Prequatization, differential cohomology and the genus integration - PowerPoint PPT Presentation

Prequatization, differential cohomology and the genus integration Rui Loja Fernandes Department of Mathematics University of Illinois at Urbana-Champaign, USA 2nd Workshop S˜ ao Paulo J. of Math. Sci. USP, November 2019

This talk is an exercise based on: ◮ Ivan Contreras & RLF, “Genus Integration, Abelianization and Extended Monodromy”, arXiv:1805.12043 . ◮ Discussions with Alejandro Cabrera on obstructions to strict deformation quantization.

This talk is an exercise based on: ◮ Ivan Contreras & RLF, “Genus Integration, Abelianization and Extended Monodromy”, arXiv:1805.12043 . ◮ Discussions with Alejandro Cabrera on obstructions to strict deformation quantization. Aim: ◮ relationship between the classical prequantization condition and integration of Lie algebroids

This talk is an exercise based on: ◮ Ivan Contreras & RLF, “Genus Integration, Abelianization and Extended Monodromy”, arXiv:1805.12043 . ◮ Discussions with Alejandro Cabrera on obstructions to strict deformation quantization. Aim: ◮ relationship between the classical prequantization condition and integration of Lie algebroids Wait!!! Wasn’t this solved a long time ago?!

This talk is an exercise based on: ◮ Ivan Contreras & RLF, “Genus Integration, Abelianization and Extended Monodromy”, arXiv:1805.12043 . ◮ Discussions with Alejandro Cabrera on obstructions to strict deformation quantization. Aim: ◮ relationship between the classical prequantization condition and integration of Lie algebroids Wait!!! Wasn’t this solved a long time ago?! • M. Crainic, Prequantization and Integrability, J. Sympl. Geom. (2004)

This talk is an exercise based on: ◮ Ivan Contreras & RLF, “Genus Integration, Abelianization and Extended Monodromy”, arXiv:1805.12043 . ◮ Discussions with Alejandro Cabrera on obstructions to strict deformation quantization. Aim: ◮ relationship between the classical prequantization condition and integration of Lie algebroids Wait!!! Wasn’t this solved a long time ago?! • M. Crainic, Prequantization and Integrability, J. Sympl. Geom. (2004) ... but this paper assumes manifold is 1-connected .

The prequantization condition • ω ∈ Ω 2 ( M ) – closed 2-form ◮ Group of periods of ω : �� � Per( ω ) := ω : σ ∈ H 2 ( M , Z ) ⊂ ( R , +) σ ◮ Group of spherical periods of ω : �� � SPer( ω ) := ω : σ ∈ π 2 ( M ) ⊂ Per( ω ) σ

The prequantization condition • ω ∈ Ω 2 ( M ) – closed 2-form ◮ Group of periods of ω : �� � Per( ω ) := ω : σ ∈ H 2 ( M , Z ) ⊂ ( R , +) σ ◮ Group of spherical periods of ω : �� � SPer( ω ) := ω : σ ∈ π 2 ( M ) ⊂ Per( ω ) σ Definition ( M , ω ) satisfies the prequantization condition if Per( ω ) ⊂ R is a discrete subgroup, i.e., if there exists a ∈ R such that Per( ω ) = a Z ⊂ R . One can also consider the weaker requirement that SPer( ω ) ⊂ R is a discrete subgroup. One of our aims is to understand the differences...

The prequantization condition Notation: S 1 a := R / a Z Note that one can have a = 0 in which case S 1 0 = R .

The prequantization condition Notation: S 1 a := R / a Z Note that one can have a = 0 in which case S 1 0 = R . Theorem (Souriau 1967, Kostant 1970) Let ω ∈ Ω 2 cl ( M ) . There exists a principal S 1 a -bundle π : P → M with connection θ ∈ Ω 1 ( P , R ) satisfying π ∗ ω = d θ if and only if Per( ω ) ⊂ a Z .

The prequantization condition Notation: S 1 a := R / a Z Note that one can have a = 0 in which case S 1 0 = R . Theorem (Souriau 1967, Kostant 1970) Let ω ∈ Ω 2 cl ( M ) . There exists a principal S 1 a -bundle π : P → M with connection θ ∈ Ω 1 ( P , R ) satisfying π ∗ ω = d θ if and only if Per( ω ) ⊂ a Z . ◮ What are the possible such principal S 1 a -bundle π : P → M with connection θ ?

The prequantization condition Notation: S 1 a := R / a Z Note that one can have a = 0 in which case S 1 0 = R . Theorem (Souriau 1967, Kostant 1970) Let ω ∈ Ω 2 cl ( M ) . There exists a principal S 1 a -bundle π : P → M with connection θ ∈ Ω 1 ( P , R ) satisfying π ∗ ω = d θ if and only if Per( ω ) ⊂ a Z . ◮ What are the possible such principal S 1 a -bundle π : P → M with connection θ ? The answer is provided by differential cohomology .

Differential cohomology (Cheeger & Simons) Definition A differential character of degree k on M relative to a Z is a group homomorphism χ : Z k ( M ) → S 1 a for which there exists a closed form ω ∈ Ω k +1 ( M ) such that: cl � χ ( ∂σ ) = ω (mod a Z ) , ∀ σ ∈ C k +1 ( M ) . σ ˆ H k ( M , S 1 a ) = { differential characters of degree k }

Differential cohomology (Cheeger & Simons) Definition A differential character of degree k on M relative to a Z is a group homomorphism χ : Z k ( M ) → S 1 a for which there exists a closed form ω ∈ Ω k +1 ( M ) such that: cl � χ ( ∂σ ) = ω (mod a Z ) , ∀ σ ∈ C k +1 ( M ) . σ ˆ H k ( M , S 1 a ) = { differential characters of degree k } ◮ ω is uniquely determined by the differential character χ and Per( ω ) ⊂ a Z : δ 1 : ˆ H k ( M , S 1 a ) → Ω k +1 a Z ( M ) , χ �→ ω.

