊗ E ⊗ and the two spin c structures s and s are isomorphic if andonly if c ( E ) = 0 . Every complex vector bundle (
V, J ) equipped with a hermitian metric h , i admits a canonical spin c structure s can = ( S can , ρ can ) with S +can = Λ , even V ∗ S − can = Λ , odd V ∗ ,ρ can ( v ) α = −√ ι v α + 1 √ h· , v i ∧ α. (2.2)As a result, the correspondence s can ⊗ E ↔ E determines a canonical bijection between isomorphismclasses of spin c structures and isomorphism classes of complex line bundles over X. Specifically, every almost-complex manifold ( X n , J ) admits a canonical spin c structure s can with S +can = Λ , even T ∗ X, S − can = Λ , odd T ∗ X, L s can = K ∗ X ,ρ can ( v ) α = −√ ι v α + 1 √ h· , v i ∧ α.
9n particular, if (
X, ω X ) is a symplectic manifold, we can choose an ω X -compatible (or ω X -tame)almost-complex structure J and define s can as above. In this case, K X = Λ n, T ∗ X is the canonicalbundle of X with respect to ω X and h· , v i = ω X ( · , i v + J v ) is the (0 , i v + J v ∈ T , X. If J is ω X -tame but not ω X -compatible, then one has to replace ω X in the definition of h· , v i with e ω X ( u, v ) = 12 (cid:0) ω X ( u, v ) + ω X ( J u, J v ) (cid:1) . Let g be a riemannian metric on X n , ∇ the corresponding Levi-Civita connection, and s ∈ S c ( X ) . A hermitian connection e ∇ on S is called a spin c connection if it is compatible with ∇ and theClifford multiplication e ∇ v ( w · Φ) = w · e ∇ v Φ + ( ∇ v w ) · Φ , ∀ Φ ∈ Γ( S ) , v, w ∈ Γ( T X ) . Every spin c connection preserves S ± . Every two such connections differ by an imaginary-valued 1-form α ∈ Ω ( X, i R ) . Moreover, the gauge group G = Maps( X, S ) acts on the space of connectionsby ( u ∗ e ∇ )Φ = u − e ∇ ( u Φ) = d uu ⊗ Φ + e ∇ Φfor Φ ∈ Γ( S ) and u ∈ Maps(
X, S ) . Every spin c connection e ∇ on the spinor bundle S is uniquely determined by ∇ and a connection A on the characteristic line bundle L of the spin c structure. This is essentially due to the fact that theunderlying principal bundle of S (with fiber Spin c (2 n )) lifts the principal tangent bundle of X (withfiber SO(2 n )) as a circle-bundle extension (corresponding to the line bundle L ). This correspondsto the lifting of w ( V ) to the integral class c ( L ); see [M, Chap. 3]. The space of connections on thecharacteristic line bundle L, denoted by A ( L ) , is an affine space with tangent space Ω ( X, i R ) . If A ∈ A ( L ) and a ∈ Ω ( X, i R ) , then the curvature F A ∈ Ω ( X, i R ) and F A + a = F A +d a. The connection A on L induces a spin c connection e ∇ = ∇ A on S and their curvatures are related by [Sal, Sec. 6.1] F A ( v, w ) = 12 n tr (cid:0) F ∇ A ( v, w ) (cid:1) . The gauge group G = Maps( X, S ) acts on A ( L ) by u ∗ A = d uu + A and leaves F A invariant. The operator /∂ A : Γ( S ± ) −→ Γ( S ∓ ) , /∂ A Φ = n X i =1 e i · e ∇ e i Φ , where e , . . . , e n form an orthonormal frame for T X, is the Dirac operator associated to A ∈ A ( L ).It is a self-adjoint operator independent of the particular choice of e , . . . , e n . If (
X, ω X , J ) is K¨ahler, thenΓ( S +can ⊗ E ) = Ω , even ( X, E ) , Γ( S − can ⊗ E ) = Ω , odd ( X, E ) , and 1 √ /∂ A = ¯ ∂ A E + ¯ ∂ ∗ A E : Ω , even ( X, E ) −→ Ω , odd ( X, E ) , (2.3)10here A E is a hermitian connection on E, ¯ ∂ A E = ∇ , A E is the ¯ ∂ -operator associated to A , and A is the induced connection on the characteristic line bundle L of the spin c structure. For example,in dimension 4, S − can = Λ , T ∗ X, det( S − can ) = K ∗ X and L = det( S − ) = det( S − can ⊗ E ) = K ∗ X ⊗ E . Therefore, any hermitian connection A on L is equivalent to a hermitian connection A E on E andwe have A = A can ⊗ A E , where A can is the holomorphic hermitian connection on K ∗ X . The identity (2.3) continues to hold in the symplectic case [Sal, Thm. 6.17]. If (
X, ω X ) is asymplectic manifold, J is compatible with ω X (resp. tames ω X ), and ∇ is the Levi-Civita connectionof the metric g = ω X ( · , J · ) (resp. g = e ω X ( · , J · )), then ∇ does not preserve Ω ,k ( X ) unless ( X, ω X , J )is K¨ahler, i.e., ∇ J = 0. However, there is a canonical hermitian connection on T X, defined by e ∇ v w .. = ∇ v w − J ( ∇ v J ) w, which gives rise to a hermitian connection on S can = Λ , ∗ T ∗ X, compatible with the Clifford multi-plication but not with ∇ . However, it is possible to modify e ∇ further to produce a spin c connection b ∇ , compatible with both the Clifford multiplication and the Levi-Civita connection ∇ b ∇ v Φ .. = e ∇ v Φ + 12 µ ( J ∇ v J )Φ , where Φ ∈ Γ( S can ) = Ω , ∗ ( X ) and µ : so ( T X ) −→ End( S can ) is characterized by[ µ ( A ) , ρ can ( v )] = ρ can ( Av ) . When X is K¨ahler, the connections b ∇ and e ∇ coincide with the Levi-Civita connection ∇ on forms[Sal, pp. 198–199]. Let X be a smooth closed connected oriented riemannian 4-manifold (with metric g) and s = ( S , ρ )a spin c structure on X. The (unperturbed) Seiberg-Witten monopole equations are a system offirst order differential equations for a pair ( A, Φ) in the configuration space C ( X, s ) = A ( L ) × Γ( S + ) , where A is a connection on the characteristic line bundle of s and Φ is a plus-spinor. The spaces A ( L ) and Γ( S + ) are completed with respect to appropriate Sobolev norms, so that we will beworking in the context of Banach spaces; see [M]. The Seiberg-Witten equations read F + A = (ΦΦ ∗ ) , /∂ A Φ = 0 , (SW )where (ΦΦ ∗ ) ∈ Γ(End ( S + )) , defined by(ΦΦ ∗ ) ( w ) = h Φ , w i Φ − | Φ | w ∀ w ∈ Γ( S + ) , is the trace-less part of ΦΦ ∗ ∈ Γ(End( S + )) and the two sides of the curvature equation in SW areidentified via the bundle isomorphismΛ , + T ∗ X ⊗ C −→ End ( S + ) , X i
