Temporal Gauge Transformation

Consider the closed oriented Riemannian split-manifold (M, g^M) and the flat Hermitian vector bundle (E,∇, h^E) with the structure group G := U(n,C) as introduced in Subsection 7.1. Denote the principal G-bundle associated toE by P. G acts onP from the right.

Consider U ∼= (−, )×N the collar neighborhood of the splitting hypersur-faceN. We view the restrictionsP|_U, P|_N asG-bundles, where the structure group can possibly be reduced to a subgroup of G.

By the setup of Subsection 7.1 the bundle structures are product over U. More precisely letπ : (−, )×∂X →∂X be the natural projection onto the second component. We have a bundle isomorphisms E|_U ∼= π^∗E|_N and for the associated principal bundles

P|_U ∼=π^∗P|_N −→^f P|_N,

where f is the principal bundle homomorphism, covering π, with the asso-ciated homomorphism of the structure groups G → G being the identity automorphism.

Now letωN denote a flat connection one-form on P|N. Then ω_U :=f^∗ω_N

gives a connection one-form onP|_U which is flat again.

In order to understand the structure of ω_U =f^∗ω_N, let {Ue_α,Φe_α} be a set of

local trivializations for P|_N. Then P|_U ∼= π^∗P|_N is equipped with a set of naturally induced local trivializations {U_α := (−, )×Ue_α,Φ_α}. The local trivializations define local sections esα and sα as follows. For any p ∈ Ueα, normal variablex∈(−, ) and fore ∈G being the identity matrix we put

es_α(p) :=Φe⁻¹_α (p, e), s_α(x, p) := Φ⁻¹_α ((x, p), e).

We use the local sections to obtain local representations for the connection one-formsω_U and ω_N:

ωe^α :=es^∗_αω_N ∈Ω¹(Ue_α,G), ω^α :=s^∗_αω_U ∈Ω¹(U_α,G),

where G denotes the Lie Algebra of G. By construction both local repre-sentations are related as follows. Let (x, y) = (x, y₁, .., y_n) denote the local coordinates on Uα = (−, )×Ueα with x∈ (−, ) being the normal coordi-nate and y denoting the local coordinates onUe_α. Then

ωe^α =

n

X

i=1

ω_i^α(y)dy_i, ω^α =ω₀^α(x, y)dx+

n

X

i=1

ω_i^α(x, y)dy_i,

with ω₀^α ≡0, and ω_i^α(x, y)≡ω_i^α(y). (7.3) We call a flat connection ω on P a connection in temporal gauge, if there exists a flat connection ω_N onP_N such that over the collar neighborhood U

ω|U =f^∗ωN.

The local properties of a connection in temporal gauge and in particular its independence of the normal variable x ∈(−, ) are discussed in (7.3). Our aim is to show that any flat connection one-form on a principal bundleP can be gauge transformed to a flat connection in temporal gauge. Recall that a gauge-transformation of P is a principal bundle automorphism g ∈ Aut(P) covering identity on M with g(p·u) = g(p)·u for any u∈Gand p∈P. A gauge transformation can be viewed interchangedly as a transformation from one system of local trivializations into another. Hence the action of a gauge transformation on a connection one-form is determined by the trans-formation law for connections under change of coordinates.

We have the following result.

Proposition 7.1. Any flat connection on the principal bundle P is gauge equivalent to a flat connection in temporal gauge.

Proof. By a partition of unity argument it suffices to discuss the problem locally over a trivializing neighborhood (U_α := (−, )×Ue_α,Φ_α).

Let ω be a flat connection on P|_U. Let g be any gauge transformation on P|_U. Denote the gauge transform of ω under g byω_g.

Over the trivializing neighborhood Uα the connections ω, ωg and the gauge tranformation g are given by local G-valued one-forms ω^α, ω^α_g and a G-automorphism g^α respectively. They are related in correspondence to the transformation law of connections as follows

ω^α_g = (g^α)⁻¹◦ω^α◦g^α+ (g^α)⁻¹dg^α,

where the action ◦ is the concatenation of matrices (G ⊂ GL(n,C)), after evaluation at a local vector field and a base point in U_α. The local one form ω^α writes as

ω^α =ω^α₀(x, y)dx+

n

X

i=1

w_i^α(x, y)dy_i.

