ricci flow for warped product manifolds - kartik … flow...ricci flow for warped product manifolds...

Ricci Flow for Warped Product Manifolds

Kartik Prabhu (05PH2001)Supervisor: Dr. Sayan Kar

Dept. of Physics & MeteorologyIndian Institute of Technology Kharagpur

May 04, 2010

Kartik Prabhu (IIT-Kgp) Ricci Flow May 04, 2010 1 / 24

Outline

1 Ricci flow: definition & motivationRF as a heat flowExample: Sphere

2 Warped manifoldsSeparable solutionScaling solution

3 RG flowSeparable solution

4 Conclusion

5 Possible Further Work

6 References

7 Q & A

Ricci flow: definition & motivation

Ricci Flow: definition

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

∂gij

∂λ= −2Rij (1)

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

∂gij

∂λ= −2Rij +

n〈R〉gij (2)

Ricci Flow: definition

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

∂gij

∂λ= −2Rij (1)

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

∂gij

∂λ= −2Rij +

n〈R〉gij (2)

Ricci Flow: motivation

• RF is like the heat equation and tends to smooth out the irregularitiesin the metric.

• Finding the best metric on a manifold to solve mathematical problemslike the Poincare Conjecture. (Perelman [4] )

• Renormalization Group flow in non-linear σ-model of string theoryleads to Ricci flow in the lowest order.

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 = −

xφ+ ∂2yφ)

and the Ricci flow Eq.(1) becomes

∂λ= 4φ (4)

where 4 = e−2φ(∂2

x + ∂2y

)is generalized Laplacian.

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

RF as heat flow

)with Ricci

xφ+ ∂2yφ)

∂λ= 4φ (4)

x + ∂2y

RF as heat flow

)with Ricci

xφ+ ∂2yφ)

∂λ= 4φ (4)

x + ∂2y

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 lm(θ, φ) then r1 ∼ e−l(l+1)λ

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 lm(θ, φ) then r1 ∼ e−l(l+1)λ

Warped manifolds

• Extra-dimensional brane world metric

ds2 = e2f (σ,λ)ηµνdxµdxν + r2c (σ, λ)dσ2 (5)

• The RF is now a system of PDEs –

(f ′′ + 4f ′2 − f ′r ′c

(f ′′ + f ′2 − f ′r ′c

Warped manifolds

• Extra-dimensional brane world metric

ds2 = e2f (σ,λ)ηµνdxµdxν + r2c (σ, λ)dσ2 (5)

• The RF is now a system of PDEs –

(f ′′ + 4f ′2 − f ′r ′c

(f ′′ + f ′2 − f ′r ′c

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 + λλc

) [exp

(± σ√

)ηµνdxµdxν + dσ2

]• Curvature scalar R = − 5/2

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

Separable solution

ds2 =(

1 + λλc

) [exp

(± σ√

Separable solution

ds2 =(

1 + λλc

) [exp

(± σ√

Warped manifolds Scaling solution

Scaling solution

• Invariance under σ → ασ and λ→ α2λ. So use variable x = 12 ln λ

and convert to ODE.

• Also use B = ex

dx= A (8)

2B2− A− 24A3B2 (9)

dx= −4AB + B + 24A2B3 (10)

Scaling solution

• Invariance under σ → ασ and λ→ α2λ. So use variable x = 12 ln λ

and convert to ODE.

• Also use B = ex

dx= A (8)

2B2− A− 24A3B2 (9)

dx= −4AB + B + 24A2B3 (10)

Scaling solution: plot

0 1 2 3 4 5

8 f H0L 1, AH0L 1, BH0L 1<

(a) f v/s x

0 1 2 3 4 5

8 f H0L 1, AH0L 1, BH0L 1<

(b) A v/s x

Scaling solution: plot

0 1 2 3 4 5

8 f H0L 1, AH0L 1, BH0L 1<

(c) rc v/s x

0 1 2 3 4 5

8 f H0L 1, AH0L 1, BH0L 1<

(d) σ2R v/s x

Scaling solution: singularities

0 1 2 3 4 50

r cH1L

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

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′

(2)ij + α′

(3)ij + α′

(4)ij + O

(α′

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′

(2)ij + α′

(3)ij + α′

(4)ij + O

(α′

RG flow

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

β(1)ij = Rij β

(2)ij =

2RiklmRj

klm (13a)

