what is logarithmic in schramm-loewner evolutions?
TRANSCRIPT
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What is logarithmic in Schramm-LoewnerEvolutions?
Kalle Kytola
Advanced Conformal Field Theory and
Applications
Workshop 1: Logarithmic Conformal Field
Theory, October 3-7, 2011
Kalle Kytola SLEs and logarithmic CFTs
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Random geometry of critical phenomena and LCFT
Geometric questions have lead to logarithmic CFT- polymers, percolation ([Saleur ’87], [Cardy ’99], . . . )- non-diagonalizable transfer matrices ([Saleur], [Saint-Aubin & Pearce &
Rasmussen ’09], . . . )Schramm-Loewner evolutions (SLE)
- random conformally invariant fractal curves ([Schramm ’00])- some rigorous results about scaling limits of interfaces incritical models of statistical mechanics
- breakthrough: Fields medals Werner (2006), Smirnov (2010)
Kalle Kytola SLEs and logarithmic CFTs
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Random curves in 2D critical phenomena
Percolation
p = pc
Ising model
T = Tc
Kalle Kytola SLEs and logarithmic CFTs
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Random curves in 2D critical phenomena
Self-avoiding walk
Loop erased random walk
Kalle Kytola SLEs and logarithmic CFTs
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Chordal SLEκ in the half-plane
Kalle Kytola SLEs and logarithmic CFTs
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Chordal SLEκ in the half-plane
In the upper half plane H = {z ∈ C : ℑm z > 0}, from 0 to ∞
Loewner’s equation for gt
d
dtgt(z) =
2gt(z)−Xt
Driving process Brownian motion
Xt =√κ Bt
Kalle Kytola SLEs and logarithmic CFTs
![Page 7: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/7.jpg)
Chordal SLEκ in the half-plane
In the upper half plane H = {z ∈ C : ℑm z > 0}, from 0 to ∞
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gt
H
γ[0, t]
Xt
H
Kt
Loewner’s equation for gt
d
dtgt(z) =
2gt(z)−Xt
Driving process Brownian motion
Xt =√κ Bt
g0(z) = z =⇒ gt : Ht → H conformal map
Kalle Kytola SLEs and logarithmic CFTs
![Page 8: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/8.jpg)
Chordal SLEκ in the half-plane
In the upper half plane H = {z ∈ C : ℑm z > 0}, from 0 to ∞
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gt
H
γ[0, t]
Xt
H
Kt
Loewner’s equation for gt
d
dtgt(z) =
2gt(z)−Xt
Driving process Brownian motion
Xt =√κ Bt
g0(z) = z =⇒ gt : Ht → H conformal map
Loewner chain generated by a curve γ : [0,∞[ → H
* γ is a non-self-crossing curve, the “chordal SLEκ trace”* Ht is the unbounded component of H \ γ[0, t]
Kalle Kytola SLEs and logarithmic CFTs
![Page 9: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/9.jpg)
Chordal SLEκ in the half-plane
In the upper half plane H = {z ∈ C : ℑm z > 0}, from 0 to ∞
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gt
H
γ[0, t]
Xt
H
Kt
Loewner’s equation for gt
d
dtgt(z) =
2gt(z)−Xt
Driving process Brownian motion
Xt =√κ Bt
g0(z) = z =⇒ gt : Ht → H conformal map
Loewner chain generated by a curve γ : [0,∞[ → H
* γ is a non-self-crossing curve, the “chordal SLEκ trace”* Ht is the unbounded component of H \ γ[0, t]
Laurent expansion at infinity (defines am, m = 2, 3, . . .)�
�
�
�gt(z) = z +
∑∞
m=2 am(t) z1−m
Kalle Kytola SLEs and logarithmic CFTs
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Some properties of chordal SLEκ
Phases: [Rohde & Schramm]
0 ≤ κ ≤ 4: The curve γ is simple (no self intersections)
4 < κ < 8: The curve γ is self-touching (but not self crossing)
κ ≥ 8: The curve γ is space filling (Peano curve)
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γ[0, t]
H HH
γ[0, t]
Kt
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
Kalle Kytola SLEs and logarithmic CFTs
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Some properties of chordal SLEκ
Phases: [Rohde & Schramm]
0 ≤ κ ≤ 4: The curve γ is simple (no self intersections)
4 < κ < 8: The curve γ is self-touching (but not self crossing)
κ ≥ 8: The curve γ is space filling (Peano curve)
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γ[0, t]
H HH
γ[0, t]
Kt
0 ≤ κ ≤ 4 4 < κ < 8 8 ≤ κ
Fractal dimensions? [Rohde & Schramm, Beffara, . . . ]
dim(γ[0,∞]) = 1 + κ/8 for 0 ≤ κ ≤ 8
dim(∂Kt) = 1 + 2/κ for κ ≥ 4
Kalle Kytola SLEs and logarithmic CFTs
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Models of statistical physics and chordal SLEs
Model SLEκ CFT?
loop-erased random walk1 κ = 2 c = −2 CFTIsing spin cluster boundary2 κ = 3 fermionic CFT, c = 1/2
free field level lines3 κ = 4 free boson, c = 1 CFTIsing FK clusters2 κ = 16
3 fermionic CFT, c = 1/2percolation cluster boudary4 κ = 6 c = 0, Cardy’s formula
uniform spanning tree1 κ = 8 c = −2 CFT
self-avoiding walk κ = 83 ? CFT of c = 0
q-Potts and FK model κ = κ(q) ?O(n) model κ = κ(n) ?
......
