Introduction to string theory

Quantum field theory is built on the idea that the various particles in nature can be modeled by mathematical point-like particles. String theory, on the other hand, extends this idea further by considering a one dimensional object, a string, as the fundamental object of the theory. What we experience as particles in nature are just the result of oscillations of these strings.

If strings are the fundamental objects describing string theory, a more extensive study of these objects is required. Let us start with a string embedded in a curved

d-dimensional target spacetime (x0_{, ..., x}d−1_{), described by a metric g}

µν. The string

worldsheet, the area that the string sweeps out as it moves in time, can be parametrised by two coordinates (τ, σ), where τ is a time coordinate and σ is a space coordinate. In order to study the kinematics of the string, as moving in the target spacetime, an action should be defined whose variational principle should

minimise the area of the string worldsheet1_{. The action is given by} S[x] = −T Z dA = −T Z dτ dσp−det(h_ab), (3.1)

where dA is the infinitesimal area element of the worldsheet, hab is the induced

metric which describes the “pullback” of the background metric on the string worldsheet and is defined as

h_ab = gµν∂axµ(τ, σ)∂bxν(τ, σ). (3.2)

The functions xµ_{(τ, σ) describe the embedding of the string in the target spacetime}

and T is the string tension. The string tension has dimension of mass per unit length and is related to the string length l_s and the Regge slope parameter α0 through T = 1 2πα0 = 1 2πl2 s . (3.3)

The action (3.1) is called the Nambu-Goto action and it describes the relativistic string. The presence of the square root in the action makes its quantisation difficult. Polyakov suggested, as a solution to the square root problem, an alternative action that is equivalent to (3.1), at a classical level, and is given by

S[x] = −T

2 Z

dτ dσp−det(γab)gµνγab∂axµ∂bxν. (3.4)

The Polyakov action introduces a new auxiliary field, the symmetric tensor γab_,

which has a physical interpretation as the worldsheet metric. It is natural that (3.4) should be supplemented with some constraints, if is to be equivalent to (3.1), coming from the equation of motion of the auxiliary tensor γab_{. These constraints}

are Tab= 02 and reflect the presence of two local symmetries of the worldsheet

action, the reparametrisation invariance and the Weyl invariance

Reparametrisation : (τ, σ) → (τ (τ0, σ0), σ(τ0, σ0)), (3.5) Weyl invariance : γab→ e2ρ(τ,σ)γab, (3.6)

1_{Greek letters µ, ν, · · · will be used for labeling target spacetime and Latin letters a, b, · · · will be} used for labeling worldsheet coordinates.

2_T

where ρ(τ, σ) is an arbitrary function. The existence of these symmetries allow us to completely gauge fix the action in the case of a string. Convenient choices of gauge can simplify the problem e.g conformal gauge.

The next natural step to be taken, is finding the equation of motion of the string by varying the Polyakov action with respect to xµ_{(τ, σ). The variation of the action}

generates some boundary terms which can only be eliminated by imposing

appropriate boundary conditions to the string endpoints. The boundary condition choice is not unique and can result in two type of strings, open and closed strings. One possible choice is

xµ(τ, 0) = xµ(τ, π) and dxµ

dσ (τ, 0) = dxµ

dσ (τ, π), (3.7)

which describes periodic boundary conditions and it corresponds to a closed string3_.

Alternative choices of boundary conditions (b.c.), which still satisfy δS = 0, can be either dxµ dσ (τ, σ) ¯ ¯ ¯ ¯ σ=0,π = 0 (Neumann b.c.) (3.8) or dxµ dτ (τ, σ) ¯ ¯ ¯ ¯ σ=0,π = 0 (Dirichlet b.c.). (3.9) The boundary conditions (3.8), (3.9) apply for the open string case. If a string obeys Neumann boundary conditions for all µ, then Poincare invariance is

preserved. If at least one of the d directions of the string obeys Dirichlet boundary conditions, then the string endpoints do not oscillate in these directions and therefore the string endpoints are fixed. Note that Dirichlet boundary conditions break Poincare invariance. Finally, the solution to the equation of motion of

xµ(τ, σ) can be found for both open and closed strings. These solutions include terms describing the center of mass position of the string xµ₀, momentum of the string pµ₀ and oscillation of the string4_.

3_{We have assumed that the closed and open string space coordinate σ can acquire values in the} interval 0 ≤ σ ≤ π.

4_{The oscillation of the string is described by a sum of the different oscillation modes which are} described by the coefficients αµ

Sector Boson/Fermion Massless fields NS-NS Boson gµν, Bµν,Φ

NS-R Fermion Ψµ,λ

R-NS Fermion Ψ0 µ,λ0

R-R Boson Ramond Ramond fields

Table 3.1: Closed string spectrum in IIB superstring theory [29]

Sector Boson/Fermion Massless fields NS Boson A_µ(x)

R Fermion Spinor

Table 3.2: Open string spectrum in superstring theory

The picture of the string given until now is purely classical. If we wish to describe a quantum field theory on the worldsheet, then the classical theory should be

quantised. The bosonic string can be quantised by following the standard canonical quantisation techniques 5_{. The outcome of this analysis reveals two very important}

features of the bosonic string. Firstly, the bosonic string requires the presence of at least twenty six dimensions and secondly the string spectrum, for closed and open strings, contains tachyonic modes. If string theory is going to be of any physical interest, it should be free from instabilities (tachyons). Furthermore, nature has many fermionic particles which somehow need to be included in string theory. A solution to the bosonic string problems is given by superstring theory. It proves that when supersymmetry is introduced in the theory, fermions are included and instabilities are cured. The critical number of dimensions required for the theory to be free from anomalies is reduced to ten. This new theory is called superstring theory6.

Superstring theory is a very extensive subject so we have chosen to restrict our discussion here to a brief presentation of the ten dimensional IIB superstring massless (α0 → 0) spectrum (see tables 3.1, 3.2) which is relevant to the following

sections. The NS-NS sector of the closed string spectrum contains: the symmetric tensor gµν which describes the background metric, Bµν which is an antisymmetric

Kalb-Ramond tensor and Φ which is a scalar field called the dilaton and which is

spacetime index.

5_{In canonical quantisation x}µ

0, pµ0, αµν, ˜αµν should be promoted to operators.

6_{There are 5 different superstring theories but for our purposes we limit our interest to the type} IIB superstring theory.

related to the string coupling through eΦ_{= g}

s. The R-R sector contains three

n-form fields C(n), specifically C(0), C(2), C(4). The fermionic sector (R-NS,NS-R) consists of Ψµ and Ψ

0

µ, which are spin 32 gravitino fields, λ and λ0 which are spin 12

dilatino fields. The open string spectrum contains a ten dimensional gauge field

A_µ(x) in the NS sector and a Spin(8) spinor in the R sector. For more details regarding superstring theory see [26–29]