The Chern Character in the Simplicial de Rham Complex Naoya Suzuki Abstract On the basis of Dupont’s work, we exhibit a cocycle in the simplicial de Rham complex which represents the Chern character. We also prove the related conjecture due to Brylinski. This gives a way to construct a cocycle in a local truncated complex.

1

Introduction

It is well-known that there is one-to-one correspondence between the characteristic classes of G-bundles and the elements in the cohomology ring of the classifying space BG. So it is important to investigate H ∗ (BG) in research on the characteristic classes. However, in general BG is not a manifold so we can not adapt the usual de Rham theory on it. To overcome this problem, a total complex of a double complex Ω∗ (N G(∗)) which is associated to a simplicial manifold {N G(∗)} is often used. In brief, {N G(∗)} is a sequence of manifolds {N G(p) = Gp }p=0,1,··· together with face operators εi : N G(p) → N G(p − 1) for i = 0, · · · , p satisfying relations εi εj = εj−1 εi for i < j (The standard definition also involves degeneracy operators but we do not need them here). The cohomology ring of Ω∗ (N G(∗)) is isomorphic to H ∗ (BG) so we can use this complex as a candidate of the de Rham complex on BG. In [5], Dupont introduced another double complex A∗,∗ (N G) on N G and showed the cohomology ring of its total complex A∗ (N G) is also isomorphic to H ∗ (BG). Then he used it to construct a homomorphism from I ∗ (G), the G-invariant polynomial ring over Lie algebra G, to H ∗ (BG) for a classical Lie group G.

1

The images of this homomorphism in Ω∗ (N G(∗)) are called the BottShulman-Stasheff forms. The main purpose of this paper is to exhibit these cocycles precisely when they represent the Chern characters. In addition, we also show that the conjecture due to Brylinski in [3] is true. This gives a way to construct a cocycle in a local truncated complex [σ

2

Review of the universal Chern-Weil Theory

In this section we recall the universal Chern-Weil theory following [6]. For ¯ and simplicial Gany Lie group G, we have simplicial manifolds N G, N G ¯ → N G as follows: bundle γ : N G q−times

}| { z N G(q) = G × · · · × G 3 (h1 , · · · , hq ) : face operators εi : N G(q) → N G(q − 1)   i=0 (h2 , · · · , hq ) εi (h1 , · · · , hq ) = (h1 , · · · , hi hi+1 , · · · , hq ) i = 1, · · · , q − 1   (h1 , · · · , hq−1 ) i = q. q+1−times

z }| { ¯ N G(q) = G × · · · × G 3 (g1 , · · · , gq+1 ) : ¯ ¯ − 1) face operators ε¯i : N G(q) → N G(q ε¯i (g0 , · · · , gq ) = (g0 , · · · , gi−1 , gi+1 , · · · , gq ) 2

i = 0, 1, · · · , q.

¯ → N G as γ(g0 , · · · , gq ) = (g0 g1 −1 , · · · , gq−1 gq −1 ). We define γ : N G For any simplicial manifold X = {X∗ }, we can associate a topological space k X k called the fat realization. Since any G-bundle π : E → M can be realized as a pull-back of the fat realization of γ, k γ k is the universal bundle EG → BG [8]. Now we construct a double complex associated to a simplicial manifold. Definition 2.1. For any simplicial manifold {X∗ } with face operators {ε∗ }, we define a double complex as follows: Ωp,q (X) := Ωq (Xp ) Derivatives are: p+1 X 0 (−1)i ε∗i , d :=

d00 := (−1)p × the exterior differential on Ω∗ (Xp ).

i=0

¯ the following holds [2] [6] [7]. For N G and N G Theorem 2.1. There exist ring isomorphisms ¯ ∼ H(Ω∗ (N G)) ∼ H(Ω∗ (N G)) = H ∗ (BG), = H ∗ (EG). ¯ mean the total complexes. Here Ω∗ (N G) and Ω∗ (N G) For example, the derivative d0 + d00 : Ωp (N G) → Ωp+1 (N G) is given as follows: Ωp (G) x −d  ε∗ −ε∗ +ε∗