12f (
X, ω X , J ) is K¨ahler, using (2.3), the SW η equations for the pair ( A, Φ) take the form2(2 F A E − η ) , = Φ Φ , i ( F A can + 2 F + A E − η ) , = ( | Φ | − | Φ | ) ω X , ¯ ∂ A E Φ + ¯ ∂ ∗ A E Φ = 0 , where A = A can ⊗ A E , Φ = (Φ , Φ ) ∈ Ω , ( X, E ) × Ω , ( X, E ) . If η ∈ Ω , ∩ Ω , + , then either Φ = 0 or Φ = 0. The latter will happen if c ( E ) · ω X < c ( K X ) · ω X , or η has a large (positive) multiple of − i ω X ; see [Sal, Sec. 12.2]. Recall that c ( K X ) · ω X ≥ b + > . By adapting the arguments in [M, Sec. 7.2], we obtain aholomorphic description of the moduli space of η -monopoles in terms of holomorphic structureson E (or, equivalently, on the characteristic line bundle L of the spin c structure) and non-zeroholomorphic sections of E or K X ⊗ E ∗ (up to constant scalar multiples), depending on the case.If Φ = 0, the unperturbed SW equations reduce to the vortex equations F , A E = 0 , i ( F A can + 2 F A E ) , = −| Φ | ω X , ¯ ∂ A E Φ = 0 . Similar results hold in the symplectic case. In the almost-complex case, the SW η equations for thepair ( A can ⊗ A E , Φ) take the form( F A can + 2 F A E ) + − η = q (Φ) , ¯ ∂ A E Φ + ¯ ∂ ∗ A E Φ = 0 . In this and the following sections, suppose (Σ , j , ω ) is a closed Riemann surface of genus g, equippedwith a complex structure j and a K¨ahler form ω. As a K¨ahler manifold of complex dimension one(real dimension two), Σ carries a canonical spin c structure ( U can , ρ can ) , where U can is a hermitianvector bundle of rank 2, and any other spin c structure ( U , ρ ) on Σ is obtained by twisting thiscanonical spin c structure by a hermitian line bundle E. The characteristic line bundle is L = det( U )and the spinor bundle U = U + ⊕ U − splits as a direct sum of two complex hermitian line bundles.We have U +can = C Σ , U − can = K ∗ Σ , and L can = K ∗ Σ , where K Σ = Λ , T ∗ Σ is the canonical line bundle of Σ and C Σ is the trivial line bundle, and U + = E, U − = K ∗ Σ ⊗ E, and L = K ∗ Σ ⊗ E . When there is a chance of confusion, we may add a subscript and, for example, denote the spinorbundles U ± or the line bundle L by U ± E or L E , respectively. Remark 2.1.
It is also possible to define a canonical spin structure on Σ by simply lifting itsprincipal SO(2)-tangent-bundle to a Spin(2)-bundle as a non-trivial double cover. One can thentensor with C to obtain a spin c structure on Σ . This is the approach taken in [N1, p. 93]. Thespinor bundles obtained in this case are U − = K − / Σ and U + = K / Σ and the characteristic linebundle is trivial. This complexification of the canonical spin structure is obviously different fromour canonical spin c structure above, which is induced by the (almost-) complex structure on Σ , butit can easily be offset by an additional twist by the line bundle K − / Σ = p K ∗ Σ . We have chosenhere to work with the canonical spin c structure, rather than the canonical spin structure, becauseof its compatibility with the overall framework of complex structures.13ow, consider the set of pairs ( A, Ψ) ∈ A ( L ) × Γ( U ) satisfying the equations F A = | Ψ + | − | Ψ − | i ω, /∂ A Ψ = 0 , (SW )where Ψ = (Ψ + , Ψ − ) is a section of U = U + ⊕ U − . The set of solutions to these equations isinvariant under the action of the gauge group G = Maps(Σ , S ) on A ( L ) × Γ( U ) . As before, we calla solution reducible if the spinor Ψ is identically zero.If A can is the holomorphic hermitian connection on L can = K ∗ Σ , then any connection on L = K ∗ Σ ⊗ E is of the form A = A can ⊗ A E , where A E ∈ A ( E ) , and the corresponding Dirac operator /∂ A onsections of U = U + ⊕ U − takes the form1 √ /∂ A = (cid:20) ∂ ∗ A E ¯ ∂ A E (cid:21) . The Dirac equation then splits into two equations ¯ ∂ A E Ψ + = 0 and ¯ ∂ ∗ A E Ψ − = 0, and the curvatureequation turns into F A can + 2 F A E = | Ψ + | − | Ψ − | i ω. Since ¯ ∂ A E ¯ ∂ A E = 0 and ¯ ∂ A E Ψ + = 0 , we conclude that A E induces a holomorphic structure on E forwhich Ψ + is a holomorphic section. Similarly, using the involution on the spin c structures inducedby complex conjugation, which sends the characteristic line bundle to its inverse, we can see thatthe equation ¯ ∂ ∗ A E Ψ − = 0 implies that Ψ − is a holomorphic section of K Σ ⊗ E ∗ , the Serre dual of E. Since line bundles of negative degree can not have non-zero holomorphic sections, we conclude thatone of Ψ + or Ψ − is identically zero, unless 0 ≤ d = deg( E ) ≤ g − . We will also be interested in another variant of the Seiberg-Witten equations F A = | Ψ + | − | Ψ − | i ω, /∂ A Ψ = 0 , and Ψ + Ψ − = i ν, (SW ν )where ν is a fixed holomorphic 1-form. Notice that Ψ + Ψ − , as a section of U + ⊗ ( U − ) ∗ = K Σ ,identifies with a holomorphic 1-form on Σ . This variant of the SW equations is closely related tothe perturbed SW equations in higher dimensions; see Lemma 2.4.
Similarly to 2-manifolds, for a closed 3-manifold Y, a spin c structure s = ( S , ρ ) consists of a rank2 hermitian bundle S and a Clifford multiplication map ρ : T Y −→ End( S ) satisfying (2.1), whichresult from a principal Spin c (3)-lifting of the principal frame bundle of T Y.
