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The hypoelliptic Dirac operator
Transcript of The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic Dirac operator
Jean-Michel Bismut
Universite Paris-Sud, Orsay
11th May 2009
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Develop the philosophy behind the hypoellipticLaplacian.
Construct a hypoelliptic deformation of the classical
Dirac operator DX = ∂X
+ ∂X∗
.
DX acts on X, its deformation acts on X total spaceof TX.
Like its de Rham counterpart, this deformationinterpolates between classical Hodge theory and thegeodesic flow.
Construction of hypoelliptic Quillen metric, andcomparison with elliptic case.
Connection with ‘physics’.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Develop the philosophy behind the hypoellipticLaplacian.
Construct a hypoelliptic deformation of the classical
Dirac operator DX = ∂X
+ ∂X∗
.
DX acts on X, its deformation acts on X total spaceof TX.
Like its de Rham counterpart, this deformationinterpolates between classical Hodge theory and thegeodesic flow.
Construction of hypoelliptic Quillen metric, andcomparison with elliptic case.
Connection with ‘physics’.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Develop the philosophy behind the hypoellipticLaplacian.
Construct a hypoelliptic deformation of the classical
Dirac operator DX = ∂X
+ ∂X∗
.
DX acts on X, its deformation acts on X total spaceof TX.
Like its de Rham counterpart, this deformationinterpolates between classical Hodge theory and thegeodesic flow.
Construction of hypoelliptic Quillen metric, andcomparison with elliptic case.
Connection with ‘physics’.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Develop the philosophy behind the hypoellipticLaplacian.
Construct a hypoelliptic deformation of the classical
Dirac operator DX = ∂X
+ ∂X∗
.
DX acts on X, its deformation acts on X total spaceof TX.
Like its de Rham counterpart, this deformationinterpolates between classical Hodge theory and thegeodesic flow.
Construction of hypoelliptic Quillen metric, andcomparison with elliptic case.
Connection with ‘physics’.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Develop the philosophy behind the hypoellipticLaplacian.
Construct a hypoelliptic deformation of the classical
Dirac operator DX = ∂X
+ ∂X∗
.
DX acts on X, its deformation acts on X total spaceof TX.
Like its de Rham counterpart, this deformationinterpolates between classical Hodge theory and thegeodesic flow.
Construction of hypoelliptic Quillen metric, andcomparison with elliptic case.
Connection with ‘physics’.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Develop the philosophy behind the hypoellipticLaplacian.
Construct a hypoelliptic deformation of the classical
Dirac operator DX = ∂X
+ ∂X∗
.
DX acts on X, its deformation acts on X total spaceof TX.
Like its de Rham counterpart, this deformationinterpolates between classical Hodge theory and thegeodesic flow.
Construction of hypoelliptic Quillen metric, andcomparison with elliptic case.
Connection with ‘physics’.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
1 The flat case
2 The hypoelliptic Dirac operator BY
3 Analytic properties of B2Y
4 B2Y deformation of usual Laplacian on X
5 The hypoelliptic Quillen metric
6 The hypoelliptic Laplacian and ‘physics’
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
4 identities
1 + 1 = 2.
(a+ b)2 = a2 + 2ab+ b2.∫Re−y
2/2 dy√2π
= 1.∫Re−iyξ−y
2/2 dy√2π
= e−ξ2/2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
4 identities
1 + 1 = 2.
(a+ b)2 = a2 + 2ab+ b2.∫Re−y
2/2 dy√2π
= 1.∫Re−iyξ−y
2/2 dy√2π
= e−ξ2/2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
4 identities
1 + 1 = 2.
(a+ b)2 = a2 + 2ab+ b2.
∫Re−y
2/2 dy√2π
= 1.∫Re−iyξ−y
2/2 dy√2π
= e−ξ2/2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
4 identities
1 + 1 = 2.
(a+ b)2 = a2 + 2ab+ b2.∫Re−y
2/2 dy√2π
= 1.
∫Re−iyξ−y
2/2 dy√2π
= e−ξ2/2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
4 identities
1 + 1 = 2.
(a+ b)2 = a2 + 2ab+ b2.∫Re−y
2/2 dy√2π
= 1.∫Re−iyξ−y
2/2 dy√2π
= e−ξ2/2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A proof of the last identity
∫R
e−iyξ−y2/2 dy√
2π= e−ξ
2/2
∫R
e−(y+iξ)2/2 dy√2π
= e−ξ2/2
∫R
e−y2/2 dy√
2π= e−ξ
2/2.
In the last identity, we made the complex translationy → y − iξ while using analyticity of e−y
2/2.
Fourier + analyticity.
For the hypoelliptic Laplacian, we will use the above ina geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A proof of the last identity
∫R
e−iyξ−y2/2 dy√
2π= e−ξ
2/2
∫R
e−(y+iξ)2/2 dy√2π
= e−ξ2/2
∫R
e−y2/2 dy√
2π= e−ξ
2/2.
In the last identity, we made the complex translationy → y − iξ while using analyticity of e−y
2/2.
Fourier + analyticity.
For the hypoelliptic Laplacian, we will use the above ina geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A proof of the last identity
∫R
e−iyξ−y2/2 dy√
2π= e−ξ
2/2
∫R
e−(y+iξ)2/2 dy√2π
= e−ξ2/2
∫R
e−y2/2 dy√
2π= e−ξ
2/2.
In the last identity, we made the complex translationy → y − iξ while using analyticity of e−y
2/2.
Fourier + analyticity.
For the hypoelliptic Laplacian, we will use the above ina geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A proof of the last identity
∫R
e−iyξ−y2/2 dy√
2π= e−ξ
2/2
∫R
e−(y+iξ)2/2 dy√2π
= e−ξ2/2
∫R
e−y2/2 dy√
2π= e−ξ
2/2.
In the last identity, we made the complex translationy → y − iξ while using analyticity of e−y
2/2.
Fourier + analyticity.
For the hypoelliptic Laplacian, we will use the above ina geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A proof of the last identity
∫R
e−iyξ−y2/2 dy√
2π= e−ξ
2/2
∫R
e−(y+iξ)2/2 dy√2π
= e−ξ2/2
∫R
e−y2/2 dy√
2π= e−ξ
2/2.
In the last identity, we made the complex translationy → y − iξ while using analyticity of e−y
2/2.
Fourier + analyticity.
For the hypoelliptic Laplacian, we will use the above ina geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A proof of the last identity
∫R
e−iyξ−y2/2 dy√
2π= e−ξ
2/2
∫R
e−(y+iξ)2/2 dy√2π
= e−ξ2/2
∫R
e−y2/2 dy√
2π= e−ξ
2/2.
