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## Elements of String Theory

In this chapter we start with a brief review of string perturbation theory and quanti-zation in 10 dimensions. The starting point is the quantization of the bosonic string and, subsequently, the requirement of having spacetime fermions in the string spectrum will ne-cessitate the passage to the superstring formalism. As in the subsequent chapters, we will be interested in closed, oriented strings.

**Brief Introduction to Bosonic String Theory**

**The Polyakov Action and its Symmetries**

In this section we will give a lightning overview of the basic elements of bosonic string theory. We will adopt a rather direct line of approach and speedily introduce only the very essential elements, without making any attempt to give a complete picture in this rather vast subject. This section will partially serve to set the conventions to be used later on.

The starting point of string theory is to replace the quantum eld-theoretic notion of point particles by extended 1-dimensional objects propagating in a D-dimensional spacetime M. With the ow of proper time, these objects trace a 2d surface , commonly referred to as the (string) worldsheet, which is embedded into the target spacetime.

One chooses local coordinates f ag = f ; tg to parametrize points on the worldsheet, which are denoted as X ( ; t). Physically, one may think of t as the proper time along the worldline of each point on the string and as the spatial parameter labeling each such point. Of course, the parametrization on the worldsheet is arbitrary and ; t are not observables of the theory. This gives rise to the requirement that any physical string theory should be invariant under reparametrizations.

Consider the case of a closed string, where the two string ends are matched at the same point. As the ring-shaped string propagates in \time » t, it traces a 2d \tube-like » surface. This can be thought of as the string-analogue of a particle line in a Feynman diagram. Likewise, the matching of two incoming closed string tubes to create a single larger tube and its subsequent breaking to two outgoing string tubes is the analogue of a tree-level interaction diagram between 4 particles in quantum eld theory. However, there are two important di erences from the eld theoretical picture. The rst is that in the string case there is no interaction vertex-point : the reason being that the string is an extended object and so, the interaction is expected to spread over the 2d surface. Thus, one might a priori expect some of the UV-divergences of eld theory to be absent or at least softer in the string case. Secondly, in the string interaction no coupling constant is introduced ‘by hand’ as in the case of eld theory but, rather, the string coupling itself is dynamical in the sense that it is identi ed with the vacuum expectation value (v.e.v.) of the dilaton. In this respect string theory dynamically determines the strength of its own interactions. Continuing along the same train of thought, one nds a similar analogy between the 2d surfaces with holes and loop diagrams in eld theory.

In this formalism the spacetime coordinates X ( ) and the induced metric gab( ) on the worldsheet become themselves dynamical elds, de ning a map X from the 2d surfaces parametrized by local coordinates a, to the target spacetime : X : a 2 ! X ( ) 2 M: (2.1)

This map also de nes the induced metric gab( ) on as the pullback of the spacetime metric G (X( )):

gab( ) = G (X)@aX @bX : (2.2)

One might then consider String theory essentially as the theory of consistent quantization of elds propagating on these 2d Riemann surfaces.

Let us motivate the structure of a general action for the elds X ; gab on a 2d surface, that is compatible with invariance under di eomorphisms and which contains up to two derivatives. The positions X ( ) on the string become scalar elds from the worldsheet point of view and we may begin with a general (Euclidean) action of the form :

S = 4 1 0 Z d2 pg gab@aX @bX G (X) + iEab@aX @bX B (X) + 0 (X)R(2)( ) + : (2.3)

One identi es the antisymmetric tensor eld B with the Kalb-Ramond eld and the scalar eld with the dilaton, both of which arise naturally in the spectrum of massless string excitations. Here, we have also considered the possibility of adding a cosmological constant term . R(2) is the Ricci scalar curvature on the worldsheet and the imaginary i arises from the Euclidean rotation. Reparametrization invariance forces Eab to be an antisymmetric tensor, proportional to the Levi-Civita antisymmetric tensor density 1.

The characteristic string scale will be de ned in terms of the Regge slope parameter 0.

