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Home , Brane, Dilaton, Graviton, Polyakov action, String theory, Supergravity

Corfu Summer Institute on Elementary , 1998

PROCEEDINGS

An introduction to perturbative and non-perturbative

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 . The outline is the following. 1. Introduction to perturbative string theory 1.1 From 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 compactifications 1.6 1.7 Effective field 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 in flat space is theory then a straight line. Similarly, the propagation of a p- 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 to describe the dy- brane, propagating in D 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 , 1998 Ignatios Antoniadis where T is the brane of dimensionality the equation of motion (1.4) becomes X¨ µ =0. ()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 : 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 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 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 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 , 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 -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 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 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 |p> of 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). 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 . 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˜). − 2 T 2 ∂+X = + (∂+X ) . (1.28) 1This convention corresponds to fixing the Regge slope p α0 ≡ 1 2πT =1. T 2Note though the factor 2 in P µ due to the change in Thus, we are left only with the transverse a−n the range of σ. that are the independent physical oscillations.

3 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

We can now derive the spectrum. Note that They can form antisymmetric, symmetric, or com- the 0-mode of eq.(1.28) reproduces the global plex representations corresponding to orthogo- hamiltonian constraint (1.25) that gives the mass nal, symplectic or unitary gauge groups, respec- formula: tively. Obviously, open strings cannot be a the- 1 − p2 = N − c. (1.29) ory of gravitation as they do not contain a mass- 4 less in the spectrum. However, in the T | The first excited state a−1 p> with N =1isa presence of , closed strings appear by − spacetime vector with D 2 transverse indepen- making string theory as a candidate for dent components. Therefore, by Lorentz invari- unification of gauge with gravitational . ance, it should be massless, implying c =1.Fur- String interactions of splitting and joining thermore, the algebra of Lorentz gen- can be introduced in analogy with the first quan- erators – or equivalently the absence of confor- tized approach of point , where the world- mal – fixes the spacetime dimensional- line can split using n-point vertices. However, ity D = 26. As a result, one obtains: unlike the particle case, the string 1 2 vertex is unique modulo world-sheet reparame- N =0 |p> -4p = -1 µ | 1 2 trizations. As a result, interactions correspond to N =1 a−1 p> -4p = 0 massless vector µ ν world-sheets with non-trivial and string a−1a−1|p> 1 2 N =2 µ -4p = 1 massive 2 , becomes a topological ex- a−2|p> (1.30) pansion in the number of handles (for oriented and so on. Note that the states of highest spin J closed strings), as well as holes and crosscaps at each mass level form a Regge trajectory: (for open strings and orientation flips of closed strings, respectively). As we will see in section − 1 2 (J 1) 1.7, string diagrams are weighted by powers of a M = 0 . (1.31) 4 α λ2g−2,withg=n+(h+c)/2 The spectrum of a closed string is obtained the genus of the surface given in terms of the by the direct product of states from left- and number of handles n,holeshand crosscaps c.An 2 2 right-movers with the condition ML = MR.At important property of string interactions is that the massless level, one thus has: (modulo reparametrizations) there is no local no- tion of interaction point, which makes string per- aµ a˜ν |p>, (1.32) −1 −1 turbation theory free of ultraviolet divergences, that can be decomposed into a spin 2 graviton that occur in point particles by products of dis- (symmetric traceless part), a scalar tributions at the same point. Thus, string theory (trace), and a 2-index (2- provides a unique mathematical framework of de- form). The automatic and unavoidable occur- scribing non-trivial particle interactions with no rence of the graviton in the massless spectrum is ultraviolet divergences. of course welcome and constitutes a strong mo- tivation for string theory as quantum theory of 1.3 Compactification on a circle and T-duality . Since the bosonic string lives in 26 dimensions, For open strings, there are no separate left- one has to compactify 22 of those on some in- and right-movers, but one has the freedom to in- ternal manifold of small (presently unobserved) troduce additional quantum numbers associated size. The simplest example of compactification is to their ends. As a result the vacuum of oscil- given by one X on a circle of radius R. lators becomes |p, ij >, where the indices i, j are 3 X must then satisfy the periodicity condition: Chan-Paton charges “living” at the two ends of the open string. Their transformation proper- X = X +2πR , (1.34) ties define a (non-abelian) gauge , and the massless states are now gauge fields: so that the solution to the equations of motion

ν 3 25 a−1|p, ij > . (1.33) Here we define X ≡ X .

4 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

(1.13) is the generic gauge group in the case of one dimen- sion is the Kaluza-Klein U(1)L × U(1)R: X = X0 + pτ + wσ (1.35) X µ 25 25 µ i 1 a−1|p>L ⊗a˜−1|p>R ,a−1|p>L ⊗a˜−1|p>R +√ {a e−in(τ +σ) +˜a e−in(τ −σ)} n n n 25 25 2 n6=0 with pL = pR =0. (1.40) p + w p − w = X0 + σ+ + σ− + oscillators. However, for the special value of the radius R =1 2 2 there is an enhanced gauge symmetry SU(2)L × Note the appearance of a new term linear in SU(2)R, due to the appearance of extra massless σ, which is allowed to be non-vanishing due to states: the periodicity condition (1.34). Its coefficient is m = n = 1 ⇒ p = 2,p =0 : quantized in units of R, w = nR, since for σ → L R |2> ⊗a˜µ |p> σ +2π we must have X → X +2nπR,where L −1 R (1.41) m = −n = 1 ⇒ p =0,p =2: nis the () of the string L R aµ |p> ⊗|  2> around the circle. On the other hand, the in- −1 L R ternal momentum is also quantized in units of This property will be generalized later on to non- ∈ 1/R, p = m/R for m Z, which follows from perturbative states. ipX the requirement that plane waves e must be For open strings, the Neumann boundary con- → univalued around the circle: X X +2πR. dition (1.21) imposes the windings to vanish, w = For closed strings, it is convenient to define 0. Thus, a T-duality is not anymore a symme- left and right momenta: try of the spectrum. Under T-duality, p → w m (σ ↔ τ), and the Neumann becomes a Dirichlet p =  nR , (1.36) L,R R boundary condition: in terms of which the mass of physical states be- ∂τ X|σ=0,π =0, (1.42) comes (in 25 dimensions): implying the ends of the string to be fixed at par- −1 2 1 2 − 1 2 − ticular points of the circle. In fact, a coordinate p = pL + NL 1= pR+NR 1. (1.37) 4 4 4 (1.22) satisfying the Neumann (N) condition can The above spectrum is invariant under the T- bewrittenintheform: duality symmetry R → 1/R with the simultane- X=X (σ+)+X (σ−), (1.43) ous interchange of momenta and windings: L R

