curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · i theorem...
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![Page 1: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/1.jpg)
Curvature of the determinant line bundle innoncommutative geometry
Masoud Khalkhali
Western University, Canada
![Page 2: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/2.jpg)
Holomorphic determinants
I Problem: define a function det : A → C which isholomorphic, detD 6= 0 iff D is invertible, and is gauge invariant.
I Holomorphic anomaly: gauge invariance and holomorphy don’t mixwell. So drop gauge invariance.
I Zeta regularized determinant: For D ∈ A, let ∆ = D∗D.
spectrum of ∆ : 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → ∞
I Spectral zeta function:
ζ∆(s) =∑ 1
λsi, Re(s) 0
![Page 3: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/3.jpg)
Holomorphic determinants
I Problem: define a function det : A → C which isholomorphic, detD 6= 0 iff D is invertible, and is gauge invariant.
I Holomorphic anomaly: gauge invariance and holomorphy don’t mixwell. So drop gauge invariance.
I Zeta regularized determinant: For D ∈ A, let ∆ = D∗D.
spectrum of ∆ : 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → ∞
I Spectral zeta function:
ζ∆(s) =∑ 1
λsi, Re(s) 0
![Page 4: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/4.jpg)
Holomorphic determinants
I Problem: define a function det : A → C which isholomorphic, detD 6= 0 iff D is invertible, and is gauge invariant.
I Holomorphic anomaly: gauge invariance and holomorphy don’t mixwell. So drop gauge invariance.
I Zeta regularized determinant: For D ∈ A, let ∆ = D∗D.
spectrum of ∆ : 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → ∞
I Spectral zeta function:
ζ∆(s) =∑ 1
λsi, Re(s) 0
![Page 5: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/5.jpg)
Holomorphic determinants
I Problem: define a function det : A → C which isholomorphic, detD 6= 0 iff D is invertible, and is gauge invariant.
I Holomorphic anomaly: gauge invariance and holomorphy don’t mixwell. So drop gauge invariance.
I Zeta regularized determinant: For D ∈ A, let ∆ = D∗D.
spectrum of ∆ : 0 ≤ λ0 ≤ λ1 ≤ λ2 ≤ · · · → ∞
I Spectral zeta function:
ζ∆(s) =∑ 1
λsi, Re(s) 0
![Page 6: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/6.jpg)
Regularized determinants
I Assume: ζ∆(s) has meromorphic extension to C and is regular at 0.
I Zeta regularized determinant:
log detζ(∆) = −ζ ′∆(0),∏
λi = e−ζ′∆(0)
I Example: ∞! = 1 · 2 · 3 · · · =√
2π
I But D √
det∆ is not holomorphic!
I Quillen’s approach: based on determinant line bundle.
![Page 7: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/7.jpg)
Regularized determinants
I Assume: ζ∆(s) has meromorphic extension to C and is regular at 0.
I Zeta regularized determinant:
log detζ(∆) = −ζ ′∆(0),∏
λi = e−ζ′∆(0)
I Example: ∞! = 1 · 2 · 3 · · · =√
2π
I But D √
det∆ is not holomorphic!
I Quillen’s approach: based on determinant line bundle.
![Page 8: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/8.jpg)
Regularized determinants
I Assume: ζ∆(s) has meromorphic extension to C and is regular at 0.
I Zeta regularized determinant:
log detζ(∆) = −ζ ′∆(0),∏
λi = e−ζ′∆(0)
I Example: ∞! = 1 · 2 · 3 · · · =√
2π
I But D √
det∆ is not holomorphic!
I Quillen’s approach: based on determinant line bundle.
![Page 9: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/9.jpg)
Regularized determinants
I Assume: ζ∆(s) has meromorphic extension to C and is regular at 0.
I Zeta regularized determinant:
log detζ(∆) = −ζ ′∆(0),∏
λi = e−ζ′∆(0)
I Example: ∞! = 1 · 2 · 3 · · · =√
2π
I But D √
det∆ is not holomorphic!
