STring Theory

Lecture 1 Master en Física Nuclear e de Partículas e as súas aplicacións Tecnolóxicas e Médicas

Alfonso V. Ramallo The relativistic point particle

ηµ ν = diag (−1, +1, · · · , +1)

xµ = xµ(τ) xµ → parametrizes the space in which the point particle is moving

τ → coordinate that parametrizes the path of the particle

x 0

x µ(τ) τ τ dτ d s x 2

x 1 Action

τ1 S = L dτ = −m ds = −m dτ −η x˙ µ x˙ ν ! ! ! µν τ0 "

dxµ x˙ µ ≡ ˙ µ ˙ ν = ˙ µ ˙ ≡ ˙ 2 dτ ηµν x x x xµ x

Lagrangian density L = −m !−x˙ 2

Conjugate momentum ∂ mx˙ µ pµ = L = ∂x˙ µ √ x˙ 2 − EOM ∂ m x˙ ∂ = 0 = ∂ µ = 0 τ ∂x˙Lµ τ 2 ! " ⇒ ! √ x˙ " − µ 2 Constraint pµ p + m = 0

Hamiltonian analysis ∂L H = x˙ µ − L = 0 canonical ∂x˙ µ Reparametrization invariance

! ! analogue of general coordinate τ → τ (τ) =⇒ dτ !−x˙ 2(τ !) = dτ!−x˙ 2(τ) invariance in 0+1 d

N 2 2 Introduce a Lagrange multiplier N H = ( p + m ) 2m

Nxµ {xµ, p } = δµ x˙ µ = xµ, H = x˙ 2 = −N 2 ν ν { } √ x˙ 2 −

N is arbitrary (gauge election) µ µ µ p N=1 x¨ = 0 x˙ (τ) = x0 + τ m Polyakov action Define a metric on the worldline 2 2 2 2 e(τ) is the einbein ds = e (τ) (dτ) = gττ (dτ) ττ −2 e = √gττ , g = e (τ)

1 −2 2 2 1 ττ 2 S = dτ e(τ) e (τ) x˙ − m = dτ det g g ∂τ x · ∂τ x − m 2 ! " # 2 ! $ " #

EOM of e: 1 x˙ 2 1 δS − dτ m2 δe = − ˙ 2 = 2 + e ! x 2 ! " e # m EOM of x:

−1 µ −1 δS = − dτ ∂τ e x˙ δxµ ∂ e x˙ µ = 0 ! " # τ ! " Using the value of e: m x˙ ∂ µ = 0 τ 2 same as before! ! √ x˙ " −

Substituting the value of e in the action:

1 −1 2 2 S = dτ e (τ) x˙ − m e(τ) = − dτ m −x˙ 2 2 ! " # ! $

The Polyakov action reduces to the original action The relativistic string Extended object that sweeps a 2d worldsheet

x 0

x !(! , #)

2 ! d $ dA x 2

0 # " ! #

1 Nambu-Goto action: x

1 SNG = −T dA T = ! 2πα!

[α!] = [L]−2 = [M]2 α! is the Regge slope Worldsheet coordinates α ξ , α = 0, 1 , (ξ0, ξ1) = (τ, σ)

Embedding: Σ → M with ξa → Xµ(ξa) Induced metric µ ν 2 ∂X ∂X ds = G dXµ(ξ) dXν (ξ) = G dξα dξβ µν µν ∂ξα ∂ξβ

Gˆ ≡ G ∂ Xµ ∂ Xν 2 ˆ α β αβ µν α β ds = Gαβ dξ dξ

= − det ˆ 2 Area element dA ! Gαβ d ξ

− − ˆ 2 Action SNG = T det Gαβ d ξ ! " In a flat spacetime Gµν = ηµν Induced metric 2 2 !2 ! 2 det Gˆαβ = Gˆ00 Gˆ11 − Gˆ01 = X˙ X − (X˙ · X )

∂Xµ ∂Xµ X! µ ≡ X˙ µ ≡ ∂σ ∂τ

! 2 2 !2 2 2 SNG = −T (X˙ · X ) − X˙ X d ξ ≡ L d ξ ! " !