1 with left- and right-moving oscillators identified, R → ,m↔n or X → X ,X →−X . R L L R R an =˜an. After the T-duality transformation (1.38) (1.38), it becomes It can be shown that T-duality is also an exact ˜ symmetry of the string interactions to all orders X = XL − XR of perturbation theory and it is conjectured to = X˜0 +2wσ (1.44) hold at the non-perturbative level, as well. Note √ X 1 +i 2 aµe−inτ sin(nσ) , that the string coupling λ ≡ λ26 should trans- n n n6=0 form, so that the lower dimensional√ coupling of the compactified theory λ25 = λ26/ R remains which is the general solution of the wave equa- inert: tion (1.13) satisfying the Dirichlet (D) condition λ λ → . (1.39) (1.42), implying that the endpoints of the string R are fixed at X˜|σ=0,π = X˜0. An important consequence of the winding Given the two ends of the open string, one modes in closed strings is the appearance of en- can choose independently the N or D condition hanced non-abelian gauge symmetries at special for each endpoint, implying three kinds of bound- values of the compactification radii. For instance, ary conditions: NN, DD and ND (for which p =

5 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis w = 0). The introduction of Dirichlet condi- that the superstring must live in D = 10 space- tions along the internal directions leads to the time dimensions. notion of D- which are subsurfaces where In order to obtain solutions to the equations open strings can end. A Dp-brane is then de- of motion, we need to discuss boundary condi- fined by imposing Neumann conditions along its tions. For both closed and open strings, there are longitudinal directions Xµ=0,..,p, and Dirichlet two possible conditions compatible with space- conditions along its transverse ones XI=p+1,..,25. time Lorentz invariance: The endpoints are thus fixed in the XI trans- µ µ verse space, while they can move freely in a p+1 ψ (τ,σ)=ψ (τ,σ +2π), (1.47) dimensional spacetime spanned by the world-vo- lume of the p-brane. corresponding to the Ramond (R) and Neveu- It is now easy to see that performing a T- Schwarz (NS) ones. The general solution can duality along a compact transverse (longitudi- then be expanded as (for instance for the left- nal) direction, a Dp brane is transformed into movers): X a D(p+1)-brane (D-(p-1)-brane). Moreover, in µ µ −ir(τ +σ) µ† µ ψ = b e ;(b)=b−,(1.48) this picture, the Chan-Paton multiplicity corre- L r r r r sponds to the multiplicity of D-branes. Thus, D- branes on top of each other, labeled by a Chan- where r is half-integer (integer) for NS (R) bound- Paton index, give a gauge group enhancement. ary conditions, and upon quantization one has the usual canonical anticommutator relations: 1.4 The Superstring: type IIA and IIB { µ ν} Two immediate problems with the bosonic string br ,bs =ηµν δr+s,0 . (1.49) are the presence of tachyon (signaling a vacuum The mass formula (1.29) now becomes: instability) and the absence of in the spectrum. Both problems can be solved in the 1 − p2 = N + N − c, (1.50) context of superstring, obtained by adding fer- 4 ψ P mions on the world-sheet. 1 where N = r : b · b− :isasumofhalf- The fermionic coordinates are Weyl-Majora- ψ 2 r r r na two-dimensional (2d) fermions ψµ (σ, τ ), integer (integer) frequencies for NS (R) fermions, L,R and c =1/2(c= 0) in the NS (R) sector. which carry a spacetime index µ. The 2d implies that the left (right) handed fer- As in the bosonic string, unitarity is manifest mions depend only on σ+ (σ−). As in the case in the light-cone gauge, where only the transverse of bosonic coordinates, the timelike components oscillators create physical states. In the super- 0 string, there are however two sectors in the spec- ψL,R generate extra negative norm states that need a new local symmetry to be removed. This trum: the NS and the R sectors, corresponding to is indeed the super-reparametrization invariance, antiperiodic and periodic world-sheet supercur- or equivalently local on the world- rent (1.46) and giving rise to spacetime sheet. In the superconformal gauge (1.10), the and fermions, respectively. In the NS sector, the | supersymmetry transformations read (for the left- ground state is still a tachyon p> of momentum − 2 − movers): p and mass given by (1/4)p = 1/2, while at the massless level there is a vector: µ − µ δXL = iRψL µ µ µ |µ; p>= b−1 2|p> . (1.51) δψL = R∂+XL (1.45) / and emerge from the 2d supercurrent In the R sector, the fermionic coordinates (1.48) µ have zero-modes which satisfy the anticommuta- TF = ψ ∂+Xµ . (1.46) L tor relations (1.49), generating a ten-dimensional Similar transformations hold for the right movers Clifford algebra: by exchanging L ↔ R and + ↔−. The cancel- µ ν µν lation of conformal anomalies imply in this case {b0 ,b0}=η . (1.52)

