Differential Equivariant Cohomology

Transcription

1 Differential Equivariant Cohomology Corbett Redden Long Island University CW Post Union College Mathematics Conference December 3, 2016 Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

2 Goal Obtain secondary invariants for G-equivariant vector bundles with connection by refining equivariant Chern Weil theory in a natural way. References R., Differential Borel equivariant cohomology via connections [arxiv: ] R., An alternate description of equivariant connections [arxiv: ] Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

3 Chern Weil Given (E, ) Bun U, (M) (E, ) Hermitian vector bundle with connection M smooth manifold Obtain H 2k (M; Z) c k (E) Ω 2k (M) closed H 2k (M; R) c k ( ) Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

4 Cheeger Chern Simons Weil Given (E, ) Bun U, (M) (E, ) Hermitian vector bundle with connection M smooth manifold Obtain č k (E, ) Ȟ 2k (M) H 2k (M; Z) c k (E) Ω 2k (M) closed H 2k (M; R) c k ( ) Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

5 Differential cohomology Definition/Theorem (Cheeger Simons) Functors Ȟ : Man op AbelianGroups satisfying: 0 0 B H n 1 (M; R/Z) H n (M; Z) H n 1 (M; R) Ȟ n (M) H n (M; R) Ω n 1 (M) Ω n 1 (M) Z d Ω n (M) Z 0 0 where diagonals are short exact sequences, Chern Weil homomorphism factors through Ȟ. Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

6 Equivariant cohomology G compact Lie group acting on smooth manifold M Borel HG n (M; ) := Hn (EG G M; ) Weil/Cartan de Rham models Ω G (M) := ( Sg Λg Ω(M) ) basic = ( Sg Ω(M) ) G S 1 g = 2, Λ 1 g = 1 Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

7 Chern Weil Given (E, ) Bun U, (M) (E, ) Hermitian vector bundle with connection M smooth manifold Obtain H 2k (M; Z) c k (E) Ω 2k (M) closed H 2k (M; R) c k ( ) Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

8 Equivariant Chern Weil (Borel, Berline Vergne) G compact Lie group Given (E, ) G-Bun U, (M) (E, ) M G-equivariant Hermitian vector bundle with G-invariant connection on G-manifold Obtain H 2k G (M; Z) c k(e G ) Ω 2k G (M) closed c k ( G ) HG 2k (M; R) Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

9 Differential Equivariant Cohomology Definition/Theorem (R. and Kübel independently) Functors Ȟ G : G-Manop AbelianGroups satisfying: 0 0 B G H n 1 G (M; R/Z) Hn G (M; Z) H n 1 G (M; R) Ȟn G (M) Hn G (M; R) Ω n 1 G (M) (M) Z Ω n 1 G d G Ω n G (M) Z 0 0 where diagonals are short exact sequences, Chern Weil homomorphism factors through Ȟ G. Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

10 Key Idea ȞG (M) is the differential cohomology of the differential quotient stack E G G M = [M / G] Goal for the remainder of the talk Explain what this means and why it is natural from a differential geometric perspective. Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

11 Stacks things that naturally pullback Definitions A groupoid is a category whose morphisms are all isomorphisms. Any set is naturally a groupoid, with only identity morphisms. A stack F Shv Gpd is a sheaf of groupoids on the site of manifolds; a functor Man op Gpd satisfying a sheaf condition. Examples M Man M Shv Gpd, by M(X) := C (X, M) Set Gpd Ω n Shv Gpd, by Ω n (X) := Ω n (X) Set B G(X) := Bun G, (X) = principal G-bundles with connection G (P, Θ) Objects: X Morphisms: bundle isomorphisms preserving connection Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

12 Differential Quotient Stack M G-Man f (P, Θ) Obj: M f is G-equivariant (E G G M)(X) = X Mor: preserve Θ, compatible with f For M = pt, then E G G pt = B G. Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

13 Maps between stacks Yoneda Examples Map(M, F) = Shv Gpd (M, F) = F(M) Stack maps generalize smooth maps between manifolds Man(M, N) = C (M, N) Map(M, N) Shv Gpd (M, N) = N(M)? =? = ω Ω n (M) is equivalent to M ω Ω n = (P, Θ) Bun G, (M) is equivalent to M (P,Θ) B G Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

14 Important special case: M = pt Differential Forms Ω n (B G) :=? Map(B G, Ω n ) What is a differential form ω Ω n (B G)? B G(X) ω Ω n (X) f f ω B G(Y ) Ω n (Y ) In other words, to any (P, Θ) X, we get ω(p, Θ) Ω n (X), if (P, Θ) = (P, Θ ), then ω(p, Θ) = ω(p, Θ ), ω(f (P, Θ)) = f (ω(p, Θ)). Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

15 Important special case: M = pt (continued) Differential Forms Ω n (B G) = Ω n (B G) := Map(B G, Ω n ) Observation Chern Weil construction gives homomorphism Ω G (pt) = (Sg ) G = Ω(B G) = Ω(E G G pt). Theorem (Freed Hopkins) This map is an isomorphism. Goldilocks Ω(BG) - too big Ω(BG) - too small (Ω n (BG) = 0 for n > 0) Ω(B G) - just right Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

16 Equivariant forms: General case Given (P, Θ) X f M, ω = αβγ ( S i g Λ j g Ω k (M) ) basic Obtain Theorem (Freed Hopkins) α(ω i ) β(θ i ) f γ Ω(P) basic = Ω(X) Ω G (M) = Ω(E G G M). Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

17 Differential equivariant cohomology Definition: (M G-Man) Ȟ n G (M) := Ȟn (E G G M) := Map(E G G M, Ȟn ) (some details being ignored here) In other words, an element ˇλ Ȟn G (M) is a construction: to any if (P, Θ) X ϕ f M (P 1, Θ 1 ) (P 2, Θ 2 ) M ϕ associate ˇλ(P, Θ, f ) Ȟn (X), X 1 X 2 ˇλ(P 1, Θ 1, f ϕ) = ϕ ˇλ(P2, Θ 2, f ) Ȟn (X 1 ). f, then Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

18 Differential Equivariant Cohomology (continued) Abbreviate E G G M by M / G, Theorem (R.) The following diagram B H n 1 (M / G; R/Z) H n (M / G; Z) H n 1 (M / G; R) Ȟ n (M / G) H n (M / G; R) Ω n 1 (M / G) Ω n 1 (M / G) Z d Ω n (M / G) Z 0 0 Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

19 Differential Equivariant Cohomology (continued) Abbreviate E G G M by M / G, Theorem (R.)... naturally isomorphic to 0 0 B G H n 1 G (M; R/Z) Hn G (M; Z) H n 1 G (M; R) Ȟn G (M) Hn G (M; R) Ω n 1 G (M) (M) Z Ω n 1 G d G Ω n G (M) Z 0 0 Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20

20 Equivariant Cheeger Chern Simons Theorem (R.) There is a natural equivalence of categories {G-equivariant (Herm vector bundles w/ conn on M)} = Map(E G G M, B U) Given (E, ) G-Bun U, (M), E G G M (E, ) G B U HG (M) = Ȟ (E G G M) (E, ) G Ȟ (B U) č k (E G, G ) refines the equivariant Chern Weil homomorphism. č k Corbett Redden (LIU Post) Differential Equivariant Cohomology December 3, / 20