∂L ∂L ∂τ + ∂σ = 0 EOM ˙ µ ∂X! µ ! ∂X " ! " Non-linear PDE Canonical analysis X˙ · X! X! − X! 2 X˙ ∂L ( ) µ µ Πµ = = −T ∂X˙ µ ˙ ! 2 ˙ 2 !2 !(X · X ) − X X Constraints ! 2 2 ! 2 Π · X = 0 Π + T ! X " = 0

Hamiltonian H = dσ X˙ · Π − L = 0 ! " #

The dynamics is governed by the constraints Polyakov action

Introduce a worldsheet metric gαβ(τ, σ)

= −T 2 − det αβ µ ν SP 2 d ξ g g ∂α X ∂β X ηµν ! "

2 1 EOM of g : δSP αβ Tαβ T = 0 ≡ −T √ det g δgαβ αβ − Since 1 − αβ δ − det g = − − det g g δgαβ δ(det g) = det g gαβ δg ! " # 2 " αβ Energy-momentum tensor 1 = · − γδ · Tαβ ∂α X ∂β X 2 gαβ g ∂γ X ∂δ X From the eom of the 2d metric 1 · = γδ · ∂α X ∂β X !2 g ∂γ X ∂δ X " gαβ induced metric

Substituting in the action

T −T − det(∂ X · ∂ X) = − gγδ ∂ X · ∂ X − det g ! α β 2 γ δ "

The Nambu-Goto and Polyakov actions coincide!!

EOM of Xµ αβ µ ∂α − det g g ∂β X = 0 ! # " Space-time coordinates are scalar fields in 2d!! Symmetries of the Polyakov action Poincare transformations

µ µ ν µ δX = ω ν X + a , δgαβ = 0

ωµ constant and ω = −ω ν µν νµ

2d reparametrizations

γ δ ! ∂f ∂f ! ξα → f α(ξ) = ξ α , g (ξ) = g (ξ ) αβ ∂ξα ∂ξβ γδ

Local Weyl transformations Φ(τ,σ) µ gαβ → e gαβ , δX = 0 The action is invariant because Φ −Φ !− det g → e !− det g , gαβ → e gαβ Taking the determinant: T T det(∂ TXaking∂ Xthe) determinan= gγδ t∂: X ∂ X det g (0.59) − − α · β − 2 γ · δ − Thus, elimina!ting g in the Polyakov action we get the NG"actionT T det(∂ X ∂ X) = gγδ ∂ X ∂ X det g (0.59) EOM of Xµ − − α · β − 2 γ · δ − Thus, elimina!ting g in the Polyakov action we get the NG"action µ EOM of X αβ µ ∂α det g g ∂β X = 0 (0.60) # − $ " αβ µ Symmetries of the Polyakov action ∂α det g g ∂β X = 0 (0.60) # − $ Poincare transformations. Global symmetries. In infinit" esimal form: Symmetries of the Polyakov action µ µ ν µ δX =Poincaω ν Xre tr+ansformaa , tions. δGlobagαβ =l symmetries.0 In infinit(0.6e1)simal form: µ δXµ = ωµ Xν + aµ , δg = 0 (0.61) with ω ν constant and ωµν = ωνµ. ν αβ 2d parametrizations (diffe−omeorphisms of the 2d worldsheet). Local symmetry with ωµ constant and ω = ω . ν µν − νµ 2d parametrizations (diffeomeorphismsγ δ of the 2d worldsheet). Local symmetry α α α ∂f ∂f ξ f (ξ) = ξ! , gαβ(ξ) = gγδ(ξ!) (0.62) → ∂ξα ∂ξβ γ δ α α α ∂f ∂f ξ f (ξ) = ξ! , gαβ(ξ) = gγδ(ξ!) (0.62) The metric is second-order tensor. → ∂ξα ∂ξβ Local Weyl transformatioThe metrns: ic is second-order tensor. Local Weyl transformations: g eΦ(τ,σ) g , δXµ = 0 (0.63) αβ → αβ g eΦ(τ,σ) g , δXµ = 0 (0.63) The action is invariant because: αβ → αβ The action is invariant because: Φ αβ Φ αβ det g e det g , g e− g (0.64) − → − Φ → αβ Φ αβ det g e det g , g e− g (0.64) This symmetry is"responsible of the" traceles−s nature→of the−energy-momentum tensor→ Gauge fixing This symmetry is"responsible of the" traceless nature of the energy-momentum tensor Gauge fixing TheThwse metrws micegtαβrichasgαthreeβ hasindtherpeendene indtecomppendonenenttcs:omponents: The ws metric gαβ has three independent components: g00 g01 , gg0100= gg0110 (0.65) g10 g11 , g = g (0.65) % & g g 01 10 % 10 11 & By using reparametrization invariance we can make the metric conformally flat: With a reparametrizationBy using rwepae rametrcan makizatioen theinvar metriciance we confcan makormalle the ymetric flat conformally flat: g = eΛ η , η = diag( 1, +1) (0.66) αβ αβ g αβ= eΛ η , η = diag( 1, +1) (0.66) Λ αβ αβ − αβ − gαβ = e ηαβ , ηαβ = diag(−1, +1) In two dimensions withIntriviatwoldimenstopologyionsthiswithistriviaalwalystoppologyssible.thisMoris alweovaeysr, wpoessible. Moreover, we can use Weyl invariance to reduce g to η . From now on we will take can use Weyαβl invarianceαβ to reduce gαβ to ηαβ. From now on we will take

