Math. Res. Lett. 13 (2006), no. 1, c International Press 2006 ENERGY IDENTITY FOR ANTISELFDUAL INSTANTONS ON C Σ.


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1 Math. Res. Lett. 3 (2006), no., 6 66 c International Press 2006 ENERY IDENTITY FOR ANTISELFDUAL INSTANTONS ON C Σ Katrin Wehrheim Abstract. We establish an energy identity for antiselfdual connections on the product C Σ of the complex plane and a Riemann surface. The energy is a multiple of a basic constant that is determined from the values of a corresponding ChernSimons functional on flat connections and its ambiguity under gauge transformations. For SU(2)bundles this identity supports the conjecture that the finite energy antiselfdual instantons correspond to holomorphic bundles over CP Σ. Such antiselfdual instantons on SU(n) and SO(3)bundles arise in particular as bubbles in adiabatic limits occurring in the context of mirror symmetry and the AtiyahFloer conjecture. Our identity proves a quantization of the energy of these bubbles that simplifies and strengthens the involved analysis considerably.. Introduction Let Σ be a Riemann surface and consider the trivial SU(2)bundle over C Σ. A connection Ξ A(C Σ) on this bundle is a form Ξ Ω (C Σ; su(2)) with values in the Lie algebra su(2). auge transformations u (C Σ) of the bundle are represented by maps u Map(C Σ, SU(2)) and act on A(C Σ) by u Ξ=u Ξu + u du. We equip C Σ with a product metric of the Euclidean metric on C and a fixed metric on Σ. Then a connection Ξ A(C Σ) is called an ASD instanton if its curvature is antiselfdual, F Ξ + F Ξ =0, where is the Hodge operator w.r.t. the metric on C Σ. The curvature 2form F Ξ =dξ+ξ Ξtransforms under gauge transformations u (C Σ) as F u Ξ = u F Ξ u, hence the antiselfduality equation is gauge invariant. Next, we equip su(2) with the SU(2)invariant inner product ξ,η = tr(ξη). Then the energy of a connection Ξ A(C Σ) is the gauge invariant quantity E(Ξ) := 2 C Σ F Ξ 2. The main purpose of this note is to establish the following energy identity. Its surprisingly simple proof is given in section 2. For the sake of simplicity we first focus our attention to SU(2)bundles. Later, we will also indicate how to generalize this result to other structure groups and nontrivial bundles over Σ. Theorem.. Let Ξ A(C Σ) be an ASD instanton. If it has finite energy E(Ξ) <, then actually E(Ξ) 4π 2 N 0. Received by the editors September 6,
2 62 KATRIN WEHRHEIM This energy quantization supports a conjectural correspondence between finite energy ASD instantons on C Σ and holomorphic bundles over CP Σ. For Σ = T 2 Biquard and Jardim [] showed that the gauge equivalence classes of ASD instantons with quadratic curvature decay are in onetoone correspondence to a class of rank 2 stable holomorphic bundles over CP T 2. Here the holomorphic structure induced by an instanton Ξ extends over { } T 2 to define a bundle E, whose second Chern number is given by the instanton energy, c 2 (E) = 8π 2 FΞ F Ξ, see [4, 2.3]. By our result this formula continues to give integer (Chern?) numbers for finite energy instantons and any surface Σ. Remark.2. Theorem. extends to ASD instantons on C P for any principal bundle P Σ with compact structure group as follows: Suppose that the Lie algebra g is equipped with a invariant metric that satisfies (H) below. Then the statement of theorem. holds with 4π 2 replaced by the constant κ g N given below. On a nontrivial bundle P the gauge transformations are represented by sections in the associated bundle P = P c (using the conjugation action on ). We can pick a invariant inner product on g (and thus on g P = P Ad g). Then the MaurerCartan 3form on each fibre of P induces a closed 3form η := 2 g dg [g dg g dg] on P. We need the following assumption. (H): There exists κ g > 0suchthat[κ g η ] H 3 ( P, R) is an integral class. This holds for example with κ so(3) = 4π 2 for any SO(3)bundle when we choose the inner product 2tr(ξη) forξ,η so(3). It can also be achieved for any simply connected compact Lie group, e.g. for the trivial SU(n)bundles. Finally, N is the least common multiple of {, 2,...,n }, where n denotes the maximal number of connected components that the centralizer of a subgroup in can have. This is finite since is compact. For SO(3) we have N SO(3) =. One source of interest in the ASD instantons on C Σ is the following adiabatic limit. Let Σ X M be a fibre bundle with dim X = 4. Consider ASD instantons Ξ ε over X with respect to metrics g M + ε 2 g Σ for a sequence ε 0. If F Ξε fibre + ε 2 F Ξε mix converges to a nonzero value, then local rescaling on M (but not in the fibre) yields an ASD instanton on C Σ in the limit. This bubbling phenomenon is a central difficulty of the limiting process. Adiabatic limits of this type have fascinating consequences from topology to mathematical physics. They were first considered by DostoglouSalamon [3], and recently by Chen [2] and Nishinou [5]. The energy quantization presented here simplifies and strengthens the bubbling analysis and results in all these cases. It can also be used for the AtiyahFloer conjecture project [6, 9]. In that case the bundle is automatically trivial and the Lie group is isomorphic to a product S... S k of simply connected, simple, and compact Lie groups S j with π 3 (S j ) = Z. So we can pick a metric on each factor S j for which [η Sj ] H 3 (S j, R) is integral.