6 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

As a result, the oscillator vacuum forms a rep- where L, R denotes left, right 10d chiralities, and resentation of this algebra and corresponds to a |β>R (|β>L) stands for type IIA (type IIB). De- spacetime |p, α> of dimension 2d/2 = 32, composing the spectrum into representations of µ on which b0 act as the 10d gamma matrices: the 10d little group SO(8), one finds the bosons √ µ| µ | | ⊗| i 2 b0 p, α>= γαβ p, β > . (1.53) µ> ν> =1+35S+28A | ⊗| IIA Moreover, in the absence of oscillators, the con- α> β> =8(1−form) +56(3−form) IIB + straint corresponding to the zero-mode of the 2d =1(0−form) +28(2−form) +35(4−form) supercurrent (1.46) generates the 10d massless (1.58) Dirac equation which reduces the dimensionality and the fermions of the lowest lying state to 16, corresponding to |α> ⊗|ν> =8L+56L a massless 10d real (Majorana) . IIA |µ> ⊗|β> =8R+56R (1.59) At this point, the problem of the tachyon IIB =8 +56 remains. However, it turns out that the above L L spectrum is not consistent with world-sheet mod- The NS-NS states 1, 35S and 28A in eq. (1.58) ular invariance, which guarantees the absence of denote the trace, the 2-index symmetric traceless global anomalies under 2d diffeomorphisms dis- and antisymmetric combinations corresponding connected from the identity that can be performed to the dilaton, graviton and antisymmetric ten- in topologically non-trivial surfaces. Consistency sor, respectively, that are part of the massless of the theory already at the one-loop level (torus spectrum of any consistent string theory. The topology) implies that one should impose the above massless spectra coincide with those of IIA (GSO) projection: and IIB N = 2 supergravities in 10 dimensions; (−)F = −1 , (1.54) note the two gravitini 56’s in eq. (1.59) with op- posite or same . where F is the fermion number operator. This Upon compactification in nine dimensions on projection eliminates the tachyon from the NS a circle of radius R, the two type II theories are sector, while in the R sector acts as spacetime equivalent under T-duality: chirality. In fact, from the anticommutator → 1 ↔ F µ R :IIAIIB . (1.60) {(−) ,b0}=0, (1.55) R 11 This can be easily seen from the transformation (−)F can be identified with the 10d γ matrix, (1.38), whose action on the fermions is ψ → ψ in the absence of oscillators: L L P and ψ →−ψ , in order to preserve the form of ∞ µ µ R R F 11 b bn (−) = γ (−) n=1 −n . (1.56) the 2d supercurrent (1.46). As a result, (−)FR = γ11 →−γ11, implying a flip of the fermion chi- It follows that at the massless level there is one R R rality in the right-movers. massless vector and one Weyl Majorana spinor, having 8 bosonic and 8 fermionic degrees of free- 1.5 Heterotic string and orbifold compact- dom. This fermion-bose degeneracy holds to all ifications massive levels and is a consequence of a resulting The heterotic string is a closed string obtained spacetime supersymmetry. by the tensor product of the superstring (for the We can now discuss the spectrum of closed left-movers) and the bosonic string (for the right- superstrings. There are two different theories movers). Since the left-moving coordinates live depending on the relative (spacetime) chirality in 10 dimensions while the right-moving ones in between left- and right-movers: the type IIA or 26, 16 of the latter are compactified on an in- type IIB corresponding to opposite or same chi- ternal momentum lattice. One-loop modular in- rality, respectively. The massless spectrum is ob- variance then implies that the corresponding mo- tained by the tensor product: I menta pR (I =1,...,16) belong to an even self- | | ⊗ | | · 0 ∈ 2 ∈ ( µ>, α>L) ( ν>, β>R,L) , (1.57) dual lattice (p~R p~R Z and p~R 2Z). There are

7 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis only two such lattices in 16 dimensions generated because modular invariance requires the presence by the roots of SO(32) and × E8 groups. of a new (twisted) sector corresponding to strings The massless spectrum of the heterotic string with center of mass localized at the orbifold fixed is given by the tensor product points. In four dimensions, the internal rotations form an SO(6) ≡ SU(4) and the condition of un- | | ⊗| ( µ>, α>) ν>=1+35S +28A +8L +56L, broken N = 1 supersymmetry amounts to divide (1.61) the torus T 6 with a discrete subgroup of SU(3) which forms the particle content of the N =1 that leaves one of the four gravitini invariant. multiplet in 10 dimensions, and by The simplest orbifold example reduces the | | ⊗| I 2 number of by 1/2 and can be ( µ>, α>L) pR;~pR =2>, (1.62) 4 studied in 6 dimensions. It is defined by T /Z2, which forms a 10d N = 1 gauge where the Z2 inverts the sign of the four internal of SO(32) or E8 × E8. coordinates Xi →−Xi(i=1,...,4) and of their Upon compactification in nine dimensions on fermionic . Since in the heterotic a circle, the two heterotic theories are equivalent string there are no right-moving superpartners, under T-duality: Z2 should also act non-trivially on the gauge de- 1 grees of freedom breaking partly the gauge sym- R → :HetSO(32) ↔ Het E8 × E8 . R metry together with the N = 4 supersymmetry. (1.63) To see the reduction of supersymmetry in the The continuous connection of the two theories massless sector, consider the (left-moving) Ra- can be easily seen using the freedom of gauge mond vacuum that forms a 10d Weyl Majorana symmetry breaking by turning on Wilson lines spinor with (−)F = −1. Its decomposition under (flux lines in the compact direction). These cor- SO(4) × SO(4), with the first factor correspond- respond to the constant values of the internal ing to the 6d little group and the second to the I (10th) component of the gauge fields A9 along internal rotations, reads the 16 Cartan generators, and break generically ψµ ψi the gauge group to its maximal abelian subgroup 16 + − (1.64) U(1) . One can then show that eq. (1.63) is − + valid at the point with SO(16) × SO(16) gauge symmetry. where the signs denote the chiralities under the The heterotic string appears to be the only two SO(4). The Z2 orbifold action on this spinor perturbative closed string theory that can de- is identical to the chirality projection in the inter- scribe our observable world. One of the main nal part and thus eliminates half of the gravitini. problems is however to get a 4-dimensional su- The Hilbert space of the theory consists in perstring which is phenomenologically viable. In two sectors: particular, in order to be chiral the spectrum • should be at most N = 1 supersymmetric. For The untwisted sector which is obtained from the Hilbert space of the toroidal compacti- a toroidal compactification, the 10d N = 1 spec- 4 trum is converted into a non-chiral N =4super- fication on T projected into the Z2 invari- symmetric spectrum in four dimensions. ant states: i i i A solution to this problem is provided by |p =0>+ , |p >+|−p > orbifold compactifications. These are obtained with even number of oscillators i i i from toroidal compactifications by identifying |p =0>− , |p >−| − p > points under some discrete subgroup of the inter- with odd number of oscillators nal rotations that remains an exact symmetry of (1.65) i the compactified theory. The resulting compact where |p =0> denote the Z2 even (+) spaces are not smooth manifolds because there and odd (−) states with vanishing inter- i are singularities associated to the fixed points. nal momentum p in which the Z2 action is String propagation however is made consistent non-trivial.