In order to gauge-transformω into temporal gauge, we need to annihilateω₀^α and thex-dependence inω^α_i. For this reason we consider the following initial value problem with parametery ∈Ue_α

∂_xg^α(x, y) = −ω^α₀(x, y)g^α(x, y),

g^α(0, y) = 1∈GL(n,C). (7.4) In order to identify the solution to (7.4) consider for any fixed y ∈ Ue_α the following x−time dependent vector field V_x,y^α , x∈(−, ) on G:

∀u∈G V_x,y^α u:=−(R_u)_∗ω^α₀(x, y) =−ω₀^α(x, y)·u,

where R_u is the right multiplication on G and the second equality follows from the fact thatG∈GL(n,C) is a matrix Lie group.

Let eg^α(x, y) be the unique integral curve of the time-dependent vector field V_x,y^α with eg^α(0, y) =1∈G. It satisfies

∂_xeg^α(x, y) =V_x,y^α eg^α(x, y) =−ω₀^α(x, y)eg^α(x, y).

Hence the integral curve eg^α(x, y) solves (7.4). By the fundamental theorem for ordinary linear differential equations (cf. [KN, Appendix 1]) we know that

the initial value problem (7.4) has a unique solution, smooth in x ∈ (−, ) and y∈Ue_α. Sinceeg^α(x, y) solves (7.4) we find that the solution is moreover G−valued.

With gauge transformation g being locally the solution to (7.4) we find for the gauge transformed connection ω_g

ω^α_g = (g^α)⁻¹◦ω^α◦g^α+ (g^α)⁻¹dg^α =

= (g^α)⁻¹◦ω₀^α◦g^αdx+

n

X

i=1

(g^α)⁻¹◦ω_i^α◦g^αdy_i+ +(g^α)⁻¹∂xg^αdx+

n

X

i=1

(g^α)⁻¹∂yig^αdyi =

=

n

X

i=1

(g^α)⁻¹◦ω_i^α◦g^αdy_i+

n

X

i=1

(g^α)⁻¹∂_yig^αdy_i.

where in the last equality we cancelled two summands due to g^α being the solution to (7.4). So far we didn’t use the fact thatω is a flat connection. A gauge transformation preserves flatness, soω_g is flat again. Put

ω^α_g =ω^α_g,0(x, y)dx+

n

X

i=1

ω_g,i^α (x, y)dy_i, where by the previous calculation

ω^α_g,0 ≡0, ω^α_g,i ≡(g^α)⁻¹◦ω_i^α◦g^α+ (g^α)⁻¹∂_yig^α. Flatness of ω_g implies

∂_xω_g,i^α (x, y) =∂_yiω_g,0^α (x, y) = 0.

Hence the gauge transformed connection is indeed in temporal gauge. This completes the proof.

A gauge transformation, viewed so far as a principal bundle automorphism on the G−principal bundle P, can equivalently be viewed as a G−valued bundle automorphism on the vector bundle E associated to P. We adopt this point of view for the forthcoming discussion.

Take the given flat connection∇on the Hermitian vector bundle (E, h^E) with the structure groupG=U(n,C) and the canonical metrich^E induced by the

standard inner product on Cⁿ. Proposition 7.1 asserts existence of a tem-poral gauge transformation g ∈ Aut_G(E) such that the gauge transformed covariant derivativeg∇g⁻¹is in temporal gauge (a covariant derivative is said to be in temporal gauge if the associated connection one-form is in temporal gauge).

The temporal gauge transformation g gives rise to a map on sections in a natural way

G: Ω^∗(M, E⊗E)→Ω^∗(M, E⊗E).

Due to the fact that g takes locally values in U(n,C) and the Hermitian metric h^E is canonically induced by the standard inner product on Cⁿ, we obtain the following result:

Proposition 7.2. G extends to a unitary transformation G:L²_∗(M, E⊗E, g^M, h^E)→L²_∗(M, E⊗E, g^M, h^E).