3 ENERY IDENTITY FOR ANTISELFDUAL INSTANTONS ON C Σ Proof of the energy identity In the following, S r C denotes the circle of radius r centered at 0. We moreover denote by D r C the disk of radius r, and we introduce polar coordinates (r, φ) (0, ) S on C = C \{0}, with S = R/2πZ. Then on C Σ we can write a connection Ξ A(C Σ) in the splitting Ξ=A(r)+R(r)dr +Φ(r)dφ with A(r): S A(Σ) and R(r), Φ(r): S Ω 0 (Σ, su(2)) for all r (0, ). The antiselfduality equation becomes in this splitting { r ( r Φ φ R +[Φ,R] ) + F A =0, r ( φ A d A Φ ) ( r A d A R ) =0. By F Ξ (r) we denote the curvature of Ξ A(C Σ) over S r Σ(butasa2form on C Σ). Then the curvature of an ASD instanton is 2 F Ξ(r) 2 = F A(r) 2 + r 2 φ A(r) d A(r) Φ(r) 2. The energy of an ASD instanton on D r Σ can be expressed in terms of the ChernSimons functional of B(r) :=A(r)+Φ(r)dφ A(S Σ), 2 D r Σ F Ξ 2 = 2 D r Σ F Ξ F Ξ = CS(B(r)). The ChernSimons functional on connections B = A +Φdφ A(S P )is CS(B) = 2 B ( F B 6 [B B]) S Σ () = 2 ϕa A + F A, Φ. S Σ For future reference we note the following identity which shows that the Chern Simons functional is continuous with respect to the W, 3 2 norm. (Note that W, 3 2 L 3 on a 3manifold.) For all B,B 0 A(S Σ) (2) CS(B) CS(B 0 )= 2 (F B + F B 0) (B B 0 ) [(B B 0 ) (B B 0 )] (B B 0 ). 2 The ChernSimons functional is not gauge invariant, but its ambiguity on gauge orbits is determined by the degree of the gauge transformations (as maps to SU(2) = S 3 ): For all B A(S Σ) and u (S Σ) (3) CS(B) CS(u B)=4π 2 deg(u) 4π 2 Z For a general (possibly nontrivial) bundle P Σ one has to fix a flat reference connection. Then connections are given by forms with values in g P and the ChernSimons functional depends on the choice of this reference connection only up to an additive constant. (The proof of theorem. will show that a flat
4 64 KATRIN WEHRHEIM connection exists.) The right hand side of (2) is then given by u η. So under the assumption (H) we have CS(B) CS(u B) κ g Z. The second point that affects the constant in the energy identity is the possible values of the ChernSimons functional on flat connections. The following result holds for SU(2) and SO(3)bundles, and we will give the argument for a general bundle P Σ, indicating how to proceed for other structure groups. Lemma 2.. For every flat connection B A flat (S Σ) there is a gauge transformation u (S Σ) such that CS(u B) = 0, and consequentially CS(B) =4π 2 deg(u) 4π 2 Z. Proof: Any flat connection B on S Σ corresponds to a holonomy representation ρ : π (Σ) and an element g S im ρ ;the holonomy around S which lies in the centralizer of im ρ. We will use parallel transport along S and a homotopy g n l S im ρ (for some n n ) to bring B into the form A +Φdφ with S independent A, for which the Chern Simons functional trivially vanishes. Then ncs(b) κ g Z by (3). More precisely, we periodically extend B to to a connection in A flat (R P ). Then there is a gauge transformation u: R (P ) such that u(0) l and u B A flat (R P ) has no dφcomponent. Thus the curvature component φ (u B) vanishes, and hence u B A 0 A flat (P ). The gauge transformation is found by parallel transport, i.e. solving φ u = Φu. So due to the periodicity of Φ we obtain the twisted periodicity u(φ + 2π) = u(φ)u(2π) for the gauge transformation. Unless u(2π) l this does not define a gauge transformation on S P. However, we know that u(2π) lies in the isotropy subgroup A 0, since u(2π) A 0 = u(2π) B(2π, ) Σ = u(0) B(0, ) Σ = A 0. If A 0 is connected, then we can multiply u with a path within A 0 from l tou(2π) to obtain the required gauge transformation w (S P ). It satisfies w B = A 0 +Φ 0 dφ with φ A 0 = 0 but possibly nonzero Φ 0. Now compare () to see that CS(w B)=0, and so CS(B) =4π 2 deg(w) by (3). For SO(3)bundles, any isotropy subgroup is connected since any centralizer (of the holonomy subgroup) in SO(3) is connected. Thus the proof is finished. For a general Lie group whose centralizers have up to n components, one finds that u(2π) n is homotopic to the identity for some integer n n.thenan nfold cover B (n) of B can be put into a gauge whose ChernSimons functional vanishes, and thus CS(B) =n CS(B (n) ) κ g n Z κ g N Z if (H) holds. For SU(2) we would have n = 2 due to the centralizer {l, l}. However, since the isotropy element u(2π) = l isaconstant,wedonotneedtogotoa cover. More generally suppose that u(2π) = exp(2πξ) for some constant ξ g. Let v(φ) := exp( φξ), then w := uv (S Σ) and w B = v A 0 v ξdφ (and both are of class W, ). Then using F A 0 =0anddξ = 0 we obtain CS(w B)= 2 v [ξ,a 0 ]v v A 0 v S Σ = ξ,a 0 A 0 = ξ,da 0 = 0. S Σ S Σ
