Orientation Problems in 7-dimensional Gauge Theory Markus Upmeier University of Oxford Talk based on: 1) M. Upmeier, A categorified excision principle for elliptic symbol families (soon) 2) D. Joyce, Y. Tanaka and M. Upmeier, On orientations for gauge-theoretic moduli spaces , arXiv:1811.01096. 3) D. Joyce and M. Upmeier, Canonical orientations for moduli spaces of G 2 -instantons with gauge group SU( m ), arXiv:1811.02405. January 28, 2019 1 / 26
Outline Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions 2 / 26
Outline Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions 3 / 26
Twisted Dirac Operators Setup 1. Compact 7 -dimensional spin manifold ( X , g ) S ց X, connection ∇ / S 2. Real spinor bundle / 3. Clifford multiplication c : TX × / S → / S 4. Lie group G 5. G-principal bundle P ց X 6. Ad P := P × G g ց X 4 / 26
Twisted Dirac Operators Setup 1. Compact 7 -dimensional spin manifold ( X , g ) S ց X, connection ∇ / S 2. Real spinor bundle / 3. Clifford multiplication c : TX × / S → / S 4. Lie group G 5. G-principal bundle P ց X 6. Ad P := P × G g ց X Definition Let ∇ P ∈ Ω 1 ( P ; g ) be a connection on P . The twisted Diracian is D ∇ Ad P : C ∞ ( / → C ∞ ( / / S ⊗ R Ad P ) − S ⊗ R Ad P ) , 7 � c ( e i , ∇ / S ⊗ Ad P s �− → s ) e i i =1 4 / 26
Example in 7D: Manifolds with G 2 -Structure Definition A topological G 2 -structure on ( X 7 , g ) is a structure of normed algebras on O := R ⊕ TX with two-sided unit 1 = (1 , 0): bilinear c : O × O − − − − → O , � v · w � = � v � · � w � . 5 / 26
Example in 7D: Manifolds with G 2 -Structure Definition A topological G 2 -structure on ( X 7 , g ) is a structure of normed algebras on O := R ⊕ TX with two-sided unit 1 = (1 , 0): bilinear c : O × O − − − − → O , � v · w � = � v � · � w � . Adjoint of c | TX is φ ∈ Ω 3 ( X ); Let ψ := ∗ φ ∈ Ω 4 ( X ). 1. Every manifold with G 2 -structure is spin / S := O . 2. Clifford multiplication is c . S = ∇ R ⊕ ∇ LC . 3. For a torsion-free G 2 -structure ∇ / 5 / 26
Example in 7D: Manifolds with G 2 -Structure Definition A topological G 2 -structure on ( X 7 , g ) is a structure of normed algebras on O := R ⊕ TX with two-sided unit 1 = (1 , 0): bilinear c : O × O − − − − → O , � v · w � = � v � · � w � . Adjoint of c | TX is φ ∈ Ω 3 ( X ); Let ψ := ∗ φ ∈ Ω 4 ( X ). 1. Every manifold with G 2 -structure is spin / S := O . 2. Clifford multiplication is c . S = ∇ R ⊕ ∇ LC . 3. For a torsion-free G 2 -structure ∇ / General 7-dimensional spin manifold, with preferred spinor. 5 / 26
Example in 7D: Continued Connection ∇ P induces d ∇ P : Ω k ( X , Ad P ) − → Ω k +1 ( X , Ad P ). Proposition D ∇ Ad P Assume ∇ LC φ = 0 . Then the twisted Diracian / equals � 0 d ∗ � ∇ P L ∇ P = d ∇ P ∗ ( ψ ∧ d ∇ P ) Ω 0 ( X , Ad P ) ⊕ Ω 1 ( X , Ad P ) − → Ω 0 ( X , Ad P ) ⊕ Ω 1 ( X , Ad P ) . 6 / 26