The connection between the string tension T , the string length (Ms) 1 and the 0 is : T = 1 ; Ms = ( 0) 1=2: (2.4)

For general curved manifolds M the above sigma model becomes non-linear and, hence, very di cult to quantize. Appart from very special cases, for example when M has the structure of a group manifold , exact quantization in the interacting case is, in general, not possible and one resorts to a low curvature expansion in ( 0=R) around a at background, with R being the characteristic curvature scale.

In this low-curvature and low-energy regime the purely stringy e ects become e ectively suppressed and one recovers the e ective eld theory description. This is a mixed blessing, however, in the sense that even though string theory does naturally reduce to the e ective eld theory description in this limit (as is desirable) but, at the same time, the novel in-herently stringy phenomena arising from the extended nature of the string (in particular phenomena related to the winding around compact dimensions) are masked and can only be probed at scales of the order of the string length 0 Ms 1. Consequently, attempts to study these interesting, inherently stringy phenomena are closely related to the problem of exact quantization in non-trivial backgrounds. This will be of major concern to us in later chapters.

The action (2.3) enjoys manifest di eomorphism and general coordinate invariance. For purposes of exact quantization we will usually consider at Minkowski backgrounds G (X) = , in which case the relevant symmetry is Poincare invariance. One may employ the reparametrization invariance in order to gauge- x the metric to covariantly at form : g^ab( ) = e2!( ) ab : (2.5)

The scale factor !( ) is the the only remaining degree of freedom of the 2d metric. In the limit = 0 one recovers the (classical) symmetry under Weyl rescalings of the metric gab( ) ! gab0( ) = e2 ( )gab( ) : (2.6)

Under this symmetry, the scale !( ) completely decouples (at least classically) from the action. In contrast to reparametrization invariance, the Weyl scaling of the metric is not a redundancy but a dynamical transformation of the metric gab and, thus, causes the distances between points to actually change. It is a highly non-trivial point about the dynamics of the theory to have an invariance also under Weyl scalings and it is a special property of 2 dimensions. The requirement that Weyl invariance is preserved at the quantum level, i.e. the requirement of cancellation of Weyl anomalies, gives rise to a set of consistency conditions on the central charge of the conformal eld theory that remains as a residual symmetry, after gauge xing. In this work we will take = 0 and, hence, restrict ourselves to ‘critical’ string constructions 3.

In general, the 2d Ricci term pgR(2) breaks Weyl invariance because pgR(2) has an explicit dependence on the scale factor ! : g^R = 2 !( ): (2.7)

However, if the scalar ( ) is constant the R(2)-term becomes a total derivative and is, then, allowed to enter the action. Actually, for a 2-dimensional surface without boundary the integral is simply the topological Gauss-Bonnet term, proportional to the Euler characteristic = 2(1 g ), with g being the genus of the surface. By separating out the constant expectation value of the dilaton, ( ) = h i + ( ), we have : h4 i Z d2 pgR(2) = h i : (2.8)

**Gauge Fixing a la Faddeev-Popov**

The path integral over the elds X ; gab naively diverges as a result of the local di eo-morphism and Weyl symmetry. One then xes the gauge by using the gauge symmetry to take the metric to conformally at form. The idea is to write the generic metric gab as the transformation of a representative g^ab : gab = T g^ab; (2.9) where T denotes the gauge transformation. Following the Фаддеев-Попов (Faddeev-Popov or simply Ф-П) procedure, we change integration variables from Dgab into DT , with T being (somewhat abstractly) the transformation parameter : Z = DX Dgab e S[X;g] = DT DX [^g] e S[X;g^] : (2.10)

The Jacobian of this change of integration variables is the Faddeev-Popov gauge invariant determinant : [^g] = @ T = det 0 1 (2.11) @ T gab P = det P :

