Index theorems on anti-self-dual orbifolds Jeff Viaclovsky August - PowerPoint PPT Presentation

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Index theorems on anti-self-dual orbifolds Jeff Viaclovsky August 6, 2012, Kyoto

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces ALE = Asymptotically Locally Euclidean Definition A space ( X 4 , g ) is ALE of order τ if there exists a finite subgroup Γ ⊂ SO(4) acting freely on R 4 \ { 0 } , and a diffeomorphism φ : X \ K → ( R 4 \ B (0 , R )) / Γ where K is a compact subset of X , satisfying ( φ ∗ g ) ij = δ ij + O ( r − τ ) , and ∂ | k | ( φ ∗ g ) ij = O ( r − τ − k ) , as r → ∞ .

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Orbifolds Definition A Riemannian orbifold ( M 4 , g ) is a topological space which is a smooth manifold of dimension 4 with a smooth Riemannian metric away from finitely many singular points. At a singular point p , M is locally diffeomorphic to a cone C on S 3 / Γ , where Γ ⊂ SO(4) is a finite subgroup acting freely on S 3 . Furthermore, at such a singular point, the metric is locally the quotient of a smooth Γ -invariant metric on B 4 under the orbifold group Γ .

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Conformal blow-up Given a compact Riemannian orbifold ( ˆ M, ˆ g ) with non-negative scalar curvature, one can use the Green’s function for the conformal Laplacian G p to associate with any point p a non-compact scalar-flat orbifold by ( M \ { p } , g p = G 2 p ˆ g ) . A coordinate system at infinity arises from using inverted normal coordinates in the metric g in a neighborhood of the point p ,

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Conformal blow-down ALE space ( X, g ) choose a conformal factor u : X → R + such that u = O ( ρ − 2 ) as ρ → ∞ . The space ( X, u 2 g ) then compactifies to a C 1 ,α orbifold. In the anti-self-dual case, there moreover exists a C ∞ -orbifold conformal compactification ( ˆ X, ˆ g ) with positive Yamabe invariant. Similarly, a scalat-flat ASD ALE space is ALE of order 2. (Tian-Viaclovsky, Streets, Ache-Viaclovsky).

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Self-dual or Anti-self-dual metrics ( M 4 , g ) oriented.   W + + R 12 I E       R = .     W − + R   E 12 I   W + = 0 is Anti-self-dual (ASD). W − = 0 is Self-dual (SD). Conformally invariant.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces ASD Deformation Complex K g D Γ( T ∗ M ) → Γ( S 2 0 ( T ∗ M )) → Γ( S 2 0 (Λ 2 − − + )) , where K g is the conformal Killing operator defined by ( K g ( ω )) ij = ∇ i ω j + ∇ j ω i − 1 2( δω ) g, with δω = ∇ i ω i , S 2 0 ( T ∗ M ) denotes traceless symmetric tensors, and D = ( W + ) ′ g is the linearized self-dual Weyl curvature operator.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Kuranishi map The cohomology groups of the above complex yield information about the local structure of the moduli space of anti-self-dual conformal classes. There is a map Ψ : H 1 ( M, g ) → H 2 ( M, g ) called the Kuranishi map which is equivariant with respect to the action of H 0 , and the moduli space of anti-self-dual conformal structures near g is locally isomorphic to Ψ − 1 (0) /H 0 . Therefore, if H 2 = 0 , the moduli space is locally isomorphic to H 1 /H 0 .

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Index The analytical index is given by Ind ( M, g ) = dim( H 0 ( M, g )) − dim( H 1 ( M, g )) + dim( H 2 ( M, g )) , The index is given in terms of topology via the Atiyah-Singer index theorem Ind ( M, g ) = 1 2(15 χ ( M ) + 29 τ ( M )) , where χ ( M ) is the Euler characteristic and τ ( M ) is the signature of M . On an orbifold, there are correction terms to this formula, depending upon the group action.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Subgroups of SU (2) • Type A n , n ≥ 1 : Γ the cyclic group Z n +1 , � exp 2 πip/ ( n +1) � 0 , 0 ≤ p ≤ n. (1) exp − 2 πip/ ( n +1) 0 • Type D n , n ≥ 3 : Γ the binary dihedral group D ∗ n − 2 of order 4( n − 2) . This is generated by e πi/ ( n − 2) and ˆ j , both acting on the left.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Subgroups of SU (2) • Type E 6 : Γ = T ∗ , the binary tetrahedral group of order 24 , double cover of A (4) . • Type E 7 : Γ = O ∗ , the binary octohedral group of order 48 , double cover of S (4) . • Type E 8 : Γ = I ∗ , the binary icosahedral group of order 120 , double cover of A (5) .