Note that, unlikedimensions 2 and 4, the spinor bundle S does not decompose into plus- and minus-spinor bundles.We are interested in the case where Y is a circle bundle over a Riemann surface Σ . In this subsection,to simplify notations, we will temporarily use the shorthand π to denote the bundle projection π = π Y, Σ : Y −→ Σ . Viewing Y as a principal U (1)-bundle, a choice of a principal U (1)-connection gives rise to adecomposition T Y ∼ = T ver Y ⊕ T hor Y, (2.6)14here the vertical summand is tangent to the S fibers ker(d π ) and the horizontal summand T hor Y ∼ = π ∗ T Σ is a lifting of the tangent space of Σ which is equivariant under the U (1) action. Wewill denote this connection by i α, where α is a U (1)-invariant real 1-form on Y whose restrictionto each fiber is d θ. Under such a decomposition, Y admits a canonical spin c structure ( S can , ρ can ) , where S can = C Y ⊕ π ∗ ( K ∗ Σ )is the pullback of the canonical spinor bundle U can = C Σ ⊕ K ∗ Σ on Σ , the Clifford map ρ can on T ver Y sends the generator ∂ θ to (cid:20) i − i (cid:21) , and ρ can on T hor Y is the pullback of the canonical ρ Σ , can as in (2.2). The isomorphism class of thisspin c structure is independent of the choice of the principal U (1)-connection and lifts the canonicalSpin c (2) structure on Σ to the canonical Spin c (3) structure on Y. Every other spin c structure on Y is obtained by tensoring with a complex line bundle on Y. Remark 2.2.
By the Gysin long exact sequence H (Σ , Z ) −→ H ( Y, Z ) −→ H (Σ , Z ) −→ , we see that, unless Σ = S , there are complex line bundles on Y that are not the pullback of acomplex line bundle on Σ. In fact, the Gysin exact sequence implies that if the Euler number ℓ ofthe circle-bundle Y −→ Σ is non-zero, then H ( Y, Z ) ∼ = Z g ⊕ (cid:0) Z /ℓ Z (cid:1) , where the torsion part is generated by the pullback π ∗ ω of the volume form on Σ . Therefore, theSW theory on Y involves a larger class of spin c structures than those on Σ . In the setup that wewill be developing for relative SW theory in Section 3.1, we will be working with complex linebundles that are defined over the entire X . Therefore we will be dealing only with complex linebundles on Y that are pullbacks of those on Σ . Unless the circle bundle Y −→ Σ is trivial, theseline bundles on Y will always be torsion. Moreover, it follows from [N1, p. 94, Fact 1] that spin c structures on Y that are not of pullback type result in trivial relative invariants, as we will see inSection 3.1. Remark 2.3.
If the degree ℓ = deg( Y ) of the U (1)-bundle Y −→ Σ is not zero and E and E ′ aretwo complex line bundles on Σ, then π ∗ ( E ) ∼ = π ∗ ( E ′ ) if and only if deg( E ) ≡ deg( E ′ ) modulo ℓ. Unless deg( E ) = deg( E ′ ), such an isomorphism does not extend to the disk bundle D over Σ whichhas Y as its boundary, where D is defined using the same U (1)-cocycles of Y −→ Σ , only with circlefibers replaced by unit disks.The general setup of Seiberg-Witten equations in dimension 3 is in most respects analogous tothose in dimensions 2 and 4. For a closed, oriented, riemannian 3-manifold Y, equipped with aspin c structure s = ( S , ρ ) , consider the following (perturbed) Seiberg-Witten equations for a pair15 B, Ψ) , consisting of a connection on the characteristic line bundle L = det( S ) and a section of thespinor bundle S F B − η = (ΨΨ ∗ ) , /∂ B Ψ = 0 , (SW η )where η ∈ Ω ( Y, i R ) is a closed 2-form and the two sides of the curvature equation are identified viathe isomorphismΛ T ∗ Y ⊗ C ∼ = −→ End ( S ) , X i 12 ( F α ,α,t − i F α ,α,t ) + β (0 , ,t (cid:1) α , − i π ∗ ν = Ψ + ( t )Ψ − ( t ) , (3.37)where α , is defined as in (2.13) and F B ( t ) = F α ,α ,t α ∧ α + F α ,α,t α ∧ α + F α ,α,t α ∧ α,β ( t ) = β α,t α + β (0 , ,t α , − β (0 , ,t α , . Applying ¯ ∂ B ( t ) to the second equation in (3.34) we get − ¯ ∂ B ( t ) ∇ ∂ t + i ∂ θ Ψ + ( t ) + √ ∂ B ( t ) ¯ ∂ ∗ B ( t ) Ψ − ( t ) = 0 . (3.38)Commuting the differential operators ∇ ∂ t + i ∂ θ and ¯ ∂ B ( t ) produces a curvature term ∇ ∂ t + i ∂ θ ¯ ∂ B ( t ) − ¯ ∂ B ( t ) ∇ ∂ t + i ∂ θ = (cid:0) β (0 , ,t + ( F α ,α,t − i F α ,α,t ) (cid:1) α , Note that λ = 0 because we are using the adiabatic connection on Y . −∇ ∂ t + i ∂ θ ¯ ∂ B ( t ) Ψ + + 2 (cid:0) Ψ + ( t )Ψ − ( t ) + i π ∗ ν (cid:1) Ψ + ( t ) + √ ∂ B ( t ) ¯ ∂ ∗ B ( t ) Ψ − ( t ) = 0 . Applying the first equation in (3.34) to the first term and dividing by √ − ∇ ∂ t + i ∂ θ ∇ ∂ t − i ∂ θ Ψ − ( t ) + √ (cid:0) Ψ + ( t )Ψ − ( t ) + i π ∗ ν (cid:1) Ψ + ( t ) + ¯ ∂ B ( t ) ¯ ∂ ∗ B ( t ) Ψ − ( t ) = 0 . (3.39)Similarly, we get − ∇ ∂ t − i ∂ θ ∇ ∂ t + i ∂ θ Ψ + ( t ) + √ (cid:0) Ψ + ( t )Ψ − ( t ) − i π ∗ ν (cid:1) Ψ − ( t ) + ¯ ∂ ∗ B ( t ) ¯ ∂ B ( t ) Ψ + ( t ) = 0 . (3.40)Define ¯ ∂ ver = ∇ ∂ t + i ∂ θ Note that the first terms in (3.39) and (3.40) are12 ¯ ∂ ver ¯ ∂ ver ∗ and 12 ¯ ∂ ver ∗ ¯ ∂ ver , respectively. Also note that the ¯ ∂ -operator associated to the connection A over the entire complexmanifold R × Y is¯ ∂ A = (cid:0) (d t − i α ) ⊗ ∇ ∂ t + i ∂ θ (cid:1) + ¯ ∂ B = (cid:0) (d t − i α ) ⊗ ∇ ∂ t + i ∂ θ (cid:1) + (cid:0) α , ⊗ ∇ ζ + i ζ (cid:1) : Ω ( O (ß)) −→ Ω , (cid:0) O (ß) (cid:1) . Therefore, if Ψ + satisfies ¯ ∂ ver Ψ + = 0 and ¯ ∂ B Ψ + = 0 , then it satisfies ¯ ∂ A Ψ + = 0. We have a similarstatement for Ψ − . As in the proof of Lemma 2.4, we want to take the inner product of, say (3.40),with Φ + over the entire R × Y and conclude that¯ ∂ ver Φ + = 0 , ¯ ∂ A Φ + = 0 , and