Ricci Flow for Warped Product Manifolds Kartik Prabhu (05PH2001) - - PowerPoint PPT Presentation

Ricci Flow for Warped Product Manifolds

Kartik Prabhu (05PH2001)

Supervisor: Dr. Sayan Kar

• Dept. of Physics & Meteorology

Indian Institute of Technology Kharagpur May 04, 2010

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Outline

1 Ricci flow: definition & motivation

RF as a heat flow Example: Sphere

2 Warped manifolds

Separable solution Scaling solution

3 RG flow

Separable solution

4 Conclusion 5 Possible Further Work 6 References 7 Q & A

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Ricci flow: definition & motivation

Ricci Flow: definition

• Ricci flow is a geometric flow defined on a manifold M with a metric
• gij. It deforms the metric along a parameter λ according to the

differential equation– ∂gij ∂λ = −2Rij (1)

• To preserve the volume the Ricci flow can be normalized to give –

∂gij ∂λ = −2Rij + 2 nRgij (2)

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Ricci flow: definition & motivation

Ricci Flow: definition

• Ricci flow is a geometric flow defined on a manifold M with a metric
• gij. It deforms the metric along a parameter λ according to the

differential equation– ∂gij ∂λ = −2Rij (1)

• To preserve the volume the Ricci flow can be normalized to give –

∂gij ∂λ = −2Rij + 2 nRgij (2)

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Ricci flow: definition & motivation

Ricci Flow: motivation

• RF is like the heat equation and tends to smooth out the irregularities

in the metric.

• Finding the best metric on a manifold to solve mathematical problems

like the Poincare Conjecture. (Perelman [4] )

• Renormalization Group flow in non-linear σ-model of string theory

leads to Ricci flow in the lowest order.

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Ricci flow: definition & motivation

Ricci Flow: motivation

• RF is like the heat equation and tends to smooth out the irregularities

in the metric.

• Finding the best metric on a manifold to solve mathematical problems

like the Poincare Conjecture. (Perelman [4] )

• Renormalization Group flow in non-linear σ-model of string theory

leads to Ricci flow in the lowest order.

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Ricci flow: definition & motivation

Ricci Flow: motivation

• RF is like the heat equation and tends to smooth out the irregularities

in the metric.

• Finding the best metric on a manifold to solve mathematical problems

like the Poincare Conjecture. (Perelman [4] )

• Renormalization Group flow in non-linear σ-model of string theory

leads to Ricci flow in the lowest order.

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Ricci flow: definition & motivation RF as a heat flow

RF as heat flow

Conformally flat 2-d manifold – ds2 = e2φ(x,y) dx2 + dy2 with Ricci curvature – Rxx = Ryy = −

• ∂2

xφ + ∂2 yφ

• (3)

and the Ricci flow Eq.(1) becomes ∂φ ∂λ = △φ (4) where △ = e−2φ ∂2

x + ∂2 y

• is generalized Laplacian.

RF is like a generalized non-linear heat/diffusion equation.

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Ricci flow: definition & motivation RF as a heat flow

RF as heat flow

Conformally flat 2-d manifold – ds2 = e2φ(x,y) dx2 + dy2 with Ricci curvature – Rxx = Ryy = −

• ∂2

xφ + ∂2 yφ

• (3)

and the Ricci flow Eq.(1) becomes ∂φ ∂λ = △φ (4) where △ = e−2φ ∂2

x + ∂2 y

• is generalized Laplacian.

RF is like a generalized non-linear heat/diffusion equation.

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Ricci flow: definition & motivation RF as a heat flow

RF as heat flow

Conformally flat 2-d manifold – ds2 = e2φ(x,y) dx2 + dy2 with Ricci curvature – Rxx = Ryy = −

• ∂2

xφ + ∂2 yφ

• (3)

and the Ricci flow Eq.(1) becomes ∂φ ∂λ = △φ (4) where △ = e−2φ ∂2

x + ∂2 y

• is generalized Laplacian.

