Higher analytic torsion

Ulrich Bunke

1. Analytic torsion forms

2. Equivariant higher analytic torsion 3. Examples and problems

B - smooth manifold H → B - flat complex vector bundle f(x) := xe^{x^{2}} define characteristic class f(H) ∈ H^{∗}(B, C) as follows :

choose hermitean metric h^{H} on H X := \frac{1}{2} ((∇^{H})^{∗} − ∇^{H} ) ∈ Ω(B, End(H)) N - denotes Z-gradings ϕ := (2πı)^{−N/2} ∈ End(Ω(B))

then f(H, h^{H}) := (2πı)^{1/2}ϕTr_{s}f(X) ∈ Ω(B) is closed f(H) := [f(H, h^{H})] ∈ H^{∗}(B, C) is real and independent of h^{H}

E →^{π} B - smooth locally trivial fibre bundle, B connected F → E - flat complex vector bundle

F - sheaf of parallel sections of F R^{i}π_{∗}F - locally constant sheaf on B H^{i} → B - associated flat complex vector bundle H := ⊕_{i}H^{i} - Z-graded flat complex vector bundle

Problem 1 Express f(H) in terms of f(F) and the topology of E → B.

T^{v}E -vertical tangent bundle e(T^{v}E) - Euler class

Theorem 2 (Bismut/Lott 95)

f(H) = ∫_{E/B} f(F)e(T^{v}E)

we recall main steps of the proof choose

• horizontal bundle T^{H}E

• vertical Riemannian metric g^{v}

• hermitean metric h^{F} on F W - infinite dimensional vector bundle with fibre Ω(E_{b}, F_{|E_{b}}) over b use decomposition T E = T^{v}E ⊕ T^{H}E in order to write

Ω(E, F) = Ω(B) ⊗ Γ(B, W)

d - de Rham differential on Ω(E, F) split d into terms of vertical degree 1, 0, −1

d = d^{v} + ∇ + i

d - flat super connection on W of degree one ∇ - connection on W d^{v} ∈ Γ(B, End^{1}(W)) i ∈ Ω^{2}(B, End^{−1}(W)) g^{v} and h^{F} induce hermitean metric h^{W} on W (fibrewise L^{2}-scalar product) form adjoint d^{∗} := (d^{v})^{∗} + ∇^{∗} + i^{∗} for t > 0 consider C_{t}, D_{t} ∈ Ω(B, End(W)) :

C_{t} := \frac{1}{2}(t^{N/2} d t^{−N/2} + t^{−N/2} d^{∗} t^{N/2})
D_{t} := \frac{1}{2}(t^{N/2} d t^{−N/2} − t^{−N/2} d^{∗} t^{N/2})

we can form (recall that f(x) = xe^{x^{2}})

f(W, h^{W} , t) := (2πı)^{1/2}ϕTr_{s}f(D_{t}) ∈ Ω(B)
f̂(W, h^{W} , t) := ϕTr_{s} \frac{N}{2} f′(D_{t}) ∈ Ω(B)

Proposition 3

df(W, h^{W}, t) = 0
\frac{d}{dt}f(W, h^{W}, t) = \frac{1}{t}df̂(W, h^{W}, t)

Hodge theory gives H ≅ ker(d^{v} + (d^{v})^{∗}) ⊂ W h^{W} induces hermitean metric h^{H}

e(T^{v}E, g^{v}) - Euler form representing e(T^{v}E) χ := ∑_{i}(−1)^{i} dim(H^{i}) χ′ := ∑_{i}(−1)^{i}i dim(H^{i})

Proposition 4 If dim E_{b} is even, then

f(W, h^{W}, t) =^{t→0} ∫_{E/B} e(T^{v}E, g^{v})f(F, h^{F}) + O(t)
f̂(W, h^{W}, t) =^{t→0} \frac{1}{4} dim(E_{b})χ .

If dim E_{b} is odd, then

f(W, h^{W}, t) =^{t→0} O(\sqrt{t})
f̂(W, h^{W}, t) =^{t→0} O(\sqrt{t})
.
f(W, h^{W}, t) =^{t→∞} f(H, h^{H}) + O(\frac{1}{\sqrt{t}})
f̂(W, h^{W}, t) =^{t→∞} \frac{χ′}{2} + O(\frac{1}{\sqrt{t}}) .