I Quillen’s approach: based on determinant line bundle.
![Page 10: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/10.jpg)
Regularized determinants
I Assume: ζ∆(s) has meromorphic extension to C and is regular at 0.
I Zeta regularized determinant:
log detζ(∆) = −ζ ′∆(0),∏
λi = e−ζ′∆(0)
I Example: ∞! = 1 · 2 · 3 · · · =√
2π
I But D √
det∆ is not holomorphic!
I Quillen’s approach: based on determinant line bundle.
![Page 11: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/11.jpg)
The determinant line
I Let λ= top exterior power functor. Given T : V →W , let
λT : λV → λW ,
detT := λT ∈ (λV )∗ ⊗ λW ← determinant line
I Goal: globalize this and construct a line bundle Det→ Fred overFredholm operators s.t.
DetT ' λ(kerT )∗ ⊗ λ(cokerT )
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The determinant line
I Let λ= top exterior power functor. Given T : V →W , let
λT : λV → λW ,
detT := λT ∈ (λV )∗ ⊗ λW ← determinant line
I Goal: globalize this and construct a line bundle Det→ Fred overFredholm operators s.t.
DetT ' λ(kerT )∗ ⊗ λ(cokerT )
![Page 13: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/13.jpg)
The determinant line bundle
I Space of Fredholm operators:
F = Fred(H0,H1) = T : H0 → H1; T is Fredholm
K0(X ) = [X ,F ]
I Theorem (Quillen) 1) There is a holomorphic line bundle DET → Fs.t.
(DET )T = λ(KerT )∗ ⊗ λ(KerT ∗)
2) There map σ : F0 → DET
σ(T ) =
1 T invertible
0 otherwise
is a holomorphic section of DET over F0.
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The determinant line bundle
I Space of Fredholm operators:
F = Fred(H0,H1) = T : H0 → H1; T is Fredholm
K0(X ) = [X ,F ]
I Theorem (Quillen) 1) There is a holomorphic line bundle DET → Fs.t.
(DET )T = λ(KerT )∗ ⊗ λ(KerT ∗)
2) There map σ : F0 → DET
σ(T ) =
1 T invertible
0 otherwise
is a holomorphic section of DET over F0.
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The determinant line bundle
I Space of Fredholm operators:
F = Fred(H0,H1) = T : H0 → H1; T is Fredholm
K0(X ) = [X ,F ]
I Theorem (Quillen) 1) There is a holomorphic line bundle DET → Fs.t.
(DET )T = λ(KerT )∗ ⊗ λ(KerT ∗)
2) There map σ : F0 → DET
σ(T ) =
1 T invertible
0 otherwise
is a holomorphic section of DET over F0.
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Sketch proof
I Open cover: Fred =⋃UF , dimF <∞,
UF = T ∈ Fred; Im(T ) + F = H
I Over UF defineDetT = λ(T−1F )∗ ⊗ λ(F )
I Fact: These glue together nicely to define a lie bundle over Fred.
0→ Ker(T )→ T−1F → F → coker(T )→ 0
shows thatDetT ' λ(kerT )∗ ⊗ λ(cokerT )
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Sketch proof
I Open cover: Fred =⋃UF , dimF <∞,
UF = T ∈ Fred; Im(T ) + F = H
I Over UF defineDetT = λ(T−1F )∗ ⊗ λ(F )
I Fact: These glue together nicely to define a lie bundle over Fred.
0→ Ker(T )→ T−1F → F → coker(T )→ 0
shows thatDetT ' λ(kerT )∗ ⊗ λ(cokerT )
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From det section to det function
I Use elliptic theory to pull back DET to a holomorphic line bundleL → A with
LD = λ(KerD)∗ ⊗ λ(KerD∗).
I If L admits a canonical global section s, then
σ(D) = det(D)s
defines a holomorphic determinant. s is defined once we have acanonical flat connection.