5 ENERY IDENTITY FOR ANTISELFDUAL INSTANTONS ON C Σ 65 In the subsequent proof of the energy identity we work with a general bundle P Σ and only for the final conclusion use the knowledge from lemma 2. on the possible values of the ChernSimons functional on flat connections. Proof of theorem.: Let B(r) A(S Σ) be given by Ξ on S r Σ, then F B(r) 2 = F A(r) 2 + ϕ A(r) d A(r) Φ(r) 2 2 r2 F Ξ (r) 2 for r, and hence r F B(r) 2 L 2 (S Σ) dr E(Ξ) <. Thus we find a sequence r i with F B(ri ) L 2 (S Σ) 0. By Uhlenbeck s weak compactness [7] we then find a further subsequence, gauge transformations u i (S P ), and a flat limit connection B A flat (S P )suchthat (4) u i B(r i ) B W,2 (S Σ) 0. More precisely, the Uhlenbeck compactness theorem (also see [8, Theorem A] with p =2 2 dim(s Σ)) provides a subsequence B i := B(r i ) that converges in the weak W,2 topology to a flat (and hence smooth) connection B A flat (S P ). Since the Sobolev embedding W,2 L 4 is compact (in dimension 3) we can moreover assume that B i B in the L 4 norm. So by the local slice theorem (e.g. [8, Theorem 8.]) one can for a further subsequence and sequence of gauge transformations achieve the additional relative Coulomb gauge condition Moreover, we have d B (u i B i B )=0. d B (u i B i B )=u i F Bi u i 2 [(u i B i B ) (u i B i B )]. So from the regularity of the Hodge decomposition of forms (e.g. [8, Theorem 5.]) one obtains the convergence (4) in the W,2 norm. NowwehaveCS(u i B(r i)) CS(B ) due to the convergence of u i B(r i) and (2). On the other hand the energy is finite, so E(Ξ) = lim F Ξ F Ξ = lim CS(B(r i )). i D ri Σ i This shows that CS(B(r i )) also converges. Now for an SU(2)bundle we have CS(B ) 4π 2 Z from lemma 2.. Thus CS(B(r i )) = CS(u i B(r i))+4π 2 deg(u i ) must converge to some value in 4π 2 Z. This proves the claim since that limit is also the energy E(Ξ). For a general bundle under the assumption (H) we know that CS(u i B(r i)) converges to a value in κ g N Z. Since CS(B(r i)) CS(u i B(r i)) κ g Z we must have E(Ξ) = lim CS(B(r i )) κ g N Z + κ g Z = κ g N Z, which proves remark.2.
6 66 KATRIN WEHRHEIM Acknowledgments I would like to thank Benoit Charbonneau, Kenji Fukaya, Marcos Jardim, Tom Mrowka, and Dietmar Salamon for valuable discussions. Support by the Swiss National Science Foundation and the US National Science Foundation is gratefully acknowledged. References [] O. Biquard and M. Jardim, Asymptotic behaviour and the moduli space of doublyperiodic instantons, J. Eur. Math. Coc. 3 (200) [2] J. Chen, Convergence of antiselfdual connections on SU(n)bundles over product of two Riemann surfaces, Comm. Math. Phys. 96 (998), no. 3, [3] S. Dostoglou and D. A. Salamon, Selfdual instantons and holomorphic curves, Ann.of Math. 39 (994) [4] M. Jardim, Nahm transform and spectral curves for doublyperiodic instantons, Comm. Math. Phys. 225 (2002), no. 3, [5] T. Nishinou, Convergence of HermitianYangMills Connections on Kähler Surfaces and mirror symmetry, preprint, math.s/ [6] D. A. Salamon, Lagrangian intersections, 3manifolds with boundary, and the Atiyah Floer conjecture, Proceedings of the ICM, Zürich 994, Vol., [7] K. K. Uhlenbeck, Connections with L p bounds on curvature, Comm. Math. Phys. 83 (982) [8] K. Wehrheim, Uhlenbeck Compactness, EMS, Zürich, [9], Lagrangian boundary conditions for antiselfdual instantons and the AtiyahFloer conjecture, to appear in J. Symplectic eom. Institute for Advanced Study, Einstein Drive, Princeton NJ address:
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