Corfu Summer Institute on Elementary Particle Physics, 1998 PROCEEDINGS An introduction to perturbative and non-perturbative string theory Ignatios Antoniadis∗ Centre de Physique Th´eorique (CNRS UMR 7644), Ecole Polytechnique, 91128 Palaiseau Cedex, France [email protected] Abstract: In these lectures we give a brief introduction to perturbative and non-perturbative string theory. The outline is the following. 1. Introduction to perturbative string theory 1.1 From point particle to extended objects 1.2 Free closed and open string spectrum 1.3 Compactification on a circle and T-duality 1.4 The Superstring: type IIA and IIB 1.5 Heterotic string and orbifold compactifications 1.6 Type I string theory 1.7 Effective field theories References 2. Introduction to non-perturbative string theory 2.1 String solitons 2.2 Non-perturbative string dualities 2.3 M-theory 2.4 Effective field theories and duality tests References 1. Introduction to perturbative string tory which minimizes the action in flat space is theory then a straight line. Similarly, the propagation of a p-brane leads to a (p+1)-dimensional world- 1.1 From point particle to extended ob- volume. The action describing the dynamics of jects a free p-brane is then proportional to the area of the world-volume and its minimization implies A p-brane is a p-dimensional spatially extended that the classically prefered motion is the one of object, generalizing the notion of point parti- minimal volume. cles (p=0) to strings (p=1), membranes (p=2), etc. We will discuss the dynamics of a free p- More precisely, in order to describe the dy- brane, propagating in D spacetime dimensions namics of a p-brane in the embedding D-dimen- (p ≤ D − 1), using the first quantized approach sional spacetime, we introduce spacetime coordi- µ − in close analogy with the case of point parti- nates X (ξα)(µ=0,1, .., D 1) depending on cles. Indeed, the propagation of a point leads the world-volume coordinates ξα (α =0,1, .., p). to a world-line. The corresponding action that The Nambu-Gotto action in a flat spacetime is describes the dynamics of a free particle is pro- then given by Z portional to the length of this line. The trajec- √ − − p+1 ∗Written in collaboration with Guillaume Ovarlez. S = T dethd ξ, (1.1) Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis where T is the brane tension of dimensionality the equation of motion (1.4) becomes X¨ µ =0. (mass)p+1,and The general solution is µ µ ν µ µ µ hαβ = ∂αX ∂βXµ ≡ ∂αX ∂βX ηµν (1.2) X (τ)=X0 +p τ, (1.7) is the induced metric on the brane. which is a straight line. On the other hand, the An equivalent but more convenient descrip- second equation of motion (1.5) leads to the con- tion is given by the covariant Polyakov action: straint X˙ 2 = −m2 , (1.8) Z √ T p+1 S = − d ξ −deth {hαβ∂ Xµ∂ X which is the on-shell condition p2 = −m2.The 2 α β µ quantization can be done following the canoni- −(p − 1)} , (1.3) cal procedure, i.e. by applying the (equal-time) where the intrinsic metric of the brane world- commutators volume hαβ is introduced as an independent vari- [Xµ, X˙ ν]=iηµν , (1.9) able. Solving the equations of motion for Xµ and µ ν µν hαβ leading to the usual relation [X0 ,p ]=iη . √ For p = 1, we have two world-sheet coordi- − αβ ∂α( dethh ∂βXµ)=0, (1.4) nates ξα =(τ,σ), where σ is a parameter along µ 1 2 the string. Using the gauge freedom, one can fix ∂αX ∂βXµ − hαβ(∂X) 2 two components of the metric and bring it into 1 + (p − 1)h =0, (1.5) the conformally flat form 2 αβ Φ(σ,τ ) 2 αβ µ hαβ(σ, τ )=e ηαβ , (1.10) with (∂X) ≡ h ∂αX ∂βXµ, one finds where the scale factor Φ is the only remaining µ 6 hαβ = ∂αX ∂βXµ ;p=1. (1.6) degree of freedom. However for p = 1, the ac- tion (1.3) has an additional symmetry under lo- hαβ is thus identified with the induced metric cal rescalings of the metric (local conformal in- (1.2), and after substitution into the Polyakov ac- variance), so that Φ decouples from both the ac- tion (1.3) we obtain back the Nambu-Gotto form tion and the equations of motion (1.4, 1.5). As (1.1). a result, there is an additional gauge freedom The symmetries of the Polyakov action are and one can again fix the metric