Using Weyl invariance 1 0 1 0 gαβ = ηαβ = − g = η = (0.67) (0.67) 0 1 αβ αβ −0 1 % & % & EOMS for the flat metric:EOMS for the flat metric:

The Polyakov action can be6 gauge-fixed to:6

τ2 σ¯ = −T αβ µ SP 2 dτ dσ η ∂α X ∂β Xµ !τ1 !0 EOM for the flat ws metric α µ ∂α ∂ X = 0 Explicitly ∂2 ∂2 − Xµ = 0 ! ∂τ 2 ∂σ2 " just Klein-Gordon in 2d !

The constraints are

1 2 !2 ! T00 = T11 = X˙ + X = 0 , T01 = T10 = X˙ · X = 0 2 ! "

They can be rewritten as:

˙ ! 2 ! X ± X " = 0 Light-cone components of the EM tensor

µ T++ = ∂+ X ∂+ Xµ (0.92)

µ T = ∂ X ∂ Xµ (0.93) −− − −

T+ = T + = 0 (0.94) − − Boundary conditions: Closed strings: take σ¯ = 2π and impose periodic boundary conditions

Xµ(τ, σ + 2π) = Xµ(τ, σ) (0.95) Open strings: take σ¯ = π. There are two possibilities: Neumann boundary conditions:

X"(τ, σ) = 0 (0.96) Noether currentsσ=0,π ! Dirichlet boundary conditions: ! Consider a symmetry transf!ormation X˙ (τ, σ) = 0 (0.97) σ=0,π →! + X is fixed at σ = 0, π: δX(σ = φ0, π) =! 0φor: δ! φ !

µ µ X σ=0 = X0 (0.98) L → L ! ∂ J α Taking ! = !(ξ) ! + α µ! µ X σ=π = Xπ (0.99) TheyTranslations:break Poincare invarXianceµ → X! µ + bµ Momentum flow: ! α The charge of Pµ : P µ = T ∂ Xµ Conservσ¯ ed current associated α α τ topµ P=oincarPµed σinvariance (0.100) "0 Associated σ¯ pµ = P µ dσ Momentum (0.101) charge τ "0 variation in time:

dp σ¯ dP τ µ = µ dσ (0.102) dτ dτ "0 Since

τ Pµ = T ∂τ Xµ (0.103)