1Lawler & Schramm & Werner : Ann. Probab. 32 no. 1B (2004).2Smirnov & Chelkak 2011(?)3Schramm & Sheffield : (2006)4Smirnov: C.R.Acad.Sci Paris 333 (2001), Camia & Newman: (2006)
Kalle Kytola SLEs and logarithmic CFTs
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Variants of SLE, SLEκ(ρ)
d
dtgt(z) =
2
gt(z)− Xt
, g0(z) = z
Kalle Kytola SLEs and logarithmic CFTs
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Variants of SLE, SLEκ(ρ)
d
dtgt(z) =
2
gt(z)− Xt
, g0(z) = z
Statistics of γ determined by the driving process Xt =√κ Bt
Kalle Kytola SLEs and logarithmic CFTs
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Variants of SLE, SLEκ(ρ)
d
dtgt(z) =
2
gt(z)− Xt
, g0(z) = z
Statistics of γ determined by the driving process Xt =√κ Bt
Variants? dXt =√κ dBt + κ (∂x logZ )(Xt ,Yt) dt
Kalle Kytola SLEs and logarithmic CFTs
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Variants of SLE, SLEκ(ρ)
d
dtgt(z) =
2
gt(z)− Xt
, g0(z) = z
Statistics of γ determined by the driving process Xt =√κ Bt
Variants? dXt =√κ dBt + κ (∂x logZ )(Xt ,Yt) dt
- driving process Xt = gt(γt), passive point Yt = gt(Y0)* SLEκ(ρ) with choice Z (x , y) = (x − y)ρ/κ
Kalle Kytola SLEs and logarithmic CFTs
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Variants of SLE, SLEκ(ρ)
d
dtgt(z) =
2
gt(z)− Xt
, g0(z) = z
Statistics of γ determined by the driving process Xt =√κ Bt
Variants? dXt =√κ dBt + κ (∂x logZ )(Xt ,Yt) dt
- driving process Xt = gt(γt), passive point Yt = gt(Y0)* SLEκ(ρ) with choice Z (x , y) = (x − y)ρ/κ
H
X0 Y0
Example variant 1:
chordal SLEκ from X0 to ∞,conditioned to visit Y0 ∈ ∂H
= SLEκ(ρ = κ− 8)
Kalle Kytola SLEs and logarithmic CFTs
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Examples of SLEκ(ρ)
H
X0 Y0
Example variant 2:
chordal SLEκ from X0 to ∞,conditioned to avoid the arc[Y0,∞) ⊂ ∂H
= SLEκ(ρ = κ− 4)
H
X0 Y0
freefree
Example variant 3:
dipolar SLEκ [Bauer & Bernard & Houdayer]
(role of Y0 and ∞ symmetric)
= SLEκ(ρ = κ−62 )
e.g. Ising interface with plus-minus-free
boundary conditions [Hongler & Kytola]
Kalle Kytola SLEs and logarithmic CFTs
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More example variants by coordinate changes
X0 Y0
0
µ
Coordinate change: [. . . , Schramm & Wilson, . . . ]
Kalle Kytola SLEs and logarithmic CFTs
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More example variants by coordinate changes
X0 Y0
0
µ
Coordinate change: [. . . , Schramm & Wilson, . . . ]
- γ = trace of SLEκ(ρ)
- µ : H → H Mobius withµ(X0) = X0, µ(Y0) = ∞, µ(∞) = Y0
The curve µ ◦ γ is the trace of an SLEκ(ρ′) with
ρ′ = κ− 6− ρ
Kalle Kytola SLEs and logarithmic CFTs
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More example variants by coordinate changes
X0 Y0
0
µ
Coordinate change: [. . . , Schramm & Wilson, . . . ]
- γ = trace of SLEκ(ρ)
- µ : H → H Mobius withµ(X0) = X0, µ(Y0) = ∞, µ(∞) = Y0
The curve µ ◦ γ is the trace of an SLEκ(ρ′) with
ρ′ = κ− 6− ρ
More example variants:
* Chordal SLEκ from X0 to Y0 isSLEκ(ρ = κ− 6)
* Chordal SLEκ from X0 to Y0 conditioned to“visit infinity” is SLEκ(ρ = 2)
* Chordal SLEκ from X0 to Y0 conditioned toavoid [Y0,∞) is SLEκ(ρ = −2)
* etc.
Kalle Kytola SLEs and logarithmic CFTs
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A typical SLE calculation: “Schramm’s formula”
z
X0
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
. Kalle Kytola SLEs and logarithmic CFTs
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A typical SLE calculation: “Schramm’s formula”
z
X0
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
. Kalle Kytola SLEs and logarithmic CFTs
![Page 24: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/24.jpg)
A typical SLE calculation: “Schramm’s formula”
z
X0
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
. Kalle Kytola SLEs and logarithmic CFTs
![Page 25: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/25.jpg)
A typical SLE calculation: “Schramm’s formula”
z
X0
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
. Kalle Kytola SLEs and logarithmic CFTs
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A typical SLE calculation: “Schramm’s formula”
z
X0
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
. Kalle Kytola SLEs and logarithmic CFTs
![Page 27: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/27.jpg)
A typical SLE calculation: “Schramm’s formula”
z
X0
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
. Kalle Kytola SLEs and logarithmic CFTs
![Page 28: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/28.jpg)
A typical SLE calculation: “Schramm’s formula”
z
gt(z)
X0
H
gt : H \ γ[0, t] → H
Xt
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
- map by gt : H \ γ[0, t] → H
. Kalle Kytola SLEs and logarithmic CFTs
![Page 29: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/29.jpg)
A typical SLE calculation: “Schramm’s formula”
z
gt(z)
X0
H
gt : H \ γ[0, t] → H
Xt
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
- map by gt : H \ γ[0, t] → H
- the future is again an SLE
. Kalle Kytola SLEs and logarithmic CFTs
![Page 30: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/30.jpg)
A typical SLE calculation: “Schramm’s formula”
z
gt(z)
X0
H
gt : H \ γ[0, t] → H
Xt
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
- map by gt : H \ γ[0, t] → H
- the future is again an SLE- martingale: P(Xt , gt(z), gt(z))
. Kalle Kytola SLEs and logarithmic CFTs
![Page 31: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/31.jpg)
A typical SLE calculation: “Schramm’s formula”
z
gt(z)
X0
H
gt : H \ γ[0, t] → H
Xt
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
- map by gt : H \ γ[0, t] → H
- the future is again an SLE- martingale: P(Xt , gt(z), gt(z))
0 =d
dtE[P(Xt , gt(z), gt(z))
]= E
[(AP)
]
where A = 2z−x
∂∂z + 2
z−x∂∂z + κ
2∂2
∂x2
. Kalle Kytola SLEs and logarithmic CFTs
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A typical SLE calculation: “Schramm’s formula”
z
gt(z)
X0
H
gt : H \ γ[0, t] → H
Xt
H
chordal SLEκ from X0 = x
Does the curve leave z ∈ H on its left or right?
P(x , z , z) = P[γ leaves z on its left]
Given a piece of the curve γ[0, t]. . .
- map by gt : H \ γ[0, t] → H
- the future is again an SLE- martingale: P(Xt , gt(z), gt(z))
0 =d
dtE[P(Xt , gt(z), gt(z))
]= E
[(AP)
]
where A = 2z−x
∂∂z + 2
z−x∂∂z + κ
2∂2
∂x2
* To find P , solve the diff. eq. AP = 0 withappropriate boundary conditions!
