Prospects from strings and branes
Alexander Sevrin Vrije Universiteit Brussel and The International Solvay Institutes for Physics and Chemistry
http://tena4.vub.ac.be/
Moriond 2004
Strings and branes… Moriond, March 24, 2004 1 References
Not-too-technical review paper, including numerous references: Strings, Gravity and Particle Physics by Augusto Sagnotti and 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 2 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 3 Branes
Solitons: 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.
12m3 mass = λ
Other example in d = 3 + 1: magnetic monopole:
1 mass ∝ 2 gYM
Strings and branes… Moriond, March 24, 2004 4 Solitons in string theory: Dirichlet branes
Besides the “conventional” fields, such as e.g.,
gµν (x)=gνµ(x): metric = graviton Φ(x): dilaton, one has RR- potentials as well.
E.g. vector potential, A µ :
Fµν = ∂µAν − ∂ν Aµ,
Aµ → Aµ − ∂µf, Fµν → Fµν . Couples to particles:
µ S = q dτ x˙ (τ)Aµ(x(τ)). Z Strings and branes… Moriond, March 24, 2004 5 E.g. 2-form potential, A µ ν = − A ν µ :
Fµνρ = ∂µAνρ + ∂ν Aρµ + ∂ρAµν ,
Aµν → Aµν − ∂µfν + ∂ν fµ,Fµνρ → Fµνρ.
Couples to strings:
µ ν S = q dτdσ x˙(τ, σ) x0(τ, σ) Aµν (x(τ, σ)). Z 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?
Strings and branes… Moriond, March 24, 2004 6 They are the Dirichlet-branes, objects on which open strings end. They are solitonic, tension:
(p 1)/2 (p+1)/2 1 Tp =(2π)− − (2πα0)− gS−
Open strings are “stuck” on Dp-branes, closed strings move freely in the bulk.
Strings and branes… Moriond, March 24, 2004 7 A nice metaphor: insects walking on water…
Strings and branes… Moriond, March 24, 2004 8 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 9 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 10 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: 1 S = dp+1x − det(η + ∂ ΦI ∂ ΦI − 2πα F ) 2πα 2g2 µν µ ν 0 µν 0 Z + derivative correctionsq
2 α0 = `string 2 p 2 (p 3)/2 g =(2π) − α0 − gS
Strings and branes… Moriond, March 24, 2004 11 Dirac-Born-Infeld action. Switch off the transversal scalars, 1 (2πα 2) S = dp+1x − F F µν − 0 F F νρF F σµ 4g2 µν 8g2 µν ρσ Z (2πα 2)³ + 0 (F F µν )2 + ··· 32g2 µν Born & Infeld: point source, ρ =´ q δ ( ~ r ) , has an ∞ energy in Maxwell. Modify Maxwell as above, E = q r 2 4 2πα0q 4π r + 4π r ³ ´ 3/2 1/2 Energy ≈ 0.349 q (2πα0)−
Strings and branes… Moriond, March 24, 2004 12 D-branes and gauge theories Worldvolume 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 vev is ` .
Strings and branes… Moriond, March 24, 2004 13 N coinciding D-branes, in leading order in α 0 , 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.
(1) + (1) + Aµ Wµ Φ Φ Aµ = (2) , Φ = (2) . W A Φ− Φ Ã µ− µ ! µ ¶
Similar for transversal scalars introduces “fuzziness” in bulk geometry. Not too much known about it: active but very difficult domain of research!
Strings and branes… Moriond, March 24, 2004 14 D-branes provide geometric realization of gauge theories. E.g. Dirac monopole:
( ) m y A ± = − , x 2 r(z ± r) ( ) m x A ± =+ , y 2 r(z ± r) ( ) Az± =0. Where we need 2 patches, A (+) is defined on θ > −ε ( ) and A − on θ < ε . Requiring them to coincide on the overlap gives the Dirac quantization condition, m ∈ Z .
B~ = ∇ × A,~ ∇ · B~ =2πmδ(~x).
Strings and branes… Moriond, March 24, 2004 15 Rigorous definition through the introduction of a scalar field, Φ , ∂aΦ = −2πα0Ba with, πα0m Φ = r View Φ as a coordinate transversal to a D3-brane bound system of 1 D3-brane with m perpendicular D1-branes (called a BIon configuration).
Strings and branes… Moriond, March 24, 2004 16 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. Abelian limit easily understood.
D-branes are a powerful tool for organizing the monopole zoo, understand the ADHM construction for instantons, explore gauge solitons in higher dimension (e.g. in octonionic analogue of Dirac and ‘t Hooft-Polyakov monopoles in d=7, octonionic instantons in d=8), …
Strings and branes… Moriond, March 24, 2004 17 AdS/CFT
Consider type IIB in flat d=9+1 space with N parallel D3-branes.
S = Sbranes + Sbulk + Sbulk/brane interactions Newton constant: (10) 6 4 2 GN =8π α0 gS
Take α 0 → 0 (low energy) , keeping g S , N , … fixed
Sbulk : non-interacting gravity theory Sbulk/brane interactions : vanishes Sbranes : reduces to n=4, d=3+1, U(N) susy Yang-Mills
Strings and branes… Moriond, March 24, 2004 18 Take D3-brane solution of IIB supergravity, near 5 horizon geometry (= low energy limit) is AdS5 × S
5 6 2 2 2 2 2 2 2 S in R : X1 + X2 + X3 + X4 + X5 + X6 = R 2,4 2 2 2 2 2 2 2 AdS5 in R : X0 + X1 − X3 − X4 − X5 − X6 = R
4 2 with R =4 π g S α 0 N .
