Strings and branes… Moriond, March 24, 2004 1

Prospects from strings and branes

Alexander Sevrin

Vrije Universiteit Brusseland

The International Solvay Institutes for Physics and Chemistry

http://tena4.vub.ac.be/

Moriond 2004

Strings and branes… Moriond, March 24, 2004 2

References

Not-too-technical review paper, including numerous references:

Strings, Gravity and Particle Physics by Augusto Sagnottiand AS

In the proceedings of 37th Rencontres de Moriond on Electroweak Interactions and Unified Theories, 2002.

e-Print Archive: hep-ex/0209011

Strings and branes… Moriond, March 24, 2004 3

Contents• Dirichlet-branes• D-branes and gauge theories

- Worldvolume point of view- AdS/CFT

• D-branes and black holes• Cosmology• Some conclusions

Strings and branes… Moriond, March 24, 2004 4

BranesSolitons: solutions of the equations of motion with a finite energy(-density) and a mass inversely proportional to the coupling constant. E.g. Scalar field in d = 1 + 1: kink.

Other example in d = 3 + 1: magnetic monopole:

mass = 12m3

λ

mass ∝ 1g2Y M

Strings and branes… Moriond, March 24, 2004 5

Solitons in string theory: Dirichlet branes

gµν(x) = gνµ(x) :

Φ(x) :

Besides the “conventional” fields, such as e.g.,

metric = gravitondilaton,

one has RR- potentials as well.

E.g. vector potential, :Fµν = ∂µAν − ∂νAµ,

Aµ → Aµ − ∂µf, Fµν → Fµν .

Couples to particles:

Aµ

S = q

Zdτ xµ(τ)Aµ(x(τ)).

Strings and branes… Moriond, March 24, 2004 6

E.g. 2-form potential, :Aµν = −Aνµ

Fµνρ = ∂µAνρ + ∂νAρµ + ∂ρAµν ,

Aµν → Aµν − ∂µfν + ∂νfµ, Fµνρ → Fµνρ.

Couples to strings:

S = q

Zdτdσ x(τ,σ)µx0(τ,σ)νAµν(x(τ,σ)).

Going on like this, one finds in type II string theory potentials coupling to p-dimensional objects with p= 0 (points), 1 (strings), 2 (membranes), 3 (blobs?), … They are called p-branes. What and where are they in string theory?

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They are the Dirichlet-branes, objects on which open strings end. They are solitonic, tension:

Tp = (2π)−(p−1)/2(2πα0)−(p+1)/2 g−1S

Open strings are “stuck” on Dp-branes, closed strings move freely in the bulk.

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A nice metaphor: insects walking on water…

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From the point of view of the worldvolume of the Dp-brane: (p+1)-dimensional effective field theory.

Degrees of freedom? Simple susy argument:

→ type II strings: 32 susy charges→ open strings: 16 susy charges→ insertion of a Dp-brane in type II

16 susy’s broken 16 Goldstinos 8 fermionic propagating degrees of freedom.

SUSY 8 bosonic propagating degrees of freedom needed…

Strings and branes… Moriond, March 24, 2004 10

Dp-brane in 9 + 1 dimensional space-time: 9 – p transversal directions 9 – p scalar fields 8 – (9 – p) = p – 1 bosonic degrees of

freedom missing vector field (U(1) gauge field) in p +

1dimensions.

In leading order described by a U(1) gauge theory.

Strings and branes… Moriond, March 24, 2004 11

Dp-brane worldvolume theory:

- U(1) gauge theory in p+1 dimensions.- 9 - p scalar fields- 16 fermions

Also: complicated couplings to bulk degrees of freedom!