This is because the variation of the metric with respect to an in nitesimal di eomorphism with parameter a = a and a Weyl transformation with parameter ! can be decomposed into two pieces :

gab = L gab + 2 ! gab = 2 !~ gab 2(P )ab ; (2.12)

where P is the operator : (P )ab = 1 (ra b + rb a gabrc c) ; (2.13) mapping vectors into traceless symmetric rank 2 tensors. The latter is independent of the variation in the scale factor ! and, hence, the – determinant reduces simply to the determinant of P . The latter can be represented in exponential form by integration over Grassmann ghost elds bab; ca. Putting everything together and dropping the in nite volume factor arising from the integration over the gauge parameters we nally nd : Z = DX Db Dc exp S[X; g^] Sgh[b; c; g^] ; (2.14) where now the ghost action is written : d2 pg^ babracb : Sgh[b; c; g^] = 2 Z (2.15)

The latter explicit form is obtained by using the symmetry properties of (P c)ab to force the b-ghost to be also symmetric. After dropping an irrelevant total derivative term, one arrives at the above ghost action.

The summation over all possible metrics must also take into account the various dif-ferent worldsheet topologies. One is, then, lead to the Polyakov prescription for the string perturbative expansion, which takes the form : topologies gs Z DX Db Dc e S[X;g^] Sgh[b;c;g^] i Z d2 i g^( i) Vi(ki; i) ; (2.16) where the string coupling gs is naturally identi ed with the expectation value of the dilaton : gs eh i : (2.17)

Note that the location i of the vertex operators Vi(k; ), which describe the external on-shell states of the process, is integrated over the worldsheet in order to yield a di eomorphism invariant amplitude.

The string loop expansion is then a topological expansion over Riemann surfaces of distinct topology, characterized by the number of handles, boundaries and crosscaps, all of which contribute to the Euler characteristic . The dilaton is singled out in this way, being the eld whose v.e.v. controls the strength of string interactions.

**The Weyl Anomaly**

Let us next return to the problem of examining the decoupling of the scale factor of the metric !( ) at the quantum level. Under a Weyl rescaling of the metric the path integral acquires a non-trivial Liouville factor : Z[ e2 g ] = exp 24 Z d2 pg gab @a @b + R(2) Z[g] : (2.18)

The anomaly coe cient c is identi ed with the central charge of the conformal eld theory (CFT) on a at worldsheet, which is the residual symmetry of the gauge- xed action :

T (z)T (w) = c=4 + 2 T (w) + 1 @T (w) + : : : ; (2.19)

(z w)4 (z w)2 z w

where T (z) is the holomorphic energy momentum tensor of the CFT in complex coordinates mapping the cylinder to the complex plane. The central charge is related to the conformal anomaly hT aai = 12c R(2) arising from the coupling of the 2d CFT to a curved metric. In terms of the Laurent modes Ln of the Laurent expansion, T (z) = P Ln , the above OPE

takes the form of the (in nite-dimensional) Virasoro algebra : [Ln; Lm] = (n m)Ln+m + c n(n2 1) n+m : (2.20)

Notice that fL 1; L0; L+1g close into an SL(2; R) Lie subalgebra. They generate translations L 1, dilatations (rescalings) L0 and special conformal transformations L1. Together with the right-moving subalgebra they form SL(2; C), which contains the globally de ned geberators on the Riemann sphere. They generate the Mobius (or rational) transformations :

az + b

z ! cz + d ; (2.21)

with a; b; c; d 2 C and ad bc = 1. Actually, this group is P SL(2; C) = SL(2; C)=Z2, because an overall change in sign leaves the transformation una ected.

Furthermore, one may show that the absence of gravitational anomalies (anomalies un-der di eomorphism invariance) requires the left-moving and right-moving central charges to match c = c. The absence of conformal (Weyl) anomaly then requires the total central charge c receiving contribution from all degrees of freedom, including the ghosts, to cancel : cmatter + cghost = 0 : (2.22)

When this condition is applied to a CFT of D free bosons X , it determines the (maximal) critical dimension, D = 26, of the bosonic string. The contribution to the central charge of a free boson is cX = +1, whereas a bc-ghost system of conformal weight (h; 1 h) contributes cbc = 3(2h 1)2 + 1. These may be easily derived by going to the at metric in conformal coordinates : z = e i 1+ 2 ; (2.23)

that map the cylinder to the complex plane in the ‘radial frame’ 4. There, the operator product expansion (OPE) of the energy-momentum tensor with itself, T (z)T (w), can be calculated in terms of free- eld OPEs for the CFT of a free boson X and similarly for the bc-ghosts. From the ghost action (2.15) one reads the conformal weights hb = 2 and hc = 1 hb = 1. The anomaly cancellation then yields c = D 26 = 0, from which the critical dimension of the bosonic string derives. Note that the maximal critical dimension of the bosonic string is xed solely by the reparametrization properties of the worldsheet, which determine the kernel P and, hence, the conformal structure of the ghost action (2.15).