RF is like a generalized non-linear heat/diffusion equation.

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Ricci flow: definition & motivation Example: Sphere

Sphere

• metric: ds2 = r2(λ)
• dθ2 + sin2 θdφ2
• RF: r ∼ (λ0 − λ)1/2 NRF: r = constant
• but if r = r0(λ) + r1 with r1 = Y l

m(θ, φ) then r1 ∼ e−l(l+1)λ

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Ricci flow: definition & motivation Example: Sphere

Sphere

• metric: ds2 = r2(λ)
• dθ2 + sin2 θdφ2
• RF: r ∼ (λ0 − λ)1/2 NRF: r = constant
• but if r = r0(λ) + r1 with r1 = Y l

m(θ, φ) then r1 ∼ e−l(l+1)λ

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Warped manifolds

Warped manifolds

• Extra-dimensional brane world metric

ds2 = e2f (σ,λ)ηµνdxµdxν + r2

c (σ, λ)dσ2

(5)

• The RF is now a system of PDEs –

˙ f = 1 r2

c

• f ′′ + 4f ′2 − f ′r′

c

rc

• (6)

˙ rc = 4 rc

• f ′′ + f ′2 − f ′r′

c

rc

• (7)

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Warped manifolds

Warped manifolds

• Extra-dimensional brane world metric

ds2 = e2f (σ,λ)ηµνdxµdxν + r2

c (σ, λ)dσ2

(5)

• The RF is now a system of PDEs –

˙ f = 1 r2

c

• f ′′ + 4f ′2 − f ′r′

c

rc

• (6)

˙ rc = 4 rc

• f ′′ + f ′2 − f ′r′

c

rc

• (7)

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Warped manifolds Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 =

• 1 + λ

λc

exp

• ±

σ √2λc

• ηµνdxµdxν + dσ2
• Curvature scalar R = − 5/2

λ+λc . Flow becomes singular at λ = −λc.

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Warped manifolds Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 =

• 1 + λ

λc

exp

• ±

σ √2λc

• ηµνdxµdxν + dσ2
• Curvature scalar R = − 5/2

λ+λc . Flow becomes singular at λ = −λc.

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Warped manifolds Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 =

• 1 + λ

λc

exp

• ±

σ √2λc

• ηµνdxµdxν + dσ2
• Curvature scalar R = − 5/2

λ+λc . Flow becomes singular at λ = −λc.

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Warped manifolds Scaling solution

Scaling solution

• Invariance under σ → ασ and λ → α2λ. So use variable x = 1

2 ln λ σ2

and convert to ODE.

• Also use B = ex

rc

df dx = A (8) dA dx = A 2B2 − A − 24A3B2 (9) dB dx = −4AB + B + 24A2B3 (10)

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Warped manifolds Scaling solution

Scaling solution

• Invariance under σ → ασ and λ → α2λ. So use variable x = 1

2 ln λ σ2

and convert to ODE.

• Also use B = ex

rc

df dx = A (8) dA dx = A 2B2 − A − 24A3B2 (9) dB dx = −4AB + B + 24A2B3 (10)

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Warped manifolds Scaling solution

Scaling solution: plot

1 2 3 4 5 1.00 1.01 1.02 1.03 1.04 x f

f 0 1, A0 1, B0 1

(a) f v/s x

1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 x A

f 0 1, A0 1, B0 1

(b) A v/s x

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Warped manifolds Scaling solution

Scaling solution: plot

1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 x r

f 0 1, A0 1, B0 1

(c) rc v/s x

1 2 3 4 5 2 4 6 8 x Σ2R

f 0 1, A0 1, B0 1

(d) σ2R v/s x

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Warped manifolds Scaling solution

Scaling solution: singularities

1 2 3 4 5 1 2 3 4 5 A1 rc1

Figure: phase diagram showing non-singular and singular flows in space of (A(0), rc(0))

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RG flow

RG flow

• Behaviour of non-linear σ-models under renormalization given by –

∂gij ∂λ = −βij (11)