8 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

• The twisted sector which contains states lo- multiplicity of D9-branes, leading to an SO(32) calized at the 24 = 16 fixed points and, gauge group. thus, are confined to live on 5d subspaces Under a T-duality along one compact direc- (in the large volume limit). tion, we get → ˜ − 1.6 Type I string theory X = XL + XR X = XL XR (1.67) Up to this point, we have seen four consistent and the action of Ω(L ↔ R) becomes closed superstring theories in ten dimensions: X˜ →−X.˜ (1.68) type IIA and IIB with two spacetime supersym- metries, and heterotic SO(32) and E8 × E8 with So the effect of a T-duality on Ω is one supersymmetry. Moreover, the two type II → R theories and the two heterotic ones are connected Ω Ω , (1.69) by T-duality upon compactification to nine di- where R is defined as R : X →−X. Therefore, mensions. Actually, there is a 5th consistent T-duality gives in 10d with N =1super- 0 symmetry, the type I theory of open and closed type I = IIB/Ω → type I =IIA/ΩR(1.70) strings; open strings provide the gauge sector, D9 − branes → D8 − branes (1.71) while closed strings provide gravity needed for 1.7 Effective field theories unitarity. 0−1/2 A consistent to construct type I At low , lower than the string scale α , theory is to “orbifolding” type IIB by the world- one can integrate out all massive string modes to sheet involution Ω that exchanges left- and right- obtain an effective field theory for the massless movers and is a symmetry of the theory: excitations of the string. There are two ways to obtain this effective action. Either by computing Ω:σ→−σ (L↔R). (1.66) the string scattering amplitudes, or by consid- ering string propagation in the presence of non- We thus obtain type I theory = IIB/Ω. As in trivial background fields. The latter is described ordinary , the spectrum consists in an by the world-sheet action: untwisted and a twisted sector. 1 R The untwisted sector contains closed strings S = − d2ξ{G (X)∂ X µ∂αX ν (1.72) 4π µν α projected by Ω (unoriented closed strings). This αβ µ ν turns to symmetrize the NS-NS sector and to +Bµν (X) ∂αX ∂βX − R(2) a µ } antisymmetrize the R-R. As a result, the two- φ(X) + Aµ(X)Ja + ... , index NS-NS antisymmetric tensor is projected a where Gµν , Bµν , φ, A ,... are backgrounds for out, together with the R-R scalar and 4-form (see µ the massless fields (metric, 2-index antisymmet- eq.(1.59)), and we are left with the scalar dila- (2) ric tensor, dilaton, gauge fields, etc), and R ton scalar φ and the symmetric tensor (graviton) denotes the 2d scalar . Note that this G from NS-NS, and a 2-form B from R-R. µν µν is the most general non-linear which The twisted sector corresponds to the “fixed is renormalizable in two dimensions. Conformal points” X(−σ, τ )=X(σ, τ ) which are equiva- invariance implies the vanishing of all beta-func- lent to the (Neumann) boundary conditions for tions which reproduce the spacetime equations of open strings: ∂ X| =0 = 0. Moreover, the σ σ ,π motion for the background fields. “fixed-point multiplicity” N corresponds to the A particular property of string theories, that Chan-Paton charges. It is determined by the tad- can be seen from the action (1.72), is that the pole cancellation condition, that plays the role constant dilaton background eφ plays the role of of modular invariance for open and closed unori- the string coupling. Indeed, a shift φ → φ + c ented strings and guarantees the absence of po- amounts to multiply the path integral by a factor tential gauge and gravitational anomalies. One finds N=32 which can also be interpreted as the e−S → e2c(g−1)e−S , (1.73)

