Perturbative String and Superstring Theories

In this section we overview some basic aspects of bosonic and fermionic strings. We focus mainly in the description of the spectrum of the theory in the light-cone gauge, the effective action, the description of spectra of the five consistent superstring theories and the perturbative Calabi-Yau compactifications (for details, precisions and further developments see for instance [3,4,5,6,7,8]). First of all consider, as usual, the action of a relativistic point particle. It is given by $S = -m \int d{\tau} \sqrt{- \dot{X}^{I}\dot{X}_{I}}$, where $X^{I}$ are $D$ functions representing the coordinates of the $(D-1,1)$-dimensional Minkowski spacetime (the target space), $\dot{X}^{I} \equiv {d X^{I} \over d \tau}$ and $m$ can be identified with the mass of the point particle. This action is proportional to the length of the world-line of the relativistic particle. In analogy with the relativistic point particle, the action describing the dynamics of a string (one-dimensional object) moving in a $(D-1,1)$-dimensional Minkowski spacetime (the target space) is proportional to the area ${\bf A}$ of the worldsheet $\Sigma$. We know from the theory of surfaces that such an area is given by ${\bf A}=\int \sqrt{-g}$, where $g = {\rm det}(g_{ab})$ is the induced metric (with signature $(-,+)$) on the worldsheet $\Sigma$. The background metric will be denoted by $\eta_{IJ}$ and $\sigma^a=(\tau,\sigma)$ with $a=0,1$ are the local coordinates on the worldsheet. $\eta_{IJ}$ and $g_{ab}$ are related by $g_{ab}= \eta_{IJ} {\partial}_a X^{I} {\partial}_b X^{J}$ with $I , J =0,1, \dots , D-1$. Thus the classical action of a relativistic string is given by the Nambu-Goto action

$$S_{NG}[X^{I}]=- T \int_{\Sigma} d\tau d\sigma \sqrt{-det({\partial}_a X^{I} {\partial}_b X^{J} \eta_{IJ})},$$ (1)

where $T= {1\over 2 \pi {\alpha}'}$ is the string tension, $X^I$ are $D$ embedding functions of the worldsheet $\Sigma$ into the target space $X$. Now introduce a metric $h$ describing the intrinsic worldsheet geometry, we get a classically equivalent action to the Nambu-Goto action. This is the Polyakov action originally proposed by Brink, di Vecchia, Howe and Zumino

$$S_P[X^I,h_{ab}] = - {1\over 4 \pi {\alpha}'} \int_{\Sigma} ... ...igma \sqrt{-h}h^{ab}\partial_a X^I \partial_b X^J \eta_{IJ},$$ (2)

where the $X^{I}$'s are $D$ scalar fields on the worldsheet. Such a fields can be interpreted as the coordinates of spacetime $X$ (target space), $h$ = det$({h}^{ab})$ and $h_{ab}={\partial}_aX^{I}{\partial}_bX^{J} \eta_{IJ}$. Polyakov action has the following symmetries: $(i)$ Poincaré invariance, $(ii)$ Worldsheet diffeomorphism invariance, and $(iii)$ Weyl invariance (rescaling invariance). The energy-momentum tensor of the two-dimensional theory is given by

$$T^{ab}:= {1 \over \sqrt{- h}} {\delta S_P \over \delta h_{a... ...h}^{ab} h^{cd} {\partial}_c X^{I} {\partial}_d X_{I} \bigg).$$ (3)

Invariance under worldsheet diffeomorphisms implies that it should be conserved i.e. ${\nabla}_aT^{ab}=0$, while the Weyl invariance gives the traceless condition, $T^a_a=0$. The equation of motion associated with Polyakov action is given by

$$\partial_a \bigg( \sqrt{-h} h^{ab} \partial_b X^{I} \bigg) = 0.$$ (4)

Whose solutions should satisfy the boundary conditions for the open string: ${\partial}_{\sigma}X^{I} {\mid}^{\ell=\pi}_0=0$ (Neumann) and for the closed string: $X^{I} (\tau , \sigma )=X^{I}(\tau , \sigma + 2 \pi)$ (Dirichlet). Here $\ell=\pi$ is the characteristic length of the open string. The variation of $S_P$ with respect to $h^{ab}$ leads to the constraint equations: $T_{ab} = 0$. From now on we will work in the conformal gauge. In this gauge: $h_{ab} = \eta_{ab}$ the equations of motion (4) reduce to the Laplace equation in the flat worldsheet whose solutions can be written as linear superposition of plane waves. truecm The Closed String For the closed string the boundary condition $X^{I} (\tau , \sigma )=X^{I}(\tau , \sigma + 2 \pi)$, leads to the general solution of Eq. (4) in the conformal gauge

$$X^{I}(\tau,\sigma) = X^{I}_0 + {1 \over \pi T} P^{I}\tau$$

$$+ {i \over 2\sqrt{\pi T}} \sum_{n \neq 0} {1\over n} \big... ...ilde{\alpha}^{I}_n exp\bigg(-i2n(\tau + \sigma )\bigg)\bigg\}$$ (5)

