Morse-Bott Framework for the 4-dim SW-Equations

Transcription

1 Morse-Bott Framework for the 4-dim SW-Equations Celso M. Doria UFSC - Depto. de Matemática august/2010 Figure 1: Ilha de Santa Catarina Figure 2: Floripa research partially supported by FAPESC 2568/

2 1 Introduction Let (X, g) be a closed riemannian 4-manifold. The space of spin c structures on X is spin c (X) = {c = α + β H 2 (X, Z) H 1 (X, Z 2 ) w 2 (X) = β mod 2}. For each c Spin c (X), c (S c, L c ) 1. S c (spinor bundle): S c = S c + Sc, ±-spinors. 2. L c (determinant bundle) such that c 1 (L c ) = α. Fix c spin c (X): 1. A c is the space of U 1 -connections on L c, 2. Γ(S c + ) is the space of sections of S c +, 2

3 3. G c = Map(X, U 1 ) is the gauge group acting on C c = A c S + c by g.(a, φ) = (A + 2g 1 dg, g 1 φ). F ix Gc = {(A, 0) C c }. For all (A, 0), the isotropy group is U 1 = {g : X U 1 g is constant}. 4. The moduli space B c = C c /G c. B c F ix Gc is a smooth -manifold. The monopole (SW c )-equations on X are F + A = σ(φ) D + A φ = 0, + σ :Γ(Sc ) End 0 (S c ) Ω 2 +(X, ir) σ(φ) = φ φ φ 2 2 I. (1) Definition: Let M c be the space of solutions for the SW c -eq. (i) (A, φ) M c is an irreducible SW c -monopole if φ 0. (ii) otherwise it is reducible (singular points for the G c -action - isotropy is U 1 ). remarks: (1) M c {φ Γ(S + c ) φ 0} =, but for a finite set of classes c spin c named basic classes. The irreducible SW c -monopoles are the ones which matter for the applications in smooth topology. Question: Find a necessary or a sufficient condition for the existence of a SW c -monopole on a smooth manifold. 3

4 There are two majors theorems about this issue; Thrm 1 (Witten) φ 2 sup x X {0, k g (x)}. So, if k g 0, then φ = 0. Thrm 2 (Taubes) X is symplectic and c is the canonical class, so there exists an unique solution (A, φ) B c, φ 0. The space M c has the following properties: 1. it is a compact, smooth, orientable manifold away from the singular set 2. dim(m c ) = 1 4 [α2 (2χ(X) + 3σ(X)]. Exploring the topology of M c, the 4-dim topologist developed a black-box in order to distinguish smooth structures on X. 4

5 2 Variational Formulation Fix c spin c. Consider SW c : C c R given by SW c (A, φ) = where k g = scalar curvature of (X, g). Facts X { 1 2 F + A 2 + A φ 2 + k g 4 φ φ 4 }dv g, (2) 1. solutions for the SW-equations are stable critical points for the SW c -functional. 2. the functional is well defined on C c because, in dimension n=4, there is the Sobolev embeddings L 4 L 1,2. 3. SW c is gauge invariant SW c : B c R 4. SW c : B c R satisfies the Palais-Smale condition. 5. Euler-Lagrange equations for SW c are d F A + d φ 2 = 0, (3) A φ + φ 2 +k g φ = 0,. (4) 4 For every c spin c, the minimum is achieved in B c. Reducible Solutions if (A, 0) solves eqt s. 5 and 6 then d F A = 0. Therefore, F A is a harmonic 2-form (α = 1 2πi [F A]), Proposition The space of reducible solutions is diffeomorphic to the Jacobian torus J X = H1 (X, R) H 1 (X, Z) S1 b 1(X)... S 1. ( SW c JX is constant) 5