For trivial bulk fields:

α0 = `2stringg2 = (2π)p−2α0(p−3)/2gS

S =1

2πα02g2

Zdp+1x

q−det(ηµν + ∂µΦI∂νΦI − 2πα0Fµν)

+ derivative corrections

Strings and branes… Moriond, March 24, 2004 12

Dirac-Born-Infeld action. Switch off the transversal scalars,

S =

Zdp+1x

³−

1

4g2FµνF

µν −(2πα02)8g2

FµνFνρFρσF

σµ

+(2πα02)32g2

(FµνFµν)2 + · · ·

´Born & Infeld: point source, , has an ∞energy in Maxwell. Modify Maxwell as above,

ρ = qδ(~r)

Er =q

4π

rr4+

³2πα0q4π

´2

Energy ≈ 0.349 q3/2(2πα0)−1/2

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D-branes and gauge theoriesWorldvolume point of view

Mass of open strings ~ minimal distance between the branes it connects.

`→ 0⇒ U(1) × U(1)→ U(2)

D-brane realization of Higgs mechanism, Higgs vevis . `

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N coinciding D-branes, in leading order in , given by d=p+1 dimensional supersymmetric U(N) Yang-Mills. Higher order corrections are under investigation. Using orientifolds SO(n) and Sp(2n) gauge groups as well.

Similar for transversal scalars introduces “fuzziness” in bulk geometry. Not too much known about it: active but very difficult domain of research!

α0

Aµ =

ÃA(1)µ W+

µ

W−µ A(2)µ

!, Φ =

µΦ(1) Φ+

Φ− Φ(2)

¶.

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D-branes provide geometric realization of gauge theories. E.g. Dirac monopole:

A(+)

A(−) θ < εθ > −εWhere we need 2 patches, is defined on

and on . Requiring them to coincide on the overlap gives the Dirac quantization condition, .m ∈ Z

~B = ∇ × ~A, ∇ · ~B = 2πmδ(~x).

A(±)x = −m

2

y

r(z ± r),

A(±)y = +m

2

x

r(z ± r),

A(±)z = 0.

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Rigorous definition through the introduction of a scalar field, ,

with,

View as a coordinate transversal to a D3-brane bound system of 1 D3-brane with m perpendicular D1-branes (called a BIonconfiguration).

∂aΦ = −2πα0Ba

Φ = πα0mr

Φ

Φ

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Charges, energies, … all work out. Dual point of view as well possible: point of view of the m D1-branes (Myers effect) → relation with non-commutative geometry.

Similarly ‘t Hooft-Polyakov monopoles: m D1-branes stretched between two parallel D3-branes. Abelianlimit easily understood.

D-branes are a powerful tool for organizing themonopole zoo, understand the ADHM construction for instantons, explore gauge solitons in higher dimension (e.g. in octonionicanalogue of Dirac and ‘t Hooft-Polyakovmonopoles in d=7, octonionic instantons in d=8), …

Strings and branes… Moriond, March 24, 2004 18

AdS/CFT

Consider type IIB in flat d=9+1 space with N parallel D3-branes.

Newton constant:

Take (low energy) , keeping , , … fixed

: non-interacting gravity theory: vanishes

: reduces to n=4, d=3+1, U(N) susyYang-Mills

S = Sbranes + Sbulk + Sbulk/brane interactions

G(10)N = 8π6α04g2S

α0 → 0 gS N

SbulkSbulk/brane interactionsSbranes

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Take D3-brane solution of IIB supergravity, near horizon geometry (= low energy limit) is AdS5 × S5

S5 in R6 : X21 +X

22 +X

23 +X

24 +X

25 +X

26 = R

2

AdS5 in R2,4 : X20 +X

21 −X

23 −X

24 −X

25 −X

26 = R

2

with .

Near horizon region decouples from bulk (free gravity theory).

Maldacena: string theory on is equivalent to d=3+1, n=4, U(N) susy Yang-Mills.

R4 = 4πgSα02N

AdS5 × S5

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R4

α02 = 4πgsN = 8π2g2Y MN À 1

Yang-Mills in itsperturbative region:

Supergravity descriptionvalid:

Supergravity calculations = SYM in deep non-perturbative regime…

→ Realization of ‘t Hooft’s holographic principle→ Operator mappings known→ Tested (anomalies, relevant + marginal

deformations, …)→ Other examples known

g2Y MN = gSN2π = R4

4πα02 ¿ 1

Strings and branes… Moriond, March 24, 2004 21

→ Full test difficult as string theory on isnot tractable yet.