. Kalle Kytola SLEs and logarithmic CFTs
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Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 83
P = 12(1− cos(θ))
Kalle Kytola SLEs and logarithmic CFTs
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Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 6
P = (hypergeom.)Kalle Kytola SLEs and logarithmic CFTs
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Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 4
P = 1π θ
Kalle Kytola SLEs and logarithmic CFTs
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Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 4
P = 1π θ
Summary: “typical SLE calculation”
Kalle Kytola SLEs and logarithmic CFTs
![Page 37: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/37.jpg)
Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 4
P = 1π θ
Summary: “typical SLE calculation”
P is a CFT correlation function
* field φ(z , z) has conformal weighth = h = 0
Kalle Kytola SLEs and logarithmic CFTs
![Page 38: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/38.jpg)
Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 4
P = 1π θ
Summary: “typical SLE calculation”
P is a CFT correlation function
* field φ(z , z) has conformal weighth = h = 0
Importance of martingales
* generator:
A = 2z−x
∂∂z + 2
z−x∂∂z + κ
2∂2
∂x2
* local martingales: Ker A
Kalle Kytola SLEs and logarithmic CFTs
![Page 39: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/39.jpg)
Schramm’s formula
P(x , z , z) = 〈ψh1,2(x) φh=h=0(z , z) ψh1,2(∞)〉
=1
2− i Γ( 4κ)√
πΓ(8−4κ2κ )
z + z − 2x
z − z2F1
(1
2,4
κ;3
2;(z + z − 2x
z − z
)2)
(x = 0, z = e iθ)
0.5 1.0 1.5 2.0 2.5 3.0Θ
0.2
0.4
0.6
0.8
1.0
P
κ = 4
P = 1π θ =
12πi log(z/z)
Summary: “typical SLE calculation”
P is a CFT correlation function
* field φ(z , z) has conformal weighth = h = 0
Importance of martingales
* generator:
A = 2z−x
∂∂z + 2
z−x∂∂z + κ
2∂2
∂x2
* local martingales: Ker ACase κ→ 4 is logarithmic
Kalle Kytola SLEs and logarithmic CFTs
![Page 40: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/40.jpg)
SLE local martingales
Say SLEκ(ρ) data at time t consists of
- positions of driving process Xt and passive point Yt
- coefficients am(t) of the map gt(z) =∑
m≥2 am(t)z1−m
[Kytola: Rev. Math. Phys. 19 (2007)]
Kalle Kytola SLEs and logarithmic CFTs
![Page 41: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/41.jpg)
SLE local martingales
Say SLEκ(ρ) data at time t consists of
- positions of driving process Xt and passive point Yt
- coefficients am(t) of the map gt(z) =∑
m≥2 am(t)z1−m
Consider a function ϕ(x , y ; a2, a3, . . .).
[Kytola: Rev. Math. Phys. 19 (2007)]
Kalle Kytola SLEs and logarithmic CFTs
![Page 42: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/42.jpg)
SLE local martingales
Say SLEκ(ρ) data at time t consists of
- positions of driving process Xt and passive point Yt
- coefficients am(t) of the map gt(z) =∑
m≥2 am(t)z1−m
Consider a function ϕ(x , y ; a2, a3, . . .). The drift is
d
dsE[ϕ(Xt+s ,Yt+s ; a2(t + s), . . .)
Z (Xt+s ,Yt+s)
∣∣∣ data at time t]=
(Aκ;ρ ϕ
Z
)
where the “generator” Aκ;ρ is a second order differential operator
[Kytola: Rev. Math. Phys. 19 (2007)]
Kalle Kytola SLEs and logarithmic CFTs
![Page 43: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/43.jpg)
SLE local martingales
Say SLEκ(ρ) data at time t consists of
- positions of driving process Xt and passive point Yt
- coefficients am(t) of the map gt(z) =∑
m≥2 am(t)z1−m
Consider a function ϕ(x , y ; a2, a3, . . .). The drift is
d
dsE[ϕ(Xt+s ,Yt+s ; a2(t + s), . . .)
Z (Xt+s ,Yt+s)
∣∣∣ data at time t]=
(Aκ;ρ ϕ
Z
)
where the “generator” Aκ;ρ is a second order differential operator
If ϕ ∈ Ker Aκ;ρ, the process ϕ(Xt ,Yt ;a2(t),...)Z(Xt ,Yt)
is a local martingale
[Kytola: Rev. Math. Phys. 19 (2007)]
Kalle Kytola SLEs and logarithmic CFTs
![Page 44: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/44.jpg)
SLE local martingales
Say SLEκ(ρ) data at time t consists of
- positions of driving process Xt and passive point Yt
- coefficients am(t) of the map gt(z) =∑
m≥2 am(t)z1−m
Consider a function ϕ(x , y ; a2, a3, . . .). The drift is
d
dsE[ϕ(Xt+s ,Yt+s ; a2(t + s), . . .)
Z (Xt+s ,Yt+s)
∣∣∣ data at time t]=
(Aκ;ρ ϕ
Z
)
where the “generator” Aκ;ρ is a second order differential operator
Aκ;ρ =κ
2
∂2
∂x2+
2
y − x
∂
∂y−
ρ(ρ + 4 − κ)/(2κ)
(y − x)2
+ 2∑
l≥2
( ∑
0≤m≤l−2
0≤r≤⌊ l−2−m2
⌋
xm(−1)r (m + r)!
m! r!
∑
k1,...kr≥2P
kj=l−m−2
ak1· · · akr
) ∂
∂al
If ϕ ∈ Ker Aκ;ρ, the process ϕ(Xt ,Yt ;a2(t),...)Z(Xt ,Yt)
is a local martingale
[Kytola: Rev. Math. Phys. 19 (2007)]
Kalle Kytola SLEs and logarithmic CFTs
![Page 45: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/45.jpg)
SLE local martingales
Say SLEκ(ρ) data at time t consists of
- positions of driving process Xt and passive point Yt
- coefficients am(t) of the map gt(z) =∑
m≥2 am(t)z1−m
Consider a function ϕ(x , y ; a2, a3, . . .). The drift is
d
dsE[ϕ(Xt+s ,Yt+s ; a2(t + s), . . .)
Z (Xt+s ,Yt+s)
∣∣∣ data at time t]=
(Aκ;ρ ϕ
Z
)
where the “generator” Aκ;ρ is a second order differential operator
Aκ;ρ =κ
2
∂2
∂x2+
2
y − x
∂
∂y−
ρ(ρ + 4 − κ)/(2κ)
(y − x)2
+ 2∑
l≥2
( ∑
0≤m≤l−2
0≤r≤⌊ l−2−m2
⌋
xm(−1)r (m + r)!
m! r!
∑
k1,...kr≥2P
kj=l−m−2
ak1· · · akr
) ∂
∂al
If ϕ ∈ Ker Aκ;ρ, the process ϕ(Xt ,Yt ;a2(t),...)Z(Xt ,Yt)
is a local martingale
Ker Aκ;ρ = “the space of SLEκ(ρ) local martingales”
[Kytola: Rev. Math. Phys. 19 (2007)]
Kalle Kytola SLEs and logarithmic CFTs
![Page 46: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/46.jpg)
A representation of Virasoro algebra
Let c = (6−κ)(3κ−8)2κ , hx = 6−κ
2κ , hy = ρ(ρ+4−κ)4κ
Kalle Kytola SLEs and logarithmic CFTs
![Page 47: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/47.jpg)
A representation of Virasoro algebra
Let c = (6−κ)(3κ−8)2κ , hx = 6−κ
2κ , hy = ρ(ρ+4−κ)4κ and
L0 =∑
l≥2
lal
∂
∂al
+ x∂
∂x+ y
∂
∂y+ hx + hy , L1 =
∑
l≥3
((l − 2)al−1
) ∂
∂al
+∂
∂x+
∂
∂y, L2 = −
∂
∂a2
+∑
l≥4
((l − 3)al−n
) ∂
∂al
L−1 =∑
l≥2
(
(l + 2)al+1 +∑
m1,m2≥2m1+m2=l+1
am1am2
) ∂
∂al
+(x2 − 3a2
) ∂
∂x+
(y2 − 3a2
) ∂
∂y+ 2xhx + 2yhy
L−2 =∑
l≥2
(
(l + 4)al+2 − 4a2al +∑
m1,m2≥2m1+m2=l+2
3am1am2
+∑
m1,m2,m3≥2m1+m2+m3=l+2
am1am2
am3
) ∂
∂al
+(x3 − 4xa2 − 5a3
) ∂
∂x+
(y3 − 4ya2 − 5a3
) ∂
∂y+ (3x2 − 4a2)hx + (3y2 − 4a2)hy −
c
2a2
Ln =1
(n − 2)!
