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Instantons on Conic 4-Manifolds: Fredholm Theory

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Instantons on Conic 4-Manifolds: Fredholm Theory J. Korean Math. Soc. 44 (2007), No. 2, pp. 275{296 INSTANTONS ON CONIC 4-MANIFOLDS: FREDHOLM THEORY Weiping Li and Shuguang Wang Reprinted from the Journal of the Korean Mathematical Society Vol. 44, No. 2, March 2007 °c 2007 The Korean Mathematical Society J. Korean Math. Soc. 44 (2007), No. 2, pp. 275{296 INSTANTONS ON CONIC 4-MANIFOLDS: FREDHOLM THEORY Weiping Li and Shuguang Wang Abstract. We study the self-duality operator on conic 4-manifolds. The self-duality operator can be identi¯ed as a regular singular operator in the sense of BrÄuningand Seeley, based on which we construct its parametriza- tions and closed extensions. We also compute the indexes. 1. Introduction The main purpose of the paper is to initiate the investigation of gauge theory on a class of manifolds that have the simplest kind of singularity. Let X0 be a compact smooth manifold with boundary Y . The associated topological conical space X is by de¯nition the gluing X0 [Y N along Y , where N = [0; 1]£Y=f0g£Y , i.e., one boundary f0g£Y collapsing into a single point c. Clearly X is a compact topological space but the manifold structure at c is singular. To best suit this type of singularity, we place the (close) conic metric 2 2 of the form dt + t gY on N away from c, where gY is any smooth Riemannian metric on Y . Conical spaces and conic metrics arise naturally in many di®erent contexts. In Algebraic Geometry, a conic singular complex surface is obtained by con- tracting a curve of negative self intersection in a smooth KÄahlersurface. The smooth KÄahlermetric descends to a conic metric on the complex quotient. The point of such a construction is that away from the singular point, the complex quotient is often topologically much simpler than the original KÄahlersurface. For example, contracting the sixteen (¡2)-curves on a Kummer surface yields simply the quotient of a torus under the natural involution. It is well-known that negative curves are di±cult to study since they are isolated and do not form any moduli. In Symplectic Geometry, for symplectic 4-manifolds with boundary a concave contact 3-manifold, the contact structure can be extended naturally into a symplectic form which is compatible with a conic metric on the end. A recent result of J. Etnyre and K. Honda [8] shows that any concave Received July 22, 2005. 2000 Mathematics Subject Classi¯cation. 57R57, 58G12, 53C05, 58G18, 58J20. Key words and phrases. gauge theory, conic 4-manifold, self-duality operator, parame- trization, closed extension, index. °c 2007 The Korean Mathematical Society 275 276 WEIPING LI AND SHUGUANG WANG contact structure is ¯llable by in¯nitely many non-isomorphic symplectic man- ifolds, thus quite di®erent from convex contact structures. In String Theory, there is a 6-dimensional conifold X with base S3£S2 introduced in Gopakumar- Vafa [9]. The cone singularity can be resolved in two ways: either blow up the S2 direction resulting in the bundle O(¡1) © O(¡1) over S2, or deform along the S3 direction resulting in the cotangent bundle T ¤S3. The conjectured duality in [9] says that the Gromov-Witten invariant of O(¡1) © O(¡1) is re- lated to a Chern-Simons type knot invariant of S3. A possible approach might be to study directly the conifold X and relate the invariants to those on the resolutions. The (close) conic metric on X de¯ned above is conformally equivalent to the 2 2 2 open conic metric ds +s gY and product metric ds +gY , where 1 < s < +1. However the geometry in the ¯rst case is considerably di®erent and harder than the latter two cases. The main reason for such a disparity is due to the fact that the open conic and product metrics are both complete whereas the close conic metric is not. Thus the traditional analytic approach for complete metrics is no longer suitable and new phenomena do emerge in close conic metric geometry (see [1, 3, 4, 11]). In the aforementioned examples, it is necessary to work with close conic met- rics, since open or product metrics can not be KÄahleror symplectic. (Of course being KÄahleror symplectic metrics is not conformally invariant.) The conic singularities and manifolds with boundary are closely related, and topological and geometric properties