Differential cohomology (Cheeger & Simons) Definition A differential character of degree k on M relative to a Z is a group homomorphism χ : Z k ( M ) → S 1 a for which there exists a closed form ω ∈ Ω k +1 ( M ) such that: cl � χ ( ∂σ ) = ω (mod a Z ) , ∀ σ ∈ C k +1 ( M ) . σ ˆ H k ( M , S 1 a ) = { differential characters of degree k } ◮ ω is uniquely determined by the differential character χ and Per( ω ) ⊂ a Z : δ 1 : ˆ H k ( M , S 1 a ) → Ω k +1 a Z ( M ) , χ �→ ω. ◮ Choose lift ˜ χ : C k ( M ) → R and define c : C k +1 ( M ) → R by: � c ( σ ) := ω − ˜ χ ( ∂σ ) . σ Then c ∈ Z k +1 ( M , a Z ) and [ c ] ∈ H k +1 ( M , a Z ) does not depend on ˜ χ : δ 2 : ˆ H k ( M , S 1 a ) → H k +1 ( M , a Z ) , χ �→ [ c ] .

Differential cohomology If r : H k +1 ( M , a Z ) → H k +1 ( M , R ) is the natural map, then: r ([ c ]) = [ ω ].

Differential cohomology If r : H k +1 ( M , a Z ) → H k +1 ( M , R ) is the natural map, then: r ([ c ]) = [ ω ]. Theorem (Cheeger & Simons, 1985) There is a short exact sequence: ( δ 1 ,δ 2 ) � R k +1 ( M , a Z ) � ˆ H k ( M , R ) / r ( H k ( M , a Z )) H k ( M , S 1 a ) where: R • ( M , a Z ) = { ( ω, u ) ∈ Ω • a Z ( M ) × H • ( M , a Z ) : [ ω ] = r ( u ) } . - Differential cohomology provides a refinement of integral cohomology and differential forms with a Z -periods. - Differential cohomology has a graded ring structure: ∗ : ˆ H k ( M , S 1 a ) × ˆ H l ( M , S 1 a ) → ˆ H k + l +1 ( M , S 1 a ) and ( δ 1 , δ 2 ) is a ring homomorphism.

Differential cohomology in degree 1

Differential cohomology in degree 1 Example π : P → M be a principal S 1 a -bundle with connection θ ∈ Ω 1 ( P , R ) and curvature ω ∈ Ω 2 ( M ): π ∗ ω = d θ.

Differential cohomology in degree 1 Example π : P → M be a principal S 1 a -bundle with connection θ ∈ Ω 1 ( P , R ) and curvature ω ∈ Ω 2 ( M ): π ∗ ω = d θ. Holonomy of the connection along a loop γ gives an element: χ ( γ ) ∈ S 1 a .

Differential cohomology in degree 1 Example π : P → M be a principal S 1 a -bundle with connection θ ∈ Ω 1 ( P , R ) and curvature ω ∈ Ω 2 ( M ): π ∗ ω = d θ. Holonomy of the connection along a loop γ gives an element: χ ( γ ) ∈ S 1 a . Extend χ to any cycle γ + ∂σ ∈ Z 1 ( M ) by: � χ ( γ + ∂σ ) := χ ( γ ) + ω (mod a Z a ) . σ

Differential cohomology in degree 1 Example π : P → M be a principal S 1 a -bundle with connection θ ∈ Ω 1 ( P , R ) and curvature ω ∈ Ω 2 ( M ): π ∗ ω = d θ. Holonomy of the connection along a loop γ gives an element: χ ( γ ) ∈ S 1 a . Extend χ to any cycle γ + ∂σ ∈ Z 1 ( M ) by: � χ ( γ + ∂σ ) := χ ( γ ) + ω (mod a Z a ) . σ This defines a differential character χ ∈ ˆ H 1 ( M , S 1 a ) with: ◮ δ 1 χ = ω ∈ Ω 2 a Z ( M ); ◮ δ 2 χ ∈ H 2 ( M , a Z ) the (integral) Chern class of the bundle.

Differential cohomology in degree 1 Example π : P → M be a principal S 1 a -bundle with connection θ ∈ Ω 1 ( P , R ) and curvature ω ∈ Ω 2 ( M ): π ∗ ω = d θ. Holonomy of the connection along a loop γ gives an element: χ ( γ ) ∈ S 1 a . Extend χ to any cycle γ + ∂σ ∈ Z 1 ( M ) by: � χ ( γ + ∂σ ) := χ ( γ ) + ω (mod a Z a ) . σ This defines a differential character χ ∈ ˆ H 1 ( M , S 1 a ) with: ◮ δ 1 χ = ω ∈ Ω 2 a Z ( M ); ◮ δ 2 χ ∈ H 2 ( M , a Z ) the (integral) Chern class of the bundle. Note: one can have δ 1 χ = δ 2 χ = 0 with χ � = 0 (e.g., if M = S 1 ).

� � � Differential cohomology in degree 1 Theorem (Cheeger & Simons, 1985) � � principal S 1 ˆ a -bundles H 1 ( M , S 1 a ) with connection ≃ � � isomorphism classes of principal S 1 a -bundles with connection