Definition 5 Analytic torsion form T (E, F, T^{H}E, g^{v}, h^{F})

:= − ∫_{0}^{∞} [f̂(W, h^{W}, t) − \frac{χ′}{2}
− (\frac{1}{4} dim(E_{b})χ − \frac{1}{2}χ′) (1 − \frac{t}{2})e^{−\frac{t}{4}} ] \frac{dt}{t}

Theorem 6 Transgression formula

dT (E, F, T^{H}E, g^{v}, h^{F})

= ∫_{E/B} e(T^{v}E, g^{v})f(F, h^{F}) − f(H, h^{H})

assume • H = 0

• dim(E_{b}) odd then T (E, F, T^{H}E, g^{v}, h^{F}) is closed and its cohomology class is independent of g^{v}, h^{F} and T^{H}E

Definition 7 Higher analytic torsion of (E, F): T (E, F) := [T (E, F, T^{H}E, g^{v}, h^{F})] ∈ H^{∗}(B, R)

T (E, F)_{0} ∈ R is analytic torsion of (E_{b}, F_{|E_{b}}) T (E, F) is a differential-topological invariant of E → B and F T (E, F) is natural, i.e. if we have a diagram

U^{∗}F  F
↓  ↓
u^{∗}E →^{U} E
↓  ↓
B′ →^{u} B

then

u^{∗}T (E, F) = T (u^{∗}E, U^{∗}F) .

Problem 8 Give a topological description of T (E, F).

variants :

assume that fibre of E → B is an odd-dimensional rational homology sphere F := E × C

choose g^{v} such that volume of fibres is 1 then T (E, F, T^{H}E, g^{v}, h^{F}) is still closed and defines a topological invariant T (E) ∈ H^{∗}(B, R) of the rational homology sphere bundle E → B special case :

V → B - hermitean vector bundle E → B - unit sphere bundle of V Proposition 9

T (E) = ∞ ; ∑ ; j=1 \frac{(4j + 1)!}{2^{4j}(2j)!}ζ_{R}(2j + 1)ch_{4j}(V) .

G - compact Lie group M - closed G-manifold P → B - G-principal bundle

choose • G-invariant Riemannian metric g^{M} on M

• connection ω on P E := P ×_{G} M F := E × C let T^{H}E be induced by ω let g^{v} be induced by g^{M} then the higher part of T (E, F, T^{H}E, g^{v}, h^{F}) is closed and its cohomology class T (P, M) ∈ H^{>}(B, C) only depends on P and the G-manifold M. Ĩ(G) - invariant power series on Lie(G) modulo constants Lott defines higher equivariant analytic torsion

T (M) ∈ Ĩ(G)

Chern-Weyl homomorphisms

CW_{ω} : Ĩ(G) → Ω^{>}(B), CW : Ĩ(G) → H^{>}(B, C)

Theorem 10 (Bismut/Lott 95)

T (E, T^{H}E, g^{v}) = CW_{ω}(T (M))

T (P, M) = CW (T (M))

G - compact connected Lie group M - closed oriented G-manifold

choose G-invariant Riemannian metric g^{M} Ĩ(G) - invariant power series on g := Lie(G) modulo constants define higher equivariant analytic torsion as follows : A := S(g^{∗}) ⊗ End(Ω(M)) - algebra X^{∗} ∈ Γ(M, T M) - fundamental vector field to X ∈ g I, E ∈ A - defined by : I(X), E(X) ∈ End(Ω(M)) inner, resp. exterior, product with X^{∗} for X ∈ g d ∈ A - de Rham differential, d^{∗} - its adjoint

D_{t} := \frac{\sqrt{t}}{2}(d^{∗} − d) + \frac{1}{2\sqrt{t}}(E + I) .