→ If conjecture accepted powerful probe for certain non-perturbative aspects of gauge theories. E.g. leads to Dijkgraaf-Vafacorrespondence…

→ Peculiar limit (pp-waves) can be studied as a string theory but corresponds to a very singular truncation of the gauge theory.

→ Remains very active field of research!

AdS5 × S5

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D-branes and black holesBlack holes are very simple objects characterized by a few parameters: their mass, angular momentum, various charges… All examples here for Reissner-Nordstrom.

ds2 =

µ1−

2GNM

r c2+GNQ

2

r2 c4

¶dt2 −µ

1−2GNM

r c2+GNQ

2

r2 c4

¶−1dr2 − r2dΩ2,

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rH = c−2³GNM +

p(GNM)2 −GNQ2

´.

TH =c3~p(GNM)2 −GNQ2

2πkB(GNM +p(GNM)2 −GNQ2 )2

,

SHkB

=π

c~GN(GNM +

p(GNM)2 −GNQ2 )

2 =1

4AH l

−2p .

Horizon:

Hawking radiation, thermal with temperature and entropy:

Boltzmann: S = kB logΣ .

Strings and branes… Moriond, March 24, 2004 24

Questions:

- Microscopic description?- Information problem.

Branes provide answers, at least for the case of extremal and near-extremal black holes.

Extremal or supersymmetric black holes, e.g. for Reissner-Nordstrom:

M → Q/√GN

Behaves like an elementary particle.

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Microstates (Strominger-Vafa): complicated composite of D-branes wrapped around the compact dimensions + strings ending on them. Macrostatecharacterized by a few quantum numbers, numerous microscopic realization. Bekenstein-Hawking reproduced!

Unitarity (Callan-Maldacena): black hole evaporation can be studied in the near-extremal case. Perfect agreement with Hawking (T and reaction rate), healthy quantum theory.

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CosmologyObservations (SN, WMAP, …) all point towards an accelerating universe. Can this be accommodated within string theory?

First some standard stuff…

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FLRW model

dτ2 = dt2 − R(t)2µ

dr2

1− kr2+ r2dθ2 + r2 sin2 θdφ2

¶.

Spatial geometry:

Observations: k ' 0

R(t)Determine using Einstein’s equations…

Rµν −1

2gµνR−λ gµν = 8πGTµν

Strings and branes… Moriond, March 24, 2004 28

R2 + k =8πG

3R2µρ+

λ

8πG

¶ρ+ (ρ+ p)

3R

R= 0

Need a relation between energy density and pressure: equation of state. Simplest: linear, time-independent.

p(t) = αρ(t) ⇒ ρ = ρ0R3(α+1)0

R3(α+1)

p = ρ/3 ⇒ ρ = ρ0R40R4

p = 0 ⇒ ρ = ρ0R30R3

p = −ρ ⇒ ρ = ρ0

Radiation:

Non-relativistic matter:

Positive cosmological constant:

Strings and branes… Moriond, March 24, 2004 29

Acceleration or deceleration?

R

R= −

4πG

3(1+ 3α)ρ+

λ

3

Acceleration if or more generally .

Observation: .

λ > 0 α < − 13

−1.62 < α < −0.74

α < −1 : phantom matter, negative kinetic energies instabilities.

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Strings and branes… Moriond, March 24, 2004 31

Positive cosmological constant.

→ Why so small?- Value:

- Compare to Planck scale: mismatch

- Compare to susy scale: mismatch

→ Positive?