Example in 7D: Continued Connection ∇ P induces d ∇ P : Ω k ( X , Ad P ) − → Ω k +1 ( X , Ad P ). Proposition D ∇ Ad P Assume ∇ LC φ = 0 . Then the twisted Diracian / equals � 0 d ∗ � ∇ P L ∇ P = d ∇ P ∗ ( ψ ∧ d ∇ P ) Ω 0 ( X , Ad P ) ⊕ Ω 1 ( X , Ad P ) − → Ω 0 ( X , Ad P ) ⊕ Ω 1 ( X , Ad P ) . Corollary The tangent space at ∇ P of the moduli space of G 2 -instantons F ∇ Ad P ∧ ψ = 0 is described by the Diracian / D ∇ Ad P . d ∇ P d ∇ P ∧ ψ d ∇ P Ω 0 ( X ; g P ) → Ω 1 ( X ; g P ) → Ω 6 ( X ; g P ) → Ω 7 ( X ; g P ) − − − − − − − − 6 / 26
Today’s Problem Let A P := { connections ∇ P on P → X } . By twisting / D using each ∇ P ∈ A P get an A P -family of differential operators on X . Questions D ∇ Ad P ◮ Equivariant orientability of � ∇ P ∈A P Det / ց A P ? − → Can be answered using index theory. ◮ How do we pick orientations, canonically, fixing perhaps some topological data on X ? 7 / 26
Today’s Problem Let A P := { connections ∇ P on P → X } . By twisting / D using each ∇ P ∈ A P get an A P -family of differential operators on X . Questions D ∇ Ad P ◮ Equivariant orientability of � ∇ P ∈A P Det / ց A P ? − → Can be answered using index theory. ◮ How do we pick orientations, canonically, fixing perhaps some topological data on X ? Theorem (Joyce–U. 2018) Let ( X , φ 3 , ψ 4 = ∗ φ φ ) be a closed G 2 -manifold. A flag structure F on X determines, for every principal SU( n ) -bundle P → X, an orientation of the moduli space M irr P of G 2 -instantons { A ∈ A irr � P | F A ∧ ψ = 0 } Aut( P ) . 7 / 26
Outline Orientation Problems for Twisted Diracians Determinants, Symbols, and Excision Canonical Orientations in Seven Dimensions 8 / 26
Determinant Line Bundle and Orientations Definition Let { D y } y ∈ Y be a Y -family of real Fredholm operators. The Quillen determinant line bundle is � Λ top (Ker D y ) ⊗ Λ top (Coker D y ) ∗ Det { D y } := ց Y . y ∈ Y 9 / 26
Determinant Line Bundle and Orientations Definition Let { D y } y ∈ Y be a Y -family of real Fredholm operators. The Quillen determinant line bundle is � Λ top (Ker D y ) ⊗ Λ top (Coker D y ) ∗ Det { D y } := ց Y . y ∈ Y Definition The orientation cover is Or { D y } y ∈ Y := (Det { D y } \ { zero section } ) � R > 0 ց Y . Represents π 1 Fred R = Z 2 9 / 26
Determinant Line Bundle and Orientations Definition Let { D y } y ∈ Y be a Y -family of real Fredholm operators. The Quillen determinant line bundle is � Λ top (Ker D y ) ⊗ Λ top (Coker D y ) ∗ Det { D y } := ց Y . y ∈ Y Definition The orientation cover is Or { D y } y ∈ Y := (Det { D y } \ { zero section } ) � R > 0 ց Y . Represents π 1 Fred R = Z 2 Today’s problem D ∇ Ad P → P ։ X 7 , trivialize Or { / Given G ֒ } ∇ P ∈A P ց A P , canonically in terms of data on X . 9 / 26
