[String&M-Theory] 1. Introduction
This article is one of M-theory.
String Theory = Quantum Mechanics + General Relativity
In quantum field theory, field at space-time points with a space-like separation should commute (or anticommute if they are fermionic). However, we doesn’t know whether or not two space-time points have a space-like separation until the metric has been computed.
1.1 Historical origins
First develope to understand strong nuclear force in the late 1960s. However, early 1970s, QCD appeared. Now we use string theory for quantum theory of gravity.
1.2 General features
String thoey is not yet fully formulated, but there is some features of it.
Gravity
General relativity is naturally incorporated in the theory.
Yang-Mills gauge theory
Yang-Mills gauge theory appears in string theory.
Supersymmetry
Symmetry between boson and fermion is needed.
However, we didn't find supersymmetry yet..
Extra dimensions of space
The theory is only consistemt in a 1-dimensional space-time.
Other dimensions ($10-4$) are compactified on an internal manifold.
This compact dimension was first discussed by kaluza and Klein in the 1920s.
The size of strings
Quantum field theory have point particle, but string theory have length. Plank length is $$l_p=\left(\frac{\hbar G}{c^3}\right)^{1/2}=1.6\times 10^{-33}cm.$$ and Plank mass is $$m_p=\left(\frac{\hbar c}{G}\right)^{1/2}=1.2\times 10^{19} GeV/c^2.$$ These plank scale characterize compact extra dimensions.
1.3 Basic string theory
String make 2-dim world sheet like world line of particle. Feynman diagram of interaction show various topology of world sheet. There are two differences from QFT.
1. The structure of interactions is uniquely determined by the free theory.
2. Since string interactions are not associated with short-distance singularities, string theory amplitudes have no ultraviolet divergences.
World-volume actions and the critical dimension
$p$-brane is an object with $p$ spatial dimensions and tension (or energy density) $T_p$. The classical motion of a $p$-brane extremizes the $(p+1)$-dimensional volume $V$ that it sweeps out in space-time. Thus there is a $p$-brane action that is given by $S_p = −T_pV$. In the case of string, $p=1$, $V$ is the area of the string world sheet and the action is called the Nambu-Goto action(string sigmamodel action): $$S_\sigma =-\frac{T}{2}\int \sqrt{-h}h^{\alpha\beta}\eta_{\mu\nu}\partial_\alpha X^\mu \partial_\beta X^\nu d\sigma d\tau,$$ where $h_{\alpha\beta}(\sigma\tau)$ is an auxiliary world-sheet metric, $h=\det h_{\alpha\beta}$, and $h^{\alpha\beta}$ is the inverse of $h_{\alpha\beta}$. The functions $X^\mu(\sigma,\tau)$ describe the space-time embedding of the string world sheet. The Euler-Lagrange equation for $h^{\alpha\beta}$ can be used to eliminate it from the action and recover the Nambu-Goto action.
In quantum, gauge fixing make spacetime dimension become 26. (For superstring, 10.)
Closed strings and open strings
The parameter $\tau$ in the embedding functions $X^\mu(\sigma,\tau)$ is the world-sheet time coordinate and $\sigma$ parametrized the string at a given world sheet time. For a closed string, there is boundary condition $$X^\mu(\sigma,\tau)=X^\mu(\sigma+\pi,\tau).$$
For open string, they satisfy either Neumann or Dirichlet boundary conditions for each $\mu$. Neumann boundary condition can be any space-time, but Dirichlet boundary condition only exists in D-brane.
Perturbation
This useful for small dimensionless coupling constant like QED.
Superstrings
The first superstring revolution began in 1984 with the discovery that quantum mechanical consistency of a ten-dimensional theory with $N = 1$ supersymmetry requires a local Yang–Mills gauge symmetry based on one of two possible Lie algebras: $SO(32)$ or $E8\times E8.$ As is explained in Chapter 5, only for these two choices do certain quantum mechanical anomalies cancel.
There are 5 types of supertring theory.
1.4 Modern developments in superstring theory
T-duality
In the late 1980s T-duality which connect two type II theory and two heterotic theory. (IIA =IIB, $ SO(32)=E8\times E8$) This duality relate external dimensionj geometry. For example, a circle of radius $R$ is equivalent to a circle of radius $2 s /R$, where (as before) $s$ is the fundamental string length scale.
S-duality
This connect coupling constant $g_s$ to $1/g_s$. Type I theory corresponde to $SO(32)$ theory, and Type IIB itself.
D-branes
$p$-brane is important in non-perturbative situation. D-brane is one of $p$-brane which tension is proportional $1/g_s$ in type I and II theory. The Yang–Mills fields arise as the massless modes of open strings attached to the D-branes.
What is M-theory?
When $g_s$ is huge for IIA and $E8\times E8$ theory, it became 11 dimension. This called M-theory, which low energy limit is 11 dimension supergravity.
F-theory
Correspondence between type IIB and M-theory is F-theory.
Flux compactifications
Why dimension became compactified? This called modulispace problem.
Black-hole entropy
AdS/CFT duality
Holographic duality
String and M-theory cosmology
Um!
Reference
Katrin Becker, Melanie Becker, and John H. Schwarz - String theory and M-theory