9 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis where g is the genus of the world-sheet, and we field theory solitons. In this section we will estab- used the Euler integral lish that string solitons are p-brane extended ob- Z jects and study their main properties. As point- 1 2 (2) d ξR =2(g−1) . (1.74) particles (0-branes) are electric sourcesR for gauge 4π µ fields (1-forms), with a coupling Aµdx , strings It follows that the dilaton shift can be absorbed (1-branes) are sources for two-index antisymmet-R µ in a rescaling of the string coupling λ → ecλ.The ric (2-forms), with a coupling Bµν dx ∧ 10d effective action can therefore by expanded in dxν as displayed in eq. (1.72)), p-branes can be powers of eφ corresponding to the perturbative seen as electric sources for (p + 1)-form poten- topological string expansion: tials, with a coupling: Z Z (p+1) µ1 ∧ ∧ µp+1 10 −2φ (10) Aµ1 ...µp+1 dx ... dx , (2.1) Seff = d X { e [R + ···] (1.75) ··· +e−φ[···]+1[···]+···}. with µi =0,1, ,p. Magnetic sources can be understood in a sim- The first term proportional to e−2φ corresponds ilar way by performing Poincar´e duality to the to the tree-level contribution associated to spher- corresponding field strengths. In fact, a (p + ical world-sheet topology (g = 0), the second 1)-form potential has a (p + 2) field strength (p+2) (p+1) term multiplying e−φ denotes the disk contribu- H = dA . Hodge duality in D dimen- tion (g =1/2), the third term proportional to sions then gives a D − (p + 2) dual form: the identity corresponds to the one-loop toroidal H˜ (D−p−2) = H(p+2) , (2.2) topology (g = 1), and so on. Closed oriented string diagrams give rise to even powers e2(g−1)φ with  the corresponding Levi-Civita totally an- with g integer, while closed unoriented and open tisymmetric tensor. This dual field strength is string diagrams introduce boundaries and cross- now associated a (D − 3 − p)-form dual poten- ( −p−2) ( −p−3) caps having g half-integer and can to odd tial H˜ D = dA˜ D , which couples to powers of eφ, as well. (D −4−p)-branes playing the role of “magnetic” sources for the initial (p + 1)-form potential. As a result, a (p + 1)-form potential has p-branes References as electric sources and (D − 4 − p)-branes as [1]M.Green,J.SchwarzandE.Witten,Super- magnetic ones. Furthermore, the analog of Theory,Vols I and II, Cambridge Univer- quantization gives the following relation between sity Press, 1987. the dual charges µp of branes: [2] D. L¨ust and S. Theisen, Lectures in String The- µp µD−4−p =2πn , n ∈ Z . (2.3) ory, Lecture Notes in Physics, 346, Springer Verlag, 1989. For instance, in D = 4 dimensions, point particles (p = 0) electrically charged are dual to [3] J. Polchinski, String Theory, Vols. I and II (Cambridge University Press, Cambridge, (point-like) magnetic monopoles, while it is also 1998). possible to have dyons carrying simultaneously non-vanishing electric and magnetic charges. Sim- [4] E. Kiritsis, Introduction to Superstring Theory, ilarly, one can have dyonic strings (p = 1) in hep-th/9709062. D =6,dyonicmembranes(p=2)inD=8,and dyonic 3-branes in D = 10. 2. Introduction to non-perturbative Let us now consider the spectra of the vari- string theory ous string theories: • In the heterotic string, there is a 2-form po- 2.1 String solitons tential Bµν . Its electric source is the fun- The first step towards a non-perturbative under- damental string, while its magnetic source standing of string theory is to study the analog of is a solitonic NS 5-brane.

10 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

• In type II strings the same result holds for algebra, and using the BPS property the mass the NS-NS antisymmetric tensor. In ad- formula receives no quantum corrections. dition, as we have seen in section 1.4 (eq. The effective action of a p-brane (neglecting (1.58), there are R-R p-form potentials aris- background fields) is: Z ing in the decomposition of two . In √ contrast to the NS-NS 2-form, the R-R po- S = −Tp − det h (2.4) tentials have no elementary (perturbative) Z(p+1) sources of either electric or magnetic type. − (p+1) µ1 ∧ ∧ µp+1 µp Aµ1...µp+1 dx ... dx , This can be understood for instance from µ the vanishing of all amplitudes containing with hαβ = ∂αX ∂βXµ. The BPS property fixes R-R fields at zero momentum since the cor- its tension Tp to be equal to its µp in 10d responding string vertices involve directly supergravity units: their field strengths. q 2 2 7 0 4 2 In type IIA, there are all possible even-form µp = 2κ10Tp 2κ10 =(2π) (α) λ . R-R field strengths: 0, 2, 4 and their duals (2.5) 6, 8, 10. They give rise to odd-form po- For Dp-branes, the values of the tension (or the tentials 1, 3, 5, 7, 9, having even p-brane charge) can be extracted from the one-loop vac- sources p = 0, 2, 4, 6, 8. uum amplitude of an open string ending on two D-branes, which can also be seen in the trans- In type IIB, there are all possible odd-form verse channel as the propagation of a closed string R-R field strengths: 1, 3, 5 and their duals between the two D-branes. the result is: 5, 7, 9, the 5-form being self-dual. They 1 give rise to even-form potentials 0, 2, 4, 6, Tp = (2.6) 0 1+p 8, having odd p-brane sources p = −1, 1, 3, λ (2π)p(α ) 2 √ − 2 0 3−p 5, 7. Note the ( 1)-branes that correspond µp = 2π(4π α ) 2 , to . The R-R p-branes are called Dp-branes be- which satisfies the Dirac quantization condition cause they interact through the emission of (2.3) with minimal charge n = 1. Notice the non- open strings with Dirichlet boundary con- perturbative factor 1/λ in the expression of the ditions in the transverse to the world-volu- D-brane tension (2.6). me directions. Besides the check of the quantization con- dition (2.3), one can derive a recursion relation • In type I theory, there is no 2-form from for the D-brane tensions by use of T-duality dis- the NS-NS sector, because of the left-right cussed in section 1.6. In fact, a T-duality trans- symmetrization of the spectrum. Similarly, formation along a compact direction longitudinal because of the antisymmetrization of the to the brane, maps a Dp-brane to a D(p − 1). R-R sector, only the 2-form (and its 6-form Wrapping the Dp-brane around a circle of ra- dual) potential survive in the spectrum cou- dius R,onegetsa(p−1)-brane with tension 4 pled only to D1 and D5-branes. Using now 2πRTp. Performing now a T-duality√ along the T-duality to type I0, one can actually gen- circle (R → α0/R, λ → λ α0/R) and using erate all types of Dp-branes. the non-perturbative dependence Tp ∼ 1/λ,one finds the relation All the branes we found above are solutions √ 0 of the supergravity effective action. They are 1/2 2πRTp → 2π α Tp ≡ Tp−1 , (2.7) BPS states breaking 1/2 of the supersymmetries, and they form short (analog of which is satisfied by the general formula (2.6). massless representations). Their mass is deter- The tension and charge of the fundamental mined by their charge due to the supersymmetry string are: 4 In fact, there are also 32 D9 branes, which is a special 1 5 0 T = µ =(2π)2α λ. (2.8) case as there is no field strength for a 10-form potential. fund 2πα0 fund

