# String Theory

## Table of Contents

## Notation

- Greek letters, e.g. , for variables or indices of variables
- Std. leteters, e.g. , for indices of
*target space* - denotes how the Lagrangian changes wrt

## Just notes

- Gauge theories are used to obtain a "larger" space in which the symmetries (e.g. Lorentz invariance) becomes
*linear*transforms Start with action

where is some constant (mass), and is the arclength

**Nambu-Goto action:**where

with , i.e. a matrix.

- We are
*pulling back the metric of the target space*

- We are
- Symmetries
*Gauge symmetries*:Reparametrizations $δ σ

^{α}= - ζ^{α}(z, σ)Std. parametrization:

where is called the

*string length*, and is called*tension*(energy of unit of length of the string)

*Global symmetries*:- (where are the
*Killing coefficients*)

- (where are the

**Polyakov action:**where

and is a symmetric metric.

- No kinetic term implies the equations of motion are , thus the symmetries become
*algebraic*!Note

for some function , which we can solve for if we know the action, but in general it can be any , and so this is an example of a

*non-trivial Gauge symmetry*

- Symmetries of
- Global:

- No kinetic term implies the equations of motion are , thus the symmetries become