In the last identity, we made the complex translationy → y − iξ while using analyticity of e−y
2/2.
Fourier + analyticity.
For the hypoelliptic Laplacian, we will use the above ina geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The harmonic oscillator
H = 12
(− ∂2
∂y2 + y2 − 1)
.
H self-adjoint, Sp (H) = N.
Ground state =e−y2/2 and eigenfunctions the weighted
Hermite polynomials (=e−y2/2× polynomials).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The harmonic oscillator
H = 12
(− ∂2
∂y2 + y2 − 1)
.
H self-adjoint, Sp (H) = N.
Ground state =e−y2/2 and eigenfunctions the weighted
Hermite polynomials (=e−y2/2× polynomials).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The harmonic oscillator
H = 12
(− ∂2
∂y2 + y2 − 1)
.
H self-adjoint, Sp (H) = N.
Ground state =e−y2/2 and eigenfunctions the weighted
Hermite polynomials (=e−y2/2× polynomials).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The harmonic oscillator
H = 12
(− ∂2
∂y2 + y2 − 1)
.
H self-adjoint, Sp (H) = N.
Ground state =e−y2/2 and eigenfunctions the weighted
Hermite polynomials (=e−y2/2× polynomials).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.Kolmogorov operator L = H + y ∂
∂x.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.
Kolmogorov operator L = H + y ∂∂x
.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.Kolmogorov operator L = H + y ∂
∂x.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.Kolmogorov operator L = H + y ∂
∂x.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.Kolmogorov operator L = H + y ∂
∂x.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.Kolmogorov operator L = H + y ∂
∂x.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator of Kolmogorov
Operator in 2 = 1 + 1 variables x, y.Kolmogorov operator L = H + y ∂
∂x.
L = 12
(− ∂2
∂y2 + y2 − 1)
+ y ∂∂x
.
L = 12
(− ∂2
∂y2 +(y + ∂
∂x
)2 − 1)− 1
2∂2
∂x2 .
If ∂∂x→ iξ,
L =1
2
(− ∂2
∂y2+ (y + iξ)2 − 1
)+
1
2ξ2.
If y → y − iξ, L becomes L given by
L = H +1
2ξ2.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A conjugation of L
M = ∂2
∂x∂yhyperbolic.
Conjugation identity
e−MLeM =1
2
(− ∂2
∂y2+ y2 − 1
)− 1
2
∂2
∂x2.
L hypoelliptic (Hormander).
e−MLeM elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A conjugation of L
M = ∂2
∂x∂yhyperbolic.
Conjugation identity
e−MLeM =1
2
(− ∂2
∂y2+ y2 − 1
)− 1
2
∂2
∂x2.
L hypoelliptic (Hormander).
e−MLeM elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A conjugation of L
M = ∂2
∂x∂yhyperbolic.
Conjugation identity
e−MLeM =1
2
(− ∂2
∂y2+ y2 − 1
)− 1
2
∂2
∂x2.
L hypoelliptic (Hormander).
e−MLeM elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A conjugation of L
M = ∂2
∂x∂yhyperbolic.
Conjugation identity
e−MLeM =1
2
(− ∂2
∂y2+ y2 − 1
)− 1
2
∂2
∂x2.
L hypoelliptic (Hormander).
e−MLeM elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A conjugation of L
M = ∂2
∂x∂yhyperbolic.
Conjugation identity
e−MLeM =1
2
(− ∂2
∂y2+ y2 − 1
)− 1
2
∂2
∂x2.
L hypoelliptic (Hormander).
e−MLeM elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Hypoellipticity
∂∂t− L hypoelliptic (existence of heat kernel).
Heisenberg commutation[∂∂y, y]
= 1 . . .
. . . implies hypoellipticity by Hormander[∂
∂y, y
∂
∂x
]=
∂
∂x.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Hypoellipticity
∂∂t− L hypoelliptic (existence of heat kernel).
Heisenberg commutation[∂∂y, y]
= 1 . . .
. . . implies hypoellipticity by Hormander[∂
∂y, y
∂
∂x
]=
∂
∂x.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Hypoellipticity
∂∂t− L hypoelliptic (existence of heat kernel).
Heisenberg commutation[∂∂y, y]
= 1 . . .
. . . implies hypoellipticity by Hormander[∂
∂y, y
∂
∂x
]=
∂
∂x.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Hypoellipticity
∂∂t− L hypoelliptic (existence of heat kernel).
Heisenberg commutation[∂∂y, y]
= 1 . . .
. . . implies hypoellipticity by Hormander[∂
∂y, y
∂
∂x
]=
∂
∂x.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Conjugation is legitimate
Here (x, y) ∈ S1 ×R.
By analyticity, y → y − iξ acts on the weightedHermite polynomials.
Hypoelliptic non self-adjoint L is isospectral to ellipticself-adjoint e−MLeM .
Sp (L) = N + 2k2π2, k ∈ Z.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Conjugation is legitimate
Here (x, y) ∈ S1 ×R.
By analyticity, y → y − iξ acts on the weightedHermite polynomials.
Hypoelliptic non self-adjoint L is isospectral to ellipticself-adjoint e−MLeM .
Sp (L) = N + 2k2π2, k ∈ Z.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Conjugation is legitimate
Here (x, y) ∈ S1 ×R.
By analyticity, y → y − iξ acts on the weightedHermite polynomials.
Hypoelliptic non self-adjoint L is isospectral to ellipticself-adjoint e−MLeM .
Sp (L) = N + 2k2π2, k ∈ Z.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Conjugation is legitimate
Here (x, y) ∈ S1 ×R.
By analyticity, y → y − iξ acts on the weightedHermite polynomials.
Hypoelliptic non self-adjoint L is isospectral to ellipticself-adjoint e−MLeM .
Sp (L) = N + 2k2π2, k ∈ Z.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Conjugation is legitimate
Here (x, y) ∈ S1 ×R.
By analyticity, y → y − iξ acts on the weightedHermite polynomials.
Hypoelliptic non self-adjoint L is isospectral to ellipticself-adjoint e−MLeM .