11 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

To find the tension and charge of the NS 5-brane, 2.2 Non-perturbative string dualities the magnetic dual of the fundamental string, we In part 1, we have found five consistent super- can use the quantization condition (2.3) with min- string theories in ten dimensions. Two type II imal charge n = 1 and the BPS relation (2.5) to with N = 2 supersymmetry, and two heterotic deduce: and the type I with N = 1 supersymmetry. The 1 1 two type II theories, as well as the two heterotic NS 5 NS 5 T = 2 5 0 3 µ = 3 0 . λ (2π) (α ) (2π) 2 α λ ones are related by T-duality upon compactifica- (2.9) tion in nine dimensions, leaving in principle three Notice that the tension of the NS 5-brane is pro- independent consistent superstring theories. In 2 portional to 1/λ , in contrast to the 1/λ factor of perturbation theory, type II theories are phe- D-branes, while the tension of the fundamental nomenologically uninteresting since they contain string is of course perturbative (of order unity). only gravity in ten dimensions and compactifica- Upon compactification to lower dimensions, tion is not sufficient to produce the rich particle on finds two important consequences: content of the , while type I the- ory appears quite complicated. This singled out • A p-brane wrapped around a p-cycle of the the heterotic string as the theory where most of compact manifold lead to a non-perturba- the effort was devoted. The surprising result of tive point-like state with string dualities is that when non-perturbative ef- fects are taken into account, all superstring the- Mass = Tension × Area of the cycle, ories are equivalent in the sense that they corre- (2.10) spond to different perturbative vacua of the same generalizing the notion of string winding underlying theory, called M-theory, which con- modes with mass an integer multiple of tains also a new vacuum described by the 11d 2πR/(2πα0). For non-trivial manifolds, if supergravity. the cycle shrinks to zero size, one obtains The S-duality inverts the coupling constant new non-perturbative massless states that λ → 1/λ and exchanges the role of perturbative can be charged under some gauge fields and states with solitons. T-duality (1.38) is thus an give rise to enhanced gauge symmetries. In S-duality from the 2d world-sheet point of view, this way, it is possible to obtain non-pertur- since the radius R is the coupling constant of batively interesting non-abelian gauge the 2d σ-model and winding modes correspond groups in type II theories. to solitons. In ten dimensions, there are two S- • A p-brane with euclidean world-volume duality conjectures: wrapped around a (p + 1)-cycle leads to 1. Type I - Het SO(32) an and thus can generate non- perturbative corrections to the effective ac- 1 λ ↔ ,BH↔ BI , (2.12) tion, proportional to λ µν µν

e−S ,S=Tension×world volume . under which we identify (2.11) − ≡ The NS 5-brane generate typical field the- D1 string Het string (2.13) 2 ory instanton corrections of order e−1/λ , D5 − brane ≡ Het NS 5 − brane . following from the expression of its tension (2.9). On the other hand, D-branes have By identifying the tension of the D1-brane tension ∼ 1/λ generating non-perturbative with the one of the heterotic string T1 = −1 0 ≡ 0 effects of order e /λ. These are typical 1/(λI 2παI) 1/(2παH), we derive the stringy in nature and are much stronger relation between the type I and heterotic than the field theory ones in the weak cou- scales: 0 0 pling limit. αH = λI αI . (2.14)

12 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

2. Type IIB is self-dual under the S-duality we deduce that the two theories are related by group SL(2, Z)S, acting on λ complexified an S-duality in D =6: with the R-R scalar (see spectrum (1.58)): 1 0 2 0 λ6 ↔ and α ↔ λ6 α , (2.20) pλ − iq λ6 λ → (2.15) irλ + s where λ6 = λ/V4 is the string coupling in six dimensions. and The two theories have indeed the same 6d      NS NS Bµν pq Bµν massless spectrum, which consists of the N =2 RR = RR (2.16) 20 Bµν rs Bµν supergravity multiplet coupled to U(1) abelian vector multiplets containing 4 × 20=80scalar with the integer parameters satisfying ps− . However there is a potential problem: qr = 1. For the particular case p = s =0 on the heterotic side, there are special points in and q = −r = 1, one finds the simple S- the with enhanced gauge symme- duality λ ↔ 1/λ and BNS ↔ BRR, which tries, while on the type IIA side there are no correspond to the exchange: perturbative states charged under the U(1)’s be- cause the latter come from the R-R sector. The ↔ − IIB string D1 string (2.17) missing states appear in fact non-perturbatively NS 5 − brane ↔ D5 − brane and correspond to D2-branes wrapped around 2- cycles of K3. As we discussed in the previous By comparing the fundamental and D1- section, these states become massless when the string tensions, we get again the transfor- cycles shrink to zero-size, giving rise to enhanced 0 ↔ 0 mation α λα . gauge symmetries.

To summarize up to this point, we have dis- 2.3 M-theory cussed the following relations: The connection between the heterotic and type T S II theories can also be understood from eleven Nsusy =1:HetE8×E8 ←→ HetSO(32) ←→ type I T S dimensions, suggesting the existence of some un- Nsusy =2:IIA←→ IIB ←→ IIB (2.18) derlying fundamental theory, called M-theory, whose low- limit is D = 11 supergravity. which lead to two independent theories accord- In this context, the various string dualities follow ing to the number of supersymmetries. The next from 11d general coordinate invariance. question is how to relate theories with different D = 11 supergravity is unique and contains number of space-time supersymmetries. The an- the metric G, the and a 3-form poten- swer is after compactification on different appro- (3) tial A . Upon on a cir- priate manifolds leading in lower dimensions to cle, it gives the D = 10 IIA supergravity with the same number of supersymmetries. the following field identification: The first non-trivial example arises in six (3) (3) dimensions. We can relate the heterotic string 11D Gµν Aµν11 G 11 11 G µ 11 A µνλ 4 (3) (2.21) compactified on T and type IIA on K3, which IIA Gµν Bµν φAµAµνλ have both N = 2 (non-chiral) supersymmetry in where B is the NS-NS 2-index antisymmetric ten- D = 6, by identifying the following branes: sor, φ the dilaton and A(p) the R-R p-form poten- Het NS 5-brane wrapped around T 4 tials. From the identification of the dilaton with ≡ IIA string G11 11 , it follows that the type IIA string coupling IIA NS 5-brane wrapped around K3 is given by the radius of the eleventh dimension: √ ≡ Het string 0 R11 = λ α . (2.22) (2.19) Comparing the branes tensions Tfund ↔ TNS 5V4 The corresponding Kaluza-Klein (KK) states with with V4 the volume of the 4d compact manifold, n/R11 ∼ 1 /λ are non-perturbative states

