String Theory

Notation

The metric has signature ( + , − , − , − ,...).

Greek letters from the middle of the alphabet (μ,ν,...) denote coordinates in spacetime. Roman letters taken from the beginning of the alphabet (a,b,...) denote coordinates on the worldsheet.

The metric is denoted g_μν. In the case where there is a metric of spacetime and a metric of the worldsheet, the worldsheet metric is denoted h_ab

A lower-case d is the number of dimensions. An upper-class D represents the number of spacial dimensions.

Introduction

String Theory as a Theory of Everything

In the past couple of decades, only recently were alternative thoughts of past, present, and future dimension thought to exist. But as more relative physical and particle data are being discovered, more and more theories of extra dimensions, in which quantum mechanists believe alternatives live in.

The History of String Theory

The Second String Theory Revolution

In the early ninties when Edward Witten unified the five String Theories and Super Gravity into the framework of M-Theory.

Supersymmetry

This chapter on supersymmetry intends to present it WITHOUT the use of Grassmann variables, preferring to use instead the formalism of Z₂ grading.

A Z₂-graded vector space is a vector space together with an assignment of an even (bosoninc) (corresponding to 0 of Z₂) and an odd (fermionic) (corresponding to 1) subspace such that the vector space is the direct sum of the even and odd subspaces.

An even vector is an element of the even subspace and an odd vector is an element of the odd subspace. A pure vector is either an even or an odd vector. Any vector can be decomposed uniquely as the sum of an even and an odd vector.

The tensor product of two Z₂-graded vector spaces is another Z₂-graded vector space.

In fact, in this book, we will take the stronger point of view that it makes no physical sense to add even and odd vectors together. From this point of view, we might as well view a Z₂-graded vector space as an ordered pair <V₀,V₁> where V₀ is the even space and V₁ is the odd space.

Similarly, a Z₂-graded algebra is an algebra A with a direct sum decomposition into an even and an odd part such that the product of two pure elements obeys the Z₂ relations. Alternatively, we can think of it as <A₀,A₁>.

A Lie superalgebra is a Z₂-graded algebra whose product [·, ·], called the Lie superbracket or supercommutator, satisfies

[x,y] = − ( − 1) ^{| x | | y |} [y,x]

and

( − 1) ^{| z | | x |} [x,[y,z]] + ( − 1) ^{| x | | y |} [y,[z,x]] + ( − 1) ^{| y | | z |} [z,[x,y]] = 0

where x, y, and z are pure in the Z₂-grading. Here, |x| denotes the degree of x (either 0 or 1).

Lie superalgebras are a natural generalization of normal Lie algebras to include a Z₂-grading. Indeed, the above conditions on the superbracket are exactly those on the normal Lie bracket with modifications made for the grading. The last condition is sometimes called the super Jacobi identity.

Note that the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the funny signs disappear, and the superbracket becomes a normal Lie bracket.

One way of thinking about a Lie superalgebra -- it's not the most symmetric way of looking at it -- is to consider its even and odd parts, L₀ and L₁ separately. Then, L₀ is a Lie algebra, L₁ is a linear rep of L₀, and there exists a symmetric L₀-intertwiner $\{.,.\}:L_1\otimes L_1\rightarrow L_0$ such that for all x,y and z in L₁,

$\left\{x, y\right\}[z]+\left\{y, z\right\}[x]+\left\{z, x\right\}[y]=0$

A supermanifold is a concept in noncommutative geometry. Recall that in noncommutative geometry, we don't look at point set spaces but instead, the algebra of functions over them. If M is a (differential) manifold and H is an (smooth) algebra bundle over M with a Grassmann algebra as the fiber, then the space of (smooth) sections of M forms a supercommutative algebra under pointwise multiplication. We say that this algebra defines the supermanifold (which isn't a point set space).

If M is a real manifold and we define an involution * over the fiber turning it into a * algebra, then the resulting algebra would define a real supermanifold.

Two-Dimensional Conformal Field Theory

Conformal Transformations

The Conformal Group

The story of string theory begins with two-dimensional conformal invariance.

Conformal transformations on a manifold preserve angles at every point, an example of such a transformation being the Mercator projection of the Earth onto an infinite cylinder. They may be defined as transformations that leave the metric invariant up to a scale.

$\underline{g}_{\mu \nu}(\underline{x}^\xi) = \Lambda(x^\xi) g_{\mu \nu}(x^\xi)$

The set of invertable conformal transformations form a group. This is the conformal group.

Let us apply this rule to a two dimensional manifold.

For this transformation to be conformal the metrics must be proportional to one another, which means,

Writing out the components, the following conditions emerge:

$\left(\frac{\partial z^0}{\partial \underline{z}^0}\right)^2 + \left(\frac{\partial z^0}{\partial \underline{z}^1}\right)^2 = \left(\frac{\partial z^1}{\partial \underline{z}^0}\right)^2 + \left(\frac{\partial z^1}{\partial \underline{z}^1}\right)^2$

$\frac{\partial \underline{x}^0}{\partial x^0}\frac{\partial \underline{x}^1}{\partial x^0}+\frac{\partial \underline{x}^0}{\partial x^1}\frac{\partial \underline{x}^1}{\partial x^1}$

These conditions turn out to be equivalent to the Cauchy-Riemann conditions for either holomorphic or antiholomorphic functions!

$\frac{\partial \underline{x}^1}{\partial x^0} = \frac{\partial \underline{x}^0}{\partial x^1}$ and $\frac{\partial \underline{x}^0}{\partial x^0} = -\frac{\partial \underline{x}^1}{\partial x^1}$ (holomorphic)

$\frac{\partial \underline{x}^1}{\partial x^0} = -\frac{\partial \underline{x}^0}{\partial x^1}$ and $\frac{\partial \underline{x}^0}{\partial x^0} = \frac{\partial \underline{x}^1}{\partial x^1}$ (antiholomorphic)

In two dimensions, therefore, the conformal group is the set of all invertable holomorphic maps, which is isomorphic to the set of all antiholomorphic maps. For this reason it is convenient to use complex coordinates when discussing two-dimensional conformal fields.

The set of all reversible holomorphic functions is the set of fractional linear transformations

$f(z) = \frac{\heartsuit z - \clubsuit }{ \spadesuit z + \diamondsuit}$

where

$\heartsuit \diamondsuit - \clubsuit \spadesuit = 1$

It is easily verified by composing two such functions that their composition is equivalent to matrix multiplication for matrices of the form

$\begin{pmatrix} \heartsuit & \clubsuit \\ \spadesuit & \diamondsuit \end{pmatrix}$

It is clear that the conformal group in two dimensions is equivalent to the group of complex invertible $2\times2$ matrices having a determinate of 1. This group is also known as $SL(2,\mathbb{C})$.

Classical Strings

The Classical String

Let us embed an action that is conformally invariant in two dimensions into a higher dimensional space. We will find that such an action generalizes the concept of the point particle.

Two-Dimensional Quantum Field Theory

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