Sp (L) = N + 2k2π2, k ∈ Z.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The deformation parameter
b > 0, and Lb = Hb2
+ 1by ∂∂x
.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.b→ 0, the resolvent of Lb concentrates on ground state
exp (−y2/2) so that we recover Sp(−1
2∂2
∂x2
).
b→ +∞, after conjugation, Lb ' y2
2+ y ∂
∂x.
y ∂∂x
generator of geodesic flow.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The deformation parameter
b > 0, and Lb = Hb2
+ 1by ∂∂x
.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.b→ 0, the resolvent of Lb concentrates on ground state
exp (−y2/2) so that we recover Sp(−1
2∂2
∂x2
).
b→ +∞, after conjugation, Lb ' y2
2+ y ∂
∂x.
y ∂∂x
generator of geodesic flow.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The deformation parameter
b > 0, and Lb = Hb2
+ 1by ∂∂x
.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.
b→ 0, the resolvent of Lb concentrates on ground state
exp (−y2/2) so that we recover Sp(−1
2∂2
∂x2
).
b→ +∞, after conjugation, Lb ' y2
2+ y ∂
∂x.
y ∂∂x
generator of geodesic flow.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The deformation parameter
b > 0, and Lb = Hb2
+ 1by ∂∂x
.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.b→ 0, the resolvent of Lb concentrates on ground state
exp (−y2/2) so that we recover Sp(−1
2∂2
∂x2
).
b→ +∞, after conjugation, Lb ' y2
2+ y ∂
∂x.
y ∂∂x
generator of geodesic flow.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The deformation parameter
b > 0, and Lb = Hb2
+ 1by ∂∂x
.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.b→ 0, the resolvent of Lb concentrates on ground state
exp (−y2/2) so that we recover Sp(−1
2∂2
∂x2
).
b→ +∞, after conjugation, Lb ' y2
2+ y ∂
∂x.
y ∂∂x
generator of geodesic flow.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The deformation parameter
b > 0, and Lb = Hb2
+ 1by ∂∂x
.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.b→ 0, the resolvent of Lb concentrates on ground state
exp (−y2/2) so that we recover Sp(−1
2∂2
∂x2
).
b→ +∞, after conjugation, Lb ' y2
2+ y ∂
∂x.
y ∂∂x
generator of geodesic flow.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Supersymmetry
Witten Laplacian H = 12
(− ∂2
∂y2 + y2 − 1)
+NΛ·(R).
Lb = Hb2
+ 1by ∂∂x
is still hypoelliptic.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.Tr [exp (t∂2/∂x2/2)] = Trs [exp (−tLb)].Proof of Poisson formula by interpolation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Supersymmetry
Witten Laplacian H = 12
(− ∂2
∂y2 + y2 − 1)
+NΛ·(R).
Lb = Hb2
+ 1by ∂∂x
is still hypoelliptic.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.Tr [exp (t∂2/∂x2/2)] = Trs [exp (−tLb)].Proof of Poisson formula by interpolation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Supersymmetry
Witten Laplacian H = 12
(− ∂2
∂y2 + y2 − 1)
+NΛ·(R).
Lb = Hb2
+ 1by ∂∂x
is still hypoelliptic.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.Tr [exp (t∂2/∂x2/2)] = Trs [exp (−tLb)].Proof of Poisson formula by interpolation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Supersymmetry
Witten Laplacian H = 12
(− ∂2
∂y2 + y2 − 1)
+NΛ·(R).
Lb = Hb2
+ 1by ∂∂x
is still hypoelliptic.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.
Tr [exp (t∂2/∂x2/2)] = Trs [exp (−tLb)].Proof of Poisson formula by interpolation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Supersymmetry
Witten Laplacian H = 12
(− ∂2
∂y2 + y2 − 1)
+NΛ·(R).
Lb = Hb2
+ 1by ∂∂x
is still hypoelliptic.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.Tr [exp (t∂2/∂x2/2)] = Trs [exp (−tLb)].
Proof of Poisson formula by interpolation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Supersymmetry
Witten Laplacian H = 12
(− ∂2
∂y2 + y2 − 1)
+NΛ·(R).
Lb = Hb2
+ 1by ∂∂x
is still hypoelliptic.
Sp (Lb) = Nb2
+ 2π2k2, k ∈ Z.Tr [exp (t∂2/∂x2/2)] = Trs [exp (−tLb)].Proof of Poisson formula by interpolation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Geometrisation of the construction
X Riemannian manifold, X total space of TX.
H → harmonic oscillator H along the fibres of X .
y ∂∂x→ generator of geodesic flow ∇Y .
− ∂2
∂x2 → X Laplacian of X.
Can H +∇Y become a deformed ‘Laplacian’ for anexotic Hodge theory?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Geometrisation of the construction
X Riemannian manifold, X total space of TX.
H → harmonic oscillator H along the fibres of X .
y ∂∂x→ generator of geodesic flow ∇Y .
− ∂2
∂x2 → X Laplacian of X.
Can H +∇Y become a deformed ‘Laplacian’ for anexotic Hodge theory?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Geometrisation of the construction
X Riemannian manifold, X total space of TX.
H → harmonic oscillator H along the fibres of X .
y ∂∂x→ generator of geodesic flow ∇Y .
− ∂2
∂x2 → X Laplacian of X.
Can H +∇Y become a deformed ‘Laplacian’ for anexotic Hodge theory?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Geometrisation of the construction
X Riemannian manifold, X total space of TX.
H → harmonic oscillator H along the fibres of X .
y ∂∂x→ generator of geodesic flow ∇Y .
− ∂2
∂x2 → X Laplacian of X.
Can H +∇Y become a deformed ‘Laplacian’ for anexotic Hodge theory?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Geometrisation of the construction
X Riemannian manifold, X total space of TX.
H → harmonic oscillator H along the fibres of X .
y ∂∂x→ generator of geodesic flow ∇Y .
− ∂2
∂x2 → X Laplacian of X.
Can H +∇Y become a deformed ‘Laplacian’ for anexotic Hodge theory?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Geometrisation of the construction
X Riemannian manifold, X total space of TX.
H → harmonic oscillator H along the fibres of X .
y ∂∂x→ generator of geodesic flow ∇Y .
− ∂2
∂x2 → X Laplacian of X.
Can H +∇Y become a deformed ‘Laplacian’ for anexotic Hodge theory?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Dolbeault complex
(X, gTX) compact complex Kahler manifold.
(E, gE) holomorphic Hermitian vector bundle on X.(Ω(0,·) (X,E) , ∂
X)
Dolbeault complex with
cohomology H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Dolbeault complex
(X, gTX) compact complex Kahler manifold.
(E, gE) holomorphic Hermitian vector bundle on X.(Ω(0,·) (X,E) , ∂
X)
Dolbeault complex with
cohomology H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Dolbeault complex
(X, gTX) compact complex Kahler manifold.
(E, gE) holomorphic Hermitian vector bundle on X.
(Ω(0,·) (X,E) , ∂
X)
Dolbeault complex with
cohomology H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Dolbeault complex
(X, gTX) compact complex Kahler manifold.