• Perturbative expansion of β in terms of α′ the inverse string tension

βij = α′β(1)

ij

+ α′2β(2)

ij

+ α′3β(3)

ij

+ α′4β(4)

ij

+ O

• α′5

(12)

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RG flow

RG flow

• Behaviour of non-linear σ-models under renormalization given by –

∂gij ∂λ = −βij (11)

• Perturbative expansion of β in terms of α′ the inverse string tension

βij = α′β(1)

ij

+ α′2β(2)

ij

+ α′3β(3)

ij

+ α′4β(4)

ij

+ O

• α′5

(12)

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RG flow

β-functions

β functions upto O(α′4) (for explicit expressions see[6, 7]) β(1)

ij

= Rij β(2)

ij

= 1 2RiklmRj klm (13a) β(3)

ij

= 1 8∇pRiklm∇pRj klm − 1 16∇iRklmp∇jRklmp + 1 2RklmpRi mlrRj kp

r − 3

8RikljRksprRl spr (13b) β(4)

ij

= − 1 16R1 + 1 48R2 − 1 16 1 2 + ζ(3)

• R3 + 1

4 (1 + ζ(3)) R4 + 1 16 13 3 − 3ζ(3)

• R5 + 1

8 2 3 − ζ(3)

• R6 + 1

4 8 3 + ζ(3)

• R7

+ 1 4

• −1

3 + ζ(3)

• R8 + 1

12R9 + 1 12R10 − 1 6R11 + 1 16 4 3 + ζ(3)

• R12 − 1

4 4 3 + ζ(3)

• R13 + higher derivatives

(13c)

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RG flow Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 = Ω(λ)

• ekσηµνdxµdxν + dσ2
• Curvature scalar R = − 4

r2

• 2f ′′ + 5f ′2

= −20k2

Ω .

• Constant negative curvature i.e. Anti deSitter (AdS) space time.

Brane world model of Randall-Sundrum. [5]

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RG flow Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 = Ω(λ)

• ekσηµνdxµdxν + dσ2
• Curvature scalar R = − 4

r2

• 2f ′′ + 5f ′2

= −20k2

Ω .

• Constant negative curvature i.e. Anti deSitter (AdS) space time.

Brane world model of Randall-Sundrum. [5]

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RG flow Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 = Ω(λ)

• ekσηµνdxµdxν + dσ2
• Curvature scalar R = − 4

r2

• 2f ′′ + 5f ′2

= −20k2

Ω .

• Constant negative curvature i.e. Anti deSitter (AdS) space time.

Brane world model of Randall-Sundrum. [5]

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RG flow Separable solution

Separable solution

• Assume separable functions – rc(σ, λ) = rc(λ) f (σ, λ) = fσ(σ) + fλ(λ)
• The equations become separable and the solution becomes –

ds2 = Ω(λ)

• ekσηµνdxµdxν + dσ2
• Curvature scalar R = − 4

r2

• 2f ′′ + 5f ′2

= −20k2

Ω .

• Constant negative curvature i.e. Anti deSitter (AdS) space time.

Brane world model of Randall-Sundrum. [5]