[[[···[L2, L1], . . .], L1
], L1
︸ ︷︷ ︸
n − 2 times ]
, L−n =1
(n − 2)!
[
L−1,[L−1, [. . . , [L−1
︸ ︷︷ ︸
n − 2 times , L−2] ···]]] for n > 2.
Kalle Kytola SLEs and logarithmic CFTs
![Page 48: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/48.jpg)
A representation of Virasoro algebra
Let c = (6−κ)(3κ−8)2κ , hx = 6−κ
2κ , hy = ρ(ρ+4−κ)4κ and
L0 =∑
l≥2
lal
∂
∂al
+ x∂
∂x+ y
∂
∂y+ hx + hy , L1 =
∑
l≥3
((l − 2)al−1
) ∂
∂al
+∂
∂x+
∂
∂y, L2 = −
∂
∂a2
+∑
l≥4
((l − 3)al−n
) ∂
∂al
L−1 =∑
l≥2
(
(l + 2)al+1 +∑
m1,m2≥2m1+m2=l+1
am1am2
) ∂
∂al
+(x2 − 3a2
) ∂
∂x+
(y2 − 3a2
) ∂
∂y+ 2xhx + 2yhy
L−2 =∑
l≥2
(
(l + 4)al+2 − 4a2al +∑
m1,m2≥2m1+m2=l+2
3am1am2
+∑
m1,m2,m3≥2m1+m2+m3=l+2
am1am2
am3
) ∂
∂al
+(x3 − 4xa2 − 5a3
) ∂
∂x+
(y3 − 4ya2 − 5a3
) ∂
∂y+ (3x2 − 4a2)hx + (3y2 − 4a2)hy −
c
2a2
Ln =1
(n − 2)!
[[[···[L2, L1], . . .], L1
], L1
︸ ︷︷ ︸
n − 2 times ]
, L−n =1
(n − 2)!
[
L−1,[L−1, [. . . , [L−1
︸ ︷︷ ︸
n − 2 times , L−2] ···]]] for n > 2.
SLEκ(ρ) local martingales form a Virasoro module
Kalle Kytola SLEs and logarithmic CFTs
![Page 49: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/49.jpg)
A representation of Virasoro algebra
Let c = (6−κ)(3κ−8)2κ , hx = 6−κ
2κ , hy = ρ(ρ+4−κ)4κ and
L0 =∑
l≥2
lal
∂
∂al
+ x∂
∂x+ y
∂
∂y+ hx + hy , L1 =
∑
l≥3
((l − 2)al−1
) ∂
∂al
+∂
∂x+
∂
∂y, L2 = −
∂
∂a2
+∑
l≥4
((l − 3)al−n
) ∂
∂al
L−1 =∑
l≥2
(
(l + 2)al+1 +∑
m1,m2≥2m1+m2=l+1
am1am2
) ∂
∂al
+(x2 − 3a2
) ∂
∂x+
(y2 − 3a2
) ∂
∂y+ 2xhx + 2yhy
L−2 =∑
l≥2
(
(l + 4)al+2 − 4a2al +∑
m1,m2≥2m1+m2=l+2
3am1am2
+∑
m1,m2,m3≥2m1+m2+m3=l+2
am1am2
am3
) ∂
∂al
+(x3 − 4xa2 − 5a3
) ∂
∂x+
(y3 − 4ya2 − 5a3
) ∂
∂y+ (3x2 − 4a2)hx + (3y2 − 4a2)hy −
c
2a2
Ln =1
(n − 2)!
[[[···[L2, L1], . . .], L1
], L1
︸ ︷︷ ︸
n − 2 times ]
, L−n =1
(n − 2)!
[
L−1,[L−1, [. . . , [L−1
︸ ︷︷ ︸
n − 2 times , L−2] ···]]] for n > 2.
SLEκ(ρ) local martingales form a Virasoro module
[Ln, Lm] = (n −m) Ln+m + c12(n
3 − n)δn+m,0
Kalle Kytola SLEs and logarithmic CFTs
![Page 50: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/50.jpg)
A representation of Virasoro algebra
Let c = (6−κ)(3κ−8)2κ , hx = 6−κ
2κ , hy = ρ(ρ+4−κ)4κ and
L0 =∑
l≥2
lal
∂
∂al
+ x∂
∂x+ y
∂
∂y+ hx + hy , L1 =
∑
l≥3
((l − 2)al−1
) ∂
∂al
+∂
∂x+
∂
∂y, L2 = −
∂
∂a2
+∑
l≥4
((l − 3)al−n
) ∂
∂al
L−1 =∑
l≥2
(
(l + 2)al+1 +∑
m1,m2≥2m1+m2=l+1
am1am2
) ∂
∂al
+(x2 − 3a2
) ∂
∂x+
(y2 − 3a2
) ∂
∂y+ 2xhx + 2yhy
L−2 =∑
l≥2
(
(l + 4)al+2 − 4a2al +∑
m1,m2≥2m1+m2=l+2
3am1am2
+∑
m1,m2,m3≥2m1+m2+m3=l+2
am1am2
am3
) ∂
∂al
+(x3 − 4xa2 − 5a3
) ∂
∂x+
(y3 − 4ya2 − 5a3
) ∂
∂y+ (3x2 − 4a2)hx + (3y2 − 4a2)hy −
c
2a2
Ln =1
(n − 2)!
[[[···[L2, L1], . . .], L1
], L1
︸ ︷︷ ︸
n − 2 times ]
, L−n =1
(n − 2)!
[
L−1,[L−1, [. . . , [L−1
︸ ︷︷ ︸
n − 2 times , L−2] ···]]] for n > 2.
SLEκ(ρ) local martingales form a Virasoro module
[Ln, Lm] = (n −m) Ln+m + c12(n
3 − n)δn+m,0
[Ln,Aκ;ρ] = qn(x , a2, . . .) Aκ;ρ where qn is polynomial
Kalle Kytola SLEs and logarithmic CFTs
![Page 51: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/51.jpg)
A representation of Virasoro algebra
Let c = (6−κ)(3κ−8)2κ , hx = 6−κ
2κ , hy = ρ(ρ+4−κ)4κ and
L0 =∑
l≥2
lal
∂
∂al
+ x∂
∂x+ y
∂
∂y+ hx + hy , L1 =
∑
l≥3
((l − 2)al−1
) ∂
∂al
+∂
∂x+
∂
∂y, L2 = −
∂
∂a2
+∑
l≥4
((l − 3)al−n
) ∂
∂al
L−1 =∑
l≥2
(
(l + 2)al+1 +∑
m1,m2≥2m1+m2=l+1
am1am2
) ∂
∂al
+(x2 − 3a2
) ∂
∂x+
(y2 − 3a2
) ∂
∂y+ 2xhx + 2yhy
L−2 =∑
l≥2
(
(l + 4)al+2 − 4a2al +∑
m1,m2≥2m1+m2=l+2
3am1am2
+∑
m1,m2,m3≥2m1+m2+m3=l+2
am1am2
am3
) ∂
∂al
+(x3 − 4xa2 − 5a3
) ∂
∂x+
(y3 − 4ya2 − 5a3
) ∂
∂y+ (3x2 − 4a2)hx + (3y2 − 4a2)hy −
c
2a2
Ln =1
(n − 2)!