β(3)ij =

8∇pRiklm∇pRj

klm − 1

16∇iRklmp∇jR

2RklmpRi

mlrRjkp

r −3

8RikljR

ksprR lspr

β(4)ij = − 1

16R1 +

48R2 −

2+ ζ(3)

4(1 + ζ(3)) R4

3− 3ζ(3)

3− ζ(3)

3+ ζ(3)

12R9 +

12R10 −

3+ ζ(3)

)R12 −

3+ ζ(3)

)R13 + higher derivatives

(13c)Kartik Prabhu (IIT-Kgp) Ricci Flow May 04, 2010 14 / 24

RG flow Separable solution

Separable solution

• The equations become separable and the solution becomes –ds2 = Ω(λ)

[ekσηµνdxµdxν + dσ2

]• Curvature scalar R = − 4

[2f ′′ + 5f ′2

]= −20k2

• Constant negative curvature i.e. Anti deSitter (AdS) space time.Brane world model of Randall-Sundrum. [5]

Separable solution

[2f ′′ + 5f ′2

]= −20k2

Separable solution

[2f ′′ + 5f ′2

]= −20k2

Separable solution

[2f ′′ + 5f ′2

]= −20k2

ODE for Ω

• Single ODE left for scale factor Ω

dλ= 1− α′k2

(α′k2

− 3 + 5ζ(3)

(α′k2

• rescale the variables as Ω = Ω|α′|k2 ; λ = 8λ

|α′|• rescaled equation will be

d λ= 1∓ Ω−1 + 2Ω−2 ∓ 3 + 5ζ(3)

2Ω−3

ODE for Ω

dλ= 1− α′k2

(α′k2

− 3 + 5ζ(3)

(α′k2

d λ= 1∓ Ω−1 + 2Ω−2 ∓ 3 + 5ζ(3)

2Ω−3

ODE for Ω

dλ= 1− α′k2

(α′k2

− 3 + 5ζ(3)

(α′k2

d λ= 1∓ Ω−1 + 2Ω−2 ∓ 3 + 5ζ(3)

2Ω−3

Solutions

• λ+ C1 = Ω

• λ+ C2 = Ω± ln∣∣Ω∓ 1

∣∣ or Ω = 1

• λ+ C3 = Ω± 12 ln

∣∣Ω2 ∓ Ω + 2∣∣− 3√

7tan−1

(2Ω∓1√

)• λ+ C4 =

Ω± a ln∣∣Ω∓ ξ4

∣∣∓ b2 ln

∣∣Ω2 ± βΩ + γ∣∣− 2c−βb√

4γ−β2tan−1

(2Ω±β√4γ−β2

Ω = ξ4 where

ξ4 ≈ 1.5636 β ≈ 0.5636 γ ≈ 2.8812 (14a)

a ≈ 0.6158 b ≈ −0.3841 c ≈ 1.7464 (14b)

Solutions: plot α′ > 0

-2 0 2 4 6 80

W0 =0.5 Α' > 0

(a) Ω0 = 0.5

-6 -4 -2 0 2 4 60

W0 =4 Α' > 0

(b) Ω0 = 4

Solutions: expansion

• Solutions for small curvature (large Ω)

λ+ C1 = Ω (15a)

λ+ C2 = Ω± ln Ω−(

)∓ 1

+ . . . (15b)

λ+ C3 = Ω± ln Ω +

+ . . . (15c)

λ+ C4 = Ω± ln Ω +

)∓ 0.7526

− 2.6699

+ . . .

• leading order correction ∼ ln Ω

Solutions: expansion

• Solutions for small curvature (large Ω)

λ+ C1 = Ω (15a)

λ+ C2 = Ω± ln Ω−(

)∓ 1

+ . . . (15b)

λ+ C3 = Ω± ln Ω +

+ . . . (15c)

λ+ C4 = Ω± ln Ω +

)∓ 0.7526

− 2.6699

+ . . .

• leading order correction ∼ ln Ω

Conclusion

Conclusions

• Conformally AdS spacetime is a solution to RG flow equations upto4th order in α′. This is same as the spacetime in the brane worldmodel 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 correctionof ∼ ln Ω, while other correction vanish in the limit.

Conclusion

Conclusions

Conclusion

Conclusions

Possible Further Work

• Extension of scaling type solutions to higher order terms inRenormalization Group flow.

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

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+ε 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

Thank You

ricci flow for warped product manifolds - kartik … flow...ricci flow for warped product manifolds...

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