(E, gE) holomorphic Hermitian vector bundle on X.(Ω(0,·) (X,E) , ∂
X)
Dolbeault complex with
cohomology H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic deformation
X total space of tangent bundle, with fibre TX.
Deformation acts on Ω(0,·) (X , π∗ (Λ· (T ∗X)⊗ E)).
Y generator of the geodesic flow, Cartan formula[dX , iY
]= LY .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic deformation
X total space of tangent bundle, with fibre TX.
Deformation acts on Ω(0,·) (X , π∗ (Λ· (T ∗X)⊗ E)).
Y generator of the geodesic flow, Cartan formula[dX , iY
]= LY .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic deformation
X total space of tangent bundle, with fibre TX.
Deformation acts on Ω(0,·) (X , π∗ (Λ· (T ∗X)⊗ E)).
Y generator of the geodesic flow, Cartan formula[dX , iY
]= LY .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic deformation
X total space of tangent bundle, with fibre TX.
Deformation acts on Ω(0,·) (X , π∗ (Λ· (T ∗X)⊗ E)).
Y generator of the geodesic flow, Cartan formula[dX , iY
]= LY .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ∂ operator on X
gdTX Hermitian metric on TX, ∇dTX associated
connection.
∂X∂ operator on X .
∂X
= ∇I′′ + ∂V
.
∇I′′ horizontal ∂ for ∇dTX , ∂V
vertical ∂.
A′′ = ∂X
is a antiholomorphic superconnection suchthat A′′2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ∂ operator on X
gdTX Hermitian metric on TX, ∇dTX associated
connection.
∂X∂ operator on X .
∂X
= ∇I′′ + ∂V
.
∇I′′ horizontal ∂ for ∇dTX , ∂V
vertical ∂.
A′′ = ∂X
is a antiholomorphic superconnection suchthat A′′2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ∂ operator on X
gdTX Hermitian metric on TX, ∇dTX associated
connection.
∂X∂ operator on X .
∂X
= ∇I′′ + ∂V
.
∇I′′ horizontal ∂ for ∇dTX , ∂V
vertical ∂.
A′′ = ∂X
is a antiholomorphic superconnection suchthat A′′2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ∂ operator on X
gdTX Hermitian metric on TX, ∇dTX associated
connection.
∂X∂ operator on X .
∂X
= ∇I′′ + ∂V
.
∇I′′ horizontal ∂ for ∇dTX , ∂V
vertical ∂.
A′′ = ∂X
is a antiholomorphic superconnection suchthat A′′2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ∂ operator on X
gdTX Hermitian metric on TX, ∇dTX associated
connection.
∂X∂ operator on X .
∂X
= ∇I′′ + ∂V
.
∇I′′ horizontal ∂ for ∇dTX , ∂V
vertical ∂.
A′′ = ∂X
is a antiholomorphic superconnection suchthat A′′2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ∂ operator on X
gdTX Hermitian metric on TX, ∇dTX associated
connection.
∂X∂ operator on X .
∂X
= ∇I′′ + ∂V
.
∇I′′ horizontal ∂ for ∇dTX , ∂V
vertical ∂.
A′′ = ∂X
is a antiholomorphic superconnection suchthat A′′2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ‘adjoint’ A′ of A′′
A′ = ∇I′ + ∂V ∗, A = A′′ + A′.
Principal symbol of the superconnection Aσ (A) = iξH ∧+ic
(ξV).
σ (A) is nilpotent horizontally, and elliptic vertically.
A2 is a second order elliptic differential operator actingfibrewise along TX.
A2 = −12∆V + . . .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ‘adjoint’ A′ of A′′
A′ = ∇I′ + ∂V ∗, A = A′′ + A′.
Principal symbol of the superconnection Aσ (A) = iξH ∧+ic
(ξV).
σ (A) is nilpotent horizontally, and elliptic vertically.
A2 is a second order elliptic differential operator actingfibrewise along TX.
A2 = −12∆V + . . .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ‘adjoint’ A′ of A′′
A′ = ∇I′ + ∂V ∗, A = A′′ + A′.
Principal symbol of the superconnection Aσ (A) = iξH ∧+ic
(ξV).
σ (A) is nilpotent horizontally, and elliptic vertically.
A2 is a second order elliptic differential operator actingfibrewise along TX.
A2 = −12∆V + . . .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ‘adjoint’ A′ of A′′
A′ = ∇I′ + ∂V ∗, A = A′′ + A′.
Principal symbol of the superconnection Aσ (A) = iξH ∧+ic
(ξV).
σ (A) is nilpotent horizontally, and elliptic vertically.
A2 is a second order elliptic differential operator actingfibrewise along TX.
A2 = −12∆V + . . .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ‘adjoint’ A′ of A′′
A′ = ∇I′ + ∂V ∗, A = A′′ + A′.
Principal symbol of the superconnection Aσ (A) = iξH ∧+ic
(ξV).
σ (A) is nilpotent horizontally, and elliptic vertically.
A2 is a second order elliptic differential operator actingfibrewise along TX.
A2 = −12∆V + . . .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The ‘adjoint’ A′ of A′′
A′ = ∇I′ + ∂V ∗, A = A′′ + A′.
Principal symbol of the superconnection Aσ (A) = iξH ∧+ic
(ξV).
σ (A) is nilpotent horizontally, and elliptic vertically.
A2 is a second order elliptic differential operator actingfibrewise along TX.
A2 = −12∆V + . . .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Koszul resolution
y ∈ TX canonical holomorphic section of TX.
TX ' TX.
Interior multiplication iy acts on π∗Λ· (T ∗X).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Koszul resolution
y ∈ TX canonical holomorphic section of TX.
TX ' TX.
Interior multiplication iy acts on π∗Λ· (T ∗X).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Koszul resolution
y ∈ TX canonical holomorphic section of TX.
TX ' TX.
Interior multiplication iy acts on π∗Λ· (T ∗X).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Koszul resolution
y ∈ TX canonical holomorphic section of TX.
TX ' TX.
Interior multiplication iy acts on π∗Λ· (T ∗X).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Dolbeault-Koszul operator
A′′Y = ∂X
+ iy.
A′′2Y = 0.
A′′Y = ∇I′′ + ∂V
+ iy.
A′′Y is not a superconnection.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Dolbeault-Koszul operator
A′′Y = ∂X
+ iy.
A′′2Y = 0.
A′′Y = ∇I′′ + ∂V
+ iy.
A′′Y is not a superconnection.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Dolbeault-Koszul operator
A′′Y = ∂X
+ iy.
A′′2Y = 0.
A′′Y = ∇I′′ + ∂V
+ iy.