13 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis from the type IIA viewpoint and can be identified pendent tensions that are given in terms of two with D0-branes. Indeed D0-branes are charged parameters, so that there is one non-trivial rela- under the R-R gauge field Aµ which is precisely tion among the tensions. Since this relation can the KK U(1) Gµ√11 . Moreover, the D0-brane ten- be understood from T-duality (2.7) within type 0 sion T0 =1/λ α =1/R11 isthesamewith IIA theory, it follows that T-duality is a conse- the mass of the lightest KK mode having the quence of 11d reparametrization in the context minimum charge. KK states become infinitely of M-theory. heavy and decouple in the type IIA weak cou- We have seen above that M-theory compact- pling limit. On the other hand, when λ →∞, ified on a circle S1 describes the strong coupling they become light and the eleventh dimension limit of type IIA. One can also argue that M- 1 opens up (R11 →∞). As a result, the strong cou- theory compactified on a line segment S /Z2 de- pling limit of type IIA string theory is described scribes the strong coupling limit of the heterotic by the D = 11 supegravity. string E8 × E8. Besides inverting the 11th co- In order to describe the rest of the type IIA ordinate X11 →−X11, Z2 changes also the sign (3) (3) spectrum, 11d supergravity should be implemen- of the 3-form A →−A . As a result, Gµ 11 ted with (BPS) sources for the 3-form potential, (3) and Aµνλ are projected out and the Z2 invariant in the context of the underlying M-theory. A (untwisted) bosonic spectrum is membrane (M2-brane) as “electric” source and a (3) ≡ ≡ M5-brane as a “magnetic” one. Upon compacti- Gµν ,Aµν11 B µν ,G11 11 φ, (2.26) 1 fication on a circle S of radius R11 , one obtains which forms the particle content of N =1super- the following identification: gravity in D = 10. This truncation of the spec- fundamental string ≡ M2-brane wrapped on S1 trum introduces a potential at the two (ten-dimensional) ends of the segment, NS 5-brane ≡ M5-brane which is cancelled by introducing twisted states ≡ D0-brane KK (2.23) localized at the two endpoints. They consist of D2-brane ≡ M2-brane two 9-brane walls with one E8 gauge factor each. D4-brane ≡ M5-brane wrapped on S1 The between the two walls is the radius D6-brane ≡ KK monopole of the 11th dimension which is related to the het- erotic string coupling by the same relation (2.22) The tensions of M-theory branes can be com- as in type IIA. puted in the effective supergravity, although they With this result, we completed the discus- are essentially determined by dimensional anal- sion on the connection of all known consistent ysis in terms of the 11d superstring theories in the context of M-theory. κ11 : 2.4 Effective field theories and duality tests   1 2 π 2 3 T 2 = (2.24) Here, we will rederive the duality transforma- M κ 2 11 tions that relate the different string theories in   2 1 2π2 3 various dimensions by studying the effective field TM5 = 2 . (2.25) 2π κ11 theories, and discuss some duality tests. The 10d effective lagrangians for heterotic They satisfy the quantization condition (2.3) for and type I theories are: n = 1, with the charges equal top the tensions 2 H 1 1 in 11d supergravity units, µp = 2κ11Tp as in −2φ10 2 LH ∼ e [ R + F + ···] (2.27) eq. (2.5). All type IIA brane tensions can be 2 4 −2 I 1 − I 1 2 derived from eqs. (2.24), (2.25), using the iden- φ10 φ10 LI ∼ e [ R + ···]+e F , (2.28) tification (2.23), in terms of 2 parameters: κ11 2 4 0 and R11 , or equivalently α and λ (determined by where φD is the D-dimensional dilaton, such that φD the dimensional reduction). In fact, the quanti- e = λD is the D-dimensional string coupling, zation condition (2.3) leaves us with three inde- and for simplicity you keep only the gravitational

14 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis and gauge kinetic terms. The difference between • Finally, in D = 4 one obtains a mixing: ! the two lagrangians comes from the difference in   1 −2φ4 −φ4 2 the topological expansion of the two theories, as e e V6 ↔ − 1 (2.36) 6 −3φ4 2 discussed in section 1.7: gauge fields appear on V e V6 the sphere in closed strings, at the same order with gravity, while they appear on the disk in For type IIA compactified on K3, the effec- open strings. tive field theory lagrangian is: Compactifying to D dimensions, the lagran- − 2φ II 1 1 2 gian is multiplied by the internal volume V10− L ∼ e 6 [ R + ···]+ F , (2.37) D II 2 4 (in string units) where the absence of dilaton dependence in gauge

L→LV10−D , (2.29) couplings is due to the fact that gauge fields are R-R states. Compactifying to D dimensions and and we define the D-dimensional dilaton going to the Einstein frame as before, we obtain the gauge coupling: −2φD −2φ10 e ≡ e V10−D . (2.30) 1 D−4 II 2 D−2 φD II 2 = e V 6− D . (2.38) In order to normalize the gravitational kinetic gD terms, we go to Einstein frame by rescaling the We can now deduce the dualities between het- metric 4 4 erotic string on T and type IIA on K3: D−2 φD Gµν → Gµν e . (2.31) • In D =6wehave The gauge kinetic terms take then the form 2 2 II − H F /4gD with eφ6 ↔ e φ 6 (2.39)

−4 H 2 D−2 φD 1/gD = e for heterotic which is an S-duality λ6 ↔ 1/λ6. More- D−6 I 1 2 D−2 φD I 2 1/gD = e (V10−D) for type I over, by identifying the Newton’s constant (2.32) on both sides: We can now deduce the duality transforma- 1 −2φ6 1 tions in various dimensions by comparing the 2 = e 0 2 , (2.40) κ6 (α ) gauge couplings in both theories: 0 2 0 we get α ↔ λ6α , and reproduce the trans- • In D =10wehave formations (2.20).

