Bosonic String Theory

String Theory
All Roads Lead to String Theory (Polchinski)
Prior to the First Superstring Revolution
Early History	S-Matrix Theory Regge Trajectory
Bosonic String Theory	Worldsheet String Bosonic String Theory String Perturbation Theory Tachyon Condensation
Supersymmetric Revolution	Supersymmetry RNS Formalism GS Formalism BPS
Superstring Revolutions
First Superstring Revolution	GSO Projection Type II String Theory Type IIB String Theory Type IIA String Theory Type I String Theory Type H String Theory Type HO String Theory Type HE String Theory
Second Superstring Revolution	T-Duality D-Brane S-Duality Horava-Witten String Theory M-Theory Holographic Principle N=4 Super-Yang-Mills Theory AdS CFT BFSS Matrix Theory Matrix String Theory (2,0) Theory Twistor String Theory F-Theory String Field Theory Pure Spinor Formalism
After the Revolutions
Phenomenology	String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology

Bosonic String Theory is the original version of String Theory. It is completely determined by the Polyakov Action, and only includes bosons among it's mass spectrum. It also contains a tachyon in it's mass spectrum as the ground state.

Nambu-Goto Action

The Nambu-Goto Action is the action principle generally associated with Classical Bosonic String Theory. However, in principle, it is possible to use it for the fully quantised theory as well, but it is much harder. The Nambu-Goto Action can easily be obtained by looking at a Worldsheet suspended in spacetime. It can be stated as:

$S=-T \int \mbox{d}^2\sigma \sqrt{-\det \gamma_{\alpha\beta} }$

Where $\gamma_{\alpha\beta}$ is the induced metric on the worldsheet when it is embedded in the background spacetime. This has an obvious geometrical meaning. However, due to the presence of the square root, this action is unsuitable for quantisation. For clarity, observe that under a suitable Diffeomorphism, i.e. reparameterisation, / \

$S=-T\int\mbox{d}^2\sigma\sqrt{\left(\dot X X'\right)^2 - \dot X^2 X'^2 }$

Which is clearly hard to quantise.

Polyakov Action

The Polyakov Action is an easier-to-quantise action for the Bosonic String Theory. It can be stated as:

Polyakov Action $S=-\frac{T}2 \int \mbox{d}^{2}\sigma \sqrt{-\det h_{ab}} h^{ab}\partial_aX^\mu\partial_bX^\nu g_{\mu\nu}$

This Polyakov Action is not so appealing geometrically, but it is easier to quantise. It can be shown that this is classically equivalent to the Nambu-Goto Action as follows (Please click here to view properly):

Proof that the Polyakov Action classically reduces to the Nambu-Goto Action

We start with the Polyakov Action: $S=-\frac{T}2 \int \mbox{d}^{2}\sigma \sqrt{-\det h_{ab}} h^{ab}\partial_aX^\mu\partial_bX^\nu g_{\mu\nu}$ We use the Euler-Lagrange Equation: $0 = \frac{\delta S}{\delta h^{ab}} = T_{ab}$ $0 = \frac{\delta S}{\delta h^{ab}} = \frac{\tau}{2} \sqrt{-h} \left( \gamma_{ab} - \frac12 h_{ab} h^{cd} \gamma_{cd} \right)$ Where $\tau$ is the trace of the Stress-Energy-Momentum Tensor and $\gamma_{ab}$ is the induced metric on the worldsheet. $0 = T_{ab} = T \left( \gamma _{ab} - \frac12 h_{ab} h^{cd} G_{cd} \right)$ Also recall that: $G_{ab} = \frac12 h_{ab} h^{cd} \gamma _{cd}$ $G = \mathrm{det} \left( \gamma_{ab} \right) = \frac14 h \left( h^{cd} \gamma _{cd} \right)^2$ From this we see that, if the intrinsic worldsheet metric $h_{ab}$ satisfies the following field equation: $\sqrt{-h} = \frac{2 \sqrt{-\gamma }}{h^{cd} \gamma _{cd}}$ Then, the Polyakov Action becomes the Nambu-Goto Action: $S = \frac{T}2 \int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} \partial_a X^\mu\partial_b X^\nu g _{\mu\nu} = \frac{T}2 \int \mathrm{d}^2 \sigma \sqrt{-h} h^{ab} \gamma _{ab} = {T \over 2}\int \operatorname{d}^2 \sigma \frac{2 \sqrt{-\gamma }}{h^{cd} \gamma _{cd}} h^{ab} \gamma _{ab} = T \int \mathrm{d}^2 \sigma \sqrt{-\gamma }$

However, as in the Polyakov Action, there is no square root over the $X$ ' s, it is much easier to quantise compared to the Nambu-Goto Action.