Tr_{s} : A → S(g^{∗}) - supertrace on End(Ω(M)) ϕ ∈ End(Ĩ(G)) - defined by ϕ(f)(X) := f(\frac{X}{2πı})

Definition 11 Higher equivariant analytic torsion

T (M) := ϕTr_{s} ∫_{0}^{∞} \frac{N}{2} f′(D_{t})\frac{dt}{t}

T (M) is independent of g^{M} T (M) is a differential-topological invariant of the G-manifold M we will give a topological formula for T (M) : Or(G) - orbit category of G :

• objects : homogeneous spaces G/H

• morphisms : G-equivariant maps [G/H] - isomorphism class of G/H ∈ Or(G) Euler group

U(G) := ⊕_{[G/H]∈Or(G)}Z[G/H]

n-dimensional G-cell - space of the form E = G/H × D^{n} G-conjugacy class of H =: t(E) - type of E n = dim(E) - dimension of E M has G-homotopy type of a finite G-CW -complex X define equivariant Euler characteristic by

χ_{G}(M) := _{E⊂X}^{∑}(−1)^{dim(E)}[G/t(E)] ∈ U(G)

Theorem 12 There is a homomorphism

T_{G} : U(G) → Ĩ(G)

s.t. for any closed oriented G-manifold M T (M) = T_{G}(χ_{G}(M)) .

we will explain T_{G} if T ⊂ G - closed subgroup then we have restrictions

res_{T}^{G} : U(G) → U(T)
res_{T}^{G} : Ĩ(G) → Ĩ(T)

consider functor Ĩ_{G} : Or(G) → C − vector spaces

Ĩ_{G}(G/T) := Ĩ(T)

S(G) - full subcategory of Or(G) of spaces G/T with T ≅ S^{1}

V (G) := lim_{S(G)} Ĩ_{G}

for [G/H] ∈ Or(G) is

τ(G/H) := {T_{T} (res_{T}^{G}[G/H])}_{G/T ∈S(G)} ∈ V (G) R_{G} : Ĩ(G) → V (G) - injective restriction T_{G} is determined by R_{G}(T_{G}([G/H])) = τ(G/H) it remains to describe T_{T} for T = S^{1}

Theorem 13

T_{S^{1}}([S^{1}/S^{1}]) = 0

If H ⊂ S^{1} is finite, then

T_{S^{1}}([S^{1}/H]) = ∞ ; ∑ ; j=1 \frac{(4j + 1)!}{2^{4j}((2j)!)^{2}}ζ_{R}(2j + 1) (\frac{|H|α}{4π^{2}})^{2j} ,

where α ∈ (s_{C}^{1})^{∗} generates weight lattice.

we will discuss steps in the proof of Theorem 12 let (M, N) - compact oriented G manifold with boundary form double M̃ := M ∪_{N} M define T (M) := \frac{1}{2}T (M̃)

Theorem 14 If (M_{1}, N) and (M_{2}, N) are compact oriented G-manifolds with boundary and

M = M_{1} ∪_{N} M_{2} ,

then

T (M) = T (M_{1}) + T (M_{2}) .

proof involves finite propagation speed techniques more general let (M, N_{1}, …, N_{r}) - compact oriented G-manifold with r possibly intersecting boundary components define T (M) using doubling construction and induction by number of boundary components Theorems 12 and 14 extend to manifolds with several boundary components (by a formal argument) it suffices to prove Theorem 12 for G = S^{1} main steps : use induction over number of orbit types and additivity of T and χ_{S^{1}} start : one orbit type H :

• H = S^{1} - trivial action and T (M) = 0, compare with χ_{S^{1}}(M) = χ(S^{1})[S^{1}/S^{1}]

• H ≠ S^{1} : M is smooth fibre bundle M → M/S^{1} – decompose bundle into trivial pieces S^{1}/H × U_{α} – T (S^{1}/H × U_{α}) = T (S^{1}/H)χ(U_{α})

– T (M) = ∑_{α} T (S^{1}/H)χ(U_{α}) = T_{S^{1}}([S^{1}/H])χ(M/S^{1}) compare with χ_{S^{1}}(M) = χ(M/S^{1})[S^{1}/H]

induction step : H is orbit type fixed point set M_{H} is embedded smooth manifold with normal bundle NM_{H} M \ NM_{H} has less orbit types T (M) = T (M \ NM_{H}) + T (N̄M_{H}) compute T (N̄M_{H}) as follows :

• M_{H} → M_{H}/S^{1} is locally trivial fibre bundle

• decompose it into trivial pieces S^{1}/H × U_{α}

• decompose N̄M_{H} into pieces S^{1} ×_{H} D̄^{n} × U_{α}

• T (S^{1} ×_{H} D̄^{n} × U_{α}) = T (S^{1} ×_{H} D̄^{n})χ(U_{α}) = T (S^{1}/H)χ(U_{α})