A′′Y is not a superconnection.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Dolbeault-Koszul operator
A′′Y = ∂X
+ iy.
A′′2Y = 0.
A′′Y = ∇I′′ + ∂V
+ iy.
A′′Y is not a superconnection.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Dolbeault-Koszul operator
A′′Y = ∂X
+ iy.
A′′2Y = 0.
A′′Y = ∇I′′ + ∂V
+ iy.
A′′Y is not a superconnection.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A′Y
A′Y = A′ + iy.
A′Y = ∇I′ + ∂V ∗
+ iy.
A′2Y = 0.
AY = A′′Y + A′Y not good enough.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A′Y
A′Y = A′ + iy.
A′Y = ∇I′ + ∂V ∗
+ iy.
A′2Y = 0.
AY = A′′Y + A′Y not good enough.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A′Y
A′Y = A′ + iy.
A′Y = ∇I′ + ∂V ∗
+ iy.
A′2Y = 0.
AY = A′′Y + A′Y not good enough.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A′Y
A′Y = A′ + iy.
A′Y = ∇I′ + ∂V ∗
+ iy.
A′2Y = 0.
AY = A′′Y + A′Y not good enough.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A′Y
A′Y = A′ + iy.
A′Y = ∇I′ + ∂V ∗
+ iy.
A′2Y = 0.
AY = A′′Y + A′Y not good enough.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY
gTX = gdTX Kahler metric on TX, with Kahler form
ωX .
B′′Y = A′′Y .
B′Y = eiωXA′Y e
−iωX.
BY = B′′Y +B′Y .
BY = ∇I + ∂V
+ ∂V ∗
+ iy+y + y∗∧.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY
gTX = gdTX Kahler metric on TX, with Kahler form
ωX .
B′′Y = A′′Y .
B′Y = eiωXA′Y e
−iωX.
BY = B′′Y +B′Y .
BY = ∇I + ∂V
+ ∂V ∗
+ iy+y + y∗∧.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY
gTX = gdTX Kahler metric on TX, with Kahler form
ωX .
B′′Y = A′′Y .
B′Y = eiωXA′Y e
−iωX.
BY = B′′Y +B′Y .
BY = ∇I + ∂V
+ ∂V ∗
+ iy+y + y∗∧.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY
gTX = gdTX Kahler metric on TX, with Kahler form
ωX .
B′′Y = A′′Y .
B′Y = eiωXA′Y e
−iωX.
BY = B′′Y +B′Y .
BY = ∇I + ∂V
+ ∂V ∗
+ iy+y + y∗∧.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY
gTX = gdTX Kahler metric on TX, with Kahler form
ωX .
B′′Y = A′′Y .
B′Y = eiωXA′Y e
−iωX.
BY = B′′Y +B′Y .
BY = ∇I + ∂V
+ ∂V ∗
+ iy+y + y∗∧.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY
gTX = gdTX Kahler metric on TX, with Kahler form
ωX .
B′′Y = A′′Y .
B′Y = eiωXA′Y e
−iωX.
BY = B′′Y +B′Y .
BY = ∇I + ∂V
+ ∂V ∗
+ iy+y + y∗∧.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The principal symbol of BY
Principal symbol of BY given by
σ (BY ) = iξH ∧+ic (ξV ) .
The horizontal part of σ (BY ) is nilpotent.
BY not elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The principal symbol of BY
Principal symbol of BY given by
σ (BY ) = iξH ∧+ic (ξV ) .
The horizontal part of σ (BY ) is nilpotent.
BY not elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The principal symbol of BY
Principal symbol of BY given by
σ (BY ) = iξH ∧+ic (ξV ) .
The horizontal part of σ (BY ) is nilpotent.
BY not elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The principal symbol of BY
Principal symbol of BY given by
σ (BY ) = iξH ∧+ic (ξV ) .
The horizontal part of σ (BY ) is nilpotent.
BY not elliptic.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A formula for B2Y
B2Y = [B′′Y , B
′Y ] given by
B2Y =
1
2
(−∆V + |Y |2 + c (ei) c (ei)
)+∇Y −∇RTXY
+1
4
⟨RTXei, ej
⟩c (ei) c (ej)
+1
2Tr[RTX
]+RE.
∇Y horizontal covariant differential in direction Y .∂∂t−B2
Y is hypoelliptic (Kolmogorov, Hormander).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A formula for B2Y
B2Y = [B′′Y , B
′Y ] given by
B2Y =
1
2
(−∆V + |Y |2 + c (ei) c (ei)
)+∇Y −∇RTXY
+1
4
⟨RTXei, ej
⟩c (ei) c (ej)
+1
2Tr[RTX
]+RE.
∇Y horizontal covariant differential in direction Y .∂∂t−B2
Y is hypoelliptic (Kolmogorov, Hormander).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A formula for B2Y
B2Y = [B′′Y , B
′Y ] given by
B2Y =
1
2
(−∆V + |Y |2 + c (ei) c (ei)
)+∇Y −∇RTXY
+1
4
⟨RTXei, ej
⟩c (ei) c (ej)
+1
2Tr[RTX
]+RE.
∇Y horizontal covariant differential in direction Y .∂∂t−B2
Y is hypoelliptic (Kolmogorov, Hormander).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A formula for B2Y
B2Y = [B′′Y , B
′Y ] given by
B2Y =
1
2
(−∆V + |Y |2 + c (ei) c (ei)
)+∇Y −∇RTXY
+1
4
⟨RTXei, ej
⟩c (ei) c (ej)
+1
2Tr[RTX
]+RE.
∇Y horizontal covariant differential in direction Y .
∂∂t−B2
Y is hypoelliptic (Kolmogorov, Hormander).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A formula for B2Y
B2Y = [B′′Y , B
′Y ] given by
B2Y =
1
2
(−∆V + |Y |2 + c (ei) c (ei)
)+∇Y −∇RTXY
+1
4
⟨RTXei, ej
⟩c (ei) c (ej)
+1
2Tr[RTX
]+RE.
∇Y horizontal covariant differential in direction Y .∂∂t−B2
Y is hypoelliptic (Kolmogorov, Hormander).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator B2Y
B2Y essentially the sum of a harmonic oscillator along
TX and of ∇Y .
B2Y = [B′′Y , B
′Y ] is called a hypoelliptic Laplacian.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator B2Y
B2Y essentially the sum of a harmonic oscillator along
TX and of ∇Y .
B2Y = [B′′Y , B
′Y ] is called a hypoelliptic Laplacian.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator B2Y
B2Y essentially the sum of a harmonic oscillator along
TX and of ∇Y .