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From det section to det function
I Use elliptic theory to pull back DET to a holomorphic line bundleL → A with
LD = λ(KerD)∗ ⊗ λ(KerD∗).
I If L admits a canonical global section s, then
σ(D) = det(D)s
defines a holomorphic determinant. s is defined once we have acanonical flat connection.
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Families of Cauchy-Riemann operators
I E → M smooth vector bundle over a compact Riemann surface.
Ωp,q(E ) (p, q) - forms with coefs. in E
I Let A= space of ∂-connections D : Ω0,0(E )→ Ω0,1(E ) on E .
D = ∂ + A, A ∈ Ω0,1(End(E ))
It is an affine space over B = Ω0,1(End(E )).
A/G ' holomorphic structures on E
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Families of Cauchy-Riemann operators
I E → M smooth vector bundle over a compact Riemann surface.
Ωp,q(E ) (p, q) - forms with coefs. in E
I Let A= space of ∂-connections D : Ω0,0(E )→ Ω0,1(E ) on E .
D = ∂ + A, A ∈ Ω0,1(End(E ))
It is an affine space over B = Ω0,1(End(E )).
A/G ' holomorphic structures on E
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Quillen’s metric
I D is an elliptic 1st order PDE and defines a Fredholm operator
D : L2(E )→W 1(Ω0,1(E )), D ∈ Fred(H0,H1)
I This defines a map f : A → Fred(H0,H1). Pull back DET along f
L := f ∗(DET )
I L is a holomorphic line bundle over A.
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Quillen’s metric
I D is an elliptic 1st order PDE and defines a Fredholm operator
D : L2(E )→W 1(Ω0,1(E )), D ∈ Fred(H0,H1)
I This defines a map f : A → Fred(H0,H1). Pull back DET along f
L := f ∗(DET )
I L is a holomorphic line bundle over A.
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Quillen’s metric on L
I Define a metric on L, using regularized determinants. Pick an o. n.basis for ker(D) and ker(D∗). Get a basis v forLD ' λ(kerD)∗ ⊗ λ(kerD∗). Let
||v ||2 = exp(−ζ ′∆(0)) = det∆.
I Prop: This defines a smooth Hermitian metric on L.
I A Hermitian metric on a holomorphic line bundle has a uniquecompatible connection. Its curvature can be computed from
∂∂ log ||s||2,
where s is any local holomorphic frame.
![Page 25: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/25.jpg)
Quillen’s metric on L
I Define a metric on L, using regularized determinants. Pick an o. n.basis for ker(D) and ker(D∗). Get a basis v forLD ' λ(kerD)∗ ⊗ λ(kerD∗). Let
||v ||2 = exp(−ζ ′∆(0)) = det∆.
I Prop: This defines a smooth Hermitian metric on L.
I A Hermitian metric on a holomorphic line bundle has a uniquecompatible connection. Its curvature can be computed from
∂∂ log ||s||2,
where s is any local holomorphic frame.
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The curvature of L
I A Hermitian metric on B = Ω0,1(EndE )
||B||2 =i
2π
∫M
TrE (B∗B)
where B = α(z)dz ,B∗ = α(z)∗dz .
I Kaehler form of A. Fix D0 ∈ A, let q(D) = ||D − D0||2, and
ω = ∂∂q.
I Theorem (Quillen): The curvature of the determinant line bundle isthe symplectic form ω.
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The curvature of L
I A Hermitian metric on B = Ω0,1(EndE )
||B||2 =i
2π
∫M
TrE (B∗B)
where B = α(z)dz ,B∗ = α(z)∗dz .
I Kaehler form of A. Fix D0 ∈ A, let q(D) = ||D − D0||2, and
ω = ∂∂q.
I Theorem (Quillen): The curvature of the determinant line bundle isthe symplectic form ω.
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The curvature of L
I A Hermitian metric on B = Ω0,1(EndE )
||B||2 =i
2π
∫M
TrE (B∗B)
where B = α(z)dz ,B∗ = α(z)∗dz .