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Index for SU (2) subgroups Theorem Let ( ˆ M, ˆ g ) be a compact anti-self-dual orbifold with a single orbifold point p with orbifold group Γ orientation-preserving conjugate to type A n with n ≥ 1 , or D n with n ≥ 3 , or E n with n = 6 , 7 , 8 . Then g ) = 1 Ind ( ˆ 2(15 χ ( ˆ M ) + 29 τ ( ˆ M, ˆ M )) − 4 n. (2)

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Hyperk¨ ahler ALE spaces Proposition Let ( X, g ) be a hyperk¨ ahler ALE space of type A n for n ≥ 1 , type D n for n ≥ 3 , or type E n for n = 6 , 7 , 8 . For − 4 ≤ ǫ < 0 , let H 1 ǫ ( X, g ) denote the space of traceless symmetric 2 -tensors h ∈ S 2 0 (( T ∗ X )) satisfying ( W + ) ′ g ( h ) = 0 , δ g ( h ) = 0 , with h = O ( ρ ǫ ) as ρ → ∞ , Then H 1 ǫ ( X, g ) = H 1 − 4 ( X, g ) , and using the isomorphism S 2 0 ( T ∗ M ) = Λ 2 + ⊗ Λ 2 − , H 1 − 4 ( X, g ) has a basis { ω I ⊗ ω − j , ω J ⊗ ω − j , ω K ⊗ ω − j } , where { ω − j , j = 1 , . . . , n = dim( H 2 ( M )) } is a basis of the space of L 2 harmonic 2 -forms.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Proof of Proposition Since Ricci-flat, use that B ′ ( h ) = ∆ 2 L h = D ∗ D h, to get that infinitesimal ASD deformations are the same as infinitesimal Einstein deformations. Identification of the decaying kernel done in Biquard, Biquard-Rollin, also implicit in earlier work of Itoh.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Index comparison Proposition Let ( X, g ) be an anti-self-dual ALE metric with group Γ at infinity, and let ( ˆ X, ˆ g ) be the orbifold conformal compactification. Then for − 2 < δ < 0 , we have Ind ( ˆ g ) = − dim( H 1 δ ( X, g )) + dim( H 2 X, ˆ − 2 − δ ( X, g )) + dim( H 0 ( R 4 / Γ)) .

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Idea of proof of Index comparison From conformal invariance, together with a removable singularity theorem of Ache-Viaclovsky, dim( H 2 − 2 − δ ( X, g )) = dim( H 2 ( ˆ X, ˆ g )) . So the identity becomes dim( H 1 δ ( X, g )) = dim( H 1 ( ˆ X, ˆ g )) + { dim( H 0 ( R 4 / Γ)) − dim( H 0 ( ˆ X, ˆ g )) } . The term in braces arises from considering solutions of � ω = δ K ω = 0 , which do not extend to global conformal Killing fields.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces Completion of proof Kawasaki shows that there must be a formula of the form Ind ( M, g ) = 1 2(15 χ ( M ) + 29 τ ( M )) + N where the corrrection term N depending only upon the oriented conjugacy class of the group action, and NOT on the space. Let ( X, g ) be a Kronheimer ALE metric with orbifold compactification ( ˆ X, ˆ g ) . Using the above index comparison, we can find N for group action orientation-reversing conjugate to ADE-type. Next, consider a football metric S 4 / Γ . Then dim H 0 ( S 4 / Γ) = 15 + N + N ′ gives the correction term for orientation-preserving conjugate to ADE-type.

Introduction Subgroups of SU (2) Cyclic quotient singularities Weighted Projective Spaces LeBrun negative mass metrics Define dr 2 1 + Ar − 2 + Br − 4 + r 2 � 2 + (1 + Ar − 2 + Br − 4 ) σ 2 � σ 2 1 + σ 2 g LB = , 3 where r is a radial coordinate, and { σ 1 , σ 2 , σ 3 } is a left-invariant coframe on S 3 = SU(2) , and A = n − 2 , B = 1 − n . Redefine the r 2 = r 2 − 1 , and attach a CP 1 at ˆ radial coordinate to be ˆ r = 0 . After taking a quotient by Z n , with action given by the diagonal action ( z 1 , z 2 ) �→ exp 2 πip/n ( z 1 , z 2 ) , 0 ≤ p ≤ n − 1 , the metric then extends smoothly over the added CP 1 , is ALE at infinity, and is diffeomorphic to O ( − n ) .