The Worldvolume Action of Kink Solitons Kurt Hinterbichler, Justin - - PowerPoint PPT Presentation

The Worldvolume Action of Kink Solitons

Kurt Hinterbichler, Justin Khoury, Burt A. Ovrut, James Stokes

The University of Pennsylvania

March 18, 2012

Outline

Motivation Scalar Kinks in d = 5 Flat Spacetime Scalar Kinks in d = 5 AdS Spacetime Scalar Kinks in d = 5 Heterotic M-theory Spacetime

2 of 17

The Universe on a Domain Wall?

Motivation from String Theory

E8 × E8 Heterotic string E 2≪α′

= ⇒ d = 11 SUGRA on R1,9 × S1

ℓsgs/Z2

with E8 SYM on the orbifold fixed planes

Compactify 6 of the 9 non-compact spacelike directions on a

Calabi-Yau 3-fold = ⇒ spacetime is a warped product of R1,3 × CY3 and S1/Z2

There exists a regime in which the universe appears 5-dimensional

(Heterotic M-theory)

5 + 1-dimensional “5-branes” in the 11D bulk =

⇒ 3 + 1-dimensional “3-branes” in 5D effective theory

The physics of the brane-bending mode π of the 3-brane has

implications for 4D cosmology (e.g. Galileons)

3 of 17

Problem

Compute the effective action for the brane bending mode of a probe domain-wall kink soliton in maximally symmetric spacetime as a precursor for the Heterotic M-theory 5-brane.

UV Toy Model

S5D =

• M5

d5x√−g 1 2κ2

5

(R − 2Λ) − 1 2g mn∂mΦ∂nΦ − V (Φ)

• ,

(1)

For Λ = 0 (flat spacetime) choose

V (Φ) = λ(Φ2 − η2)2 (probe brane approximation)

“Kink” solution describes a static

domain wall of width: ℓ =

1 η √ 2λ

Φ = ηφ(0), φ(0) = tanh(η √ 2λy) (2)

l l 1 1 y Φ

4 of 17

Procedure

Consider classical fluctuations about the classical background

configuration which depend on both the internal space y and the tangential 3-brane coordinates σµ

The fluctuation of the position of the zero-locus of Φ is described

by a field π = π(σµ). What is its effective action?

Expect on general grounds that operators in the effective action

• rganize into geometric invariants

√ −h

dim = 0

, √ −hK

dim = 1

, √ −hR, √ −hK 2

• dim = 2

, . . . (3)

Define the effective Lagrangian by integrating out the internal

space

Compute Lagrangian by expanding in powers of the wall thickness ℓ 5 of 17

Gauss Codazzi formalism (Carter and Gregory hep-th/9410095)

The dynamics of the wall is fully described by the scalar wave

equation (gmn = (1, ηµν) flat)

g mn∇m∇nΦ − 4λΦ(Φ2 − η2) = 0 (4)

This is equivalent to

L2

nΦ + KLnΦ + g mnDmDnΦ − 4λΦ(Φ2 − η2) = 0

(5)

where

hmn = gmn − nmnn, Kmn = ∇mnn (6)

are subject to the constraints

Lnhmn = 2Kmn, LnKmn = KmpK p

n

(7)

Work in gaussian normal coordinates adapted to the worldvolume

(σµ, y) so Ln = ∂/∂y

6 of 17

Rescalings

Let L = characteristic length of wall fluctuation, ℓ = wall width, ǫ = ℓ/L Rescale worldvolume quantities (e.g., K, σµ) by L and quantities

transverse to the brane (i.e, y) by ℓ

EOM in dimensionless variables

h′

mn = 2ǫκmn

(8) κ′

mn = ǫκmpκp n

(9) 0 = φ′′ + ǫκφ′ − 2φ(φ2 − 1) + ǫ2 ˆ Dm ˆ Dmφ (10)

Expand φ, hmn, κmn in dimensionless small parameter ǫ,

φ = φ(0) + ǫφ(1) + ǫ2 2 φ(2) + O(ǫ3), (11) hmn = h(0)mn + ǫh(1)mn + ǫ2 2 h(2)mn + O(ǫ3), (12) κmn = 1 ǫ κ(0)mn + κ(1)mn + ǫ 2κ(2)mn + ǫ2 6 κ(3)mn + O(ǫ3) (13)

7 of 17

ǫ-expansion

At zeroth the Gauss-Codazzi equations are trivially solved by

h(0)mn = ˆ h(0)mn(σ), κ(0)mn = 0, φ(0) = tanh(u) (14)