−1 H 1 I e 2φ10 ↔ e 2 φ10 , (2.33) • In D =4wehave

H II − 2φ 4 which is an S-duality λ ↔ 1/λ.Onthe V2 = e , (2.41) other hand, by identifying the Newton’s which is a U-duality λ4 ↔ 1/R,withR constant on both sides: 2 defined by V2 ≡ R .

1 −2φ10 1 2 = e 0 4 , (2.34) κ10 (α ) Let us now start from 11 dimensions with the supergravity lagrangian we get α0 ↔ λα0, thus reproducing the 1 transformations (2.12) and (2.14). L ∼ R +(dA(3))2 + ··· (2.42) M 2 • In D =6wehave Compactifying to 10 dimensions, with the identi- (3) 2 1 ≡ 11 11 ≡ 2 −φ6 fication (2.21) Aµν11 B µν and G R 11 ,and V4 ↔e , (2.35) rescaling the metric Gµν → Gµν /R11 ,weobtain which is a U-duality λ ↔ 1/R2,withR L∼ 1 1 R 2 1 2 ≡ 4 3 [ +(dB) + F ] . (2.43) defined by V4 R . R11 2 4

15 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

Putting back the 11d mass units and defining expression should produce the perturbative and 9/2 l11 ≡ κ 11 , we find non-perturbative corrections on the other side, providing an explicit duality test. R11 3 λ =( ) 2 A similar but much more non-trivial exam- l 11 ple arises in D = 4, for string vacua with N =2 l 3 α 0 = 11 , (2.44) supersymmetry, obtained by compactifying het- R 11 erotic or type I on K3×T 2 and type II on Calabi- which reproduce the relation (2.22). Yau. In this case, one has vector multiplets con- One consequence of the relations obtained in taining a vector, two scalars and a Dirac fermion, this section is that although S-dualities are non- and hypermultiplets. Moreover, for neutral hy- perturbative in higher dimensions, they may be- permultiplets, supersymmetry implies a no-mix- come “perturbative” after compactification which ing at the level of low-energy lagrangian. Now, the heterotic 4d dilaton belongs to a vector while makes possible to perform various duality tests 2 by comparing appropriate couplings in the ef- the size of T belongs to a hyper. The oppo- fective action. In addition, non- site is true on the type II side, while heterotic – type II duality (2.41) maps the dilaton of the one theorems due to extended supersymmetry make 2 in some cases possible to obtain exact results. theory to the T size of the other. As a result, LN=2 The first non-trivial example arises in D = 4, vectors can be computed exactly by a tree-level 6, for string vacua with N = 1 supersymme- computation on type II side and should produce try, obtained by compactification of heterotic or the perturbative (which stops at one-loop) and type I theories on K3. In this case, there are non-perturbative expansion of the heterotic the- two types of (massless) supermultiplets containg ory. Finally, on the type I side, one has to define scalars: the tensor multiplet with one scalar, a linear combinations and duality with heterotic self-dual 2-form and a Weyl fermion, and the (2.36) is more involved. Both vectors and hypers hypermultiplet, containing 4 scalars and a Weyl can receive type I string corrections. spinor. Moreover, for neutral hypermultiplets, supersymmetry implies that the low-energy (two- Acknowledgments derivative) effective lagrangian is a direct sum of This work was supported in part by the EEC two pieces, one describing the interactions of ten- under TMR contract ERBFMRX-CT96-0090. sors and the other describing the interactions of hypers: References N =1 N=1 N=1 L6 = L6 tensors + L6 hypers . (2.45) , , [1] J. Harvey, Magnetic monopoles, duality, and su- persymmetry, hep-th/9603086. It turns out that on the heterotic side, the 6d H [2] J.H. Schwarz, Lectures on Superstring and M dilaton φ6 belongs to a tensor multiplet while H Theory Dualities, hep-th/9607201. the compactification volume V4 belongs to a hy- permultiplet. The opposite is true on the type I [3] J. Polchinski, TASI Lectures on D-Branes,hep- side, while duality (2.35) maps the dilaton of the th/9611050. one theory to the compactification volume of the [4] P.K. Townsend, Four Lectures on M-theory, other. Since tensors cannot mix with neutral hy- hep-th/9612121. LN=1 pers, it follows that 6, tensors can be computed [5] C. Vafa, Lectures on Strings and Dualities,hep- exactly by a tree-level computation on type I, th/9702201. LN=1 while 6, hypers can be computed exactly by a [6] S.E. Hjelmeland, U. Lindstr¨om, Duality for the tree-level computation on heterotic. Higher or- Non-Specialist, hep-th/9705122. der (perturbative or non-perturbative) cor- [7] I. Antoniadis, H. Partouche, T.R. Tay- rections are forbidden because they depend on lor, Lectures on Heterotic - Type I Duality, the string coupling and would bring a dilaton Nucl.Phys.Proc.Suppl. 61A (1998) 58-71; hep- dependence and therefore a mixing. The exact th/9706211.

16 Corfu Summer Institute on Elementary Particle Physics, 1998 Ignatios Antoniadis

[8] A. Sen, An Introduction to Non-perturbative String Theory, hep-th/9802051. [9] A. Dabholkar, Lectures on Orientifolds and Du- ality, hep-th/9804208. [10] C.P. Bachas, Lectures on D-branes,hep- th/9806199.

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