to a constant, (i) global spacetime Lorentz invariance and (ii) h = η , as in the case p = 0. The equations local world-volume reparametrization invariance αβ αβ of motion for Xµ and constraints then read: under an arbitrary change of coordinates ξα → 0 α β µ α µ ξ (ξ )withX and hαβ transforming as a (p + ∂α∂ X =0, (1.11) 1)-dimensional scalar and symmetric tensor, re- 1 ∂ Xµ∂ X = η (∂X)2 . (1.12) spectively. The local symmetry needs a gauge α β µ 2 αβ fixing: we should impose p + 1 gauge conditions, The above equations are further simplified by so that only the D − (p + 1) transverse to the defining light-cone variables τ,σ → σ = τ σ. brane oscillations are physical. The equations of motion (1.11) then become The equations of motion (1.4, 1.5) are gener- µ ally complicated and simplify only for the cases ∂+∂−X =0, (1.13) of particle (p = 0) and string (p = 1). Indeed, µ and the general solution for X (σ) is separated to solve the equations, we apply the gauge-fixing into left- and right-movers: conditions on the elements of the p × p symmet- 1 µ µ µ ric matrix hαβ, so that there remain 2 p(p + 1) X (σ)=XL(σ+)+XR(σ−). (1.14) free components. These functions are subject to the constraints For p = 0, one can fix the metric to a con- 2 (1.12): stant, hαβ = −m . Furthermore, there is only α 2 2 one world-line coordinate, the time ξ = τ,and (∂+XL) =(∂−XR) =0. (1.15) 2 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis In order to write explicit solutions, we need to 1.2 Free closed and open string spectrum impose boundary conditions that correspond in By inspection of the commutation relations general to two kind of strings: closed and open. µ µ (1.19, 1.20), one can identify an, a˜n with n>0 Closed strings satisfy the periodicity condi- µ µ (a− ,a˜− ) with the annihilation (creation) oper- tion n n ators. Without loss of generality, let us first re- strict to the holomorphic (left-moving) part. As Xµ(τ,σ)=Xµ(τ,σ +2π), (1.16) usual, we can define a vacuum |p> of momentum µ leading to the general solution p annihilated by an: µ µ µ aµ|p>=0 ; Pµ|p>= pµ|p> . (1.23) X = X0 + P τ (1.17) n X √i 1 { µ −in(τ +σ) µ −in(τ −σ)} + ane +˜ane , The physical states are then created by the action n µ 2 n6=0 of a−n’s on this vacuum after imposing the con- straints (1.15). Expanding the latter into Fourier in mass units of T = 1 .1 The first two terms 2π decomposition, one should require: on the r.h.s. describe the motion of the closed X string center of mass, while the remaining terms 1 L ≡ : a − · a := 0 (1.24) m 2 m n n in the sum correspond to the string oscillations n∈Z of the left- and right-movers which are subject to the constraints (1.15). Reality of the coordi- for m> 0, and µ µ † µ Z nates X imply (an) = a−n and similar for the 2π 1 2 right-movers. Applying the canonical quantiza- L0 ≡ dσ(∂+X) (1.25) 2π 0 tion procedure: X 1 2 1 1 2 = p + : a−n · an := p + N = c, µ ν 0 0 µν 4 2 4 [X (τ,σ),X˙ (τ,σ )] = 2iπδ(σ − σ )η , (1.18) n6=0 one obtains the commutation relations where N is the number operator and the constant c appears due to the normal ordering. These con- µ ν µ ν µν [am,an]=[˜am,a˜n]=mδm+n;0η ,(1.19) straints are in fact necessary to eliminate the neg- µ ν µ ν µν ative norm states generated by the action of the [˜am,an]=0 , [X0,P ]=iη . (1.20) time component of creation operators due to the || µ | || For open strings, the only Lorentz-invariant commutator (1.19); for instance a−m p> = boundary condition is the Neumann one: mηµµ < 0forµ=0. Unitarity is manifest in the light-cone gauge µ ∂σX |σ=0,π =0, (1.21) which fixes the residual symmetry of the covari- ant gauge fixed equations (1.13) and (1.15) under implying that the ends of the string propagate 0 0 σ+ → σ+(σ+)andσ− →σ−(σ−). Defining with the speed of light. The general solution (1.14) becomes in this case X = X0 XD−1 , (1.26) µ µ µ X (σ, τ )=X0 +2P τ (1.22) this invariance can be used to set: √ X 1 − + i 2 aµe inτ cos(nσ) , + + + n n X = X0 + p τ, (1.27) n6=0 while the constraints can be solved for X− in and can be obtained from the closed string ex- terms of the transverse coordinates XT : pression (1.17) by identifying left- and right-mo- 2 vers (a ≡ a˜).