[[[···[L2, L1], . . .], L1
], L1
︸ ︷︷ ︸
n − 2 times ]
, L−n =1
(n − 2)!
[
L−1,[L−1, [. . . , [L−1
︸ ︷︷ ︸
n − 2 times , L−2] ···]]] for n > 2.
SLEκ(ρ) local martingales form a Virasoro module
[Ln, Lm] = (n −m) Ln+m + c12(n
3 − n)δn+m,0
[Ln,Aκ;ρ] = qn(x , a2, . . .) Aκ;ρ where qn is polynomial
Ln preserve Ker Aκ;ρ: if Aκ;ρ ϕ = 0 then also Aκ;ρ (Lnϕ) = 0
Kalle Kytola SLEs and logarithmic CFTs
![Page 52: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/52.jpg)
Virasoro algebra
* Virasoro algebra Vir: basis . . . , L−2, L−1, L0, L1, L2, . . . ,C
[Ln, Lm] = (n −m) Ln+m + n3−n12 δn+m,0 C , [C , Ln] = 0
* anti-involution † (“adjoint”) : L†n = L−n, C† = C
* Vir± = span {Ln : ±n > 0} positive/negative modes
* U universal enveloping algebra (similarly U±)
Kalle Kytola SLEs and logarithmic CFTs
![Page 53: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/53.jpg)
Virasoro algebra
* Virasoro algebra Vir: basis . . . , L−2, L−1, L0, L1, L2, . . . ,C
[Ln, Lm] = (n −m) Ln+m + n3−n12 δn+m,0 C , [C , Ln] = 0
* anti-involution † (“adjoint”) : L†n = L−n, C† = C
* Vir± = span {Ln : ±n > 0} positive/negative modes
* U universal enveloping algebra (similarly U±)
Conformal Field Theory:
- C acts as c1 , where c is the central charge- L0 ∼ energy
Kalle Kytola SLEs and logarithmic CFTs
![Page 54: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/54.jpg)
Virasoro algebra
* Virasoro algebra Vir: basis . . . , L−2, L−1, L0, L1, L2, . . . ,C
[Ln, Lm] = (n −m) Ln+m + n3−n12 δn+m,0 C , [C , Ln] = 0
* anti-involution † (“adjoint”) : L†n = L−n, C† = C
* Vir± = span {Ln : ±n > 0} positive/negative modes
* U universal enveloping algebra (similarly U±)
Conformal Field Theory:
- C acts as c1 , where c is the central charge- L0 ∼ energy
bounded from below?diagonalizable?
Kalle Kytola SLEs and logarithmic CFTs
![Page 55: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/55.jpg)
Virasoro algebra
* Virasoro algebra Vir: basis . . . , L−2, L−1, L0, L1, L2, . . . ,C
[Ln, Lm] = (n −m) Ln+m + n3−n12 δn+m,0 C , [C , Ln] = 0
* anti-involution † (“adjoint”) : L†n = L−n, C† = C
* Vir± = span {Ln : ±n > 0} positive/negative modes
* U universal enveloping algebra (similarly U±)
Conformal Field Theory:
- C acts as c1 , where c is the central charge- L0 ∼ energy
bounded from below?diagonalizable? not in logarithmic CFT!
Kalle Kytola SLEs and logarithmic CFTs
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Virasoro algebra
* Virasoro algebra Vir: basis . . . , L−2, L−1, L0, L1, L2, . . . ,C
[Ln, Lm] = (n −m) Ln+m + n3−n12 δn+m,0 C , [C , Ln] = 0
* anti-involution † (“adjoint”) : L†n = L−n, C† = C
* Vir± = span {Ln : ±n > 0} positive/negative modes
* U universal enveloping algebra (similarly U±)
Conformal Field Theory:
- C acts as c1 , where c is the central charge- L0 ∼ energy
bounded from below?diagonalizable? not in logarithmic CFT!
Appropriate representations: M such that
- C acts as c1
- M[h] := {v ∈ M | (L0 − h)N v = 0 for N ≫ 0} finite dim.
- M[h] = 0 for h < h0
Kalle Kytola SLEs and logarithmic CFTs
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Virasoro algebra
* Virasoro algebra Vir: basis . . . , L−2, L−1, L0, L1, L2, . . . ,C
[Ln, Lm] = (n −m) Ln+m + n3−n12 δn+m,0 C , [C , Ln] = 0
* anti-involution † (“adjoint”) : L†n = L−n, C† = C
* Vir± = span {Ln : ±n > 0} positive/negative modes
* U universal enveloping algebra (similarly U±)
Conformal Field Theory:
- C acts as c1 , where c is the central charge- L0 ∼ energy
bounded from below?diagonalizable? not in logarithmic CFT!
Appropriate representations: M such that
- C acts as c1
- M[h] := {v ∈ M | (L0 − h)N v = 0 for N ≫ 0} finite dim.
- M[h] = 0 for h < h0
- contragredient M∗ =⊕
h(M[h])∗, with 〈Xv∗, v〉 = 〈v∗,X †v〉
Kalle Kytola SLEs and logarithmic CFTs
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Highest weight modules
Highest weight modules: H a h.w. module of highest weight h,w ∈ H (highest weight vector)
(i) L0 w = h w (and C w = c w ′)
(ii) Ln w = 0 for all n > 0
(iii) U w = H
Kalle Kytola SLEs and logarithmic CFTs
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Highest weight modules
Highest weight modules: H a h.w. module of highest weight h,w ∈ H (highest weight vector)
(i) L0 w = h w (and C w = c w ′)
(ii) Ln w = 0 for all n > 0
(iii) U w = HH[h+m] = span {L−n1 · · · L−nkw : n1 ≥ · · · ≥ nk ≥ 1,
∑j nj = m}
and in particular dim (H[h+m]) ≤ p(m)
Kalle Kytola SLEs and logarithmic CFTs
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Highest weight modules
Highest weight modules: H a h.w. module of highest weight h,w ∈ H (highest weight vector)
(i) L0 w = h w (and C w = c w ′)
(ii) Ln w = 0 for all n > 0
(iii) U w = HH[h+m] = span {L−n1 · · · L−nkw : n1 ≥ · · · ≥ nk ≥ 1,
∑j nj = m}
and in particular dim (H[h+m]) ≤ p(m)
Verma module Vc,h: A h.w.module with h.w.vector vc,h where
L−n1 · · · L−nk vc,h are linearly independent, dim(V[h+m]) = p(m).