on manifolds with conic singularities may be viewed as a rami¯cation of special boundary value problems with non-local boundary conditions. The conic singularity is interesting in its own right and was initially studied by Cheeger [3, 4]. In the current paper, we introduce Yang-Mills theory on the conic 4-mani- folds. This leads to two major changes from the standard theory in smooth 4-manifolds (e.g. [5]). (1) Since the metric is not di®erentiable at the cone sin- ¤ gular point, the slice condition and dA operator require a modi¯cation similar to [6]. (2) The functional-analytic framework and the Fredholm theory need di®erent setups than [5, 6]. In particular, the self-duality operator on conic manifolds may have many di®erent closed extensions which are all Fredholm but with di®erent domains and indices. Our main results can be summarized as follows. + ¤ Theorem. (1) The self-duality operator dA © dA on a conic 4-manifold X can @ ¡1 be identi¯ed with a ¯rst order regular singular operator PA = @t + t P0 + P1 in the neighborhood of the cone point. In particular it is of (weak) Fuchs type. iD (2) Let PA be the restriction of PA;max to the ideal Dirichlet boundary con- iD dition. The operator PA is closed and Fredholm. iD (3) The index of PA is given by Z 1 X 2¡2k c~ (adP ) ¡ (´ + m ) + Res ´ (2k); 2 2 P0 1 2k(2k + 1) 1 P0 X k¸1 CONIC SINGULARITY 277 where ´P0 (z) is the eta function of the operator P0 and m1 is the multiplicity of eigenvalue 1=4 of the Laplacian operator ¢1 on coclosed ­1(Y ) with the metric gY . The paper is organized as follows. Section 2 gives a basic set-up and shows part (1) of the Theorem in two lemmas. The parametric construction, following from the method in [1, 2] (or the more general treatment in [11]), is given in section 3. Then we prove part (2) in Theorem 3.5. Unlike the smooth manifold case, there are di®erent closed extensions of the operator PA. By applying the results in [1, 2], we obtain the index formula part (3) in the last section. 2. A basic set-up Let X be a closed, oriented conical 4-manifold with a cone point c. (The analysis can be naturally extended to the case of a ¯nite number of cone points.) Let N be a neighborhood of c in X, N = (0;"] £ Y , where the 3-manifold Y has a metric gY . Note that the 4-dimensional Yang-Mills theory is conformally unchanged. Thus up to conformal changes, N can be equipped with the metric 2 2 dt + t gY : The metric satis¯es the conformally conic manifold de¯nition 1.1.1 of [11] (with h = 1; c = 0 > ¡1 and any ±). The manifold is a product near the singularity. Our point of view is to transfer the geometric singularity into a singularity of a geometric di®erential operator on a manifold with boundary. The natural di®erential operators involving the conic metric are di®erential operators of regularly singular operators [1, 2] or Fuchs type operators [11]. Let P be an SU(2)-principal bundle over X. Let A(P ) be the space of SU(2)-connections on the principal bundle P , and G(P ) be the automorphism 1 group of P . Note that A(P )jN is an a±ne space modeled on ­ (N; adP jN ). We 1 1 0 can naturally identify ­ (N; adP jN ) with ­ (Y; adPY ) © ­ (Y; adP jY ). More generally and in order to perform further calculations, we give an identi¯cation between p-forms on N and p-forms, (p ¡ 1)-forms on Y : p p¡1 p (1) fp : ­ (Y; adPY ) © ­ (Y; adP jY ) ! ­ (N; adP jN ) p¡3=2 ¤ ¡1 ¤ (wp; wp¡1) 7! t (¼ wp + t ¼ wp¡1 ^ dt); where ¼ : N = (0;") £ Y ! Y is the projection on the second factor. Note that one has 2 kf (w ; w )k 2 p p p p¡1 L (­ (N;adP jN )) Z " 2 2 = fkw k 2 p + kw k 2 p¡1 gdt: p L (­ (Y;adPY )) p¡1 L (­ (Y;adP jY )) 0 278 WEIPING LI AND SHUGUANG WANG 2 2 With respect to the metric gN = dt + t gY , ¤gN fp(wp; wp¡1) p¡3=2 ¤ ¡1 ¤ = ¤gN (t (¼ wp + t ¼ wp¡1 ^ dt)) 3=2¡p ¤ 5=2¡p ¤ 4¡p = t ¼ (¤gY wp) ^ dt + t ¼ ((¡1) ¤gY wp¡1) (4¡p)¡3=2 ¤ 4¡p ¡1 ¤ = t (¼ ((¡1) ¤gY wp¡1) + t ¼ (¤gY wp) ^ dt) 4¡p = f4¡p((¡1) ¤gY wp¡1; ¤gY wp): Hence we obtain the Hodge-star operator ¤gN acting on the identi¯cation (1): 4¡p (2) ¤gN fp(wp; wp¡1) = f4¡p((¡1) ¤gY wp¡1; ¤gY wp): The restricted gauge group over N is denoted by G(P )jN . For any connection A 2 A(P )jN , there is a (temporal) gauge transformation in G(P )jN under which A does not contain the covariant derivative in t-direction, i.e., A = a(t), where each a(t) is a connection on Y and t 2 (0;").
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