• sum over α gives

T (N̄M_{H}) = T_{S^{1}}([S^{1}/H]) _{α}^{∑} χ(U_{α}) = T_{S^{1}}([S^{1}/H])χ(M_{H}/S^{1}) compare with χ_{S^{1}}(N̄M_{H}) = χ(M_{H}/S^{1})[S^{1}/H] we will work out examples :

(ρ, V) - n-dimensional unitary representation of G S^{2n−1} - unit sphere in V

T ⊂ G maximal torus res_{T}^{G} : Ĩ(G) → Ĩ(T) is injective t := Lie(T) - Lie algebra of T Δ(ρ) ⊂ t_{C}^{∗} - weights of ρ for α ∈ Δ(ρ) let m_{α} denote multiplicity and T_{α} ⊂ T stabilizer of weight vector assume that 0 ∉ Δ(ρ)

Proposition 15 χ_{T}(S^{2n−1}) = _{α∈Δ(ρ)}^{∑} m_{α}[T/T_{α}]+higher dimensional staff

T (S^{2n−1}) = ∞ ; ∑ ; j=1 \frac{(4j + 1)!}{2^{8j}((2j)!)^{2}π^{2j}} ζ_{R}(2j + 1) _{α∈Δ(ρ)}^{∑} m_{α}α^{2j}

(ρ, V) - 2n-dimensional orthogonal representation RP^{2n−1} - real projective space of V

Δ(ρ) ⊂ t_{C}^{∗} - weights of ρ_{C}

Proposition 16 χ_{T}(RP^{2n−1}) = \frac{1}{2} _{α∈Δ(ρ)}^{∑} m_{α}[T/T_{α}]+higher dimensional staff

T (RP^{2n−1}) = ∞ ; ∑ ; j=1 \frac{(4j + 1)!}{2^{8j+1}((2j)!)^{2}π^{2j}} ζ_{R}(2j+1) _{α∈Δ(ρ)}^{∑} m_{α}α^{2j}

G/K - irreducible compact symmetric space choose T such that S := T ∩ K is maximal torus of K assume that dim(T) − dim(S) = 1 W_{G}(T), W_{K}(T) - Weyl groups for w ∈ W_{G}(T)/W_{K}(T) put S^{w} := wSw^{−1} α ∈ (s_{C}^{1})^{∗} be generator of weight lattice α_{w} ∈ t_{C}^{∗} - pull-back of α under T → T/S^{w}

Proposition 17 χ_{T}(G/K) = _{w∈W_{G}(T)/W_{K}(T)}^{∑}[T/S^{w}]+higher dimensional staff

T (G/K) = ∞ ; ∑ ; j=1 \frac{(4j + 1)!}{2^{8j}((2j)!)^{2}π^{2j}} ζ_{R}(2j+1) _{w∈W_{G}(T)/W_{K}(T}^{∑} α_{w}^{2j}

If dim(T) − dim(S) ≠ 1, then T (G/K) = 0.

RP^{2n−1} is rational homology sphere have characteristic class T (E) for bundles E → B with fibre RP^{2n−1} this class is nontrivial in dimensions 4j, 0 < 2j ≤ n 4j-component gives nontrivial homomorphism

T_{4j} : π_{4j−1}(Diff(RP^{2n−1})) → R

have map q : O(2n) → Diff(RP^{2n−1}) we show that q is injective on rational homotopy groups let p : S^{4j−1} → O(2n) represent a rationally nontrivial class

[p] ∈ π_{4j−1}(O(2n))

form corresponding vector bundle V → S^{4j} and RP^{2n−1} -bundle E → S^{4j} Pontrjagin class p_{4j}(V) ≠ 0 T_{4j}(q_{∗}[p]) = ⟨T (E), [S^{4j}]⟩ = ⟨CW (T (RP^{2n−1})), [S^{4j}]⟩ = c⟨p_{4j}(V), [S^{4j}]⟩ ≠ 0

Problem 18 Compute T_{4j}!

A similar construction goes with S^{2n−1}.