Φ + Φ − = i π ∗ ν. (3.41)Towards this goal, we will work over [ − T , T ] × Y and then let T i −→ ∞ . To find the boundaryterms, consider the complex (2 , − i (cid:10) Φ + , ¯ ∂ ver Φ + (cid:11) (d t + i α ) ∧ π ∗ ω. Since R × Y is a holomorphic manifold, we havedΩ = ¯ ∂ Ω = 2 (cid:16) (cid:10) ¯ ∂ ver Φ + , ¯ ∂ ver Φ + (cid:11) − (cid:10) Φ + , ¯ ∂ ver ∗ ¯ ∂ ver Φ + (cid:11) (cid:17) d t ∧ dvol Y . By the same reasoning as in (2.18), and integration by parts corresponding to the 3-form above,we get0 = Z [ − T ,T ] × Y (cid:28) Φ + , 12 ¯ ∂ ver ∗ ¯ ∂ ver Φ + + ¯ ∂ ∗ B ( t ) ¯ ∂ B ( t ) Φ + + √ + Φ − − i π ∗ ν )Φ − (cid:29) d t ∧ dvol Y = Z [ − T ,T ] × Y (cid:16) | ¯ ∂ ver Φ + | + | ¯ ∂ B ( t ) Φ + | + √ | Φ + Φ − − i π ∗ ν | (cid:17) d t ∧ dvol Y − Z { T }× Y (cid:10) Φ + , ¯ ∂ ver Φ + (cid:11) t = T dvol Y + 14 Z {− T }× Y (cid:10) Φ + , ¯ ∂ ver Φ + (cid:11) t = − T dvol Y . (3.42)39he integral in the second line is non-negative and is an increasing function of each T i . The integralsin the third line converge to 0 as T i −→ ∞ . Thus the integral in the second line must be zero. Weconclude that ¯ ∂ ver Φ + = 0 , ¯ ∂ B ( t ) Φ + = 0 , and Φ + Φ − = i π ∗ ν, from which (3.41) follows. Moreover,the curvature of A will be a (1 , A defines a holomorphic structure. Notemeanwhile that the boundary terms in the third line turn out to be zero, because ¯ ∂ ver Φ + = 0. Corollary 3.9. Unless b − = b + , the moduli space M η ν ( P − Σ , s ) is empty. If b + = b − = b, whichhappens when a = 0 , then M η ν ( P − Σ , s ) ∼ = S b ( ν ) consists of fiber-wise constant solutions.Proof. The zero sets of Φ + and Φ − represent the homology classes[ a Σ − + b + F ] and [ − a Σ − + (2 g − − b + ) F ] , respectively. Therefore, we must have a = 0. If a = 0, the restrictions of Φ + and Φ − to each fiberof P are constant sections. Therefore, both are pullbacks from Σ. Remark 3.10. In the case of M ( P − Σ , s ), if a > 0, then Φ − = 0 and if a < 0, then Φ + = 0.Furthermore, if a > 0, then 0 ≤ b ± ≤ g − 1, and if a < 0, then g − ≤ b ± ≤ g − As we mentioned in the introduction and Sec. 2.5, the moduli spaces M ∗ ( X − Σ , s ) (where ∗ meanseither a compact perturbation or an adapted perturbation) are not necessarily compact. Becauseof the tunneling phenomenon (see [KM, Sec. 16] or [N, Sec. 4.4.2]), a sequence of monopoles in M ( X − Σ , s ) will, after passing to a sub-sequence, “converge” to a finite ordered set of monopoleswhere the first one is a monopole on X − Σ and the rest are non-trivial monopoles on the cylinder R × Y . For an adapted perturbation η ν , with ν = 0 , M η ν ( X − Σ , s ) is compact by Corollary 3.9.For a compact perturbation η o , however, the relative moduli spaces M η o ( X − Σ , s ) have the sameassociated tunneling spaces as in the unperturbed case and these tunneling spaces can be non-trivial.In the following, we will assume that Σ · Σ = 0 to ensure that the spin c structure on Y is torsion andthus CSD is real-valued and increasing along the flow lines on M ( Y, s Y ) , as discussed in Sec. 2.5.The case of Σ · Σ = 0 is slightly different and simpler and is already discussed in [MST].Define M η o ( X − Σ , s ) to be the space of all level- k broken trajectories for s in the following sense.A level- k element of M η o ( X − Σ , s ) is a tuple (cid:16) [ A , Φ ]; J A , Φ K ; . . . ; J A k , Φ k K (cid:17) (3.43)in the fiber product M η o ( X − Σ , s X − Σ ) ev × ev (cid:16) M ( R × Y, s R × Y ) / C ∗ (cid:17) . . . ev × ev (cid:16) M ( R × Y, s R × Y ) / C ∗ (cid:17) , (3.44)such that(1) the C ∗ -action on M ( R × Y, s R × Y ) is given by the translation symmetry on R and S -rotationsymmetry on Y ; 402) the evaluation map ev at the ends takes value in a divisor space Div m (Σ) as in (1.3), if thelimit is irreducible, or in J ∼ = T g (or T g +1 ), if the limit is reducible;(3) the limit at + ∞ of the last monopole belongs to Div d ( s ) (Σ).It is known that M η o ( X − Σ , s ) has a natural sequential convergence topology that is Hausdorff andcompact (e.g. see [KM, Sec. 16]). Compared to the compactification in [KM], the only differencehere is that we take the quotient by the action of C ∗ instead of R , therefore our compactificationhas no boundary.For a broken trajectory as in (3.43), let C i denote the component of the limit at + ∞ of [ A i , Φ i ]for i ∈ { , . . . , k } . Therefore, the limit at −∞ of [ A i , Φ i ] belongs to C i − for i ∈ { , . . . , k } and C k = Div d ( s ) (Σ). The C i ’s are in fact components of the decomposition M ( Y, s Y ) = J ∪ [ m M ( Y, s Y ) m . (3.45)Using the notation of Sec. 2.5, for i ∈ { , . . . , k } , each J A i , Φ i K is an element of M ( C i − , C i ) / C ∗ = ˘ M ( C i − , C i ) /S . Recall from Sec. 2.5 that the components C , . . . , C k are ordered by the value of CSD in the sensethat CSD( C ) < CSD( C ) < · · · < CSD( C k ) . If none of these limits belongs to J , it follows from Lemma 3.2 that the expected dimension ofsuch level- k configurations is 2 k lower than the expected dimension of M η o ( X − Σ , s ).On the other hand, depending on sign(Σ · Σ), the reducible component J appears in two ways: • If Σ · Σ < 0, then CSD takes its minimum on J ; • If Σ · Σ > 0, then CSD takes its maximum on J .To see this, first note that we can assume CSD( J ) = 0 by taking our background connection tobe a flat connection on L Y . To evaluate CSD at an irreducible solution ( B, Ψ), recall that it isthe pullback of a monopole on Σ , so the holonomy of B is exp(2 π i c/ℓ ) , where c = deg( L Σ ) and ℓ = deg( Y ) = − Σ . Σ . Therefore, for some flat connection B on L Y , we have B − B = ( c/ℓ ) i α, where i α is the connection 1-form associated to decomposition (2.6). Since i π d( i α ) and ℓ.