0 1 2 −− −→ Ωp−1 (N G(2)) Ωp−1 (G) −− x  d

Ωp−2 (N G(2)) ... Ω1 (N G(p)) x (−1)p d  Pp+1

(−1)i ε∗

i Ω0 (N G(p)) −−i=0 −−−−−→ Ω0 (N G(p + 1))

3

Remark 2.1. Let π : P → M be a principal G-bundle and {gαβ : Uαβ → G} be the transition functions of it. Then we can pull-back the cocycle ˇ in Ω∗ (N G) to the Cech-de Rham complex of M by {gαβ }. When κ is the characteristic class which corresponds to the cocycle in Ω∗ (N G), the image ∗ ∗ of gαβ in HCech−deRham (M ) is the characteristic class κ(P ) of π : P → M . ˇ For more details, see for instance [7]. There is another double complex associated to a simplicial manifold. Definition 2.2 ([5]). A simplicial n-form on a simplicial manifold {Xp } is a sequence {φ(p) } of n-forms φ(p) on ∆p × Xp such that ∗

(εi × id) φ(p) = (id × εi )∗ φ(p−1) . Here εi is the canonical i-th face operator of ∆p . Let Ak,l (X) be the set of all simplicial (k + l)-forms on ∆p × Xp which are expressed locally of the form X ai1 ···ik j1 ···jl (dti1 ∧ · · · ∧ dtik ∧ dxj1 ∧ · · · ∧ dxjl ) where (t0 , t1 , · · · , tp ) are the barycentric coordinates in ∆p and xj are the local coordinates in Xp . We call these forms (k, l)-form on ∆p × Xp and define derivatives as: d0 := the exterior differential on ∆p d00 := (−1)k × the exterior differential on Xp . Then (Ak,l (X), d0 , d00 ) is a double complex. Let A∗ (X) denote the total complex of A∗,∗ (X). We define a map I∆ : A (X) → Ω∗ (X) as follows: Z I∆ (α) := (α|∆p ×Xp ). ∗

∆p

Then the following theorem holds [5]. Theorem 2.2. I∆ induces a natural ring isomorphism ∗ I∆ : H(A∗ (X)) ∼ = H(Ω∗ (X)).

4

Let G denote the Lie algebra of G. A connection on a simplicial G-bundle π : {Ep } → {Mp } is a sequence of 1-forms {θ} on {Ep } with coefficients G such that θ restricted to ∆p × Ep is a usual connection form on a principal G-bundle ∆p × Ep → ∆p × Mp . ¯ on γ : N G ¯ → Dupont constructed a canonical connection θ ∈ A1 (N G) N G in the following way: θ|∆p ×N G(p) := t0 θ0 + · · · + tp θp . ¯ ¯ Here θi is defined by θi = pr∗i θ¯ where pri : ∆p × N G(p) → G is the ¯ ¯ projection into the i-th factor of N G(p) and θ is the Maurer-Cartan form of ¯ on γ as: G. We also obtain its curvature Ω ∈ A2 (N G) 1 Ω|∆p ×N G(p) = dθ|∆p ×N G(p) + [θ|∆p ×N G(p) ¯ , θ|∆p ×N G(p) ¯ ]. ¯ ¯ 2 Let I∗ (G) denote the ring of G-invariant polynomials on G. For P ∈ ¯ to each ∆p × N G(p) ¯ I ∗ (G), we restrict P (Ω) ∈ A∗ (N G) → ∆p × N G(p) and apply the usual Chern-Weil theory then we have a simplicial 2k-form P (Ω) on N G. Now we have a canonical homomorphism w : I∗ (G) → H(Ω∗ (N G)) which maps P ∈ I ∗ (G) to w(P ) = [I∆ (P (Ω))].