**Mode Expansions**

It will be useful to give the mode expansion of the free scalar eld X. It is obtained by solving the Laplace equation subject to the relevant boundary conditions. The solution is decomposed into an holomorphic (left-moving) and an anti-holomorphic (right-moving) contribution, X(z; z) = XL(z) + XR(z). For the closed, oriented string, which is the case of interest in this manuscript, one imposes periodicity in . In fact, periodicity is imposed up to a possible winding term, when the scalar X is compact, i.e. taking values in S1 : X( + 2 ; t) = X( ; t) + 2 nR; (2.24) where n 2 Z is the winding quantum number, counting the number of times the string encircles the compact dimension of radius R.

In the radial frame, the Fourier expansion turns into an expansion in Laurent modes, centered at z = 0. The (left-moving) free boson propagator is not a well-de ned conformal object since its short distance behaviour is logarithmic, rather than a power-law singularity : XL(z)XL(w) = 2 log (z w) + : : : (2.25)

The holomorphic current, however, i@X(z) is a well-de ned (1; 0) conformal tensor and has a well-de ned mode expansion. By integrating the latter, one obtains :

XL(z) = x20

XR(z) = x20

i 2 PL log z + i r n=0 n zn ; (2.26)

0 X 1 n

PR log z + ir 6

i 2 20 n=0 n zn : (2.27)

The average position is x0 and, in general, the left- and right- moving momenta PL;R need not be equal, unless the dimension parametrized by X is non-compact. The total momentum P = 12 (PL + PR) satis es the usual quantization : 1 (PL + PR) = m ; (2.28) where m 2 Z is the momentum quantum number. The periodicity condition for X = XL+XR gives : X(e2 iz; e 2 iz) = X(z; z) + 2 0 (PL PR) ; (2.29) which, when compared with (2.24) yields : 1 (PL PR) = nR : (2.30)

One then obtains the quantization of the momentum zero modes PL; PR in terms of the momentum and winding numbers : PL;R = m nR : (2.31)

In contrast to the point particle result, the string has tension ( 0) 1 which tends to force the string to shrink in size (counterbalanced, in turn, by quantum uctuations). The second term in (2.31) simply expresses the fact that it costs energy for the string to wrap p around the compact dimension. In the limit of large radius R= 0 1 the Kaluza-Klein modes 1=R are so densely packed that their mass spectrum becomes continuous, whereas the winding modes R become supermassive and essentially decouple, n = 0. One then recovers PL = PR, with the momentum now describing the centre-of-mass momentum of the string propagating in a non-compact dimension. Notice that the zero modes PL;R and, hence, the whole string spectrum is invariant under the inversion R ! 0=R, together with the simultaneous exchange of momenta and windings m $ n. This is the rst encounter with T -duality, the simplest perturbative duality which actually holds at all levels in perturbation theory.

In terms of the currents J(z) = i@X(z), J(z) = i@X(z) : X Jr J(z) = r2Z z r+1 the zero modes are expressed as : p J0 = p20 R + 0

This is the normalization that will appear later on in lattice sums, when we calculate the contribution to the torus partition function of compact scalars.