B2Y = [B′′Y , B
′Y ] is called a hypoelliptic Laplacian.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY is ‘self-adjoint’
B′Y is not the adjoint of B′′Y with respect to aHermitian product.
B′Y is the adjoint of B′′Y with respect to a Hermitianform η.
r : (x, y)→ (x,−y).
η (s, s′) =⟨r∗eiΛs, eiΛs′
⟩L2 .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY is ‘self-adjoint’
B′Y is not the adjoint of B′′Y with respect to aHermitian product.
B′Y is the adjoint of B′′Y with respect to a Hermitianform η.
r : (x, y)→ (x,−y).
η (s, s′) =⟨r∗eiΛs, eiΛs′
⟩L2 .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY is ‘self-adjoint’
B′Y is not the adjoint of B′′Y with respect to aHermitian product.
B′Y is the adjoint of B′′Y with respect to a Hermitianform η.
r : (x, y)→ (x,−y).
η (s, s′) =⟨r∗eiΛs, eiΛs′
⟩L2 .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY is ‘self-adjoint’
B′Y is not the adjoint of B′′Y with respect to aHermitian product.
B′Y is the adjoint of B′′Y with respect to a Hermitianform η.
r : (x, y)→ (x,−y).
η (s, s′) =⟨r∗eiΛs, eiΛs′
⟩L2 .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator BY is ‘self-adjoint’
B′Y is not the adjoint of B′′Y with respect to aHermitian product.
B′Y is the adjoint of B′′Y with respect to a Hermitianform η.
r : (x, y)→ (x,−y).
η (s, s′) =⟨r∗eiΛs, eiΛs′
⟩L2 .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The analysis of the hypoelliptic Laplacian (-,G.
Lebeau)
The operator B2Y has a discrete spectrum, which is
conjugation invariant.
The Hodge theorem almost holds.
Heat kernel is smoothing and trace class.
Heat kernel has a local index theory.
As t→ 0, ‘local supertrace’ converges toTd(TX, gTX
)ch(E, gE
).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The analysis of the hypoelliptic Laplacian (-,G.
Lebeau)
The operator B2Y has a discrete spectrum, which is
conjugation invariant.
The Hodge theorem almost holds.
Heat kernel is smoothing and trace class.
Heat kernel has a local index theory.
As t→ 0, ‘local supertrace’ converges toTd(TX, gTX
)ch(E, gE
).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The analysis of the hypoelliptic Laplacian (-,G.
Lebeau)
The operator B2Y has a discrete spectrum, which is
conjugation invariant.
The Hodge theorem almost holds.
Heat kernel is smoothing and trace class.
Heat kernel has a local index theory.
As t→ 0, ‘local supertrace’ converges toTd(TX, gTX
)ch(E, gE
).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The analysis of the hypoelliptic Laplacian (-,G.
Lebeau)
The operator B2Y has a discrete spectrum, which is
conjugation invariant.
The Hodge theorem almost holds.
Heat kernel is smoothing and trace class.
Heat kernel has a local index theory.
As t→ 0, ‘local supertrace’ converges toTd(TX, gTX
)ch(E, gE
).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The analysis of the hypoelliptic Laplacian (-,G.
Lebeau)
The operator B2Y has a discrete spectrum, which is
conjugation invariant.
The Hodge theorem almost holds.
Heat kernel is smoothing and trace class.
Heat kernel has a local index theory.
As t→ 0, ‘local supertrace’ converges toTd(TX, gTX
)ch(E, gE
).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The analysis of the hypoelliptic Laplacian (-,G.
Lebeau)
The operator B2Y has a discrete spectrum, which is
conjugation invariant.
The Hodge theorem almost holds.
Heat kernel is smoothing and trace class.
Heat kernel has a local index theory.
As t→ 0, ‘local supertrace’ converges toTd(TX, gTX
)ch(E, gE
).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Replacing y by y/b2
Replace y by y/b2.
BY,b ' ∇I + 1b
(∂V
+ ∂V ∗
+ iy+y + y∗∧)
.
After conjugation,
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
H the horizontal part of BY,b.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Replacing y by y/b2
Replace y by y/b2.
BY,b ' ∇I + 1b
(∂V
+ ∂V ∗
+ iy+y + y∗∧)
.
After conjugation,
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
H the horizontal part of BY,b.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Replacing y by y/b2
Replace y by y/b2.
BY,b ' ∇I + 1b
(∂V
+ ∂V ∗
+ iy+y + y∗∧)
.
After conjugation,
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
H the horizontal part of BY,b.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Replacing y by y/b2
Replace y by y/b2.
BY,b ' ∇I + 1b
(∂V
+ ∂V ∗
+ iy+y + y∗∧)
.
After conjugation,
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
H the horizontal part of BY,b.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Replacing y by y/b2
Replace y by y/b2.
BY,b ' ∇I + 1b
(∂V
+ ∂V ∗
+ iy+y + y∗∧)
.
After conjugation,
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
H the horizontal part of BY,b.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The kernel of the vertical part
The vertical part is ∂V
+ iy + ∂V ∗
+ y∗∧.
Fibrewise kernel 1-dimensional, in ‘degree’ 0, spannedby β = exp
(− |Y |2 /2 + iω
).
Here ω = −iwi ∧ wi.P fibrewise orthogonal projection on kernel.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The kernel of the vertical part
The vertical part is ∂V
+ iy + ∂V ∗
+ y∗∧.
Fibrewise kernel 1-dimensional, in ‘degree’ 0, spannedby β = exp
(− |Y |2 /2 + iω
).
Here ω = −iwi ∧ wi.P fibrewise orthogonal projection on kernel.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The kernel of the vertical part
The vertical part is ∂V
+ iy + ∂V ∗
+ y∗∧.
Fibrewise kernel 1-dimensional, in ‘degree’ 0, spannedby β = exp
(− |Y |2 /2 + iω
).
Here ω = −iwi ∧ wi.P fibrewise orthogonal projection on kernel.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The kernel of the vertical part
The vertical part is ∂V
+ iy + ∂V ∗
+ y∗∧.
Fibrewise kernel 1-dimensional, in ‘degree’ 0, spannedby β = exp
(− |Y |2 /2 + iω
).
Here ω = −iwi ∧ wi.
P fibrewise orthogonal projection on kernel.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The kernel of the vertical part
The vertical part is ∂V
+ iy + ∂V ∗
+ y∗∧.
Fibrewise kernel 1-dimensional, in ‘degree’ 0, spannedby β = exp
(− |Y |2 /2 + iω
).