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RG flow Separable solution

ODE for Ω

• Single ODE left for scale factor Ω

1 8k2 dΩ dλ = 1 − α′k2 Ω + 2 α′k2 Ω 2 − 3 + 5ζ(3) 2 α′k2 Ω 3

• rescale the variables as ¯

Ω =

Ω |α′|k2

; ¯ λ = 8λ

|α′|

• rescaled equation will be

d ¯ Ω d¯ λ = 1 ∓ ¯ Ω−1 + 2¯ Ω−2 ∓ 3 + 5ζ(3) 2 ¯ Ω−3

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RG flow Separable solution

ODE for Ω

• Single ODE left for scale factor Ω

1 8k2 dΩ dλ = 1 − α′k2 Ω + 2 α′k2 Ω 2 − 3 + 5ζ(3) 2 α′k2 Ω 3

• rescale the variables as ¯

Ω =

Ω |α′|k2

; ¯ λ = 8λ

|α′|

• rescaled equation will be

d ¯ Ω d¯ λ = 1 ∓ ¯ Ω−1 + 2¯ Ω−2 ∓ 3 + 5ζ(3) 2 ¯ Ω−3

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RG flow Separable solution

ODE for Ω

• Single ODE left for scale factor Ω

1 8k2 dΩ dλ = 1 − α′k2 Ω + 2 α′k2 Ω 2 − 3 + 5ζ(3) 2 α′k2 Ω 3

• rescale the variables as ¯

Ω =

Ω |α′|k2

; ¯ λ = 8λ

|α′|

• rescaled equation will be

d ¯ Ω d¯ λ = 1 ∓ ¯ Ω−1 + 2¯ Ω−2 ∓ 3 + 5ζ(3) 2 ¯ Ω−3

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RG flow Separable solution

Solutions

• ¯

λ + C1 = ¯ Ω

• ¯

λ + C2 = ¯ Ω ± ln

• ¯

Ω ∓ 1

• r ¯

Ω = 1

• ¯

λ + C3 = ¯ Ω ± 1

2 ln

• ¯

Ω2 ∓ ¯ Ω + 2

• −

3 √ 7 tan−1 2¯ Ω∓1 √ 7

• ¯

λ + C4 = ¯ Ω ± a ln

• ¯

Ω ∓ ξ4

• ∓ b

2 ln

• ¯

Ω2 ± β ¯ Ω + γ

• −

2c−βb

√

4γ−β2 tan−1

• 2¯

Ω±β

√

4γ−β2

• r

¯ Ω = ξ4 where ξ4 ≈ 1.5636 β ≈ 0.5636 γ ≈ 2.8812 (14a) a ≈ 0.6158 b ≈ −0.3841 c ≈ 1.7464 (14b)

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RG flow Separable solution

Solutions: plot α′ > 0

• 0 0.5 Α' 0

(a) ¯ Ω0 = 0.5

• 0 4 Α' 0

(b) ¯ Ω0 = 4

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RG flow Separable solution

Solutions: expansion

• Solutions for small curvature (large ¯

Ω) λ + C1 = ¯ Ω (15a) λ + C2 = ¯ Ω ± ln ¯ Ω − 1 ¯ Ω

• ∓ 1

2 1 ¯ Ω 2 − 1 3 1 ¯ Ω 3 + . . . (15b) λ + ˜ C3 = ¯ Ω ± ln ¯ Ω + 1 ¯ Ω

• ± 3

2 1 ¯ Ω 2 + 1 3 1 ¯ Ω 3 + . . . (15c) λ + ˜ C4 = ¯ Ω ± ln ¯ Ω + 1 ¯ Ω

• ∓ 0.7526

1 ¯ Ω 2 − 2.6699 1 ¯ Ω 3 + . . . (15d)

• leading order correction ∼ ln ¯

Ω

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RG flow Separable solution

Solutions: expansion

• Solutions for small curvature (large ¯

Ω) λ + C1 = ¯ Ω (15a) λ + C2 = ¯ Ω ± ln ¯ Ω − 1 ¯ Ω

• ∓ 1

2 1 ¯ Ω 2 − 1 3 1 ¯ Ω 3 + . . . (15b) λ + ˜ C3 = ¯ Ω ± ln ¯ Ω + 1 ¯ Ω

• ± 3

2 1 ¯ Ω 2 + 1 3 1 ¯ Ω 3 + . . . (15c) λ + ˜ C4 = ¯ Ω ± ln ¯ Ω + 1 ¯ Ω

• ∓ 0.7526

1 ¯ Ω 2 − 2.6699 1 ¯ Ω 3 + . . . (15d)

• leading order correction ∼ ln ¯

Ω

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Conclusion

Conclusions

• Conformally AdS spacetime is a solution to RG flow equations upto

4th order in α′. This is same as the spacetime in the brane world model of the Universe proposed by Randall and Sundrum. [5]

• Apart from flat space, two soliton solutions exist at orders 2 and 4.

