the hypoelliptic dirac operator
TRANSCRIPT
![Page 1: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/1.jpg)
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
![Page 2: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/2.jpg)
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
![Page 3: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/3.jpg)
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
![Page 4: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/4.jpg)
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
![Page 5: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/5.jpg)
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
![Page 6: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/6.jpg)
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
![Page 7: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/7.jpg)
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
![Page 8: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/8.jpg)
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
![Page 9: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/9.jpg)
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
![Page 10: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/10.jpg)
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
![Page 11: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/11.jpg)
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
![Page 12: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/12.jpg)
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
![Page 13: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/13.jpg)
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
![Page 14: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/14.jpg)
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
![Page 15: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/15.jpg)
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
![Page 16: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/16.jpg)
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
![Page 17: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/17.jpg)
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
![Page 18: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/18.jpg)
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
![Page 19: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/19.jpg)
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
![Page 20: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/20.jpg)
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
![Page 21: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/21.jpg)
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
![Page 22: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/22.jpg)
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
![Page 23: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/23.jpg)
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
![Page 24: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/24.jpg)
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
![Page 25: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/25.jpg)
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
![Page 26: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/26.jpg)
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
![Page 27: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/27.jpg)
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
![Page 28: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/28.jpg)
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
![Page 29: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/29.jpg)
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
![Page 30: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/30.jpg)
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
![Page 31: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/31.jpg)
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
![Page 32: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/32.jpg)
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
![Page 33: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/33.jpg)
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
![Page 34: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/34.jpg)
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
![Page 35: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/35.jpg)
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
![Page 36: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/36.jpg)
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
![Page 37: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/37.jpg)
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
![Page 38: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/38.jpg)
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
![Page 39: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/39.jpg)
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
![Page 40: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/40.jpg)
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
![Page 41: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/41.jpg)
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
![Page 42: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/42.jpg)
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
![Page 43: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/43.jpg)
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
![Page 44: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/44.jpg)
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
![Page 45: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/45.jpg)
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
![Page 46: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/46.jpg)
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
![Page 47: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/47.jpg)
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
![Page 48: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/48.jpg)
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
![Page 49: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/49.jpg)
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
![Page 50: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/50.jpg)
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
![Page 51: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/51.jpg)
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
![Page 52: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/52.jpg)
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
![Page 53: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/53.jpg)
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
![Page 54: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/54.jpg)
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
![Page 55: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/55.jpg)
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
![Page 56: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/56.jpg)
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
![Page 57: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/57.jpg)
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
![Page 58: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/58.jpg)
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
![Page 59: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/59.jpg)
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
![Page 60: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/60.jpg)
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
![Page 61: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/61.jpg)
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
![Page 62: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/62.jpg)
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
![Page 63: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/63.jpg)
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
![Page 64: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/64.jpg)
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
![Page 65: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/65.jpg)
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
![Page 66: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/66.jpg)
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
![Page 67: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/67.jpg)
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
![Page 68: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/68.jpg)
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
![Page 69: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/69.jpg)
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
![Page 70: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/70.jpg)
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
![Page 71: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/71.jpg)
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
![Page 72: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/72.jpg)
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
![Page 73: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/73.jpg)
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
![Page 74: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/74.jpg)
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
![Page 75: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/75.jpg)
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
![Page 76: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/76.jpg)
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
![Page 77: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/77.jpg)
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
![Page 78: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/78.jpg)
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
![Page 79: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/79.jpg)
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
![Page 80: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/80.jpg)
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
![Page 81: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/81.jpg)
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
![Page 82: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/82.jpg)
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
![Page 83: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/83.jpg)
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
![Page 84: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/84.jpg)
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
![Page 85: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/85.jpg)
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
![Page 86: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/86.jpg)
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
![Page 87: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/87.jpg)
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
![Page 88: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/88.jpg)
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
![Page 89: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/89.jpg)
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
![Page 90: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/90.jpg)
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
![Page 91: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/91.jpg)
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
![Page 92: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/92.jpg)
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
![Page 93: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/93.jpg)
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
![Page 94: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/94.jpg)
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
![Page 95: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/95.jpg)
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
![Page 96: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/96.jpg)
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