Here ω = −iwi ∧ wi.P fibrewise orthogonal projection on kernel.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The compression of the horizontal part
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
Fundamental identity of operators on Ω(0,·) (X,E),
PHP = ∂X
+ ∂X∗.
H =(wi∧+ iwi
)∇wi
+ (wi ∧ −iwi)∇wi
.
Kahler identities for H = ∂ − ∂∗ + ∂ + ∂∗
give H2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The compression of the horizontal part
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
Fundamental identity of operators on Ω(0,·) (X,E),
PHP = ∂X
+ ∂X∗.
H =(wi∧+ iwi
)∇wi
+ (wi ∧ −iwi)∇wi
.
Kahler identities for H = ∂ − ∂∗ + ∂ + ∂∗
give H2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The compression of the horizontal part
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
Fundamental identity of operators on Ω(0,·) (X,E),
PHP = ∂X
+ ∂X∗.
H =(wi∧+ iwi
)∇wi
+ (wi ∧ −iwi)∇wi
.
Kahler identities for H = ∂ − ∂∗ + ∂ + ∂∗
give H2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The compression of the horizontal part
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
Fundamental identity of operators on Ω(0,·) (X,E),
PHP = ∂X
+ ∂X∗.
H =(wi∧+ iwi
)∇wi
+ (wi ∧ −iwi)∇wi
.
Kahler identities for H = ∂ − ∂∗ + ∂ + ∂∗
give H2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The compression of the horizontal part
BY,b = H +1
b
(∂V
+ iy + ∂V ∗
+ y∗∧).
Fundamental identity of operators on Ω(0,·) (X,E),
PHP = ∂X
+ ∂X∗.
H =(wi∧+ iwi
)∇wi
+ (wi ∧ −iwi)∇wi
.
Kahler identities for H = ∂ − ∂∗ + ∂ + ∂∗
give H2 = 0.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Convergence of B2Y,b to X (-, Lebeau)
DX,2 = X =[∂X, ∂
X∗].
In every possible sense, as b→ 0, B2Y,b → X .
For t > 0, exp(−tB2
Y,b
)→ P exp
(−tX
)P .
Note that 1b∇Y ultimately produces X .
Illustration of 1 + 1 = 2 ( ∂∂x→ ∂2
∂x2 , Ito calculus).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Convergence of B2Y,b to X (-, Lebeau)
DX,2 = X =[∂X, ∂
X∗].
In every possible sense, as b→ 0, B2Y,b → X .
For t > 0, exp(−tB2
Y,b
)→ P exp
(−tX
)P .
Note that 1b∇Y ultimately produces X .
Illustration of 1 + 1 = 2 ( ∂∂x→ ∂2
∂x2 , Ito calculus).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Convergence of B2Y,b to X (-, Lebeau)
DX,2 = X =[∂X, ∂
X∗].
In every possible sense, as b→ 0, B2Y,b → X .
For t > 0, exp(−tB2
Y,b
)→ P exp
(−tX
)P .
Note that 1b∇Y ultimately produces X .
Illustration of 1 + 1 = 2 ( ∂∂x→ ∂2
∂x2 , Ito calculus).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Convergence of B2Y,b to X (-, Lebeau)
DX,2 = X =[∂X, ∂
X∗].
In every possible sense, as b→ 0, B2Y,b → X .
For t > 0, exp(−tB2
Y,b
)→ P exp
(−tX
)P .
Note that 1b∇Y ultimately produces X .
Illustration of 1 + 1 = 2 ( ∂∂x→ ∂2
∂x2 , Ito calculus).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Convergence of B2Y,b to X (-, Lebeau)
DX,2 = X =[∂X, ∂
X∗].
In every possible sense, as b→ 0, B2Y,b → X .
For t > 0, exp(−tB2
Y,b
)→ P exp
(−tX
)P .
Note that 1b∇Y ultimately produces X .
Illustration of 1 + 1 = 2 ( ∂∂x→ ∂2
∂x2 , Ito calculus).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Convergence of B2Y,b to X (-, Lebeau)
DX,2 = X =[∂X, ∂
X∗].
In every possible sense, as b→ 0, B2Y,b → X .
For t > 0, exp(−tB2
Y,b
)→ P exp
(−tX
)P .
Note that 1b∇Y ultimately produces X .
Illustration of 1 + 1 = 2 ( ∂∂x→ ∂2
∂x2 , Ito calculus).
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The limit b→ +∞
As b→ +∞, after rescaling,
B2Y,b '
1
2|Y |2 +∇Y .
∇Y vector field generating the geodesic flow.
The corresponding traces localize on closed geodesics.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The limit b→ +∞
As b→ +∞, after rescaling,
B2Y,b '
1
2|Y |2 +∇Y .
∇Y vector field generating the geodesic flow.
The corresponding traces localize on closed geodesics.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The limit b→ +∞
As b→ +∞, after rescaling,
B2Y,b '
1
2|Y |2 +∇Y .
∇Y vector field generating the geodesic flow.
The corresponding traces localize on closed geodesics.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The limit b→ +∞
As b→ +∞, after rescaling,
B2Y,b '
1
2|Y |2 +∇Y .
∇Y vector field generating the geodesic flow.
The corresponding traces localize on closed geodesics.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The zeta function
Fix b = 1.
Set λ = detH(0,·) (X,E).
Operator B2Y has zeta function.
By ‘self-adjointness’, there is a ‘hypoelliptic Quillenmetric’ ‖ ‖λ,h on λ, with sign ±1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The zeta function
Fix b = 1.
Set λ = detH(0,·) (X,E).
Operator B2Y has zeta function.
By ‘self-adjointness’, there is a ‘hypoelliptic Quillenmetric’ ‖ ‖λ,h on λ, with sign ±1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The zeta function
Fix b = 1.
Set λ = detH(0,·) (X,E).
Operator B2Y has zeta function.
By ‘self-adjointness’, there is a ‘hypoelliptic Quillenmetric’ ‖ ‖λ,h on λ, with sign ±1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The zeta function
Fix b = 1.
Set λ = detH(0,·) (X,E).
Operator B2Y has zeta function.
By ‘self-adjointness’, there is a ‘hypoelliptic Quillenmetric’ ‖ ‖λ,h on λ, with sign ±1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The zeta function
Fix b = 1.
Set λ = detH(0,·) (X,E).
Operator B2Y has zeta function.
By ‘self-adjointness’, there is a ‘hypoelliptic Quillenmetric’ ‖ ‖λ,h on λ, with sign ±1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A comparison formula
‖ ‖λ usual ‘elliptic’ Quillen metric.
Gillet-Soule additive R genus,
R (x) =∑n oddn≥1
(2ζ ′ (−n) +
n∑j=1
1
jζ (−n)
)xn
n!.