Virasoro Algebra

The L-operator in Bosonic String Theory is given by:

${{{\hat{L}}}_{m}}=\frac{1}{2}\sum\limits_{n=-\infty }^{\infty }{{{{\hat{\alpha }}}_{m-n}}\cdot {{{\hat{\alpha }}}_{n}}}$

When $m = 0$, this results in $\hat L_0=\hat H$,.

The L-operators form an an algebra, known as the Virasoro Algebra. This is actually a quantisation of the Witt Algebra (also known as the Classical Virasoro Algebra, and is isomorphic to the Conformal Algebra) through a central extension.

The multiplication; of this algebra is given, by:

$\left[ {{{\hat{L}}}_{m}},{{{\hat{L}}}_{n}} \right]=\left( m-n \right){{{\hat{L}}}_{m+n}}+\frac{D}{12}m(m-1)(m+1){{\delta }_{m,-n}}$

This algebra is actually useful in removing the negative norm states, as we will see.

This algebra also requires the Virasoro Constraints, which are stated simply as:

$\left. \begin{matrix} {{{\hat{L}}}_{0}}\left| \psi \right\rangle =a\left| \psi \right\rangle \\ {{{\hat{L}}}_{m>0}}\left| \psi \right\rangle =0 \\ \end{matrix} \right\}$

Imposing the Virasoro Constrainsts

$\begin{align} & 0={{{\hat{L}}}_{1}}\left| \Phi \right\rangle \\ & \text{ }={{{\hat{L}}}_{1}}{{{\hat{L}}}_{-1}}\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left[ {{{\hat{L}}}_{-1}},{{{\hat{L}}}_{1}} \right]\left| {{\chi }_{1}} \right\rangle +{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{1}}\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left[ {{{\hat{L}}}_{-1}},{{{\hat{L}}}_{1}} \right]\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=2{{{\hat{L}}}_{0}}\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=2{{c}_{0}}\left( a-1 \right)\left| {{\chi }_{1}} \right\rangle \\ \end{align}$

Thus, $a = 1$/.

$\begin{align} & \left[ {{{\hat{L}}}_{1}},{{{\hat{L}}}_{-2}}+k{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right]\left| \psi \right\rangle =\left( 3{{{\hat{L}}}_{-1}}+2k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}}+2k{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{0}} \right)\left| \psi \right\rangle \text{ } \\ & \text{ }=\left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}}\text{ }\left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}}\text{ } \\ & 0={{{\hat{L}}}_{1}}\left| \psi \right\rangle ={{{\hat{L}}}_{1}}\left( {{{\hat{L}}}_{-2}}+k{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right)\left| {{\chi }_{1}} \right\rangle =\left( \left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{0}}{{{\hat{L}}}_{-1}} \right)\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left( \left( 3-2k \right){{{\hat{L}}}_{-1}}+4k{{{\hat{L}}}_{-1}}\left( {{{\hat{L}}}_{0}}+1 \right) \right)\left| {{\chi }_{1}} \right\rangle \\ & \text{ }=\left( 3-2k \right){{{\hat{L}}}_{-1}}\left| {{\chi }_{1}} \right\rangle \\ & 2k=3 \\ & k=\frac{3}{2} \\ \end{align}$

$\begin{align} & {{{\hat{L}}}_{2}}\left| \Phi \right\rangle =0 \\ & {{{\hat{L}}}_{2}}\left( {{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right)\left| {{\chi }_{2}} \right\rangle =0 \\ & \left[ {{{\hat{L}}}_{2}},{{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right]\left| {{\chi }_{2}} \right\rangle +\left( {{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right){{{\hat{L}}}_{2}}\left| {{\chi }_{2}} \right\rangle =0 \\ & \left[ {{{\hat{L}}}_{2}},{{{\hat{L}}}_{-2}}+\frac{3}{2}{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}} \right]\left| {{\chi }_{2}} \right\rangle =0 \\ & \left( 13{{{\hat{L}}}_{0}}+9{{{\hat{L}}}_{-1}}{{{\hat{L}}}_{-1}}+\frac{D}{2} \right)\left| {{\chi }_{2}} \right\rangle =0 \\ & \frac{D}{2}=13 \\ & D=26 \\ \end{align}$

Therefore, $D = 26$/. -

Mass Spectrum

In the Classical Bosonic String Theory, the mass spectrum can be given by $m=\sqrt N$; however, in the quantised Bosonic String Theory, it is given by:

$m=\sqrt{N-a}$

Since $a = 1$,

$m = \sqrt{N-1}$ .

The first equation is still true in superstring theories, though the second is not, since $a$ is different. Note that this is only for open bosonic strings. For closed bosonic strings, it becomes:

$m= \sqrt{N+\tilde N-a-\tilde a} =\sqrt{N+\tilde N-2 }$ p

Where we took $a = \tilde a = 1$ because both the left- (Without the tilde) and right- (with the tilde) moving sectors are from the Bosonic String Theory.