The symplectic geometry of the vector fields tangent to I and of the diffeomorphisms leaving I invariant.

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Dynamics in cotangent space.

In this section, we use the terminology and notation of section 1 of Chapter 2. We investigate the asymptotic behaviour of the lifted flow T⋆λ. Decomposition in stable and unstable sets. We interpret C,Cρ as stable and unstable sets for the lifted flow T⋆etρ in cotangent space. We work locally, let p ∈ I and Vp a neighborhood of p in M, we fix a chart (x, h) : Vp 7→ Rn+d in which ρ = hj ∂ ∂hj .
Proposition 3.1.1 The flow T∗etρ lifted to the cotangent cone T•Vp has the following property: lim t→+∞ T∗etρ(p) ∈ (Cρ ∩ T•Vp) (3.1) lim t→−∞ T∗etρ(p) ∈ (C ∩ T•Vp).
• generically ξ 6= 0, then (x, eth; k, e−tξ) ∼ (x, eth; etk, ξ) (because it is a cotangent cone) converges to (x, 0; 0, ξ), it is immediate to deduce {(x, 0; 0, ξ)|ξ 6= 0} = (TI)⊥ = C is the stable set of the flow. Notice the conormal bundle is an intrinsic geometric object and does not depend on the choice of vector field ρ.
• Otherwise ξ = 0, (x, λh; k, 0) → (x, 0; k, 0), the limit must lie in {(x, 0; k, 0)|k 6= 0} ⊂ Cρ which we will later see belongs to the unstable set. Conversely if t → ∞:
• generically k 6= 0, then (x, eth; k, e−tξ) converges to (x, 0; k, 0), it is immediate to deduce {(x, h; k, 0)|k 6= 0} = Cρ is the unstable cone.

Main theorem.

In this section, we prove the main theorem of this chapter which gives a sufficient condition to control the wave front set of the extension t. The condition is as follows: Let t ∈ Es(M \ I) and assume WF(t) satisfies the soft landing condition, and assume that λ−stλ is bounded in D′􀀀 where 􀀀 = S λ∈(0,1]WF(tλ). Then our theorem claims that WF(t) ⊂ WF(t) ∪ C for the extension t.
Theorem 3.2.1 Let s ∈ R such that s+d > 0, V be a ρ-convex neighborhood of I and t ∈ D′(V\I). Assume that WF(t) satisfies the soft landing condition and that λ−stλ is bounded in D′􀀀 (V \ I) where 􀀀 = S λ∈(0,1]WF(tλ) ⊂ T• (M \ I). Then the wave front set of the extension t of t given by Theorem We saw in Chapter 2 that the hypothesis that WF(t) satisfies the soft landing condition is equivalent to the requirement that 􀀀|I ⊂ C in particular, this implies that 􀀀 ∩ Cρ = ∅ in a sufficiently small neighborhood of I and WF(t)|I ⊂ 􀀀|I ⊂ C. Hence we have the relation WF(t) ⊂ WF(t) ∪ C = WF(t) ∪ C.

Proof of the main theorem.

For the proof, it suffices to work in flat space Rn+d with coordinates (x, h) ∈ Rn × Rd where I = {h = 0} and ρ = hj ∂ ∂hj , since the hypothesis of the theorem and the result are local and open properties.
Proof —We denote by the setWF(t)∪C. The weight function (1+|k|+|ξ|) is denoted by θ. In order to establish the inclusion WF(t) ⊂ , it suffices to prove that for all p = (x0, h0; k0, ξ0) /∈ , there exists χ s.t. χ(x0, h0) 6= 0, V a closed conic neighborhood of (k0, ξ0) such that ktkN,V,χ < +∞ for all N. Let p = (x0, h0; k0, ξ0) /∈ , then: Either h0 6= 0, and we choose χ in such a way that χ = 0 on I thus tχ = tχ and we are done since ktkN,V,χ = ktkN,V,χ < +∞.

Estimates for the product of a distribution and a smooth function.

Theorem 3.3.1 Let m ∈ N and 􀀀 ⊂ T•(Rd). Let V be a closed cone in Rd\ 0 and χ ∈ D(Rd). Then for every N and every closed conical neighborhood W of V such that (supp χ ×W)∩􀀀 = ∅, there exists a constant C such that for all ϕ ∈ D(Rd) and for all t ∈ D′􀀀 (Rd) such that kθ−m b tχkL∞ < +∞: ktϕkN,V,χ 6 Cπ2N,K(ϕ)(ktkN,W,χ + kθ−m b tχkL∞). (3.9).
Proof — We denote by θ the weight function ξ 7→ (1 + |ξ|) and eξ := x 7→ e−ix.ξ the Fourier character. If the cone V is given, we can always define a thickening W of the cone V such that W is a closed conic neighborhood of V : W = {η ∈ Rd \ {0}|∃ξ ∈ V, | ξ |ξ| − η |η|| 6 δ},