G - Lie group (possibly infinite-dimensional) g - Lie algebra of G

(C^{∗}(G, C), d) - complex of differentiable cochains cohomology H_{c}^{∗}(G, C) - continuous group cohomology (C^{∗}(g, C), d) - complex of continuous cochains cohomology H_{c}(g, C) - continuous Lie algebra cohomology linearization λ : H_{c}^{∗}(G, C) → H_{c}^{∗}(g, C) examples

• G - semisimple linear Lie group: K ⊂ G - maximal compact subgroup G^{d} - compact dual group H_{c}^{∗}(G, C) ≅ H^{∗}(G^{d}/K, C)

• G = Diff(S^{1})^{0}, then H_{c}^{∗}(G, C) = ⟨α, β|β^{2} = 0⟩, |α| = |β| = 2, λ(β) = 0 H_{c}^{∗}(g, C) = ⟨λ(α), γ⟩, |γ| = 3

M - rational homology sphere

Met_{1}(M) - manifold of volume-one Riemannian metrics on M

E := Met_{1}(M) × M →^{pr_{1}} Met_{1}(M)

T^{H}E := T Met_{1}(M) ⊕ 0 g^{v} - tautological vertical Riemannian metric T (E, T^{H}E, g^{v}) ∈ Ω(Met_{1}(M)) - higher analytic torsion form is closed and Diff(M)-invariant choose g ∈ Met_{1}(M) define c ∈ C^{p}(Diff(M), C) as follows:

• f_{0}, …, f_{p} ∈ Diff(M)

• f_{0}(g), …, f_{p}(g) span p-simplex Δ in Met(M)

• Δ̃ - projection of Δ to Met_{1}(M)

• c(f_{0}, …, f_{p}) := ∫Δ̃ T (E, T^{H}E, g^{v}) c is differentiable invariant cocycle it represents T_{p} ∈ H_{c}^{p}(Diff(M), C) T_{p} is independent of choice of g

Theorem 19 The classes T_{4j} ∈ H_{c}^{4j}(Diff(S^{2n−1}), C), 0 < 4j ≤ 2n, are nontrivial.

pull back to H^{4j}(Diff(S^{2n−1}), C) is nontrivial, too consider S^{2n−1} as geodesic boundary of complex hyperbolic space CH^{n} SU(n, 1) acts on S^{2n−1} let Γ ⊂ SU(n, 1) be torsion-free cocompact subgroup we show that restriction of T_{4j} to Γ is nontrivial

• let Y := Γ\CH^{n} ∼_{h} BΓ

• identify H^{∗}(Y, C) = H^{∗}(Γ, C)

• i : CH^{n} ↪ Met_{1}(S^{2n−1}) - embedding of the SU(1, n)-orbit of round metric

• observe that i^{∗}E ≅ SY - unit sphere bundle of T Y

• (T_{4j})_{|Γ} = T (SY)_{4j} = c ch_{4j}(T Y)

• CP^{n} is compact dual space, ch_{4j}(T CP^{n}) ≠ 0, 0 < 4j ≤ 2n • by proportionality principle ch_{4j}(T Y) ≠ 0

\underline{Remark :} For RP^{2n−1} on would use discrete subgroup Γ of SL(2n, R) but restriction of T to Γ vanishes in case of S^{1} one can consider flat hermitean line bundle F → S^{1} holonomy of F is z ∈ U(1) obtain classes T_{j}(z) ∈ H_{c}^{j}(Diff(S^{1})^{0}, C) Diff(S^{1})^{0} ∼_{h} S^{1} hypercohomology spectral sequence associated to Ω(Diff(S^{1})^{0}) gives

H_{top}^{1}(S^{1}, C) ≅ H_{top}^{1}(Diff(S^{1})^{0}, C) →^{δ} H_{c}^{2}(Diff(S^{1})^{0}, C)

ω ∈ H^{1}(S^{1}, C), ⟨ω, [S^{1}]⟩ = 1 β := δ(ω)

Theorem 20

T_{2}(z) = \frac{Γ(5/4)}{2^{7/2}π^{7/2}} (Li_{2}(1/z) − Li_{2}(z)) β

note that λ(T_{2}(z)) = 0 Assume that M is odd-dimensional and F → M is flat hermitean vector bundle such that H^{∗}(M, F) = 0. Then we can define classes T_{2j}(F) ∈ H_{c}^{2j}(Diff(M)^{0}, C).

Problem 21 For S^{1} is T_{j}(z) = 0 for j > 2? Is λ(T) = 0 for all rational homotopy spheres? Is λ(T (F)) = 0? Give a topological interpretation of T (F)!