I Kaehler form of A. Fix D0 ∈ A, let q(D) = ||D − D0||2, and
ω = ∂∂q.
I Theorem (Quillen): The curvature of the determinant line bundle isthe symplectic form ω.
![Page 29: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/29.jpg)
A holomorphic determinant
I Modify the metric to get a flat connection:
||s||2f = e||D−D0||2 ||s||2
I Get a flat holomorphic global section of norm 1. This gives aholomorphic determinant function
det(D,D0) : A → C
It satisfies|det(D,D0)|2 = e||D−D0||2detζ(D∗D)
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A holomorphic determinant
I Modify the metric to get a flat connection:
||s||2f = e||D−D0||2 ||s||2
I Get a flat holomorphic global section of norm 1. This gives aholomorphic determinant function
det(D,D0) : A → C
It satisfies|det(D,D0)|2 = e||D−D0||2detζ(D∗D)
![Page 31: Curvature of the determinant line bundle in noncommutative … · 2020. 9. 18. · I Theorem (Quillen):The curvature of the determinant line bundle is the symplectic form !. A holomorphic](https://reader035.vdocuments.us/reader035/viewer/2022071401/60ed720367c367398e17a992/html5/thumbnails/31.jpg)
Cauchy-Riemann operators on Aθ
I Complex structure on Aθ: a complex number τ in the upper halfplane.
I A spectral triple
Aθ, H0 ⊕H0,1, D0 =
(0 ∂∗
∂ 0
)with ∂ = δ1 + τδ2.
I Cauchy-Riemann operators on Aθ:
Aθ, H0 ⊕H0,1, DA =
(0 ∂∗ + α∗
∂ + α 0
),
with α ∈ Aθ.
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Cauchy-Riemann operators on Aθ
I Complex structure on Aθ: a complex number τ in the upper halfplane.
I A spectral triple
Aθ, H0 ⊕H0,1, D0 =
(0 ∂∗
∂ 0
)with ∂ = δ1 + τδ2.
I Cauchy-Riemann operators on Aθ:
Aθ, H0 ⊕H0,1, DA =
(0 ∂∗ + α∗
∂ + α 0
),
with α ∈ Aθ.
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Cauchy-Riemann operators on Aθ
I Complex structure on Aθ: a complex number τ in the upper halfplane.
I A spectral triple
Aθ, H0 ⊕H0,1, D0 =
(0 ∂∗
∂ 0
)with ∂ = δ1 + τδ2.
I Cauchy-Riemann operators on Aθ:
Aθ, H0 ⊕H0,1, DA =
(0 ∂∗ + α∗
∂ + α 0
),
with α ∈ Aθ.
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Connes’ Pseudodifferential Calculus
I Symbols of order m: smooth maps σ : R2 → A∞θ with
||δ(i1,i2)∂(j1,j2)σ(ξ)|| ≤ c(1 + |ξ|)m−j1−j2 ,
and there exists a smooth map k : R2 → A∞θ such that
limλ→∞
λ−mσ(λξ1, λξ2) = k(ξ1, ξ2).
The space of symbols of order m is denoted by Sm(Aθ).
I To a symbol σ of order m, one associates an operator
Pσ(a) =
∫ ∫e−is·ξσ(ξ)αs(a) ds dξ.
The operator Pσ is said to be a pseudodifferential operator of orderm.
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Connes’ Pseudodifferential Calculus
I Symbols of order m: smooth maps σ : R2 → A∞θ with
||δ(i1,i2)∂(j1,j2)σ(ξ)|| ≤ c(1 + |ξ|)m−j1−j2 ,
and there exists a smooth map k : R2 → A∞θ such that
limλ→∞
λ−mσ(λξ1, λξ2) = k(ξ1, ξ2).
The space of symbols of order m is denoted by Sm(Aθ).
I To a symbol σ of order m, one associates an operator
Pσ(a) =
∫ ∫e−is·ξσ(ξ)αs(a) ds dξ.