### Equations of Motion for the Background Fields

The Weyl anomaly in T aa considered above was calculated under the implicit assumption that the 2d curved worldsheet metric is coupled to a free, exact CFT. Moreover, it was seen that the Gauss-Bonnet term in (2.3) containing the dilaton breaks Weyl invariance, even at the naive classical level, unless the dilaton eld is constant. For more general backgrounds one has to explicitly calculate the anomaly by using weak eld perturbation theory. The trace of the energy-momentum tensor for the sigma model (2.3) becomes renormalized 5 as : T aa = 1 ( G gab + i B Eab ) @aX @bX R(2); (2.34) where G; B and are the beta functions containing the information about the dependence of amplitudes on the scale factor. They are obtained as a perturbative expansion in 0 : G = R 1 H H + 2r r + O( 0);

1 2 H +O( 0);

B = e0

2cr 1

= + 4(r )2 4r2 R(D) + H2

12 8 12

+O( 0)2 ; (2.35)

where H = dB is the 3-form eld strength associated to the antisymmetric tensor :

More correctly, (2.35) should be considered a derivative expansion, since the dimensionless p

parameter controlling the perturbative expansion is R= 0, with R being the characteristic radius of curvature. The constant term in the beta function for the dilaton is given by the central charge de cit c, which induces curvature in the form of a cosmological constant term.

Thus, the absence of Weyl anomalies for general G ; B ; backgrounds requires the vanishing of the sigma model beta functions G = B = = 0. In fact, it is straightforward to show that if G = B = 0, the sigma model describes a CFT with central charge c = 12 =constant. This is consistent because the Bianchi identities give : @r 4(r )2 4r2 R + 12H2 = + 2r r 4H H = 0 ; (2.36) and, hence, is indeed a c-number.

The above conditions for the absence of Weyl anomaly provide a set of equations of motion for the background elds. They can be reproduced from the variation of the string frame action : Z dDx p Ge 2 R + 4(r )2 12 H2 + 3(0)1 + O( 0); (2.37) S=2 2 1 1 2 c where is the D-dimensional Newton’s constant. The full action for the background elds receives 0-corrections from higher loops in the sigma model, as well as gs-corrections from higher genuses in which the dilaton appears as exp ( ).

Let us brie y motivate the presence of the central charge de cit term 23 c in the e ective action (2.37). As mentioned above, is a constant proportional to the total central charge of the CFT on the at worldsheet and the dilaton equation of motion = 0 implies cmatter = 26. However, the constant term in is only the at contribution to the central charge, with the curved contribution arising from the operator valued O( 0) contribution. Let us be a bit more speci c. Consider the target space to be the direct product MD K, where MD is a D-dimensional 6 space of Lorentzian signature and K is an internal (compacti ed) space of dimension N. The naive concept of dimension would imply D + N = 26 and this is indeed true for the contribution to the central charges of the spacetime and internal CFTs in the at limit. We will take the spacetime and internal space to be parametrized by coordinates X and XI , respectively. Independently, the central charges of each of the two CFTs can be expressed as the anomalies of the two sigma models :

cM = 12 M = D + 32 0 4(r )2 4r2 R(D) + 12H2 (X ) + O( 0)2;

3 0 2 2 (D) 1 2 I 2

cK = 12 K=N+ 4(r ) 4r R + H (X )+O( 0) ; (2.38)

where we explicitly include the dependence on the spacetime and internal coordinates X , XI , explicitly. Since cK is the central charge of the internal CFT, we can write the curved contribution to the internal central charge as cK = cK N. The vanishing of the total central charge implies that the curved contributions cancel each other cK = cM. Then, the total beta function (including the ghost contribution) becomes :

= cK cghost

M 0 K 12 1

+

2 2 (D) 2

= + 4(r ) 4r R + H

12 8 12

(X )+O( 0)2 : (2.39)

Thus, the central charge de cit arises by coupling the curved ‘spacetime’ (geometrical) sigma model to an internal, curved CFT. This mechanism can be used, for example, to generate tree-level cosmological solutions in string theory, with the role of the internal CFT being played by a gauged WZW model, as in [2].