Kalle Kytola SLEs and logarithmic CFTs
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Highest weight modules
Highest weight modules: H a h.w. module of highest weight h,w ∈ H (highest weight vector)
(i) L0 w = h w (and C w = c w ′)
(ii) Ln w = 0 for all n > 0
(iii) U w = HH[h+m] = span {L−n1 · · · L−nkw : n1 ≥ · · · ≥ nk ≥ 1,
∑j nj = m}
and in particular dim (H[h+m]) ≤ p(m)
Verma module Vc,h: A h.w.module with h.w.vector vc,h where
L−n1 · · · L−nk vc,h are linearly independent, dim(V[h+m]) = p(m).
Any h.w.m. H is isomorphic to a quotient of Vc,h by asubmodule.
Kalle Kytola SLEs and logarithmic CFTs
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Submodules of the Verma module
w′′
grade r′s′
grade r′′s′′
REDUCIBLE VERMA MODULES κ /∈ Q
REDUCIBLE VERMA MODULES κ ∈ Q
grade rs
vc,h
vc,h
w′
vc,hvc,h
w′
IRREDUCIBLE VERMA MODULES
grade r′s′
Chain Braid
w′
Feigin-Fuchs theorem: submodules of Vc,h?
Kalle Kytola SLEs and logarithmic CFTs
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Submodules of the Verma module
w′′
grade r′s′
grade r′′s′′
REDUCIBLE VERMA MODULES κ /∈ Q
REDUCIBLE VERMA MODULES κ ∈ Q
grade rs
vc,h
vc,h
w′
vc,hvc,h
w′
IRREDUCIBLE VERMA MODULES
grade r′s′
Chain Braid
w′
Feigin-Fuchs theorem: submodules of Vc,h?
- all submodules generated by singular vectors
Kalle Kytola SLEs and logarithmic CFTs
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Submodules of the Verma module
w′′
grade r′s′
grade r′′s′′
REDUCIBLE VERMA MODULES κ /∈ Q
REDUCIBLE VERMA MODULES κ ∈ Q
grade rs
vc,h
vc,h
w′
vc,hvc,h
w′
IRREDUCIBLE VERMA MODULES
grade r′s′
Chain Braid
w′
Feigin-Fuchs theorem: submodules of Vc,h?
c = c(κ) = 13− 6 κ4 − 6 4
κ⇒ irreducible unless h = hr ,s(κ) for some r , s ∈ Z+
hr ,s(κ) =κ/44 (r2 − 1)− 1
2 (rs − 1) + 4/κ4 (s2 − 1)
- all submodules generated by singular vectors
Kalle Kytola SLEs and logarithmic CFTs
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Submodules of the Verma module
w′′
grade r′s′
grade r′′s′′
REDUCIBLE VERMA MODULES κ /∈ Q
REDUCIBLE VERMA MODULES κ ∈ Q
grade rs
vc,h
vc,h
w′
vc,hvc,h
w′
IRREDUCIBLE VERMA MODULES
grade r′s′
Chain Braid
w′
Feigin-Fuchs theorem: submodules of Vc,h?
c = c(κ) = 13− 6 κ4 − 6 4
κ⇒ irreducible unless h = hr ,s(κ) for some r , s ∈ Z+
hr ,s(κ) =κ/44 (r2 − 1)− 1
2 (rs − 1) + 4/κ4 (s2 − 1)
- all submodules generated by singular vectors
- rank of singular Sw : maximal # factors in S = Sr · · · S1 ∈ U−
Kalle Kytola SLEs and logarithmic CFTs
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Example of the space of local martingales
Chordal SLE in H from 0 to ∞:[Bauer & Bernard: Phys. Lett. B 557 (2003) and Ann. Henri Poincare 5 (2004)]
Kalle Kytola SLEs and logarithmic CFTs
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Example of the space of local martingales
Chordal SLE in H from 0 to ∞:[Bauer & Bernard: Phys. Lett. B 557 (2003) and Ann. Henri Poincare 5 (2004)]
Ker Aκ∼= Q∗
κ;1,2
* Qκ;1,2 = Vc(κ),h1,2(κ)/Vc(κ),h1,2(κ)+2
* Aκ, Ln operators in C[x , a2, a3, a4, . . .]
Kalle Kytola SLEs and logarithmic CFTs
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Example of the space of local martingales
Chordal SLE in H from 0 to ∞:[Bauer & Bernard: Phys. Lett. B 557 (2003) and Ann. Henri Poincare 5 (2004)]
Ker Aκ∼= Q∗
κ;1,2
* Qκ;1,2 = Vc(κ),h1,2(κ)/Vc(κ),h1,2(κ)+2
* Aκ, Ln operators in C[x , a2, a3, a4, . . .]
Note: Q∗κ;1,2 not always highest weight module!
Kalle Kytola SLEs and logarithmic CFTs
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Example of the space of local martingales
Chordal SLE in H from 0 to ∞:[Bauer & Bernard: Phys. Lett. B 557 (2003) and Ann. Henri Poincare 5 (2004)]
Ker Aκ∼= Q∗
κ;1,2
* Qκ;1,2 = Vc(κ),h1,2(κ)/Vc(κ),h1,2(κ)+2
* Aκ, Ln operators in C[x , a2, a3, a4, . . .]
Note: Q∗κ;1,2 not always highest weight module!
Example: κ = 6
- Z = 1 ∈ Ker Aκ
(constant martingale)generates irreducibleh.w.m. c = 0, h = 0
- Ξ = x ∈ Ker Aκ
(Brownian motion)generates the entire Q∗
κ;1,2
h = 0 = h1,2
1 = h1,4
2 = h3,1
5 = h1,6
7 = h5,1
Z
Ξ
Q1,2 Q∗1,2
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module S
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ S
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Pick ξ = ι(wL) and η ∈ π−1(CwR)
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Pick ξ = ι(wL) and η ∈ π−1(CwR)
* ωn = Ln η ∈ HL (for n = 1, 2) determine action of Vir+ on η
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Pick ξ = ι(wL) and η ∈ π−1(CwR)
* ωn = Ln η ∈ HL (for n = 1, 2) determine action of Vir+ on η
* ω0 = (L0 − hR)η ∈ HL is a nonzero singular vector
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Pick ξ = ι(wL) and η ∈ π−1(CwR)
* ωn = Ln η ∈ HL (for n = 1, 2) determine action of Vir+ on η
* ω0 = (L0 − hR)η ∈ HL is a nonzero singular vector- e.g. hL = hR and ω0 = ξ highest weigh vector- e.g. hL = hr ,s and hR = hL + rs and ω0 = Sξ prime singular- etc.
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Pick ξ = ι(wL) and η ∈ π−1(CwR)
* ωn = Ln η ∈ HL (for n = 1, 2) determine action of Vir+ on η
* ω0 = (L0 − hR)η ∈ HL is a nonzero singular vector- e.g. hL = hR and ω0 = ξ highest weigh vector- e.g. hL = hr ,s and hR = hL + rs and ω0 = Sξ prime singular- etc.