The operator Pσ is said to be a pseudodifferential operator of orderm.
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Classical symbols
I Product formula:
σ(PQ) ∼∑
`=(`1,`2)≥0
1
`!∂`(σ(ξ))δ`(σ′(ξ)).
I Classical symbol of order α ∈ C : f or any N and each 0 ≤ j ≤ Nthere exist σα−j : R2\0 → Aθ positive homogeneous of degreeα− j , and a symbol σN ∈ S<(α)−N−1(Aθ), such that
σ(ξ) =N∑j=0
χ(ξ)σα−j(ξ) + σN(ξ) ξ ∈ R2.
We denote the set of classical symbols of order α by Sαcl (Aθ) andthe associated classical pseudodifferential operators by Ψα
cl(Aθ).
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Classical symbols
I Product formula:
σ(PQ) ∼∑
`=(`1,`2)≥0
1
`!∂`(σ(ξ))δ`(σ′(ξ)).
I Classical symbol of order α ∈ C : f or any N and each 0 ≤ j ≤ Nthere exist σα−j : R2\0 → Aθ positive homogeneous of degreeα− j , and a symbol σN ∈ S<(α)−N−1(Aθ), such that
σ(ξ) =N∑j=0
χ(ξ)σα−j(ξ) + σN(ξ) ξ ∈ R2.
We denote the set of classical symbols of order α by Sαcl (Aθ) andthe associated classical pseudodifferential operators by Ψα
cl(Aθ).
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Noncommutative residue
I The Wodzicki residue of a classical pseudodifferential operator Pσ isdefined as
Res(Pσ) = ϕ0 (res(Pσ)) ,
where res(Pσ) :=∫|ξ|=1
σ−2(ξ)dξ.
I t is evident from its definition that Wodzicki residue vanishes ondifferential operators and on non-integer order classicalpseudodifferential operators.
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Noncommutative residue
I The Wodzicki residue of a classical pseudodifferential operator Pσ isdefined as
Res(Pσ) = ϕ0 (res(Pσ)) ,
where res(Pσ) :=∫|ξ|=1
σ−2(ξ)dξ.
I t is evident from its definition that Wodzicki residue vanishes ondifferential operators and on non-integer order classicalpseudodifferential operators.
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A cutoff integral
I Any pseudo of order < −2 is trace-class with
Tr(P) = ϕ0
(∫R2
σP(ξ)dξ
).
I For ord(P) ≥ −2 the integral is divergent, but, assuming P isclassical, one has an asymptotic expansion as R →∞∫
B(R)
σ(ξ)dξ ∼∞∑
j=0,α−j+26=0
αj(σ)Rα−j+2 + β(σ) logR + c(σ),
where β(σ) =∫|ξ|=1
σ−2(ξ)dξ.
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A cutoff integral
I Any pseudo of order < −2 is trace-class with
Tr(P) = ϕ0
(∫R2
σP(ξ)dξ
).
I For ord(P) ≥ −2 the integral is divergent, but, assuming P isclassical, one has an asymptotic expansion as R →∞∫
B(R)
σ(ξ)dξ ∼∞∑
j=0,α−j+26=0
αj(σ)Rα−j+2 + β(σ) logR + c(σ),
where β(σ) =∫|ξ|=1
σ−2(ξ)dξ.
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The Kontsevich-Vishik trace
I The cut-off integral of a symbol σ ∈ Sαcl (Aθ) is defined to be theconstant term in the above asymptotic expansion, and we denote itby∫−σ(ξ)dξ.
I The canonical trace of a classical pseudo P ∈ Ψαcl(Aθ) of
non-integral order α is defined as
TR(P) := ϕ0
(∫−σP(ξ)dξ
).
I Theorem: The functional TR is the analytic continuation of theordinary trace on trace-class pseudodifferential operators.
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The Kontsevich-Vishik trace
I The cut-off integral of a symbol σ ∈ Sαcl (Aθ) is defined to be theconstant term in the above asymptotic expansion, and we denote itby∫−σ(ξ)dξ.