3

The Chern character in the double complex

In this section we exhibit a cocycle in Ω∗,∗ (N G) which represents the Chern character. Throughout this section, G = GL(n; C) and chp means the p-th Chern character. Note that the diagram below is commutative, since I∆ acts only on the differential forms on ∆∗ , and so does γ ∗ on differential forms on each N G(∗). ∆ ¯ −−I− ¯ A∗,∗ (N G) → Ω∗,∗ (N G) x x  γ ∗ γ∗ 

I

∆ A∗,∗ (N G) −−− → Ω∗,∗ (N G)

5

¯ − q))(0 ≤ q ≤ p − 1) which corWe first give the cocycle in Ωp+q (N G(p responds to the p-th Chern character by restricting (1/p!) tr ((−Ω/2πi)p ) ∈ ¯ to Ap−q,p+q (∆p−q ×N G(p−q)) ¯ A2p (N G) and integrating it along ∆p−q . Then we give the cocycle in Ωp+q (N G(p − q)) which hits to it by γ ∗ . Since [θi , θj ] = θi ∧ θj + θj ∧ θi for any i, j, Ω|∆p−q ×N G(p−q) =− ¯

p−q X

X

dti ∧ (θ0 − θi ) −

i=1

ti tj (θi − θj )2 .

0≤i
Now dti ∧ (θ0 − θi ) = dti ∧ {(θ0 − θ1 ) + (θ1 − θ2 ) + · · · + (θi−1 − θi )} and for any G-valued differential forms α, β, γ and any integer 0 ≤ ∀x ≤ p − q − 1, the equation α ∧ (dti ∧ (θx − θx+1 )) ∧ β ∧ (dtj ∧ (θx − θx+1 )) ∧ γ = −α ∧ (dtj ∧ (θx − θx+1 )) ∧ β ∧ (dti ∧ (θx − θx+1 )) ∧ γ holds,
so the terms of the p forms above cancel with each other in −Ω|∆p−q ×N G(p−q) . Then we see: ¯

−Ω|∆p−q ×N G(p−q) ¯


p

=

p−q X

!p dti ∧ (θi−1 − θi ) +

i=1

X

ti tj (θi − θj )2

.

0≤i
Now we obtain the following theorem. Theorem 3.1. We set: X S¯p−q = (sgn(σ))(θσ(1) − θσ(1)+1 ) · · · (θσ(p−q−1) − θσ(p−q−1)+1 ) σ∈Sp−q−1

¯ − q)) (0 ≤ q ≤ p − 1) which corresponds to Then the cocycle in Ωp+q (N G(p the p-th Chern character chp is 1 p! tr



X

1 2πi

p

(−1)(p−q)(p−q−1)/2 ×

¯ q (S¯p−q ) × (p(θ0 − θ1 )) ∧ H

!

Z

Y ¯ ¯ (ti tj )aij (Hq (Sp−q )) dt1 ∧ · · · ∧ dtp−q

∆p−q i
¯ q (S¯p−q ) means the terms that (θi − θj )2 (1 ≤ i < j ≤ p − q + 1) are Here H put q -times between (θk−1 − θk ) and (θl − θl+1 ) in S¯P p−q permitting overlaps; 2 ¯ ¯ aij (Hq (Sp−q )) means the number of (θi − θj ) in it. means the sum of all such terms. 6

.

¯ − q)) which corresponds to chp is given by Proof. The cocycle in Ωp+q (N G(p  p  Z −Ω|∆p−q ×N G(p−q) ¯ 1 tr 2πi ∆p−q p! !p !  p Z p−q X X 1 1 = dti ∧ (θi−1 − θi ) + ti tj (θi − θj )2 . tr p! 2πi p−q ∆ i=1 0≤i
Here hi is the i-th factor of N G(∗). A straightforward calculation shows that γ ∗ tr(ϕi1 ϕi2 · · · ϕip−1 ϕip ) = tr(θi1 −1 − θi1 )(θi2 −1 − θi2 ) · · · (θip −1 − θip ). From the above, we conclude: Theorem 3.2. We set: Rij = (ϕi + ϕi+1 + · · · + ϕj−1 )2 (1 ≤ i < j ≤ p − q + 1) X Sp−q = sgn(σ)ϕσ(1)+1 · · · ϕσ(p−q−1)+1 . σ∈Sp−q−1

