## FANDOM

118 Pages

String Theory
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
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

A String is a 1-Dimensional weakly-coupled object in String Theory. In some old, canonical String Theoryies, it is the Fundamental "F1" String. In M-Theory, however, the M2-Brane and M5-Branes are fundamental.

## Boundary Conditions on the StringEdit this section

### In Bosonic String TheoryEdit this section

#### Closed Boundary ConditionsEdit this section

$X\left(\sigma,\tau\right)=X\left(\sigma+\ell_s,\tau\right)$

#### Newmann Boundary ConditionsEdit this section

Newmann Boundary conditions are as follows.

${{\left. \left( \frac{\partial X}{\partial \sigma } \right) \right|}_{\sigma =0}}={{\left. \left( \frac{\partial X}{\partial \sigma } \right) \right|}_{\sigma ={{\ell }_{s}}}}$

Newmann Boundary Conditions basically state that the String becomes orthogonal to the walls at it's endpoints. When the walls are drawn vertically, the ends of the String appear to look horizontal.

The String may, however, transverse across these fake walls. It is thus, obviously, not fixed.

(Note: Under T-Duality, Newmann Boundary Conditions become Dirchilet Boundary Conditions, which are discussed in the section below.)

#### Dirchilet Boundary ConditionsEdit this section

Dirchilet Boundary Conditions are as follows.

$X\left( 0,\tau \right)={{X}_{0}};X\left( {{\ell }_{s}},\tau \right)={{X}_{{{\ell }_{s}}}}$

Dirchilat Boundary Conditions basically state that the String is fixed onto walls ("D-Branes". D = Dirchilet).

The String may, however, make any angle with it's walls.

(Note: Under T-Duality, Dirchilet Boundary Conditions become Newmann Boundary Conditions, which are discussed in the section above.)

### In Superstring Theory (the RNS Formalism)Edit this section

In the RNS Formalism, in addition to the above boundary conditions, there are also boundary conditions on the fermionic fields. The Boundary Conditions for the fermionic field actually divide the entire Formalism of RNS into two different sectors.

#### Ramond Boundary ConditionsEdit this section

The Ramond Boundary Condition is as follows:

$\psi_+\left(\ell_s,\tau\right)=\psi_- \left(\ell_s,\tau\right)$

I.e. Periodic.

#### Neveu-Schwarz Boundary ConditionsEdit this section

The Neveu-Schwarz Boundary Condition is as follows:

$\psi_+\left(\ell_s,\tau\right)=-\psi_- \left(\ell_s,\tau\right)$

I.e. Anti-Periodic.

## Field Equations for the StringEdit this section

### The Bosonic StringEdit this section

One may set the variation of the Polyakov Action to 0, obtaining the Field Equation. And then, one may apply the relevant boundary conditions to obtain the explicit form of the Equations of Motion.

The Field Equation is:

$\frac{\partial^2X}{\partial\sigma^+\partial\sigma^-}=0$

#### Closed StringEdit this section

The Mode Expansion (Explicit Equation of Motion) is:

#### Open String with Newmann Boundary ConditionsEdit this section

The Mode Expansion (Explicit Equation of Motion) is:

#### Open String with Dirchilet Boundary ConditionsEdit this section

The Mode Expansion (Explicit Equation of Motion) is:

### The (RNS) SuperstringEdit this section

In addition to the previously discussed mode expansions and field equations (the ones for the Bosonic String, in the RNS Formalism, there are further more, field equations and mode expansions for the fermionic field $\psi$. The Field Equation is:

\left[ \begin{matrix} {{\partial }_{+}} & {{\partial }_{-}} \\ \end{matrix} \right]\left[ \begin{align} & {{\psi }_{-}} \\ & {{\psi }_{+}} \\ \end{align} \right]=0

#### Ramond Boundary ConditionsEdit this section

The mode expansion for these can be shown to be:

{{\psi }^{\mu }}=\frac{1}{\sqrt{2}}\sum\limits_{n\in \mathbb{Z}}^{{}}{d_{n}^{\mu }\left[ \begin{align} & {{e}^{-in{{\sigma }^{-}}}} \\ & {{e}^{-in{{\sigma }^{+}}}} \\ \end{align} \right]}

#### Neveu-Schwarz Boundary ConditionsEdit this section

The mode expansion for these can be shown to be:

{{\psi }^{\mu }}=\frac{1}{\sqrt{2}}\sum\limits_{n\in \mathbb{Z}+\frac12 }^{{}}{d_{n}^{\mu }\left[ \begin{align} & {{e}^{-in{{\sigma }^{-}}}} \\ & {{e}^{-in{{\sigma }^{+}}}} \\ \end{align} \right]}