At first order in ǫ the induced metric and extrinsic curvature are simply

h(1)mn = 2uˆ κ(1)mn, κ(1)mn = ˆ κ(1)mn(σ) (15)

The first-order scalar equation φ′′

(1) − 2(3φ2 (0) − 1)φ(1) + ˆ

κ(1)(σ)φ′

(0) = 0

can be solved with the ansatz φ(1)(σ, u) = ˆ κ(1)(σ)f (u)

Taking boundary conditions to be Φ

u→±0

− → 0, Φ

u→±∞

− → ±η gives Φ = η tanh y ℓ

• + ηℓ K(σ)f

y ℓ

• + O(ℓ2)

(16) where f (u) = ±1 2 tanh(u) − 1 2 + 2 3 ± u 2

• 1

cosh2(u) − 1 6e∓2u . (17)

8 of 17

Effective action in R1,4

S4D =

• M4

d4σ √ −hL4D(σ) (18) where L4D(σ) ≡

• dy J L5D(σ, y),

J = √−g √ −h (19) In Gaussian normal coordinates (σµ, y), J = 1 + y K + 1 2y 2 R(4) − R(5) + R(5)

mnnmnn

+ · · · (20) where K, R(4), R(5), . . . are evaluated on the brane world volume (y = 0) and R(5)

mn = 0 in flat spacetime. It follows that to order ℓ2

L4D = −4η2 3ℓ

• 1 + CIR(4) + CIIK 2 + · · ·
• (21)

where CI = III II ℓ2 2 = π2 − 6 24 ℓ2, CII = −IIII II ℓ2 2 = −1 3ℓ2 . (22) II = +∞

−∞

du φ′ 2

(0) = 4

3, III = +∞

−∞

du u2φ′ 2

(0) = π2 − 6

9 , IIII = +∞

−∞

du f φ′

(0) = 8

9 . (23)

9 of 17

Gauss Codazzi formalism (Hinterbichler, Khoury, Ovrut, JS, To

Appear)

The dynamics of the wall is fully described by the scalar wave

equation (gmn = (e2y/R, ηµν) AdS)

g mn∇m∇nΦ − 4λΦ(Φ2 − η2)−4 √ 2λ R

• η2 − Φ2

= 0 . (24)

This is equivalent to

L2

nΦ + KLnΦ + DmDmΦ − 4λΦ(Φ2 − η2)−4

√ 2λ R

• η2 − Φ2

= 0 , (25)

where

hmn = gmn − nmnn, Kmn = ∇mnn (26)

are subject to the constraints

Lnhmn = 2Kmn, LnKmn = KmpK p

n −R(5) rspqnsnqhr mhp n = KmpK p n + 1

R2 hmn

Work in gaussian normal coordinates adapted to the worldvolume

(σµ, y) so Ln = ∂/∂y

10 of 17

ǫ-expansion

At zeroth the Gauss-Codazzi equations are solved by (δ = ℓ/R)

h(0)mn = e2δuˆ h(0)mn(σ), κ(0)mn = δh(0)mn, φ(0) = tanh(u) (27)

At first order in ǫ the induced metric and extrinsic curvature are simply

h(1)mn = 1 δ

• κ(1)mn − ˆ

κ(1)mn(σ)

• ,

κ(1)mn = e2δuˆ κ(1)mn(σ) (28)

The first-order scalar equation

φ′′

(1) + κ(0)φ′ (1) + κ(1)φ′ (0) − 2

• 3φ2

(0) − 1

• φ(1) + 8δφ(0)φ(1) = 0 can be

solved with the ansatz φ(1)(σ, u) = ˆ κ(1)(σ)F(u)

Taking boundary conditions to be Φ

u→±0

− → 0, Φ

u→±∞

− → ±η gives Φ = η tanh y ℓ

• + ηℓ ˆ

K(σ)F y ℓ

• + O(ℓ2),

ˆ K(σ) = K(σ) − 4 R (29) where F(u) can be determined numerically

11 of 17

Effective action in AdS5

In Gaussian normal coordinates (σµ, y), J = 1 + y K + 1 2y 2

• R(4) + 16

R2

• + · · ·

(30) where K and R(4) are evaluated on the brane world volume (y = 0). It follows that to order ℓ2 L4D = −4η2 3ℓ

• 1−6δ2(IIII − III)
• 1 + C0K + CIR(4) + CIIK 2

(31) where

C0 = 3

• Iǫδ + IIII
• 1 − 6δ2(IIII − III )

ℓδ, CI = III

• 1 − 6δ2(IIII − III )