Index Theory Definition 1. For D Fredholm, ind D := dim R Ker D − dim R Coker D ∈ Z 10 / 26
Index Theory Definition 1. For D Fredholm, ind D := dim R Ker D − dim R Coker D ∈ Z 2. For Y -family { D y } y ∈ Y , ind D ∈ KO 0 ( Y ). Up to isomorphism w 1 (ind D ) = [Or { D y } ] ∈ H 1 ( Y ; Z 2 ) 10 / 26
Index Theory Definition 1. For D Fredholm, ind D := dim R Ker D − dim R Coker D ∈ Z 2. For Y -family { D y } y ∈ Y , ind D ∈ KO 0 ( Y ). Up to isomorphism w 1 (ind D ) = [Or { D y } ] ∈ H 1 ( Y ; Z 2 ) Properties ◮ Natural in Y ◮ If { D t } t ∈ [0 , 1] : D 0 ≃ D 1 through Fred , then ind D 0 = i ∗ 0 ind D = i ∗ 1 ind D = ind D 1 ◮ ind( D 1 ⊕ D 2 ) = ind D 1 + ind D 2 ◮ ind D † = − ind D 10 / 26
Elliptic Symbol Families Definition Family of Elliptic ( ψ )DOs D y : C ∞ ( X , E y ) → C ∞ ( X , F y ) on X determined, up to convex choice, by elliptic symbol family ∼ = p λ · ξ, y = λ m p ξ, y , p ξ, y = σ ξ, y ( D ): E y − − → F y , for all 0 � = ξ ∈ T ∗ X , y ∈ Y , λ > 0. Here m ∈ R is the order. 11 / 26
Elliptic Symbol Families Definition Family of Elliptic ( ψ )DOs D y : C ∞ ( X , E y ) → C ∞ ( X , F y ) on X determined, up to convex choice, by elliptic symbol family ∼ = p λ · ξ, y = λ m p ξ, y , p ξ, y = σ ξ, y ( D ): E y − − → F y , for all 0 � = ξ ∈ T ∗ X , y ∈ Y , λ > 0. Here m ∈ R is the order. Example E ց X × Y vector bundle, X 7 spin, c : TX ⊗ / S → / S Clifford multiplication. Let p ξ, y := c ξ ⊗ id E y for y ∈ Y . 11 / 26
Elliptic Symbol Families Definition Family of Elliptic ( ψ )DOs D y : C ∞ ( X , E y ) → C ∞ ( X , F y ) on X determined, up to convex choice, by elliptic symbol family ∼ = p λ · ξ, y = λ m p ξ, y , p ξ, y = σ ξ, y ( D ): E y − − → F y , for all 0 � = ξ ∈ T ∗ X , y ∈ Y , λ > 0. Here m ∈ R is the order. Example E ց X × Y vector bundle, X 7 spin, c : TX ⊗ / S → / S Clifford multiplication. Let p ξ, y := c ξ ⊗ id E y for y ∈ Y . Further properties ◮ ind p = ind D well-defined, for any σ ( D ) = p ◮ i : U ֒ → X open embedding, p compactly supported on U = ⇒ i ! (ind p ) = ind i ! p 11 / 26
Categorical Index Calculus For Y -family of elliptic symbols p = { p ξ, y } y ∈ Y on X have object Cov gr Or p := σ ( D )= p Or D ց Y lim in Z 2 ( Y ) 12 / 26
Categorical Index Calculus For Y -family of elliptic symbols p = { p ξ, y } y ∈ Y on X have object Cov gr Or p := σ ( D )= p Or D ց Y lim in Z 2 ( Y ) Properties become structure maps in Cov gr Z 2 ( Y ) 1. { p t } : p 0 ≃ p 1 = ⇒ Or p 0 → Or q 1 2. Or( p ⊕ q ) → (Or p ) ⊗ (Or q ), Or p † → (Or p ) ∗ 3. If φ : X − → X + diffeomorphism with φ ∗ p + = p − , then Or( φ ): Or p − − → Or p + 4. For i : U ֒ → X open embedding, p compactly supported on U i ! : Or( p ) − → Or( i ! p ) 12 / 26
Recommend
More recommend