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Table of contents :

1 The extension of distributions. 
1.1 Introduction.
1.2 Extension and renormalization.
1.2.1 Notation, definitions.
1.2.2 From bounded families to weakly homogeneous distributions.
1.3 Extension of distributions.
1.3.1 Removable singularity theorems.
1.4 Euler vector fields.
1.4.1 Invariances
1.5 Appendix.
2 A prelude to the microlocal extension. 
2.0.1 Introduction.
2.1 Geometry in cotangent space.
2.2 Geometric and metric topological properties of 􀀀.
2.3 The counterterms are conormal distributions.
2.4 Counterexample.
2.5 Appendix.
3 The microlocal extension. 
3.1 Dynamics in cotangent space.
3.1.1 Definitions.
3.2 Main theorem.
3.2.1 Proof of the main theorem.
3.2.2 The renormalized version of the main theorem.
3.3 Appendix
3.3.1 Estimates for the product of a distribution and a smooth function.
4 Stability of the microlocal extension. 
4.1 Notation, definitions.
4.2 The product of distributions.
4.2.1 Approximation and coverings.
4.2.2 The product is bounded.
4.2.3 The soft landing condition is stable by sum.
4.3 The pull-back by diffeomorphisms.
4.3.1 The symplectic geometry of the vector fields tangent to I and of the diffeomorphisms leaving I invariant.
4.3.2 The pull-back is bounded.
4.3.3 The action of Fourier integral operators.
4.4 Appendix.
5 The two point function h0|φ(x)φ(y)|0i. 
5.1 The flat case.
5.1.1 The Poisson kernel, the Wick rotation and the subordination identity.
5.1.2 Oscillatory integral.
5.2 The holomorphic family􀀀 (x0 + i0)2 − Pn i=1(xi)2 s.
5.3 Pull-backs and the exponential map.
5.3.1 The wave front set of the pull-back.
5.3.2 The pull back of the phase function.
5.4 The construction of the parametrix.
5.4.1 The meaning of the asymptotic expansions.
5.4.2 The invariance properties of the Beltrami operator g and of gradient vector fields.
5.4.3 The function 􀀀 and the vectors ρ1, ρ2.
5.4.4 The main theorem.
6 The recursive construction of the renormalization. 
6.0.5 Introduction.
6.1 Hopf algebra, T product and ⋆ product.
6.1.1 The polynomial algebra of fields.
6.1.2 Comparison of our formalism and the classical formalism from physics textbooks.
6.1.3 Hopf algebra bundle over Mn.
6.1.4 Deformation of the polynomial algebra of fields.
6.1.5 The construction of ⋆.
6.1.6 The associativity of ⋆.
6.1.7 Wick’s property.
6.1.8 Recovering Feynman graphs.
6.2 The causality equation.
6.2.1 Definition of the time-ordering operator
6.2.2 The Causality theorem.
6.2.3 Consistency condition
6.3 The geometrical lemma for curved space time.
6.4 The recursion.
6.4.1 Polarized conic sets.
6.4.2 Localization and enlarging the polarization.
6.4.3 We have 􀀀 WF 􀀀 λρxi∗+ T T•U2 ⊂ (−E+ q ) × E+ q .
6.4.4 The scaling properties of translation invariant conic sets.
6.4.5 Thickening sets.
6.4.6 The μlocal properties of the two point function.
6.4.7 Pull-back of good cones.
6.4.8 The wave front set of the product tn is contained in a good cone 􀀀n.
6.4.9 We define the extension tn and control WF(tn).
7 A conjecture by Bennequin. 
7.1 Parametrizing the wave front set of the extended distributions.
7.2 Morse families and Lagrangians.
7.3 A conjectural formula.
8 Anomalies and residues. 
8.1 Introduction.
8.2 Currents and renormalisation.
8.2.1 Notation and definitions.
8.2.2 From Taylor polynomials to local counterterms via the notion of moments of a compactly supported distribution T.
8.2.3 The results of Chapter 1.
8.3 Renormalization, local counterterms and residues.
8.3.1 The ambiguities of the operator Rε and the moments of a distribution T.
8.3.2 The geometric residues.
8.3.3 Stability of geometric residues.
9 The meromorphic regularization. 
9.1 Introduction.
9.2 Fuchsian symbols.
9.2.1 Constant coefficients Fuchsian operators.
9.2.2 Fuchsian symbols currents.
9.2.3 The solution of a variable coefficients Fuchsian equation is a Fuchsian symbol.
9.2.4 Stability of the concept of approximate Fuchsians.
9.3 Meromorphic regularization as a Mellin transform.
9.3.1 The meromorphic extension.
9.4 The Riesz regularization.
9.5 The log and the 1-parameter RG.


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