π ∗ ω bothrepresent the pullback of the (integral) Euler class of Y, a straight-forward calculation shows thatCSD( B, Ψ) is a positive multiple of c /ℓ and the two items above follow. As a result, • If Σ · Σ < 0, then only C can be equal to J ; • If Σ · Σ > 0, then none of C , . . . , C k is equal to J .In the first case, where C = J , the expected dimension of level- k configurations is 2 k + 1 lowerthan the expected dimension of M η o ( X − Σ , s ) by lemmas 3.2 and 3.3. To summarize, we have thefollowing theorem. Theorem 3.11. The moduli space M η o ( X − Σ , s ) naturally has a Hausdorff compactification M η o ( X − Σ , s ) with a “boundary” of expected real codimension at least two. The moduli space M η ν ( X − Σ , s ) is compact. emark 3.12. It may happen that (3.45) has only one irreducible component for topologicalreasons. If further Σ · Σ > 0, this implies that M η o ( X − Σ , s ) is compact. For example, ifΣ · Σ > g − 2, since different m ’s in (3.45) differ by a multiple of Σ · Σ and M ( Y, s Y ) m is emptyunless | m | ≤ g − , we conclude that M η o ( X − Σ , s ) is compact. This point has been used in [OS]to show that certain decompositions of 4-manifolds cannot happen. Remark 3.13. Subject to the regularity of the tunneling spaces, we get a gluing map M η o ( X − Σ , s X − Σ ) ev × ev (cid:16) M ( R × Y, s R × Y ) / C ∗ (cid:17) . . . ev × ev (cid:16) M ( R × Y, s R × Y ) / C ∗ (cid:17) × ( S ) k ∆ k −→ M η o ( X − Σ , s X − Σ )as in (2.27), where ∆ ⊂ C is a sufficiently small disk and the fiber product with the i -th copy of∆ is with respect to the S -actions on ∆ and the i -th tunneling space. This gluing map gives thecompactified moduli space M η o ( X − Σ , s ) the structure of a closed C -manifold in which level- k configurations in (3.44) are embedded as a codimension 2 k (or 2 k + 1) submanifold.We finish this section by some comments on the contribution of the component M η o ( X − Σ , s X − Σ , J ) , consisting of monopoles ending at reducibles, to the relative invariants (1.5) and the resulting sumformula (1.10).By the discussion above, if Σ · Σ > 0, all the monopoles appearing in M η o ( X − Σ , s ) are automaticallyirreducible. Therefore, whenever the invariants are defined, the condition b + X − Σ > X, Σ η o ( s ; − ) to be independent of the choice of generic η o and the cylindricalmetric. However, the component M η o ( X − Σ , s X − Σ , J ) may appear in two ways in this paper.Firstly, if Σ · Σ < 0, it contributes to the level- k strata of the compactified moduli space M η o ( X − Σ , s ),where [ A , Φ ] ∈ M η o ( X − Σ , s X − Σ , J ), J A , Φ K ∈ M ( J, C ) / C ∗ , and the rest of the tunnelings in(3.43) begin and end at irreducibles. The expected codimension of such a configuration is 2 k +1 ≥ M η o ( X − Σ , s X − Σ , J ) would appear in the proof of the alternative sumformula (1.10) in the following way. Proving (1.10) revolves around the following observation. Let X be the connected sum of X and X along Σ and s be a spin c structure on X. Consider the fiberproduct M η ( X − Σ , s | X − Σ ) × M ( Y, s Y ) M η ( X − Σ , s | X − Σ ) (3.46)over M ( Y, s Y ) via the limiting maps (1.2), where η and η are generic compact perturbations on X − Σ and X − Σ, respectively, and η = η + η is the resulting perturbation on X. The fiber product (3.46) decomposes into a union of main components, indexed by various pairsof spin c structures ( s , s ) on T X ( − log Σ) and T X ( − log Σ), respectively, which restrict to( s | X − Σ , s | X − Σ ) and have the same degree on Σ, [ s s = s M η ( X − Σ , s ) × Div d ( s d ( s (Σ) M η ( X − Σ , s ) , (3.47)and the fiber product M η ( X − Σ , s | X − Σ , J ) × J M η ( X − Σ , s | X − Σ , J ) . (3.48) Here, X can be thought of as any of the glued manifolds X T in Theorem 2.6 for some sufficiently large T, with η T = η. M η ( X, s ) by the Gluing Theorem 2.6. Assuming that(3.48) also embeds into M η ( X, s ), as in [OS], we can conclude that M η ( X, s ) is the same as (3.46)except for a subset of real codimension 2 (consisting of broken trajectories involving non-trivialtunnelings). This follows from a convergence result similar to the one used in the construction ofthe compactification M η o ( X − Σ , s ) above; see further below.By Lemma 3.2, the expected dimension of each component in (3.47) matches that of M η ( X, s ) . However, by Lemma 3.3, the expected dimension of (3.48) is 1 less than the expected dimension of M η ( X, s ) . This explains why (3.48) does not contribute to the sum formula (1.10). This observationis used in [OS] to prove their main theorem.With notation as above, in general, a sequence of monopoles [ A Ti , Φ Ti ] in M ( X Ti , s ), wherelim i →∞ T i = ∞ , will, after passing to a sub-sequence, “converge” to a finite chain of monopoleswhere the first one is a monopole on X − Σ, the last one is a monopole on X − Σ, and the restare non-trivial monopoles on the cylinder R × Y . The proof is more or less identical to the proofof [KM, Thm. 16.1.3] in the following sense. Every compact sub-domain K of X i − Σ can beidentified with a sub-domain of X T for T > T K . Restricted to K , after passing to a sub-sequence, { [ A Ti , Φ Ti ] } T i >T K converges to a monopole [ A ,K , Φ ,K ]. By considering a sequence of exhaustingcompact sets K , using unique continuity and a diagonal argument, we get a monopole [ A , Φ ]over X − Σ whose restriction to any K is [ A ,K , Φ ,K ]. Similarly, we get a monopole over X − Σ.However, as T i −→ ∞ , the energy of [ A Ti , Φ Ti ] may concentrate at several locations along the ex-panding cylinder [0 , T i ] × Y ⊂ X Ti . The same proof as in [KM, Thm. 16.1.3] gives us the connectingtunneling monopoles between the monopoles on X − Σ and X − Σ.More precisely, define M η ∞ ( X ∞ , s ) to be the space of all level- k broken trajectories ( k ≥ 0) for s in the following sense. A level- k element of M η ∞ ( X ∞ , s ) is a tuple (cid:16) [ A , Φ ]; J A , Φ K ; . . . ; J A k , Φ k K ; [ A k +1 , Φ k +1 ] (cid:17) (3.49)in the fiber product M η ( X − Σ , s X − Σ ) ev × ev (cid:16) M ( R × Y, s R × Y ) C ∗ (cid:17) . . . ev × ev M η ( X − Σ , s X − Σ ) , (3.50)such that(1) the C ∗ -action on M ( R × Y, s R × Y ) is given by the translation symmetry on R and S -rotationsymmetry on Y ;(2) the evaluation map ev at the ends takes values in a divisor space Div m (Σ) as in (1.3), if thelimit is irreducible, or in J ∼ = T g (or T g +1 ), if the limit is reducible.An analogous construction exists for adapted perturbations η ν , in which the intermediate tunnelingspaces are empty and we simply have the fiber product of M η i ( X i − Σ , s Xi − Σ ) . With notation as above, we have the following compactness theorem. Theorem 3.14. The union [ T ∈ [0 , ∞ ] M η T ( X T , s ) has a natural sequential convergence topology that is Hausdorff and compact. Replacing C ∗ with R , the same theorem holds for a decomposition of X along an arbitrary 3-dimensional manifold Y . 43 .7 Conclusion; proofs of Theorems A–C Except for the orientability of the relative moduli spaces M ∗ ( X − Σ , s ) , so far we have discussedall the steps that go into proving Theorems A–C. In this section, we wrap up the proofs with somecomments on the orientation problem. The SW moduli spaces of closed 4-manifolds are orientable;a choice of orientation on H ( X ) and H ( X ) determines an orientation on the moduli space.Unfortunately, a simple statement like this for the SW moduli spaces of cylindrical-end manifoldsdoes not exist in the literature; we refer to [N, Sec. 4.4.3] and [KM, Sec. 20] for a rather lengthydiscussion of the problem. The problem is on the cylindrical part, where it is not clear if all theoperators e H t in (3.20) are Fredholm. In Section 4, we introduce a different setup for constructingrelative moduli spaces which, among other things, could provide a short answer to the orientationproblem as well; see the discussion after Definition 4.4. Proof of Theorem A. The fact that, for generic η o , M η o ( X − Σ , s ) is a smooth manifold andthe evaluation map (1.3) is a submersion is proved in [N]; see Remark 2.5. The dimension formula(1.4) is derived in Lemma 3.2 by simplifying Nicolaescu’s formula [N1, (3.27)]. Since all of thenon-standard analysis happens on the cylindrical part of X − Σ, the orientability of M η o ( X − Σ , s )can be proved in the same way as in [KM, Cor. 20.4.1]. Here, for the sake of completeness, wepresent an ad hoc way of proving orientability. We show that M η o ( X − Σ , s ) embeds in the modulispace M η o ( X, s ′ ) of the closed-up manifold X with a related classical spin c structure s ′ on T X. Therefore, since X is closed, M η o ( X, s ′ ) can be oriented by a choice of homology orientation on H ( X, Z ) ⊕ H ( X, Z ) , which in turn induces an orientation on its submanifold M η o ( X − Σ , s ) ofthe same dimension.In the context of the gluing map (2.27), consider the natural decomposition of X = X T to thecylindrical-end manifolds M + = X − Σ and M − = N , where N is the normal bundle of Σ ⊂ X, asdefined in Sec. 3.1. With notation as in Sec. 3.5, the cylindrical-end manifold N can also be seen as P − Σ + . Recall that N has a canonical spin c structure determined by its almost-complex structure,and any spin c structure s N on N can be obtained by twisting this canonical spin c structure with E, where E is the pullback of a line bundle of degree d on Σ . We will be interested in the modulispace M ( N , s N ) of the cylindrical-end manifold N , where s N is determined by the line bundle E with d = d ( s ) . By gluing the pair s + = s on M + and s − = s N on M − = N , we get a spin c structure s ′ on X . If M ( N , s N ) is regular, the gluing map (2.27) then gives an embedding M η o ( X − Σ , s ) × M ( Y, s Y,E ) ∼ =Div d (Σ) M ( N , s N ) −→ M η T ( X, s ′ ) . If we show that M ( N , s N ) has a component isomorphic to Div d (Σ), for that component, theleft-hand side of the map above is simply M η o ( X − Σ , s ). Therefore, we get an embedding ofequi-dimensional manifolds M η o ( X − Σ , s ) −→ M η T ( X, s ′ ) . The following lemma finishes the proof. Lemma 3.15. For the spin c structure s N defined by E as above, where < | d − g + 1 | ≤ g − , themoduli space M ( N , s N ) is non-empty and has a regular component diffeomorphic to the irreduciblecomponent M ( Y, s Y,E ) ∼ = Div d (Σ) of M ( Y, s Y ) . Proof. As we have seen, if 0 < | d − g + 1 | ≤ g − , then M ( Y, s Y,E ) ∼ = Div d (Σ) is an irreduciblecomponent of M ( Y, s Y ) , and if d = g − , we get the reducible component. Let us start by con-structing a family of metrics on Y as follows. Recall that T Y decomposes as in (2.6), where the44ertical sub-bundle is given as the kernel of a real 1-form α on Y and the horizontal sub-bundleidentifies with π ∗ T Σ . Let g Σ denote the K¨ahler metric on Σ corresponding to the K¨ahler form ω. We can now use α and g Σ to define a family of metrics g λ = ( λα ) ⊗ ( λα ) + π ∗ g Σ on Y for any λ > . Observe meanwhile that, according to the descent process discussed in Section 2.4, if ( A, Ψ) is anirreducible monopole on Σ , then its pullback ( B, Ψ) via Y −→ Σ is also an irreducible monopole on Y, regardless of which of the metrics g λ is used on Y. Now, we can view N topologically as a quotient of [0 , ∞ ) × Y, where { }× Y is identified with Σ viathe bundle projection Y π −→ Σ . Using the family of metrics defined above, we can equip N with acylindrical-end riemannian metric as follows. Start by considering the following family of metricsg r on the slices Y r = { r }× Y of N , r > , g r = α r ⊗ α r + π ∗ g Σ , α r = β ( r ) α, where β : (0 , ∞ ) −→ [0 , 1] is a smooth increasing function such that β ( r ) = r for r ≤ / β ( r ) = 1for r ≥ . In other words, g r = g λ for λ = β ( r ) . As a result, this family of metrics has shrinkingfibers on Y as r → r = g for r ≥ Y r is compatible with the metric g r by construction. Now define a “radial” metric g on N − Σ byg = d r + g r = d r ⊗ d r + α r ⊗ α r + π ∗ g Σ . (3.51)An elementary local calculation in polar coordinates [P, Ch. 1, Sec. 3.4] shows that g has a limitas r → N . In aneighborhood of r ≈ , we can formally write g = d r + r d θ + g Σ , where the metric g collapsesto d r + g Σ at r = 0 as the slices Y r collapse to Σ . The metric g in a neighborhood of Σ ⊂ N isthe K¨ahler metric associated to the K¨ahler form (3.2). Moreover, g is clearly cylindrical for r ≥ , where we can formally write g = d r + d θ + g Σ . We will now construct a monopole on N from an arbitrary smooth irreducible monopole ( A, Ψ) onΣ . To begin with, let us pull ( A, Ψ) back via π : Y −→ Σ to obtain a smooth irreducible monopole( B, Ψ) on Y, which is invariant under the circle action on Y. We can extend this to a configurationon [0 , ∞ ) × Y as a constant family ( B, Ψ) on each slice Y r , which in turn will descend to a smoothconfiguration ( ˜ B, ˜Ψ) on the quotient N . We will show that ( ˜ B, ˜Ψ) satisfies the SW equations on N . On (0 , ∞ ) × Y, the spinor ˜Ψ is certainly harmonic, because it is harmonic on Y and constant in theradial direction (0 , ∞ ) . Moreover, the spinor bundles on the slices Y r are identified with the plus-and minus-spinor bundles on (0 , ∞ ) × Y via Clifford multiplication by the unit cotangent vector d r and, since we are using the adiabatic connection on each slice, the curvature F ˜ B has no componentin the radial direction and the curvature equation in SW reduces to the curvature equation on Y. We conclude that ( ˜ B, ˜Ψ) satisfies the SW equations on N − Σ . Therefore, by smoothness, it willsatisfy the equations over the entire N . We have just showed that M ( N , s N ) is non-empty, so we can consider the limiting map ∂ : M ( N , s N ) −→ M ( Y, s Y,E ) ∼ = Sym d (Σ)for the cylindrical-end manifold N . We have in fact constructed a right inverse to the limiting mapin the previous paragraph, so Sym d (Σ) embeds into M ( N , s N ) . A calculation using Lemma 3.245hows that the expected dimension of M ( N , s N ) is equal to 2 d, which matches the dimension ofSym d (Σ) . We conclude that at least a component of M ( N , s N ) is diffeomorphic to Sym d (Σ) viathe limiting map and the lemma follows. Proof of Theorem B. As before, the fact that M η ν ( X − Σ , s ) is a smooth manifold for a genericchoice of the compactly-supported part η o is proved in [N]; see Remark 2.5. The dimension for-mula (1.7) is proved in Lemma 3.4. Compactness, as stated in Theorem 3.11, is derived fromCorollary 3.9. Orientability can be proved as above. The fact that, for different such η ν and η ν ′ , M η ν ( X − Σ , s ) and M η ν ′ ( X − Σ , s ) are smoothly cobordant follows from considering a 1-parameterfamily of perturbations connecting η ν and η ν ′ , as in the classic case. Proof of Theorem C. The sum formula is a direct consequence of the gluing theorem andconvergence/compactness argument at the end of Section 3.6. In other words, for sufficiently large T , the gluing map [ [ s ]= s s M η ,ν ( X − Σ , s ) × S d ( s = S d ( s M η ,ν ( X − Σ , s ) −→ M η T ,ν ( X T , s )which is an embedding by Theorem 2.6, is also onto by Theorem 3.14, thus is an identification ofclosed manifolds. Here η ,ν and η ,ν are adapted perturbations on X − Σ and X − Σ correspondingto a holomorphic 1-form ν on Σ and η T ,ν is the resulting perturbation term on X T .To see the surjectivity of the gluing map, note that X T ’s can be identified with each other in anatural way by re-scaling the neck in the time direction. Therefore, a given monopole in M η ν ( X, s )can be identified with a monopole in M η T ,ν ( X T , s ) for any T ; these monopoles are compatible witheach other and have the same energy on the neck. As T −→ ∞ , these monopoles converge to anelement of the fiber product on the left, which is the desired inverse image of the original monopoleon X ≃ X T under the gluing map above.As mentioned before, the ± signs ε s , s ,q ∈ {± } in (1.9) depend on the choices of orientations. Fora meticulous discussion of how to orient the fiber products to be consistent with the gluing, see[KM, 20.5]. In this section, we explain our idea