Corollary 7.3. The odd-signature operators B = B(∇) and B^j = B^j(∇_j), j = 1,2are spectrally equivalent toB(g∇g⁻¹)andB^j(g∇g⁻¹|_Mj), j = 1,2 respectively.

The statement of the corollary above follows from invariance of minimal and maximal extensions under unitary transformations and from the fact that unitary transformations preserve spectral properties of operators, compare also Proposition 3.2 and Corollary 3.3.

The statement of the corollary implies that in the setup of this section (for unitary vector bundles) the assumption of temporal gauge is done without loss of generality, which we do henceforth. In this particular geometric setup we obtain the following specific result for refined analytic torsion.

Proposition 7.4. Let T^RS(∇)e and T^RS(∇e_j), j = 1,2denote the scalar ana-lytic torsions associated to the complexes (D,e ∇),e (De_j,∇e_j), respectively. Fur-thermore let ρ_Γ(M, E) and ρ_Γ(M_j, E) denote the associated refined torsion elements in the sense of (6.7) for λ= 0. Then we have

ρ_an(∇) = 1

T^RS(∇)e ·exp [−iπη(B_even) +iπrk(E)η(B_trivial)]×

×exp

−iπm−1

2 dim kerB_even+iπrk(E)m

2 dim kerB_trivial

ρ_Γ(M, E), ρ_an(∇_j) = 1

T^RS(∇e_j)·exp

−iπη(B^j_even) +iπrk(E)η(B_trivial^j )

×

×exp

−iπm−1

2 dim kerB^j_even+iπrk(E)m

2 dim kerB^j_trivial

ρ_Γ(M_j, E).

Proof. Recall from the definition of refined analytic torsion in Corollary 6.26 ρ_an(∇) =e^ξλ(∇)exp[−iπ(η(B^(λ,∞)_even ) +ξ_λ⁰(∇))]×

exp[+iπrk(E)(η(B_trivial) +ξ⁰(∇_trivial))]·ρ_[0,λ], ρ_an(∇_j) =e^ξλ(∇j)exp[−iπ(η(B_even^j,(λ,∞)) +ξ_λ⁰(∇_j))]×

exp[+iπrk(E)(η(B_trivial^j ) +ξ⁰(∇_j,trivial))]·ρ^j_[0,λ], j = 1,2.

The assumption of product metric structures and the temporal gauge allow a reduction to closed double manifolds, as performed explicitly in Theorem 6.30. This yields by similar arguments, as in [BK2, Proposition 6.5]:

ξ_λ⁰(∇) = 1 2

m

X

k=0

(−1)^k+1·k·dimDe^k_[0,λ], ξ⁰_λ(∇_j) = 1

2

m

X

k=0

(−1)^k+1·k·dimDe^k_j,[0,λ]. Now, via Lemma 6.25 we obtain

ξ⁰_λ(∇)≡ m

2 dimDe_[0,λ]^even mod 2Z, ξ_λ⁰(∇_j)≡ m

2 dimDe_j,[0,λ]^even mod 2Z. Similar arguments show

ξ⁰(∇_trivial)≡ m

2 dim kerB_trivial mod 2Z, ξ⁰(∇_j,trivial)≡ m

2 dim kerB^j_trivial mod 2Z. Fixλ= 0 and observe for j = 1,2 from (6.21):

η(B_even^(0,∞)) +ξ₀⁰(∇)≡η(Beven) + m−1

2 dim kerBeven mod 2Z, η(B_even^j,(0,∞)) +ξ⁰₀(∇j)≡η(B^j_even) + m−1

2 dim kerB^j_even mod 2Z.

Now the statement of the proposition follows from the fact that flatness of the Hermitian metrich^E implies equality between the squared odd-signature operator and the Laplacians of the corresponding complexes, and hence

e^ξ0(∇)= 1

T^RS(∇)e , e^ξ0(∇j)= 1 T^RS(∇e_j). This proves the proposition.

Im Dokument The Analytic Torsion of Manifolds with Boundary and Conical Singularities (Seite 160-166)