But these have large curvature scales and are non-perturbative effects.

• Higher order terms in the RG flow provide a leading order correction
• f ∼ ln ¯

Ω, while other correction vanish in the limit.

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Conclusion

Conclusions

• Conformally AdS spacetime is a solution to RG flow equations upto

4th order in α′. This is same as the spacetime in the brane world model of the Universe proposed by Randall and Sundrum. [5]

• Apart from flat space, two soliton solutions exist at orders 2 and 4.

But these have large curvature scales and are non-perturbative effects.

• Higher order terms in the RG flow provide a leading order correction
• f ∼ ln ¯

Ω, while other correction vanish in the limit.

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Conclusion

Conclusions

• Conformally AdS spacetime is a solution to RG flow equations upto

4th order in α′. This is same as the spacetime in the brane world model of the Universe proposed by Randall and Sundrum. [5]

• Apart from flat space, two soliton solutions exist at orders 2 and 4.

But these have large curvature scales and are non-perturbative effects.

• Higher order terms in the RG flow provide a leading order correction
• f ∼ ln ¯

Ω, while other correction vanish in the limit.

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Possible Further Work

Possible Further Work

• Extension of scaling type solutions to higher order terms in

Renormalization Group flow.

• Proving that AdS spacetime is a solution in all orders in α′. But

higher order terms have not been computed beyond order 4.

• Nature of solutions, solitons for even higher order expansion...
• Studying such properties from general non-perturbative principles.

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Possible Further Work

Possible Further Work

• Extension of scaling type solutions to higher order terms in

Renormalization Group flow.

• Proving that AdS spacetime is a solution in all orders in α′. But

higher order terms have not been computed beyond order 4.

• Nature of solutions, solitons for even higher order expansion...
• Studying such properties from general non-perturbative principles.

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Possible Further Work

Possible Further Work

• Extension of scaling type solutions to higher order terms in

Renormalization Group flow.

• Proving that AdS spacetime is a solution in all orders in α′. But

higher order terms have not been computed beyond order 4.

• Nature of solutions, solitons for even higher order expansion...
• Studying such properties from general non-perturbative principles.

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Possible Further Work

Possible Further Work

• Extension of scaling type solutions to higher order terms in

Renormalization Group flow.

• Proving that AdS spacetime is a solution in all orders in α′. But

higher order terms have not been computed beyond order 4.

• Nature of solutions, solitons for even higher order expansion...
• Studying such properties from general non-perturbative principles.

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References

References

• B. Chow and D. Knopf, The Ricci flow: an introduction, Mathematical Surveys and Monographs Vol. 110, AMS,

Providence, 2004. R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom. 17, 255 (1982).

• D. Friedan, Nonlinear Models in 2+ǫ Dimensions, Annals of Physics 163, 318 (1985).
• G. Perelman, The entropy formula for the Ricci flow and its geometric applications, Preprint math.DG/0211159.
• L. Randall and R. Sundrum, An alternative to compactification, Phys.Rev.Lett.83,4690 (1999) ibid. A large mass

hierarchy from a small extra dimension, Phys.Rev.Lett. 83, 3370 (1999)

• A. Sen, Phys. Rev. Lett. 55, 1846 (1985); C. G. Callan, E. T. Martinec, M. T. Perry and D. Friedan, Nucl. Phys. B262,

593 (1985)

• I. Jack, D. R. T. Jones, and N. Mohammedi, Nuc. Phys. B322 (1989), 431-470
• S. Das, K. Prabhu and S. Kar, Ricci flow of unwarped and warped product manifolds, arXiv:0908.1295(to appear in

IJGMMP)

• K. Prabhu, S. Das and S. Kar, On higher order geometric and renormalisation group flows, arXiv:1002.3464

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Q & A

Q & A

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Q & A

Thank You

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