for a direct construction of relative SW moduli spaces to bypassthe issues related to working with non-closed manifolds. We will define logarithmic SW equationsby replacing T X with T X ( − log Σ) and considering “logarithmic connections”. This constructioncan easily be generalized to the normal crossings case, where it is hard to work with X − Σ as acylindrical-end manifold. Through this construction, it should be possible to address the orienta-tion problem more systematically.Let ( X, Σ) be as in the previous sections and s = ( S , ρ ) be a spin c structure on T X ( − log Σ). If W −→ X is a vector bundle over X , a connection ∇ on W is a bi-linear map ∇ : Γ( X, T X ) × Γ( X, W ) −→ Γ( X, W ) , ( ξ, ζ ) −→ ∇ ξ ζ, that is tensorial in the first input and satisfies the Leibniz rule in the second input. In the classicaltheory, the construction of SW moduli space involves a riemannian metric on T X , a hermitianmetric on S , and compatible connections on S and T X. The latter is usually fixed to be the Levi-Civita connection. 46 efinition 4.1. Let ( X, Σ) be a pair of a closed oriented 4-manifold X and a closed oriented2-dimensional submanifold Σ ⊂ X . For any vector bundle W −→ X , a logarithmic connection ∇ on W is a bi-linear map ∇ : Γ( X, T X ( − log Σ)) × Γ( X, W ) −→ Γ( X, W ) , ( ξ, ζ ) −→ ∇ ξ ζ, (4.1)that is tensorial in the first input and satisfies the Leibniz rule in the second input.As in the classical case, in any local trivialization, we have ∇ = d ι ( · ) + Θ, where ι is the homomor-phism in (3.6) and Θ is a matrix of logarithmic 1-forms. Therefore, globally, every two logarithmicconnections ∇ and ∇ ′ on W differ by an End( W )-valued logarithmic 1-formΘ ∈ Γ (cid:0) X, T ∗ X (log Σ) ⊗ End( W ) (cid:1) . Definition 4.2. In the presence of a metric h· , ·i on W , we say ∇ in (4.1) is compatible with themetric if d ι ( ξ ) h ζ , ζ i = h∇ ξ ζ , ζ i + h ζ , ∇ ξ ζ i . With the definition above, for ( X, Σ) and a spin c structure s = ( S , ρ ) on T X ( − log Σ) , we needlogarithmic connections ∇ and ∇ S on T X ( − log Σ) and S which are compatible with the metricson T X ( − log Σ) and W , respectively, as well as with the Clifford multiplication: ∇ S (cid:0) ξ · Φ (cid:1) = ξ · ∇ S Φ + ( ∇ ξ ) · Φ . As in the classical case, a compatible connection ∇ S on S is uniquely determined by ∇ and alogarithmic connection A on the characteristic line bundle L s .Associated to a logarithmic connection ∇ S as above we define the logarithmic Dirac operator to be /∂ log : Γ( S ± ) −→ Γ( S ∓ ) , /∂ log Φ = X i =1 e i · ∇ S e i Φ , (4.2)where e , . . . , e is an orthonormal basis for T x X ( − log Σ). The metric considered on T X ( − log Σ)is the one described before (3.8): on a neighborhood D in N , identified with a neighborhood of Σin X using the map Υ , the metric is the direct sum of the pullback of the K¨ahler metric on T Σand the standard riemannian metric on C D via the identificationΥ ∗ T X ( − log Σ) = π ∗ T Σ ⊕ C D . We take ∇ to be the direct sum connection on D and the Levi-Civita connection outside a largerneighborhood (1 + ε ) D and splice them in the middle using a convex combination with suitablesmooth coefficients. For each fixed-radius circle bundle Y ⊂ D , the restriction of ∇ to T Y is theadiabatic connection mentioned in Section 2.4. Therefore, restricted to X − Σ, via the identification(3.8), ∇| T ( X − Σ) is the connection considered in the definition of relative moduli spaces M ( X − Σ , s ) . As expected, the logarithmic Dirac operator (4.2) is not elliptic on the entire X in the classicalsense of the word. The principal symbol of (4.2), which is a function on the cotangent bundle T ∗ X, is zero on the dual space T Σ ⊥ ⊂ T ∗ X | Σ of T Σ , while it is non-zero everywhere else. This isessentially due to the fact that the homomorphism ι maps ∂ log z to z∂ z in T X, which is zero alongΣ . We expect though that a logarithmic elliptic theory could be developed for /∂ log that paves the47ay for working directly over T X ( − log Σ) instead. In such a theory, the principal symbol of /∂ log would rather be a function on the logarithmic cotangent bundle T ∗ X (log Σ) , which, by analogy, isClifford multiplication by that cotangent vector.Next, we define the curvature of a logarithmic connection (4.1). Lemma 4.3. For ξ , ξ ∈ Γ( X, T X ( − log Σ)) , there exists a unique ξ .. = [ ξ , ξ ] ∈ Γ( X, T X ( − log Σ)) such that ι ( ξ ) = [ ι ( ξ ) , ι ( ξ )] . (4.3) Proof. Fix local holomorphic coordinates ( z, w ) : V −→ C around a point of Σ with Σ ∩ V = ( z ≡ a = 1 , 2, if ξ a = f a, ∂ log z + f a, ∂ log z + f a, ∂ w + f a, ∂ w then ι ( ξ a ) = f a, z∂ z + f a, z∂ z + f a, ∂ w + f a, ∂ w . We have [ f , z∂ z , f , z∂ z ] = (cid:16) f , ∂ ( zf , ) ∂z − f , ∂ ( zf , ) ∂z (cid:17) z∂ z , [ f , z∂ z , f , z∂ z ] = (cid:16) f , ∂f , ∂z z (cid:17) z∂ z − (cid:16) f , ∂f , ∂z z (cid:17) z∂ z , [ f , ∂ w , f , z∂ z ] = (cid:16) f , ∂f , ∂w (cid:17) z∂ z − (cid:16) f , ∂f , ∂z z (cid:17) ∂ w , [ f , ∂ w , f , z∂ z ] = (cid:16) f , ∂f , ∂w (cid:17) z∂ z − (cid:16) f , ∂f , ∂z z (cid:17) ∂ w . Similarly, we see that (4.3) holds for the rest of the terms. Uniqueness follows from continuity andthe fact that (4.3) is the same as ordinary bracket away from Σ . It follows from Lemma 4.3 that given a logarithmic connection ∇ as in (4.1), the curvature equation F ∇ ( ξ , ξ ) ζ = ∇ ξ ∇ ξ ζ − ∇ ξ ∇ ξ ζ − ∇ [ ξ ,ξ ] ζ is well-defined. 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