Then the cocycle in Ωp+q (N G(p − q)) (0 ≤ q ≤ p − 1) which represents the p-th Chern character chp is  p 1 1 (−1)(p−q)(p−q−1)/2 × (p − 1)! 2πi ! Z Y X tr ϕ1 ∧ Hq (Sp−q ) × (ti−1 tj−1 )aij (Hq (S)) dt1 ∧ · · · ∧ dtp−q . ∆p−q i
Here Hq (Sp−q ) means the term that Rij (1 ≤ i < j ≤ p − q + 1) are put q -times between ψk and ψP l in Sp−q permitting overlaps; aij (Hq (Sp−q )) means the number of Rij in it. means the sum of all such terms. 7

Proof. We can easily check that the cocycle in Theorem 3.2 is mapped to the ¯ − q)). The cochain in Theorem 3.1 by γ ∗ : Ωp+q (N G(p − q)) → Ωp+q (N G(p statement folllows from this. Remark 3.1. The coefficients in theorem 3.2 are calculated using the following famous formula. Z b0 ! b1 ! · · · br ! . tb00 t1b1 · · · tbrr dt1 ∧ · · · ∧ dtr = (b0 + b1 + · · · + br + r)! ∆r Corollary 3.1. The cochain ωp in Ω2p−1 (N G(1)) which corresponds to the p-th Chern character is given as follows:  p 1 1 1 tr(h−1 dh)2p−1 . ω1 = p! 2πi C 2p−1 p−1

Corollary 3.2. The cochain ωp in Ωp (N G(p)) which corresponds to the p-th Chern character is given as follows:    p X 1 1 ωp = (−1)p(p−1)/2 tr ϕ1 ∧ sgn(σ)ϕσ(1)+1 · · · ϕσ(p−1)+1  . p!(p − 1)! 2πi σ∈S p−1

Example 3.1. The cocycle which represents the second Chern character ch2 in Ω4 (N G) is the sum of the following C1,3 and C2,2 : 0 x  00 d d0

C1,3 ∈ Ω3 (G) −−−→

Ω3 (N G(2)) x  00 d d0

C2,2 ∈ Ω2 (N G(2)) −−−→ 0  C1,3 =

1 2πi

2

1 tr(h−1 dh)3 , 6

 C2,2 =

8

1 2πi

2

−1 −1 tr(dh1 dh2 h−1 2 h1 ). 2

Corollary 3.3. The cocycle which represents the second Chern class c2 in Ω4 (N G) is the sum of the following c1,3 and c2,2 : 0 x  00 d d0

c1,3 ∈ Ω3 (G) −−−→

Ω3 (N G(2)) x  00 d d0

c2,2 ∈ Ω2 (N G(2)) −−−→ 0 

1 2πi

2

−1 tr(h−1 dh)3 6



1 2πi

2

1 −1 tr(dh1 dh2 h−1 2 h1 ) − 2

c1,3 = c2,2 =



1 2πi

2

1 −1 tr(h−1 1 dh1 )tr(h2 dh2 ). 2

Example 3.2. The cocycle which represents the 3rd Chern character ch3 in Ω6 (N G) is the sum of the following C1,5 , C2,4 and C3,3 : 0 x  00 d d0

C1,5 ∈ Ω5 (G) −−−→

Ω5 (N G(2)) x  00 d d0

C2,4 ∈ Ω4 (N G(2)) −−−→

Ω4 (N G(3)) x  00 d d0

C3,3 ∈ Ω3 (N G(3)) −−−→ 0 

C1,5

1 = 3!

C2,4

−1 = 3!

1 2πi



3

1 2πi

1 tr(h−1 dh)5 10

3

1 ( tr(dh1 h1 −1 dh1 h1 −1 dh1 dh2 h2 −1 h1 −1 ) 2 9

1 + tr(dh1 dh2 h2 −1 h1 −1 dh1 dh2 h2 −1 h1 −1 ) 4 1 + tr(dh1 dh2 h2 −1 dh2 h2 −1 dh2 h2 −1 h1 −1 )) 2

C3,3

−1 = 3!