BRST Quantization and the No-Ghost Theorem

Before ending this section, we will very brie y mention an important ingredient that plays a major role in the construction of consistent string amplitudes. This is the BRST quantiza-tion, which leads to a formulation of the No-Ghost theorem. A more general application of the – procedure of gauge xing would be to implement the gauge- xing condition in terms of auxiliary elds (Lagrange multipliers) as follows. Imagine gauge transformations closing an algebra [ ; ] = f and a set of gauge- xing condition in the form F A(g) = 0. The gauge xing can be imposed through inserting a Dirac delta function (g g^), which has a simple exponential representation in terms of new dummy integration variables BA, now acting as Lagrange multipliers : Z DX DgabDBADbADc exp ( S[X; g] Sghost[g; b; c] Sgauge x[g; B; F ] ): (2.40)

The rst two terms in the exponent are the gauge invariant sigma model action and the – ghost action : Z Sghost = bA F A(g) c : (2.41)

The third term is the gauge- xing action : Z Sgauge x = i BAF A(g) : (2.42)

The full action now enjoys a symmetry under the following BRST transformation [3], due to Becchi, Rouet and Stora :

BRST(gab) = i c gab ;

BRST(BA) = 0 ;

BRST(bA) = BA;

BRST(c ) = 1 i f c c : (2.43)

BRST symmetry can be considered a remnant of the original symmetry that is preserved after gauge xing. Its importance arises from the observation that physical states must be BRST invariant. This can be seen demanding that physical amplitudes are una ected by a small variation in the gauge- xing condition F : 0 = h j (Sghost + Sgauge x) j 0i = h j fQB; bA F Ag j 0i ; (2.44) where the BRST charge QB is the generator of the BRST transformations. The requirement that this hold for arbitrary variation F leads to the physical state condition : QB j i = 0 : (2.45)

Next, the requirement that the BRST charge itself remains conserved even after the va-riation F or, equivalently, the requirement that QB itself commutes with the variation of the Hamiltonian, leads to its nilpotency, Q2B = 0. This has important consequences for the structure of physical states. Evidently, any state that is BRST-exact, QBj i, is also automa-tically BRST-closed, i.e. annihilated by the BRST charge, and so it is physical. However, such states are null, in the sense that they are orthogonal to all other physical states (including themselves !) and, hence, decouple from physical amplitudes.

Since physical states di ering by null states have equal amplitudes with any other physical state, one is lead to the notion of equivalence classes. The Hilbert space of physical states, therefore, will be de ned as the cohomology of QB :

Hphysical = Hclosed : (2.46)

Hexact

The No-Ghost theorem (see, for example, [4]) is precisely the statement that the ‘transverse’ Hilbert space, which does not contain any longitudinal X0; X1; b; c excitations (and, hence, has a positive-de nite inner product) is isomorphic to the BRST cohomology, Hphysical = HBRST. The BRST construction re ects another important fact. Namely, su cient gauge symmetry must be present in the rst place, if negative norm states are to decouple from string amplitudes. This is also particularly clear in light-cone quantization, where the gauge symmetry (reparametrization invariance) is used to go to the lightcone gauge and, hence, xing the longitudinal modes.

We conclude this brief introduction to bosonic string theory by giving the explicit form for the BRST current : jB(z) = cTmatter(z) + 1 : cTghost : (z) + 3 @2c(z): (2.47)

The BRST charge Q is then the zero mode : I 2 i jB(z) ; QB=I 2 i jB(z) (2.48) dz dz with the anti-holomorphic BRST current being de ned analogously. The conservation of the BRST charge is highly sensitive to anomalies in the gauge symmetries. As a result, a conformal anomaly in the theory will arise as a failure of the nilpotency of the BRST charge. In this case the jBjB-OPE will contain a non vanishing simple-pole term : jB(z)jB(w) = : : : cmatter 26 1 c@3c(w) + : : : ; (2.49) which contributes a non-vanishing value to the anticommutator fQB; QBg. Similarly, the T jB OPE contains a fourth order pole and, thus, jB(z) is not a primary eld unless cmatter = 26.