9 In this case a = 1 (0.240) and the tachyon has mass: 1 M 2 = (0.241) −α! D = 26 is called the critical dimension of the bosonic string Excited states. The states with N = 2 are:

i i j α 2 0 > , α 1 α 1 0 > (0.242) − | − − | The mass is M 2 = 1 . The number of states in D dimensions is: α! 1 D(D 1) D 2 + (D 2)(D 1) = − 1 (0.243) Lorentz− transf2 ormation− − 2 − This is the number of states of a symmetric traceless second-rank tensor represen- δXµ = ωµ Xν ωµ constant and ω = −2 ω tation of SO(D 1). It corresνponds to a particleν of spin 2 and massµν M =νµ1/α!. In general the−spectrum is a tower of states with increasing mass and spin. The maximun spin J for mass M is given by

Conserved current 2 J α! M + 1 (0.244) ≤ which corresponds to havingJRµeggν =e traT jectorXµie∂s withXν α−! bXeingν ∂theXRµegge slope. α ! α α " Notice that the massive states become infinitely massive when α! 0 (or ls 0). This is the field theory limit. → → An important issue in the light-cone gauge is wheter or not Lorentz symmetry is realized Generatorwithout anomalies of Lorinentzthe quantransftumormationstheory. The generator of Lorentz transformations in string theory is:

J µν = dσ J µν = T dσ Xµ X˙ ν Xν X˙ µ (0.245) τ − ! ! " # In terms of modes, it can be written (for the closed string) as:

J µν = xµ pν xν pµ + Eµν + E¯µν , (closed strings) (0.246) − where Eµν (E¯µν) is the contribution of the α-modes (α¯-modes), which are given by:

∞ µν 1 µ ν ν µ E = i α n α n α n α n , − n − − − − − n=1 $ % & ∞ ¯µν 1 µ ν ν µ E = i α¯ n α¯ n α¯ n α¯ n (0.247) − n − − − − − n=1 $ % & For open strings we have the same expression but with only one type of oscillators:

J µν = xµ pν xν pµ + Eµν , (open strings) (0.248) − 25 Boundary conditions

Let us compute a general variation of the action:

τ2 σ¯ τ2 σ¯ α µ α µ δSP = T dτ dσ ∂ ∂αX δXµ − T dτ dσ ∂α ∂ X δXµ !τ1 !0 !τ1 !0 " #

Since δXµ! = 0 in the variational procedure: !τ=τ1,2

τ2 σ¯ τ2 σ=σ¯ α µ !µ δSP = T dτ dσ ∂ ∂αX δXµ − T dτ X δXµ =0 !τ1 !0 !τ1 " #σ

The vanishing of the first term implies the eom of X We would require also

σ=σ¯ !µ ! X δXµ ! = 0 !σ=0

For closed strings

Take σ¯ = 2π and impose periodic boundary conditions

Xµ(τ, σ + 2π) = Xµ(τ, σ)

The boundary requirement is automatically satisfied Open strings Take σ¯ = π There are two possibilities

Neumann boundary conditions ! ! X (τ, σ)! = 0 !σ=0,π

Dirichlet boundary conditions

! δX(σ = 0, π) = 0 X˙ (τ, σ)! = 0 !σ=0,π

Xµ! = Xµ X is fixed at σ = 0, π !σ=π π Xµ! = Xµ !σ=0 0 breaks Poincare invariance Momentum flow

σ¯ σ¯ dP τ = τ dpµ = µ pµ Pµ dσ dσ !0 dτ !0 dτ

Since: dP τ P τ = T ∂ X µ = T ∂2 X = T ∂2 X µ τ µ dτ τ µ σ µ

Total momentum variation

σ¯ dp ! ! µ = T ∂2 X dσ = T X (τ, σ = σ¯) − X (τ, σ = 0) dτ ! σ µ µ µ 0 " # The momentum flow vanishes for:

Closed strings Open strings with Neumann boundary conditions (No momentum flow off the ends of the string) Free boundary conditions

Open strings with Dirichlet boundary conditions end on some object that absorbs the momentum

D-branes