6 Proposition B c retracts into J X. Moreover, π k (B c, b 0 ) = H 1 (S k, Z) H 1 (X, Z) = H k (J X, Z). Question: Find a condition for J X to be an unstable critical point of SW c. Question: Is the Morse-Bott index of J X finite? 3 Pertubed SW-Functional η Ω 2 +(X, ir), dη = 0. The perturbed SWc η -eq are F + A η = σ(φ), D + A φ = SW η c : C c R, SWc η (A, φ) = X X { 1 2 F + A 2 + A φ k g φ φ 4 }dv g ] [< F +A σ(φ), η > +12 η 2 dv g. 2. the functional SWc η : C c R is gauge invariant. 3. It follows from the identity SWc η (A, φ) = { D A φ 2 + F + A σ(φ) η 2 }dx, that a SW η c -monopole is a stable critical point. X 6

7 4. The Euler-Lagrange equations for the SW η c -functional are d F A d η + d φ 2 = 0, (5) A φ + φ 2 +k g φ + η.φ = 0,. (6) 4 5. (Main Estimate) If (A, φ) is a solution for the SW η c -functional, then φ η + 4 η 2 k g. (7) 6. SW η c -functional satisfies Palais-Smale Condition. 7

8 3.1 η-reducible Solutions 1. there exists finite number of η-basic classes α Spin c (X). 2. If (A, 0) is a solution for the E-L equations, then d F A = d η. (8) Proposition Assume η Ω 2 +(X, ir) is closed. d ω = d η has a unique solution such that dw = 0. So, the equation Proof. The equation d F A = d η implies F A = dd η. Because Imag(d) is orthogonal to the harmonic subspace H, the last equation admits an unique solution in H. Let ω Ω 2 (X, ir) be such solution, so d dw + dd w = dd η d(w η) 2 L + 2 d (w η) 2 L2= 0. (9) Hence, dw = 0 and d w = d η. Proposition Let θ Ω 1 (X, ir) and b 1 (X) = dim R H 1 (X, Z). The space of solutions for d F A = d η, module the G c -action, is diffeomorphic to the Jacobian Torus J X = H1 (X, R) H 1 (X, Z) Rb 1(X) /Z b 1(X). 8

9 3.2 2nd - Variation Formula The linear operator H : T (A,φ) C c T (A,φ) C c induced by the bilinear form H SW (A,φ)((θ, V ), (Λ, W )) =< (θ, V ), H(Λ, W ) > is a hermitian bounded operator because the induced quadratic form is bounded; < (θ, V ), H(θ, V ) >= dθ 2 + θ(φ) < A φ, θ(v ) > < A V, θ(φ) > + A V k g+ φ 2. V < φ, V >2 + < η.v, V >. At (A, 0), < (θ, V ), H(θ, V ) >= dθ 2 + A V 2 + k g 4. V 2 + < η.v, V >. By defining the linear operator L A,η (V ) = A V + k g V + η.v, (10) 4 H is writen at (A, 0) as ( ) d H = d 0, (11) 0 L A,η 9

10 3.3 The Morse-Bott Index of Reducible Solutions The tangent space to the orbit O (A,φ) = {g.(a, φ) g G c }, at (A, φ), is T (A,φ) O (A,φ) = Imag(T ), where T : Ω 0 (X, ir) Ω 1 (X, ir) Γ(S + c ), T (λ) = (dλ, λ.φ). Since T has closed range the local slice for the space B c, at (A, φ), is given by Ker(T ); T : Ω 1 (X, ir) Γ(S + c ) Ω 0 (X, ir), T (θ, V ) = d θ < V, φ >. So, ker(t ) = ker(d ) φ, d : Ω 1 Ω 0. Because (d ) 2 = 0, it can be further decomposed into ker(d : Ω 1 Ω 0 ) = imag(d : Ω 2 Ω 1 ) H 1, where H 1 = {θ Ω 1 (X, ir) dθ = d θ = 0} is the subspace of harmonic 1-forms and is also the tangent space to the Jacobian torus J X at (A, 0). Therefore, ker(h) = T (A,0) J X ker(l A,η ). The Morse-Bott index of (A, 0) J X is equal to the dimension of the largest negative eigenspace of the operator L A,η. Let V λ T (A,0) A c G Γ (S + c ) be the eigenspace associated to the eigenvalue λ. Since L A is an elliptic operator, V λ has finite dimension for all λ; 10