**Bosonic String Spectrum**

In section 2.1 we gave a speedy overview of some elements of bosonic string theory. Let us have a cursory look at its massless spectrum. The vertex operators of the closed bosonic string take the general form : gs dzdz V(z; z) ; (2.50)

and can, subsequently, be used to perturb the sigma model. In this way, the various exci-tations of the string act as perturbations whose coherent contribution ‘builds up’ the back-ground. In this sense, string theory is in principle expected to determine dynamically its own background, though in practice one is usually forced to treat small perturbations around some xed setup. The unintegrated vertex operators V(z; z), are elds of de nite conformal weight (1; 1) so that the integrated insertions to the path integral are conformally invariant. Consider the theory in D = 26 at dimensions. The lowest mass states are then : : eik X : (0; 0)j0i : (2.51)

Their conformal weight is picked from the double pole in the T (z)V(w; w) OPE :

T (z)V(w; w) = : : : + h @V(w; w) + 1 V(w; w) + : : : ; (2.52)

(z w)2 z w

where the free boson energy-momentum tensor is : T (z) = 1 : @X @X : (z) : (2.53)

A straightforward calculation yields : 0k2 0k2 (h; h) = 4 ; 4 : (2.54)

Throughout this manuscript we will adopt a standard ‘CFT convention’. Unless otherwise stated, in CFT calculations involving vertex operators on the sphere (genus 0), we will set 0 = 2 so that the chiral vertex operator eiqX(z), carrying a de nite U(1) charge q under J(z) = i@X(z), will have conformal weight (q2=2; 0).

With this convention, the mass of eik X is found to be : m2 mL2 + mR2 = k2 = 2 : (2.55)

The separation into (equal) left- and right- moving masses m2L = m2R will be seen to arise na-turally in the torus partition function. There, the value m2L = m2R = 1 will be precisely the mass level of the ground state, which is entirely determined by the (super-)reparametrization properties of the worldsheet theory, as they are encoded in the (super-)ghost structure.

The mass square in (2.55) is negative and the ground state of the bosonic string is, hence, tachyonic. This is a considerable embarrassment for the theory, because its presence signals an IR instability, with the tachyon rolling down its potential away from the unstable point. We will come back to this point later on, when we discuss the Hagedorn problem.

We next move to the rst excited states, which can be constructed from linear combina-tions of the vertex operators : @X e ik X (0; 0)j0i : (2.56)

In what follows we will suppress the normal ordering symbol. The conformal weight of these operators 7 is (1 + k22 ; 1 + k22 ) and, hence, leads to massless states. Representation-wise, these states are tensor products of two vector representations of the SO(1; D 1) Lorentz group. Decomposition into irreducible representations is straightforward and leads to a traceless, symmetric tensor of spin 2 (the graviton G ), an antisymmetric tensor (the Kalb-Ramond eld B ), and the trace (the scalar dilaton ). This is expected from the beginning, since the massless excitations can be used as sources in the sigma model, with their uctuations determining the background.

Higher mass levels can be constructed in a similar fashion.

#### Superstring Theory

**Motivation and Worldsheet Fermions**

Bosonic string theory su ers from several de ciencies, one of which is the presence of the tachyonic excitation. Within the framework of a string eld theory, one should be able to trace the roll-down of the tachyon to a new stable vacuum, provided the latter exists. However, to date, it remains unclear whether bosonic string theory can exist as a theory in its own right. A far more serious problem is the total absence of spacetime fermions in the bosonic string spectrum. Clearly, states built out of oscillators transform under tensor (rather than spinor) representations of the Poincare group. We present here the conventional approach to remedy this situation, in terms of the Ramond-Neuveu-Schwarz (RNS) formulation of the superstring.

In the RNS formulation, spacetime fermions arise from a degenerate ground state, j0i , carrying a spinorial index of the spacetime little group. Assuming the presence of such an operator (z), its action on the vacuum state should produce the Dirac gamma matrix representation : (z)j0i = p2 ; z ! 0 : (2.57)

The degeneracy can only arise as long as the eld (z) has zero modes. It is also necessary for (z) to carry a spacetime vector index, in order to reproduce the gamma matrix structure.