Near horizon region decouples from bulk (free gravity theory). 5 Maldacena: string theory on AdS 5 × S is equivalent to d=3+1, n=4, U(N) susy Yang-Mills.
Strings and branes… Moriond, March 24, 2004 19 Yang-Mills in its 4 2 gS N R perturbative region: g N = = 2 ¿ 1 YM 2π 4πα0 Supergravity description R4 2 2 valid: 2 =4πgsN =8π g N À 1 α0 YM
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
Strings and branes… Moriond, March 24, 2004 20 5 → Full test difficult as string theory on AdS 5 × S is not tractable yet. → If conjecture accepted powerful probe for certain non-perturbative aspects of gauge theories. E.g. leads to Dijkgraaf-Vafa correspondence… → 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!
Strings and branes… Moriond, March 24, 2004 21 D-branes and black holes
Black holes are very simple objects characterized by a few parameters: their mass, angular momentum, various charges… All examples here for Reissner- Nordstrom.
2G M G Q2 ds2 = 1 − N + N dt2 − rc2 r2 c4 µ ¶ 1 2G M G Q2 − 1 − N + N dr2 − r2dΩ2, rc2 r2 c4 µ ¶
Strings and branes… Moriond, March 24, 2004 22 Horizon:
2 2 2 rH = c− GN M + (GN M) − GN Q . ³ p ´ Hawking radiation, thermal with temperature and entropy: 3 2 2 c ~ (GN M) − GN Q TH = , 2 2 2 2πkB (GN Mp+ (GN M) − GN Q ) SH π 2 2 2 1 2 = (GN M +p (GN M) − GN Q ) = AH lp− . kB c~GN 4 p
Boltzmann: S = kB log Σ .
Strings and branes… Moriond, March 24, 2004 23 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.
Strings and branes… Moriond, March 24, 2004 24 Microstates (Strominger-Vafa): complicated composite of D-branes wrapped around the compact dimensions + strings ending on them. Macrostate characterized 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.
Strings and branes… Moriond, March 24, 2004 25 Cosmology
Observations (SN, WMAP, …) all point towards an accelerating universe. Can this be accommodated within string theory?
First some standard stuff…
Strings and branes… Moriond, March 24, 2004 26 FLRW model dr2 dτ 2 = dt2 − R(t)2 + r2dθ2 + r2 sin2 θdφ2 . 1 − kr2 µ ¶ Spatial geometry: Observations: k ' 0
Determine R ( t ) using Einstein’s equations… 1 R − g R−λ g =8πGT µν 2 µν µν µν
Strings and branes… Moriond, March 24, 2004 27 8πG λ R˙ 2 + k = R2 ρ + 3 8πG µ ¶ 3R˙ ρ˙ +(ρ + p) =0 R Need a relation between energy density and pressure: equation of state. Simplest: linear, time- independent. R3(α+1) p(t)=αρ(t) ⇒ ρ = ρ 0 0 R3(α+1) R4 Radiation: p = ρ/3 ⇒ ρ = ρ 0 0 R4 R3 Non-relativistic matter: p =0 ⇒ ρ = ρ 0 0 R3 Positive cosmological constant: p = −ρ ⇒ ρ = ρ0
Strings and branes… Moriond, March 24, 2004 28 Acceleration or deceleration?
R¨ 4πG λ = − (1 +3α)ρ + R 3 3
1 Acceleration if λ > 0 or more generally α < − 3 . Observation: − 1 . 62 < α < − 0 . 74 .
α < −1 : phantom matter, negative kinetic energies instabilities.
Strings and branes… Moriond, March 24, 2004 29 Strings and branes… Moriond, March 24, 2004 30 Positive cosmological constant. → Why so small? 3 4 - Value: ρvac ∼ 10− eV - Compare to Planck¡ scale:¢ 1 0 19 GeV ⇒ O ( 1 0 124 ) mismatch - Compare to susy scale: 1 0 3 GeV ⇒ O ( 1 0 60 ) mismatch
→ Positive?
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 31 Inflationary expansion Exponential evolution towards a de Sitter universe
Seems not possible in string or M theory…. ij R¨(t)=−R00 = −4π(T00 + g Tij )
Strong energy condition (SEC): R00 ≥ 0 d = 10, 11 supergravity satisfy SEC… Gibbons; Maldacena-Nuñez: compactify on a smooth, compact manifold, 2 2 2 2 ds11,10 = ω (y)ds4(x)+ds7,6(y) (11,10) (4) R00 ≥ 0 ⇒ R00 ≥ 0.
Strings and branes… Moriond, March 24, 2004 32 Ways out?
→ It gets cured by higher order ( α 0 , g S ) corrections. We cannot say anything sensible about this. → Reanalyze the premises of the no-go theorem and find a loophole. E.g. scalar field,
φ¨ = −V (φ)0 − 3Hφ˙,H≡ R/R˙ 1 ˙2 1 ˙2 ρ = 2 φ + V, p = 2 φ − V 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).
Strings and branes… Moriond, March 24, 2004 33 Typical potential: 2 a φ V( φ)=be− ,b>0 √ Hyperbolic: √1 Generic behaviour: eternal acceleration not possible. Strings and branes… Moriond, March 24, 2004 34 Inflation? While always big-bang singularity where, R(t) ∝ t1/3 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, …). Strings and branes… Moriond, March 24, 2004 35 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 36 - 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 37 Stability issue: e.g. 2 D1-branes at an angle will recombine and end up as 2 parallel D1-branes… Picture: thanks to Taylor & Hashimoto. Strings and branes… Moriond, March 24, 2004 38 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 39 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: ??? Strings and branes… Moriond, March 24, 2004 40