I The canonical trace of a classical pseudo P ∈ Ψαcl(Aθ) of
non-integral order α is defined as
TR(P) := ϕ0
(∫−σP(ξ)dξ
).
I Theorem: The functional TR is the analytic continuation of theordinary trace on trace-class pseudodifferential operators.
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The Kontsevich-Vishik trace
I The cut-off integral of a symbol σ ∈ Sαcl (Aθ) is defined to be theconstant term in the above asymptotic expansion, and we denote itby∫−σ(ξ)dξ.
I The canonical trace of a classical pseudo P ∈ Ψαcl(Aθ) of
non-integral order α is defined as
TR(P) := ϕ0
(∫−σP(ξ)dξ
).
I Theorem: The functional TR is the analytic continuation of theordinary trace on trace-class pseudodifferential operators.
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I Let A ∈ Ψαcl(Aθ) be of order α ∈ Z and let Q be a positive elliptic
classical pseudodifferential operator of positive order q. We have
Resz=0TR(AQ−z) =1
qRes(A).
I Proof: For the holomorphic family σ(z) = σ(AQ−z), z = 0 is a polefor the map z 7→
∫−σ(z)(ξ)dξ whose residue is given by
Resz=0
(z 7→
∫−σ(z)(ξ)dξ
)= − 1
α′(0)
∫|ξ|=1
σ−2(0)dξ
= − 1
α′(0)res(A).
Taking trace on both sides gives the result.
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I Let A ∈ Ψαcl(Aθ) be of order α ∈ Z and let Q be a positive elliptic
classical pseudodifferential operator of positive order q. We have
Resz=0TR(AQ−z) =1
qRes(A).
I Proof: For the holomorphic family σ(z) = σ(AQ−z), z = 0 is a polefor the map z 7→
∫−σ(z)(ξ)dξ whose residue is given by
Resz=0
(z 7→
∫−σ(z)(ξ)dξ
)= − 1
α′(0)
∫|ξ|=1
σ−2(0)dξ
= − 1
α′(0)res(A).
Taking trace on both sides gives the result.
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I Prop: We have TR(AB) = TR(BA) for any A,B ∈ Ψcl(Aθ),provided that ord(A) + ord(B) /∈ Z.
I z-derivatives of a classical holomorphic family of symbols is notclassical anymore. So we introduce log-polyhomogeneous symbolswhich include the z-derivatives of the symbols of the holomorphicfamily σ(AQ−z).
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I Prop: We have TR(AB) = TR(BA) for any A,B ∈ Ψcl(Aθ),provided that ord(A) + ord(B) /∈ Z.
I z-derivatives of a classical holomorphic family of symbols is notclassical anymore. So we introduce log-polyhomogeneous symbolswhich include the z-derivatives of the symbols of the holomorphicfamily σ(AQ−z).
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Logarithmic symbols
I Log-polyhomogeneous symbols:
σ(ξ) ∼∑j≥0
∞∑l=0
σα−j,l(ξ) logl |ξ| |ξ| > 0,
with σα−j,l positively homogeneous in ξ of degree α− j .
I Example: logQ where Q ∈ Ψqcl(Aθ) is a positive elliptic
pseudodifferential operator of order q > 0.
logQ := Qd
dz
∣∣∣∣z=0
Qz−1 = Qd
dz
∣∣∣∣z=0
i
2π
∫C
λz−1(Q − λ)−1dλ.
I Wodzicki residue:
res(A) =
∫|ξ|=1
σ−2,0(ξ)dξ.
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Logarithmic symbols
I Log-polyhomogeneous symbols:
σ(ξ) ∼∑j≥0
∞∑l=0
σα−j,l(ξ) logl |ξ| |ξ| > 0,
with σα−j,l positively homogeneous in ξ of degree α− j .