ρvac ∼¡10−3 eV

¢41019GeV ⇒ O(10124)

103GeV ⇒ O(1060)

α 6= −1 R(t) ∝ t2/3(α+1)

Radiation: R ∝√t

Dust: R ∝ t2/3

α = −1 R(t) ∝ e√λt/√3

Strings and branes… Moriond, March 24, 2004 32

Inflationary expansion

Exponential evolution towards a de Sitter universe

Seems not possible in string or M theory….

Strong energy condition (SEC):

d = 10, 11 supergravity satisfy SEC…

Gibbons; Maldacena-Nuñez: compactify on a smooth, compact manifold,

R00 ≥ 0

ds211,10 = ω2(y)ds24(x) + ds27,6(y)

R(11,10)00 ≥ 0⇒ R

(4)00 ≥ 0.

R(t) = −R00 = −4π(T00 + gijTij)

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Ways out?

→ It gets cured by higher order ( , ) corrections. We cannot say anything sensible about this.

→ Reanalyze the premises of the no-go theorem andfind a loophole. E.g. scalar field,

String/supergravity: there is no stationary point with V > 0. V > 0 and not stationary compact space is time dependent…

→ Two related approaches: hyperbolic compactifications and flux compactifications (e.g. d=11 supergravity on T7, 4-form flux).

α0 gS

φ = −V (φ)0 − 3Hφ, H ≡ R/R

ρ = 12φ2 + V, p = 1

2φ2 − V

Strings and branes… Moriond, March 24, 2004 34

Typical potential:

Hyperbolic:Flux:

Initially:

Field rolls up the potential till which is a transitional period of acceleration. Field rolls back down deceleration & decompactification.

V( φ) = b e−2 a φ, b > 0

1 < a <√3

a ≥√3

R ∝ t1/3, eφ ∝ t−1/√3

φ = 0

Generic behaviour: eternal acceleration not possible.

Strings and branes… Moriond, March 24, 2004 35

Inflation?

While always big-bang singularity where,

Previously discussed model not very well suited to describe initial inflation. Typically only a few e-foldings instead of 70 or more… Seems generic, e.g. systematic analysis of combined flux/hyperbolic scenarios (Chen-Ho-Neupane-Ohta-Wang; Wolfarth).

Supergravity/string based analysis (mainly at Stanford: Kachru, Kallosh, Linde, …) of inflation is encouraging (dilaton/volume stabilization, …).

R(t) ∝ t1/3

Strings and branes… Moriond, March 24, 2004 36

Conclusions- Perfect tool for the study of diverse aspects of

gauge theories.

- Cosmological applications:→ time-dependent compactifications→ study of the initial singularities→ inflation

- Phenomenology: flux-compactifications→ fix moduli→ break susy→ cosmological acceleration → but… no chiral fermions… more general fluxes are being studied…

Strings and branes… Moriond, March 24, 2004 37

- Phenomenology: intersecting brane-worlds (e.g. stacks of D6-branes intersecting in 3 dimensions)→ easy to get correct gauge group + 3 families

of quarks and leptons → many possibilities…→ generic features: right-handed neutrino, 2 or

more Higgs→ models generically non-supersymmetric

stability (NS-tadpoles)?→ hierarchy problem (at least for toroidal and

orbifold compactifications) hunt for the MSSM (additional angle

constraints) construction of the low energy effective action

identification of generic qualitative features

Strings and branes… Moriond, March 24, 2004 38

Picture: thanks to Taylor & Hashimoto.

Stability issue: e.g. 2 D1-branes at an angle will recombine and end up as 2 parallel D1-branes…

Strings and branes… Moriond, March 24, 2004 39

New insights are urgently needed, e.g. concerning the lifting of the huge vacuum degeneracy…

1984 – 1985: anomaly cancellation, heterotic strings

1995: D-branes

2005: ???

Strings and branes… Moriond, March 24, 2004 40

New insights are urgently needed, e.g. concerning the lifting of the huge vacuum degeneracy…

Or on an LHC timescale:

1984 – 1985: anomaly cancellation, heterotic strings

1995: D-branes

2007: ???