![Page 97: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/97.jpg)
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
![Page 98: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/98.jpg)
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
![Page 99: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/99.jpg)
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
![Page 100: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/100.jpg)
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
![Page 101: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/101.jpg)
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
![Page 102: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/102.jpg)
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
![Page 103: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/103.jpg)
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
![Page 104: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/104.jpg)
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
![Page 105: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/105.jpg)
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
![Page 106: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/106.jpg)
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
![Page 107: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/107.jpg)
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
![Page 108: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/108.jpg)
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
![Page 109: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/109.jpg)
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
![Page 110: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/110.jpg)
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
![Page 111: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/111.jpg)
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
![Page 112: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/112.jpg)
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
![Page 113: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/113.jpg)
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
![Page 114: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/114.jpg)
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
![Page 115: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/115.jpg)
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
![Page 116: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/116.jpg)
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
![Page 117: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/117.jpg)
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
![Page 118: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/118.jpg)
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
![Page 119: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/119.jpg)
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
![Page 120: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/120.jpg)
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
![Page 121: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/121.jpg)
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
![Page 122: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/122.jpg)
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
![Page 123: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/123.jpg)
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
![Page 124: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/124.jpg)
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
![Page 125: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/125.jpg)
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
![Page 126: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/126.jpg)
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
![Page 127: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/127.jpg)
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
![Page 128: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/128.jpg)
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
![Page 129: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/129.jpg)
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
![Page 130: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/130.jpg)
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
![Page 131: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/131.jpg)
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
![Page 132: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/132.jpg)
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
![Page 133: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/133.jpg)
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
![Page 134: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/134.jpg)
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
![Page 135: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/135.jpg)
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
![Page 136: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/136.jpg)
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
![Page 137: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/137.jpg)
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
![Page 138: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/138.jpg)
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
![Page 139: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/139.jpg)
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
![Page 140: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/140.jpg)
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
![Page 141: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/141.jpg)
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
![Page 142: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/142.jpg)
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
![Page 143: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/143.jpg)
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
![Page 144: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/144.jpg)
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
![Page 145: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/145.jpg)
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
![Page 146: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/146.jpg)
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
![Page 147: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/147.jpg)
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
![Page 148: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/148.jpg)
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
![Page 149: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/149.jpg)
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
![Page 150: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/150.jpg)
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
![Page 151: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/151.jpg)
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
![Page 152: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/152.jpg)
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
![Page 153: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/153.jpg)
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
![Page 154: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/154.jpg)
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
![Page 155: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/155.jpg)
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
![Page 156: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/156.jpg)
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
![Page 157: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/157.jpg)
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
![Page 158: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/158.jpg)
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
![Page 159: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/159.jpg)
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
![Page 160: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/160.jpg)
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
![Page 161: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/161.jpg)
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
![Page 162: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/162.jpg)
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
![Page 163: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/163.jpg)
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
![Page 164: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/164.jpg)
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
![Page 165: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/165.jpg)
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
![Page 166: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/166.jpg)
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
![Page 167: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/167.jpg)
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
![Page 168: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/168.jpg)
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
![Page 169: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/169.jpg)
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
![Page 170: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/170.jpg)
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
![Page 171: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/171.jpg)
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
![Page 172: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/172.jpg)
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
![Page 173: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/173.jpg)
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
![Page 174: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/174.jpg)
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
![Page 175: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/175.jpg)
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
![Page 176: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/176.jpg)
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
![Page 177: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/177.jpg)
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
![Page 178: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/178.jpg)
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
![Page 179: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/179.jpg)
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
![Page 180: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/180.jpg)
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
![Page 181: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/181.jpg)
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
![Page 182: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/182.jpg)
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
![Page 183: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/183.jpg)
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
![Page 184: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/184.jpg)
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
![Page 185: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/185.jpg)
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
![Page 186: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/186.jpg)
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
![Page 187: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/187.jpg)
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
![Page 188: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/188.jpg)
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
![Page 189: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/189.jpg)
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
![Page 190: The hypoelliptic Dirac operator](https://reader030.vdocument.in/reader030/viewer/2022020705/61fb82f92e268c58cd5f0a1b/html5/thumbnails/190.jpg)
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