I Example: logQ where Q ∈ Ψqcl(Aθ) is a positive elliptic
pseudodifferential operator of order q > 0.
logQ := Qd
dz
∣∣∣∣z=0
Qz−1 = Qd
dz
∣∣∣∣z=0
i
2π
∫C
λz−1(Q − λ)−1dλ.
I Wodzicki residue:
res(A) =
∫|ξ|=1
σ−2,0(ξ)dξ.
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Logarithmic symbols
I Log-polyhomogeneous symbols:
σ(ξ) ∼∑j≥0
∞∑l=0
σα−j,l(ξ) logl |ξ| |ξ| > 0,
with σα−j,l positively homogeneous in ξ of degree α− j .
I Example: logQ where Q ∈ Ψqcl(Aθ) is a positive elliptic
pseudodifferential operator of order q > 0.
logQ := Qd
dz
∣∣∣∣z=0
Qz−1 = Qd
dz
∣∣∣∣z=0
i
2π
∫C
λz−1(Q − λ)−1dλ.
I Wodzicki residue:
res(A) =
∫|ξ|=1
σ−2,0(ξ)dξ.
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Variations of LogDet and the curvature form
I Recall: for our canonical section
‖σ‖2 = e−ζ′∆α
(0)
I Consider a holomorphic family of Cauchy-Riemann operatorsDw = ∂ + αw , and compute
∂∂ log ‖σ‖2 = δwδwζ′∆(0) = δwδw
d
dzTR(∆−z)|z=0.
I The first variational formula is given by
δwζ(z) = δwTR(∆−z) = TR(δw∆−z) = −zTR(δw∆∆−z−1).
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Variations of LogDet and the curvature form
I Recall: for our canonical section
‖σ‖2 = e−ζ′∆α
(0)
I Consider a holomorphic family of Cauchy-Riemann operatorsDw = ∂ + αw , and compute
∂∂ log ‖σ‖2 = δwδwζ′∆(0) = δwδw
d
dzTR(∆−z)|z=0.
I The first variational formula is given by
δwζ(z) = δwTR(∆−z) = TR(δw∆−z) = −zTR(δw∆∆−z−1).
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Variations of LogDet and the curvature form
I Recall: for our canonical section
‖σ‖2 = e−ζ′∆α
(0)
I Consider a holomorphic family of Cauchy-Riemann operatorsDw = ∂ + αw , and compute
∂∂ log ‖σ‖2 = δwδwζ′∆(0) = δwδw
d
dzTR(∆−z)|z=0.
I The first variational formula is given by
δwζ(z) = δwTR(∆−z) = TR(δw∆−z) = −zTR(δw∆∆−z−1).
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I Using the properties of TR, we have
δwζ′(0) =
d
dzδwζ(z)
∣∣∣∣z=0
=
−ϕ0
(∫−σ(δw∆∆−1)− 1
2res (δw∆∆−1 log ∆)
).
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The curvature of the determinant line bundle
I For a holomorphic family of Cauchy-Riemann operators Dw , thesecond variation of ζ ′(0) is given by :
δwδwζ′(0) =
1
2ϕ0
(δwDδw res(log ∆D−1)
).
I Using the symbol calculus and properties of the canonical trace weprove:Theorem (A. Fathi, A. Ghorbanpour, MK.) The curvature of thedeterminant line bundle for the noncommutative two torus is givenby
δwδwζ′(0) =
1
4π=(τ)ϕ0 (δwD(δwD)∗) .
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The curvature of the determinant line bundle
I For a holomorphic family of Cauchy-Riemann operators Dw , thesecond variation of ζ ′(0) is given by :
δwδwζ′(0) =
1
2ϕ0
(δwDδw res(log ∆D−1)
).
I Using the symbol calculus and properties of the canonical trace weprove:Theorem (A. Fathi, A. Ghorbanpour, MK.) The curvature of thedeterminant line bundle for the noncommutative two torus is givenby
δwδwζ′(0) =
1
4π=(τ)ϕ0 (δwD(δwD)∗) .