* freedom in choice: η′ = η + u with u ∈ HL
[hR ]
Kalle Kytola SLEs and logarithmic CFTs
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Modules for logarithmic conformal field theory
logarithmic correlation functions =⇒ L0 not diagonalizable
Simplest (and quite common) case: Staggered module SS contains a highest weight submodule HL ⊂ Sthe quotient by this is another h.w.mod. S/HL = HR
0 −→ HL ι−→ S π−→ HR −→ 0
Pick ξ = ι(wL) and η ∈ π−1(CwR)
* ωn = Ln η ∈ HL (for n = 1, 2) determine action of Vir+ on η
* ω0 = (L0 − hR)η ∈ HL is a nonzero singular vector- e.g. hL = hR and ω0 = ξ highest weigh vector- e.g. hL = hr ,s and hR = hL + rs and ω0 = Sξ prime singular- etc.
* freedom in choice: η′ = η + u with u ∈ HL
[hR ]
According to the singular vector structure of HL,HR andcorresp. Vermas, such extensions are parametrized by anaffine subspace of a vector space of dim. 0, 1 or 2 [KK & Ridout 2009]
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Staggered modules of length two
[K.K. & D. Ridout 2009]
Given two h.w.m HL and HR, classify non-isomorphic S such that
0 −→ HL −→ S −→ HR −→ 0 , L0 not diagonalizable
η
HL HR
ω0
ξ
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Staggered modules of length two
[K.K. & D. Ridout 2009]
Given two h.w.m HL and HR, classify non-isomorphic S such that
0 −→ HL −→ S −→ HR −→ 0 , L0 not diagonalizable
η
HL HR
ω0
ξ Necessary conditions:
* ℓ = hR − hL ∈ N a grade of singular0 6= ω0 = Sξ ∈ HL, denote rank r
* if SwR = 0 ∈ HR rank r , then Sω0 = 0
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Staggered modules of length two
[K.K. & D. Ridout 2009]
Given two h.w.m HL and HR, classify non-isomorphic S such that
0 −→ HL −→ S −→ HR −→ 0 , L0 not diagonalizable
η
HL HR
ω0
ξ Necessary conditions:
* ℓ = hR − hL ∈ N a grade of singular0 6= ω0 = Sξ ∈ HL, denote rank r
* if SwR = 0 ∈ HR rank r , then Sω0 = 0
Notation:
* b = # singulars of rank r− 1 in HL
* a = # non-zero sing. of rank r+ r− 1 in HL
* d = # indep. null vectors of rank r in HR
invariants β ∈ Cb and affine func. α(β) ∈ C
a×d
* β uniquely identifies S if it exists
* S exists iff α(β) = 0
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Example staggered module: local martingales of SLEκ(κ−42 )
SLEκ(ρ) local martingales, ρ = (κ− 4)/2
* partition function Z = (x − y)(κ−4)/2κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 8−κ
16 = h0,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
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Example staggered module: local martingales of SLEκ(κ−42 )
SLEκ(ρ) local martingales, ρ = (κ− 4)/2
* partition function Z = (x − y)(κ−4)/2κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 8−κ
16 = h0,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
* Λ = (x − y)(κ−4)/2κ log(x − y) ∈ Ker Aκ;ρ
- local martingale t 7→ log(Xt − Yt)- (L0 − hZ )Λ = Z , and LnΛ = 0 for n > 0
⇒ UΛ ⊂ Ker Aκ;ρ is a staggered module with hL = hR = hZ
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Example staggered module: local martingales of SLEκ(κ−42 )
SLEκ(ρ) local martingales, ρ = (κ− 4)/2
* partition function Z = (x − y)(κ−4)/2κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 8−κ
16 = h0,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
* Λ = (x − y)(κ−4)/2κ log(x − y) ∈ Ker Aκ;ρ
- local martingale t 7→ log(Xt − Yt)- (L0 − hZ )Λ = Z , and LnΛ = 0 for n > 0
⇒ UΛ ⊂ Ker Aκ;ρ is a staggered module with hL = hR = hZ- at κ = 4: SLEκ(
κ−42 ) ∼ chordal SLE4 and the local martingale
Λ/Z is essentially the one which gives Schramm’s formula
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Example staggered module: local martingales of SLEκ(κ−42 )
SLEκ(ρ) local martingales, ρ = (κ− 4)/2
* partition function Z = (x − y)(κ−4)/2κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 8−κ
16 = h0,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
* Λ = (x − y)(κ−4)/2κ log(x − y) ∈ Ker Aκ;ρ
- local martingale t 7→ log(Xt − Yt)- (L0 − hZ )Λ = Z , and LnΛ = 0 for n > 0
⇒ UΛ ⊂ Ker Aκ;ρ is a staggered module with hL = hR = hZ- at κ = 4: SLEκ(
κ−42 ) ∼ chordal SLE4 and the local martingale
Λ/Z is essentially the one which gives Schramm’s formula
Structure of the staggered module UΛ?hL = hR: no β-invariants to be determined
hL = h0,1(κ): generically the Verma modules irreducible!
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Specialization to κ = 8: local mgales of SLEκ(κ−42 )
at κ = 8 (uniform spanning tree, c = −2)SLEκ(
κ−42 ) = SLEκ(κ− 6) and h0,1(κ) = h1,1(κ) = h1,3(κ)
X0 Y0
0
µ
h = 0
h = 1
h = 3
h = 6
Zρ=2
[Gurarie ’93] [Gaberdiel & Kausch ’96]
* χU Z (q) =
1 + q2 + q3 + 2q4 + 2q5 + 4q6 + · · · = χQ1,1(q)
* χU Λ(q) =
2 + 1q + 3q2 + 3q3 + 6q4 + 7q5 + 12q6 + · · ·
0 −→ Q1,1 −→ U Λ −→ Q1,3 −→ 0
Kalle Kytola SLEs and logarithmic CFTs
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Example staggered module: local martingales of SLEκ(−2)
SLEκ(ρ) local martingales, ρ = −2
* partition function Z = (x − y)−2/κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 0 = h1,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
Kalle Kytola SLEs and logarithmic CFTs
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Example staggered module: local martingales of SLEκ(−2)
SLEκ(ρ) local martingales, ρ = −2
* partition function Z = (x − y)−2/κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 0 = h1,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
* Λ = 4−κκ (x − y)1−
2κ log(x − y) + 4−κ
8 (x − y)−2κ (x + y)
in kernel of Aκ;ρ
- (L0 − 1)Λ = L−1Z = 4−κκ (x − y)1−
2κ
⇒ UΛ ⊂ Ker Aκ;ρ is a staggered module with hL = 0, hR = 1
Kalle Kytola SLEs and logarithmic CFTs
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Example staggered module: local martingales of SLEκ(−2)
SLEκ(ρ) local martingales, ρ = −2
* partition function Z = (x − y)−2/κ ∈ Ker Aκ;ρ
- constant martingale- L0Z = hZ Z with hZ = 0 = h1,1(κ)- LnZ = 0 for n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
* Λ = 4−κκ (x − y)1−
2κ log(x − y) + 4−κ
8 (x − y)−2κ (x + y)
in kernel of Aκ;ρ
- (L0 − 1)Λ = L−1Z = 4−κκ (x − y)1−
2κ
⇒ UΛ ⊂ Ker Aκ;ρ is a staggered module with hL = 0, hR = 1
Structure of the staggered module UΛ?ω0 = (L0 − 1)Λ = L−1Z is a rank one singular vector
- (at most) one β-invariant: L1Λ = βZ with β = 1− κ4
Kalle Kytola SLEs and logarithmic CFTs
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Specialization to κ = 8: local mgales of SLEκ(κ−42 )
at κ = 8 (uniform spanning tree, c = −2):hL = h1,1(κ) = h1,3(κ), h
R = h2,1(κ) = h1,5(κ).
h = 0
h = 1
h = 3
h = 6
Zρ=−2
[Gaberdiel & Kausch ’96]
* χU Zρ=−2(q) = 1+ q+2q2 +2q3 +4q4 +5q5 +8q6 + · · · = χQ1,3(q)
* χU Λ(q) = 1 + 2q + 3q2 + 4q3 + 7q4 + 10q5 + 14q6 + · · ·* 0 −→ Q1,3 −→ U Λ −→ Q1,5 −→ 0
* invariant β of the module: L1 Λ = β Zρ=−2, here β = −1.