where $X^{I}_0$ and $P^{I}$ are the position and momentum of the center-of-mass of the string and $\alpha^{I}_n$ and $\widetilde{\alpha}^{I}_n$ satisfy the conditions ${\alpha}_n^{ I *}={\alpha}^{I}_{-n}$ (left-movers) and $\widetilde{\alpha}_n^{I *}=\widetilde{\alpha}^{I}_{-n}$ (right-movers). truecm The Open String For the open string the corresponding boundary condition is ${\partial}_{\sigma}X^{I} {\mid}^{\ell=\pi}_0=0$ (this is the only boundary condition which is Lorentz invariant) and the solution is given by

$$X^{I}(\tau , \sigma )= X^{I}_0 + {1 \over \pi T}P^{I}\tau +... ... \over n} {\alpha}^{I}_nexp \big(-in\tau\big) \cos (n\sigma )$$ (6)

with the matching condition ${\alpha}^{I}_n = \widetilde{\alpha}^{I}_{-n}.$ truecm Quantization The quantization of the closed bosonic string can be carried over, as usual, by using the Dirac prescription to the center-of-mass and oscillator variables in the form

$$[X^{I}_0,P^{J}]=i{\eta}^{IJ},$$

$$[{\alpha}^{I}_m,{\alpha}^{J}_n]= [\widetilde{\alpha}^{I}_m,\widetilde{\alpha}^{J}_n]=m{\delta}_{m+n,0}{\eta}^{IJ},$$

$$[{\alpha}^{I}_m,\widetilde{\alpha}^{J}_n]=0.$$ (7)

One can identify $({\alpha}^{I}_n,\widetilde{\alpha}^{I}_n)$ with the annihilation operators and the corresponding operators $({\alpha}^{I}_{-n},\widetilde{\alpha}^{I}_{-n})$ with the creation ones. In order to specify the physical states we first denote the center of mass state given by $\vert P^{I}\rangle$. The vacuum state is defined by ${\alpha}^{I}_m \vert,P^{I} \rangle=0$ with $m > 0$ and $P^{I}\vert,P^{I} \rangle =p^{I}\mid 0,P^{I} \rangle$ and similar for the right movings (here $\vert,P^{I} \rangle = \vert P^{I}\rangle \otimes \vert \rangle$). For the zero modes these states have negative norm (ghosts). However one can choice a suitable gauge where ghosts decouple from the Hilbert space when $D=26$. truecm Light-cone Quantization Now we turn out to work in the so called light-cone gauge. In this gauge it is possible to solve explicitly the Virasoro constraints: $T_{ab} = 0$. This is done by removing the light-cone coordinates $X^{\pm} = {1\over \sqrt{2}}(X^0\pm X^D)$ leaving only the transverse coordinates $X^i$ representing the physical degrees of freedom (with $i,j =1, 2, \dots , D-2$). In this gauge the Virasoro constraints are explicitly solved. Thus the independent variables are $(X_0^-,P^+,X^j_0,P^j, \alpha_n^j, \widetilde{\alpha}_n^j)$. Operators $\alpha_n^-$ and $\widetilde{\alpha}_n^-$ can be written in terms of $\alpha^j_n$ and $\widetilde{\alpha}^j_n$ respectively as follows: ${\alpha}^-_n={1\over \sqrt{2 \alpha '}P^+}(\sum_{m= - \infty}^{\infty} :{\alpha}^i_{n-m}{\alpha}^i_m:-2A{\delta}_n)$ and $\widetilde{\alpha}_n^- = {1\over \sqrt{2 \alpha '}P^+}(\sum_{m= - \infty}^{\infty} :\widetilde{\alpha}^i_{n-m} \widetilde{\alpha}^i_m:-2A{\delta}_n$). For the open string we get ${\alpha}^-_n={1\over 2\sqrt{2 \alpha '}P^+}(\sum_{m = -\infty}^{\infty} : {\alpha}^i_{n-m} {\alpha}^i_m:-2A{\delta}_n)$. Here $: \cdot :$ stands for the normal ordering and $A$ is its associated constant. In this gauge the Hamiltonian is given by

$$H={1\over 2}(P^i)^2+N-A \ {\rm (open \ string),} \ \ \ H=(P^i)^2+ N_L + N_R-2A \ {\rm (closed \ string)}$$ (8)

where $N$ is the operator number, $N_L = \sum_{m = - \infty}^{\infty} : \alpha_{-m} \alpha_m:$, and $N_R = \sum_{m=- \infty}^{\infty}: \widetilde{\alpha}_{-m} \widetilde{\alpha}_m:.$ The mass-shell condition is given by $\alpha ' M^2= (N-A)$ (open string) and $\alpha ' M^2=2(N_L + N_R-2A)$ (closed string). For the open string, Lorentz invariance implies that the first excited state is massless and therefore $A=1$. In the light-cone gauge $A$ takes the form $A = - {D-2 \over 2} \sum_{n=1}^{\infty} n$. From the fact $\sum_{n=1}^{\infty}n^{-s}=\zeta(s),$ where $\zeta$ is the Riemann's zeta function (which converges for $s>1$ and has a unique analytic continuation at $s=-1$, where it takes the value $-{1\over 12}$) then $A=-{D-2\over 24}$ and therefore $D=26$. truecm