Wednesday, 29 May 2013

Classical String Theory

                                Before we consider free strings let us first take a look at the description of a free massless relativistic point-particle:
\begin{displaymath}S=\int d\tau \frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}\eta_{\mu\nu}, \end{displaymath}(3.1)

where $\tau$ is the parameter along the wordline of the particle, $x^{\mu}$ maps this worldline into target space, in this case Minkowski space. Variation of this action gives us the equation of motion of a free massless particle:

\begin{displaymath}\frac{d^2 x^\mu}{d\tau^2} =0 \end{displaymath}


Strings are the one-dimensional generalisation of particles, so they sweep out a two dimensional worldsheet in space-time. The dynamics of the string is descibed by its coordinate $X^\mu$ which is parametrized by $\{\tau,\sigma\}$ where $\tau$ is the time-evolution parameter and $\sigma$ the parameter along the length of the string.
The Action describing a free bosonic string moving in flat space then becomes:
\begin{displaymath} S=\frac{1}{2\alpha'} \int d\tau d\sigma \sqrt{-h} h^{ab} \partial _a X^\mu \partial _b X^\nu \eta_{\mu\nu}, \end{displaymath}(3.2)

where $h^{ab}$ is the metric on the worldsheet. The dimensional parameter $\alpha'$ is the characteristic length of the string (or inverse string tension) which makes the integrand dimensionless.
This string action has the following symmetries: it is invariant under reparametrization of the worldsheet (ofcourse):

\begin{displaymath}\tau , \sigma \to \xi^\tau(\tau,\sigma),\xi^\sigma(\tau,\sigma) \end{displaymath}


and it is invariant under conformal rescaling of the internal metric:

\begin{displaymath}h^{ab} \to \lambda (\tau,\sigma) h^{ab}, \end{displaymath}


this is called Weyl invariance.
We can generalise this to a string moving in curved space-time. The action for a bosonic string propagating in a gravitational background then becomes:
\begin{displaymath} S = \frac{1}{2\alpha'} \int d\tau d\sigma \sqrt{-h}h^{ab}\partial ^a X^\mu \partial ^b X^\nu g_{\mu\nu}(X). \end{displaymath}(3.3)

This bosonic string is the simplest example of string theory, but it is not realy a realistic theory for the desciption of physical phenomina. For instance its spectrum of particles includes a tachyon and no fermions. But there are more complicated string theories that have more realistic features. The supersymmetric version of string theory, superstrings, for instance solves the tachyon problem. There are now many different consistent string theories known, all with different worldsheet symmetries or different worldsheet topologies. Next to that we can change the background in which the string propagates.
For instance the action of a bosonic string moving in a general background is described by the generalised sigma model:
\begin{displaymath} S = \frac{1}{2\alpha'} \int d\tau d\sigma \sqrt{h} \left\{ g... ...^{\mu} \partial _b X^{\nu} + \alpha' R^{(2)} \Phi(X) \right\}, \end{displaymath}(3.4)

where the backgroundfields $g_{\mu\nu}$, $B_{\mu\nu}$ and $\Phi$ are the metric, the anti-symmetric tensor field and Dilaton field respectivily. $R^{(2)}$ is the Curvature-scalar on the 2-dimensional worldsheet.
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