3ℓ2 8 , CII = − IIII

• 1 − 6δ2(IIII − III )

3ℓ2 8 . (32) II = +∞

−∞

du φ′ 2

(0) = 4

3 , Iǫδ = +∞

−∞

du u

• φ(0) − 1

3 φ3

(0)

• ,

III = +∞

−∞

du u2φ′ 2

(0) = π2 − 6

9 , IIII = +∞

−∞

du F φ′

(0) 12 of 17

Evaluation of Iǫδ

Note that Iǫδ =

+∞

−∞ du u

• φ(0) − 1

3φ3 (0)

• is naively quadratically

divergent since the argument of the integrand is even and unbounded

The Gaussian normal coordinate patch is only defined up to the

point where the geodesics converge, which is determined by the minimum of L or R

After introducing the appropriate cut-off, the integral is finite and

can be estimated to be of order Iǫδ ∝    1/ǫ2 if δ < ǫ 1/δ2 if ǫ < δ (33)

13 of 17

0.0 0.1 0.2 0.3 0.4 0.5 0.6 ∆ 0.2 0.4 0.6 0.8 1.0 1.2 1 6∆2IIIIIII 0.1 0.2 0.3 0.4 0.5 0.6 ∆ 2 4 6 8 10 C0l 0.0 0.1 0.2 0.3 0.4 0.5 0.6 ∆ 0.05 0.10 0.15 CIl2 0.1 0.2 0.3 0.4 0.5 0.6 ∆ 0.12 0.10 0.08 0.06 0.04 0.02 CIIl2

Figure: Numerical calculation of

• 1 − 6δ2(IIII − III)
• , C0/l, CI/l2 and CII/l2

as functions of δ. Of the four coefficients, only C0/l depends on Iǫδ and, hence, on the value of the cut-off ratio R

L = ǫ δ. Therefore, to evaluate C0/l

we must specify a value for ǫ. In Figure 2(B), we choose ǫ = 0.2. Note that C0/l is defined piecewise and changes behavior at δ ∼ ǫ

.

14 of 17

Heterotic M-theory (Lukas, Ovrut, Stelle, Waldram, hep-th/9806051)

In the absence of any 3-branes, the bosonic part of the action for d = 5, N = 1 supersymmetric heterotic M-theory S = − 1 2κ2

5

{

• M5

d4xdy√−g 1 2R + 1 4gmn∂mφ∂nφ + 1 3α2e−2φ +

• M4

d4x√−g2αe−φ −

• M4

d4x√−g2αe−φ} , (34) ds2 = e2A(y)dxµdxνηµν + e2B(y)dy2, φ = φ(y) (35) For SUSY preserving backgrounds A and B satisfy the BPS equations e−BA′ = −α 3 e−φ, e−Bφ′ = −2αe−φ (36)

15 of 17

Modeling the 3-brane (Antunes, Copeland, Hindmarsh, Lukas

hep-th/0208219; Hinterbichler, Ovrut, JS, To Appear)

S = − 1 2κ2

5

{

• M5

d4xdy√−g 1 2R + 1 4g mn∂mφ∂nφ + 1 2e−φg mn∂mχ∂nχ +V (φ, χ)

• +
• M4

d4x√−g2W −

• M4

d4x√−g2W } , (37) where W = e−φω(χ), V (φ, χ) = 1 3e−2φω2 + 1 2e−φdω dχ 2 . (38) ω = α + mχ

• v 2 − 1

3χ2 . (39) in the limit MP → ∞ the BPS equations simplify to e−BA′ = −α 3 e−φ, e−Bφ′ = −2αe−φ

• pure heterotic geometry without 3-branes

, e−Bχ′ = m

• v 2 − χ2
• χ=v tanh(mvy)

. (40)

16 of 17

Galileon structure from the worldvolume

The worldvolume action for the kink in AdS spacetime contains the

• perators

√ −h, √ −hK, √ −hR(4) which correspond to linear combinations of the conformal Galileons L2, L3, L4 in the small-field (non-relativistic) limit

The action contains an explicitly non-Galileon interaction

√ −hK 2 at O(ǫ2), but since this is proportional to the zeroth order equation

• f motion (K = 0), it can be eliminated without affecting the

action up to and including O(ǫ2)

Field-redefinition-invariant observables such as scattering

amplitudes arising from the AdS kink are therefore dominated by the conformal Galileons

Conversely, the physical brane bending mode is explicitly

non-Galileon for any range of momenta

This method can be generalized to probe branes in more

complicated geometries such as Heterotic M-theory

17 of 17