1 2πi

3

1 ( tr(dh1 dh2 dh3 h3 −1 h2 −1 h1 −1 ) 2 1 − tr(dh1 h2 dh3 h3 −1 h2 −1 dh2 h2 −1 h1 −1 )). 2

4

The Chern-Simons form

We briefly recall the notion of the Chern-Simons form in [4]. Let π : E → M be any principal G-bundle and θ, Ω denote its connection form and the curvature. For any P ∈ Ik (G), we define the (2k − 1)-form T P (θ) on E as: Z 1 T P (θ) := k P (θ ∧ φk−1 )dt. t 0 1 t(t 2

Here φt := tΩ + − 1)[θ, θ]. Then the equation d(T P (θ)) = P (Ωk ) holds and T P (θ) is called the Chern-Simons form of P (Ωk ). When the bundle is flat, its curvature vanishes and hence d(T P (θ)) = P (Ωk ) = 0. Now we put the simplicial connection into T P and using the same argu¯ ment in section 3, then we obtain the Chern-Simons form in Ω2p−1 (N G). Proposition 4.1. The Chern-Simons form in Ω3 (N U (n)) which corresponds to the second Chern class c2 is the sum of the following T c0,3 , T c1,2 : 0 x  00 d d0

T c0,3 ∈ Ω3 (U (n)) −−−→

Ω3 (N U (n)(1)) x  00 d d0

T c1,2 ∈ Ω2 (N U (n)(1)) −−−→ Ω2 (N U (n)(2))

10

2 1 1 T c0,3 = tr(g −1 dg)3 2πi 6  2   1 1 1 −1 −1 −1 −1 T c1,2 = tr(g0 dg0 g1 dg1 ) − tr(g0 dg0 )tr(g1 dg1 ) . 2πi 2 2 

Remark 4.1. The term it to SU (n).

5


1 2 1 tr(g0−1 dg0 )tr(g1−1 dg1 ) 2πi 2

vanishes when we restrict

Formulas for a cocycle in a truncated complex

In this section, we prove the conjecture due to Brylinski in [3]. At first, we introduce the filtered local simplicial de Rham complex. ∗,∗ Definition 5.1 ([3]). The filtered local simplicial de Rham complex F p Ωloc (N G) over a simplicial manifold N G is defined as follows: ( s r lim −→1∈V ⊂G Ω (V ) if s ≥ p F p Ωr,s (N G) = loc 0 otherwise.

Let F p Ω∗ (N G) be a filtered complex ( Ωs (N G(r)) if s ≥ p F p Ωr,s (N G) = 0 otherwise and [σ

if s ≥ p otherwise.

Then there is an exact sequence: 0 → F p Ω∗ (N G) → Ω∗ (N G) → [σ

Let ω1 + · · · + ωp , ωp−q ∈ Ωp+q (N G(p − q)) be the cocycle in Ω2p (N G) which represents the p-th Chern character. By using this cocycle, Brylinski constructed a cochain η in [σ

Now we are ready to state the theorem whose statement is conjectured by Brylinski [3]. Theorem 5.1. η := η0 + · · · + ηp−1 is a cocycle in [σ

X

(−1)i ε∗i ηl = (−1)2p−l+1 dηl−1 + ω2p−l .

i=0

The left side of this equation is equal to Z m−1 Z X X m i ∗ (−1) (−1) (fm,q ◦ εi ) ωm + m+q=2p−l, m≥1

∆q

∆q

i=0

12

m+q

X i=m

! (−1)i (fm,q ◦ εi )∗ ωm

.

We can check that m−1 X

∗

i

(−1) (fm,q ◦ εi ) ωm =

i=0

hence by using the cocycle relation see the following holds: Z

m−1 X

Z

∗

i

(−1) (fm,q ◦ εi ) ωm =

∆q i=0

∆q

m−1 X ∗ fm+1,q ( (−1)i ε∗i ωm ) i=0

Pm+1 i=0

(−1)i ε∗i ωm = (−1)m dωm+1 , we can

∗ (−1)m dfm+1,q ωm+1

  Z Z ∗ m+1 ∗ m (εm ◦ fm+1,q ) ωm + (−1) (εm+1 ◦ fm+1,q ) ωm . − (−1) ∆q

∆q

Note that ∆q (εm+1 ◦fm+1,q )∗ ωm = 0 for q ≥ 1 and ∆q (εm+1 ◦fm+1,q )∗ ωm = ω2p−l if q = 0. We can also check that Z m+q Z m+q X X i ∗ (−1) (fm,q ◦ εi ) ωm = (−1)i (fm,q+1 ◦ εi−m+1 )∗ ωm . R