Comparison formula
log
(‖ ‖2
λ,h
‖ ‖2λ
)=
∫X
Td (TX)R (TX) ch (E) .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A comparison formula
‖ ‖λ usual ‘elliptic’ Quillen metric.
Gillet-Soule additive R genus,
R (x) =∑n oddn≥1
(2ζ ′ (−n) +
n∑j=1
1
jζ (−n)
)xn
n!.
Comparison formula
log
(‖ ‖2
λ,h
‖ ‖2λ
)=
∫X
Td (TX)R (TX) ch (E) .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A comparison formula
‖ ‖λ usual ‘elliptic’ Quillen metric.
Gillet-Soule additive R genus,
R (x) =∑n oddn≥1
(2ζ ′ (−n) +
n∑j=1
1
jζ (−n)
)xn
n!.
Comparison formula
log
(‖ ‖2
λ,h
‖ ‖2λ
)=
∫X
Td (TX)R (TX) ch (E) .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
A comparison formula
‖ ‖λ usual ‘elliptic’ Quillen metric.
Gillet-Soule additive R genus,
R (x) =∑n oddn≥1
(2ζ ′ (−n) +
n∑j=1
1
jζ (−n)
)xn
n!.
Comparison formula
log
(‖ ‖2
λ,h
‖ ‖2λ
)=
∫X
Td (TX)R (TX) ch (E) .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Brownian motion and geodesics
Brownian motion x = w observed at microscopic level.
Its calculus, the Ito calculus, is of order 2 = 1 + 1.
Geodesics x = 0 (Hamiltonian dynamics) observed atmacroscopic level.
The number of dots is also 2 = 1 + 1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Brownian motion and geodesics
Brownian motion x = w observed at microscopic level.
Its calculus, the Ito calculus, is of order 2 = 1 + 1.
Geodesics x = 0 (Hamiltonian dynamics) observed atmacroscopic level.
The number of dots is also 2 = 1 + 1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Brownian motion and geodesics
Brownian motion x = w observed at microscopic level.
Its calculus, the Ito calculus, is of order 2 = 1 + 1.
Geodesics x = 0 (Hamiltonian dynamics) observed atmacroscopic level.
The number of dots is also 2 = 1 + 1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Brownian motion and geodesics
Brownian motion x = w observed at microscopic level.
Its calculus, the Ito calculus, is of order 2 = 1 + 1.
Geodesics x = 0 (Hamiltonian dynamics) observed atmacroscopic level.
The number of dots is also 2 = 1 + 1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Brownian motion and geodesics
Brownian motion x = w observed at microscopic level.
Its calculus, the Ito calculus, is of order 2 = 1 + 1.
Geodesics x = 0 (Hamiltonian dynamics) observed atmacroscopic level.
The number of dots is also 2 = 1 + 1.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Langevin equation
Langevin tried to reconcile Newton’s law withBrownian motion (infinite energy).
Langevin equation mx = −x+ w in R4.
For m = 0, x = w, for m = +∞, x = 0.
If we make m = b2, the hypoelliptic Laplacian (withy = x) describes the dynamics of the Langevinequation in a geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Langevin equation
Langevin tried to reconcile Newton’s law withBrownian motion (infinite energy).
Langevin equation mx = −x+ w in R4.
For m = 0, x = w, for m = +∞, x = 0.
If we make m = b2, the hypoelliptic Laplacian (withy = x) describes the dynamics of the Langevinequation in a geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Langevin equation
Langevin tried to reconcile Newton’s law withBrownian motion (infinite energy).
Langevin equation mx = −x+ w in R4.
For m = 0, x = w, for m = +∞, x = 0.
If we make m = b2, the hypoelliptic Laplacian (withy = x) describes the dynamics of the Langevinequation in a geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Langevin equation
Langevin tried to reconcile Newton’s law withBrownian motion (infinite energy).
Langevin equation mx = −x+ w in R4.
For m = 0, x = w, for m = +∞, x = 0.
If we make m = b2, the hypoelliptic Laplacian (withy = x) describes the dynamics of the Langevinequation in a geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The Langevin equation
Langevin tried to reconcile Newton’s law withBrownian motion (infinite energy).
Langevin equation mx = −x+ w in R4.
For m = 0, x = w, for m = +∞, x = 0.
If we make m = b2, the hypoelliptic Laplacian (withy = x) describes the dynamics of the Langevinequation in a geometric context.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic Laplacian
The dynamics for the operator Lb is just
b2x = −x+ w.
The parameter b2 is a mass.
The interpolation property is exactly the one suggestedby Langevin equation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic Laplacian
The dynamics for the operator Lb is just
b2x = −x+ w.
The parameter b2 is a mass.
The interpolation property is exactly the one suggestedby Langevin equation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic Laplacian
The dynamics for the operator Lb is just
b2x = −x+ w.
The parameter b2 is a mass.
The interpolation property is exactly the one suggestedby Langevin equation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic Laplacian
The dynamics for the operator Lb is just
b2x = −x+ w.
The parameter b2 is a mass.
The interpolation property is exactly the one suggestedby Langevin equation.
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The operator A as a local limit of the elliptic
Dirac operator
DX = ∂X
+ ∂X∗
elliptic Dirac operator.
Pt (x, y) heat kernel of exp(−tDX,2
).
Asymptotics of Pt (x, x) via change of coordinates(blowup) at each x.
A2 is the ‘limit’ under a sophisticate rescaling of tDX,2
when t→ 0.
How to make the fibre TX ‘walk again’ along X?
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
The hypoelliptic deformation
X total space of tangent bundle, with fibre TX.
Deformation acts on Ω(0,·) (X , π∗ (Λ· (T ∗X)⊗ E)).
Y generator of the geodesic flow, Cartan formula[dX , iY
]= LY .
Jean-Michel Bismut The hypoelliptic Dirac operator
The flat caseThe hypoelliptic Dirac operator BY
Analytic properties of B2Y
B2Y deformation of usual Laplacian on X
The hypoelliptic Quillen metricThe hypoelliptic Laplacian and ‘physics’
Classical Hodge theory
∂X∗
formal adjoint of ∂X
.
DX = ∂X
+ ∂X∗
.
X = DX,2 =[∂X, ∂
X∗]
Hodge Laplacian.
X elliptic, self-adjoint ≥ 0.
H = ker X = ker ∂X ∩ ker ∂
X∗the harmonic forms.
By Hodge theory, H ∼ H(0,·) (X,E).
Jean-Michel Bismut The hypoelliptic Dirac operator