11 Proposition [D-] The Morse index of a redutible solution (A, 0) is finite. Proof. Because L A is an elliptic, self-adjoint operator and its lower eigenvalue λ l is bounded below by [ ( ) ] A V 2 + kg η V 2 dv X 4 g inf V Γ(S c + ) V X 2 dv g λ l, From the spectral theory, the spectrum of L A has the following features; (i) discrete, (ii) each eigenvalue has finite multiplicity, (iii) there are but a finite number of eigenvalues below any given number. Proposition [D-] There exists ν Ω 2 + such that ker(l A,ν ) = {0}. 11

12 4 Kronheimer-Mrowka Blow-Up B c is not a manifold Let V = Γ(S + c ), S(V ) = {φ V ; φ = 1} f σ :A c (0, ) S(V ) A c (V {0}), f σ (A, t, φ) = (A, tφ) f σ diffeo Now, define the space C σ c = [A c (V {0})] f [A c [0, ) S(V )], C σ c = A c S(V ) (12) Note that (f σ ) 1 (0) = C σ c. Gauge Action 1. G c acts freely on Cc σ by the action g.(a, s, φ) = (A + 2g 1 dg, s, g 1 φ) 2. U 1 = {g G c g constant} acts freely on Cc σ = {A} S(V ). 3. Therefore, Bc σ = Cc σ /G c is a manifold and B σ c = (A c /G c ) CP htpy J X CP. Consider the SW σ c -equations F + A s2 σ(φ) = 0 D + A φ = 0. 12

13 Definition: SW σ c : B σ c R given by SWc σ (A, s, φ) = = F + A 2 2 X X [ ] 1 2 F + A s2 σ(φ) 2 + D + A φ 2 dv g = s 2 < F + A, σ(φ) > +s4 8 φ 4 + < φ, A φ + k g 4 φ F + A.φ > dv g }{{} DA 2 So, SWc σ (A, s, φ) = Therefore, X [ F + ] A 2 s 2 < F + A, σ(φ) > +s4 2 8 φ 4 + < φ, (D + A )2 φ > dv g 1. if (A, s, φ) is a critical point of SW σ c, then (A, sφ) is a critical point of SW c. 2. if (A, 0, φ) A c {0} S(V ) is a critical point of SW σ c, then it satisfies the eq. d F A = 0 D A φ = 0, D A = D A D+ A Let r : B σ c J X be a retraction and f : J X R be a Morse function. Define g = f r and perturb the functional; (SW σ c ) g (A, s, φ) = SW σ c (A, s, φ) + g(a, sφ). (re- The critical point set C decomposes into a boundary component C ducibles) and a interior component C 0 (irreducibles). Moreover, C = C 0 C is finite. 13

14 There two kind of critical points in B σ c : (i) C s = -stables (ii) C u = -unstables. C = C 0 C = C 0 C s C u. Definition: 1. Consider the free groups C 0 = a C 0Z < a >, C s = a C sz < a > C u = a C uz < a >. 2. C 0 k = ind(a)=k Z < a >, C s k = ind(a)=k Z < a > C u k = ind(a)=k Z < a >. 3. Č k = C 0 k C s k, Ĉ k = C 0 k C s k, C = C s k C u k+1 Now consider the free generated groups Ĉ k = C 0 k C u k, Č k = C 0 k C s k, Ck = C s k C u k+1 So far, there are technical difficulties to define the boundary operator. Indeed, Bc σ being non compact the number of gradient flow lines connecting critical points a, b, ind(a) ind(b) = 1, may be infinite. 2 = 0? 14

15 Figure 3: X-Ray * * * 15

16 Hong-Schabrun 16