Kalle Kytola SLEs and logarithmic CFTs
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Example staggered module: local mgales of SLEκ(−κ−4
2 )
SLEκ(ρ) local martingales, ρ = −κ−42
Partition function Z = (x − y)−12− 2
κ ∈ Ker Aκ;ρ
- L0Z = hZ Z with hZ = 3κ−816 = h2,1(κ), and LnZ = 0 ∀n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
Kalle Kytola SLEs and logarithmic CFTs
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Example staggered module: local mgales of SLEκ(−κ−4
2 )
SLEκ(ρ) local martingales, ρ = −κ−42
Partition function Z = (x − y)−12− 2
κ ∈ Ker Aκ;ρ
- L0Z = hZ Z with hZ = 3κ−816 = h2,1(κ), and LnZ = 0 ∀n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
Λ =(κ− 4)(κ − 2)(κ+ 4)
2κ2
((x − y)
3κ−42κ log(x − y)
+ (x − y)−κ−42κ
4κ x2 − (4κ + κ2) y2
16
)
- (L0 − (h2,1 + 2))Λ = (L2−1 − κ4L−2)Z
⇒ UΛ ⊂ Ker Aκ;ρ is a stagg. mod. hL = h2,1(κ), hR = hL + 2
Kalle Kytola SLEs and logarithmic CFTs
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Example staggered module: local mgales of SLEκ(−κ−4
2 )
SLEκ(ρ) local martingales, ρ = −κ−42
Partition function Z = (x − y)−12− 2
κ ∈ Ker Aκ;ρ
- L0Z = hZ Z with hZ = 3κ−816 = h2,1(κ), and LnZ = 0 ∀n > 0
⇒ UZ ⊂ Ker Aκ;ρ a highest weight submodule
Λ =(κ− 4)(κ − 2)(κ+ 4)
2κ2
((x − y)
3κ−42κ log(x − y)
+ (x − y)−κ−42κ
4κ x2 − (4κ + κ2) y2
16
)
- (L0 − (h2,1 + 2))Λ = (L2−1 − κ4L−2)Z
⇒ UΛ ⊂ Ker Aκ;ρ is a stagg. mod. hL = h2,1(κ), hR = hL + 2
Structure of the staggered module UΛ?- (generically) one β-invariant: (L21 − κ
4L2)Λ = βZ withβ = − 1
16 (κ− 4)(κ− 2)(κ + 4).
Kalle Kytola SLEs and logarithmic CFTs
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Curious observations
SLE is another geometric example with logarithmic features
Kalle Kytola SLEs and logarithmic CFTs
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Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
Kalle Kytola SLEs and logarithmic CFTs
![Page 99: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/99.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
Kalle Kytola SLEs and logarithmic CFTs
![Page 100: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/100.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κ
Kalle Kytola SLEs and logarithmic CFTs
![Page 101: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/101.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
Kalle Kytola SLEs and logarithmic CFTs
![Page 102: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/102.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
- . . . and each is contragredient to itself
Kalle Kytola SLEs and logarithmic CFTs
![Page 103: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/103.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
- . . . and each is contragredient to itself- one can construct modules of the percolation story of [Mathieu &
Ridout ’07] about Cardy’s formula [Cardy ’92] with SLE6 variants
Kalle Kytola SLEs and logarithmic CFTs
![Page 104: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/104.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
- . . . and each is contragredient to itself- one can construct modules of the percolation story of [Mathieu &
Ridout ’07] about Cardy’s formula [Cardy ’92] with SLE6 variants- function space realization of Virasoro fusion?
Kalle Kytola SLEs and logarithmic CFTs
![Page 105: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/105.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
- . . . and each is contragredient to itself- one can construct modules of the percolation story of [Mathieu &
Ridout ’07] about Cardy’s formula [Cardy ’92] with SLE6 variants- function space realization of Virasoro fusion?
some β-invariants easy to calculate, continuous in κ- 0 → HL
h1,1→ S → HR
h1,1+1 → 0 with β = 1− κ4
- 0 → HL
h2,1→ S → HR
h2,1+2 → 0, β = − 116 (κ− 4)(κ− 2)(κ+4)
Kalle Kytola SLEs and logarithmic CFTs
![Page 106: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/106.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
- . . . and each is contragredient to itself- one can construct modules of the percolation story of [Mathieu &
Ridout ’07] about Cardy’s formula [Cardy ’92] with SLE6 variants- function space realization of Virasoro fusion?
some β-invariants easy to calculate, continuous in κ- 0 → HL
h1,1→ S → HR
h1,1+1 → 0 with β = 1− κ4
- 0 → HL
h2,1→ S → HR
h2,1+2 → 0, β = − 116 (κ− 4)(κ− 2)(κ+4)
- in [Mathieu & Ridout ’08] rational functions for β-invariants, agree.
Kalle Kytola SLEs and logarithmic CFTs
![Page 107: What is logarithmic in Schramm-Loewner Evolutions?](https://reader033.vdocument.in/reader033/viewer/2022042623/6264d4e538ab322ac01a04f9/html5/thumbnails/107.jpg)
Curious observations
SLE is another geometric example with logarithmic features
some very basic SLE calculations can not be done in ordinaryCFT (say, minimal models)
certain SLEκ(ρ) local martingales reproduce known staggeredVirasoro modules at specific values of κ, i.e. specific c
some SLEκ(ρ) local mgales form staggered modules at all κall the staggered modules found so far coincide with fusionproducts Qκ;1,2 ⊠ Vρ(ρ+4−κ)/4κ
- . . . and each is contragredient to itself- one can construct modules of the percolation story of [Mathieu &
Ridout ’07] about Cardy’s formula [Cardy ’92] with SLE6 variants- function space realization of Virasoro fusion?
some β-invariants easy to calculate, continuous in κ- 0 → HL
h1,1→ S → HR
h1,1+1 → 0 with β = 1− κ4
- 0 → HL
h2,1→ S → HR
h2,1+2 → 0, β = − 116 (κ− 4)(κ− 2)(κ+4)
- in [Mathieu & Ridout ’08] rational functions for β-invariants, agree.
still no examples of staggered modules with two β-invariants
Kalle Kytola SLEs and logarithmic CFTs
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X0 Y0
η
HL HR
ω0
ξ
Thank you!
Kalle Kytola SLEs and logarithmic CFTs