R

∆q i=m

∆q i=m

We set j = i − m + 1, then we see that is equal to

R ∆q

Pm+q

i i=m (−1) (fm,q+1

◦ εi−m+1 )∗ ωm

 Z Z q+1  X j+m−1 j ∗ m−1 (−1) (fm,q+1 ◦ ε ) ωm − (−1) (εm ◦ fm+1,q )∗ ωm ∆q

j=0

∆q

since εm ◦ fm+1,q = fm,q+1 ◦ ε0 . P2p−l (−1)i ε∗i ηl is equal to From above, we can see that i=0 Z

X

ω2p−l +

m+q=2p−l, m≥1

∆q

∗ dfm+1,q ωm+1 +

q+1 X

(−1)j−1

Z

! (fm,q+1 ◦ εj )∗ ωm

.

∆q

j=0

On the other hand, for any (m0 , q 0 ) which satisfies m0 + q 0 = 2p − (l − 1) the following equation holds: 0

q0

Z

(−1) d ∆q 0

∗ fm 0 ,q 0 ωm0 =

Z ∆q 0

∗ dfm 0 ,q 0 ωm0 −

q Z X j=0

13

∆q0 −1

∗

∗ (−1)j εj fm 0 ,q 0 ωm0 .

Therefore (−1)2p−l+1 dηl−1 is equal to 0

X

Z

m0 +q 0 =2p−l+1, m0 ≥1

∆q 0

∗ dfm 0 ,q 0 ωm0

−

q Z X j=0

∆q0 −1

! ∗ ∗ (−1)j εj fm 0 ,q 0 ωm0

.

This completes the proof. Remark 5.1. Let me explain Brylinski’s motivation in [3] to introduce these complexes and the conjecture briefly. Let LU be the free loop group of a contractible open set U ⊂ G containing 1 and ev : LU × S 1 → U be the evaluation i.e. for γ ∈ LU and θ ∈ S 1 , ev(γ, θ) is defined as R map, γ(θ). Then S 1 ev∗ maps η1 ∈ Ω1 (U 2p−2 ) to a cochain in Ω0 (LU 2p−2 ). This 2p−2 cochain defines a cohomology class in local cohomology group Hloc (LU, C). 2p−2 Brylinski constructed a natural map from Hloc (LU, C) to the the Lie algebra cohomology H 2p−2 (LG, C). Then as a special case p = 2, he used the cocycle in the local truncated complex [σ<2 Ω3loc (N G)] to construct the standard KacMoody 2-cocycle. He treated not only the free loop group but also the gauge group Map(X, G) for a compact oriented manifold X. Acknowledgments. I am indebted to Professor H. Moriyoshi for helpful discussion and good advice. I would like to thank the referee for his/her several suggestions to improve this paper.

References [1] R. Bott, On the Chern-Weil homomorphism and the continuous cohomology of the Lie group, Adv. in Math. 11 (1973), 289-303. [2] R. Bott, H. Shulman, J. Stasheff, On the de Rham Theory of Certain Classifying Spaces, Adv. in Math. 20 (1976), 43-56. [3] J-L. Brylinski, Differentiable math.DG/0011069.

cohomology

of

gauge

groups,

[4] S.S. Chern and J. Simons, Characteristic forms and geometric invariants, Ann. of Math. 99 (1974) 48-69.

14

[5] J.L. Dupont, Simplicial de Rham cohomology and characteristic classes of flat bundles, Top. Vol 15(1976),233-245, Perg Press. [6] J.L. Dupont, Curvature and Characteristic Classes, Lecture Notes in Math. 640, Springer Verlag, 1978. [7] M. Mostow and J. Perchick, Notes on Gel’fand-Fuks Cohomology and Characteristic Classes (Lectures by Bott).In Eleventh Holiday Symposium. New Mexico State University, December 1973. ´ [8] G. Segal, Classifying spaces and spectral sequences. Inst.Hautes Etudes Sci.Publ.Math.No.34 1968 105-112. Graduate School of Mathematics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya-shi, Aichi-ken, 464-8602, Japan. e-mail: [email protected]

15