Then, the antisymmetrized action with a second operator (z) must reproduce the Lorentz algebra : (z)j0i = [ ; ] j0i = j0i ; z ! 0 : (2.58)

We assume the singular part has been subtracted via some normal ordering scheme. The point now is that the antisymmetric combination of zero modes in the l.h.s. vanishes identically unless the elds (z) are fermionic and, hence, classically anticommuting. Therefore, one is lead naturally to the introduction of fermion elds (z) (necessarily carrying a Lorentz index) on the worldsheet. The idea is then that if worldsheet fermions can satisfy periodic boundary conditions, ( + 2 ; t) = + ( ; t), their expansion will contain zero modes which can then give rise to the vacuum degeneracy structure required for spacetime fermions.

Therefore, the rst ingredient we should include in a generalization of the bosonic sigma model action (2.3) is the addition of worldsheet fermions . One could also add ‘internal’ worldsheet fermions I , transforming according to some isometry of the internal space, but their zero-mode oscillators will give rise to spinors with respect to the internal group, rather than spacetime fermions. However, one immediately has to face an additional complication. The introduction of the new objects carrying a Lorentz index also leads to new states with negative norm. We, therefore, need to introduce a new gauge symmetry that will permit us to gauge away and decouple the new unphysical states, with a similar BRST cohomology construction as the one used for the bosonic string.

It turns out that the relevant new symmetry is super-reparametrization invariance, arising from the consistent – gauge xing of 2d worldsheet supergravity. The starting point is the

N = 1 supergravity action in 2d, coupled to D super elds fX ; g :

S = 4 0 Z d2 pg gab@aX @bX + a@a a b a @bX 4 b : (2.59)

For simplicity we assume a at target space background. The auxiliary eld of the o – shell scalar multiplet are eliminated by their equations of motion. The would-be kinetic term a abcrb c of the worldsheet gravitino a vanishes identically in two dimensions and only the ‘gamma-traceless’ (helicity 32 ) component of the gravitino, b a b, appears in the action. The spin connection does not enter explicitly in the derivative of the Majorana fermions , because of the Majorana spin- ip property.

**Table of contents :**

**1 Introduction **

**2 Elements of String Theory **

2.1 Brief Introduction to Bosonic String Theory

2.2 Bosonic String Spectrum

2.3 Superstring Theory

2.4 Superstring Vertex Operators

2.5 Vacuum Amplitude on the Torus

2.6 The Heterotic Superstring

2.7 The Type I Superstring

**3 Compactication **

3.1 Toroidal Compactication

3.2 T-Duality

3.3 Orbifold Compactication

3.4 Tensor Products of Free CFTs : Fermionic Models

3.5 Gepner Models

**4 Spacetime vs Worldsheet Supersymmetry **

4.1 Conditions for N4 = 1 Supersymmetry

4.2 N4 = 2 and N4 = 4 Spacetime Supersymmetry

**5 Supersymmetry Breaking **

5.1 Explicit Breaking

5.2 Spontaneous Breaking

5.3 Scherk-Schwarz Breaking

**6 MSDS Constructions : Algebra & Vacuum Classication **

6.1 High Curvature and High Temperature Phases of String Theory

6.2 Maximally Symmetric MSDS Vacua

6.3 ZN2 -Orbifolds with MSDS Structure

**7 Deformations and Thermal Interpretation of MSDS Vacua **

7.1 Finite Temperature

7.2 Hagedorn Instability in String Theory

7.3 Resolution of the Hagedorn Instability

7.4 Deformations and Stability

7.5 Tachyon-Free MSDS Orbifold Constructions

**8 Non-Singular String Cosmology **

8.1 `Big Bang’ Cosmology

8.2 String Cosmology

8.3 Hybrid Models in 2d

8.4 Non-Singular Hybrid Cosmology

**9 Emergent MSDS algebras & Spinor-Vector Duality **

9.1 Spinor-Vector Duality Map

9.2 Spinor-Vector Duality from Twisted N = 4 SCFT Spectral-Flow

A Theta Functions and Lattice Identities

A.1 Theta Functions

A.2 Double Poisson Resummation

A.3 OPEs involving SO(N) Spin-Fields

B Equivalence between Fermionic Constructions and ZN2 -type Orbifolds

C Unfolding the Fundamental Domain

C.1 Decomposition into Modular Orbits

D The `Abstrusa’ Identity of Jacobi

D.1 A `direct’ proof