Types of String Theory

String Theory is a theory of quantum gravity of one-dimensional objects, that encompasses all the theories which explain how the universe functions. The idea behind the string theory is that matter does not consist of point particles but rather it is replaced by strings. These strings can be either open or closed depending on the type of the particle that is being replaced. The string length(l_s) is 10^-33cm. Also, in string theory, one replaces Feynman diagrams by surfaces, and wordlines become worldsheets

S = ∫d ² σα_string

Overview

• String theory is a theory of quantum gravity. Later on, We will find out that string theory is actually a finite theory of quantum gravity. Since open string theories always contain closed strings.

Types of String Theories

1. 1
   Bosonic string theory

   :

   The Strings correspond to Bosons. It requires 26 dimensions for its spacetime.

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2. 2
   There are 5 other consistent string theories that include fermions

   :

    

   1. Type I
   2. Type II A
   3. Type II B
   4. Heterotic SO(32)
   5. Heterotic E8 � - E8 .

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All of these theories use supersymmetry. Thus, for every boson, there exists its superpartner fermion and vice versa.

Types of Strings

1. 1
   Open strings often contain non-abelian gauge fields and chiral fermions, Both of these are important ingredients for a standard model

   .

   String theory naturally exists in D > 4. It can be readily supersymmetric. So that one might hope that string theory could be the UV completion of the standard model.

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2. 2
   All closed string theories contain a massless spin-2 particle

   .

   The general argument says, that the only consistent couplings of such particles are those of gravitations.

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Let us recall Quantum field theory. For example, a toy model of scalars

that interactions λ_k with:

• positive mass dimensions are super renormalizable.
• zero mass dimensions are renormalizable.
• negative mass dimensions are non- renormalizable.

General relativity badly behaves in the ultraviolet. We can understand this intuitively. Consider a high-energy collision between two point particles in GR + QFT

This high energy collision can produce a microscopic black hole. That black hole can be nonlinear. Let us consider:

Let us define the target space "M" to be a spacetime with metric g_μw and coordinate X^μ. The string worldsheet Σ is a two-dimensional spacetime with coordinates:

ξ1= T  and ξ2= σ, 

Where we treat T as time direction. Now let us assume, Minkowski signature for both spaces though will soon rotate into Euclidean coordinates on the worldsheet.

The natural measure for the size of the worldsheet is the usual volume,

An Analogy to the Relativistic point particle:

For a point particle we would like to consider the word line with single parameter ξ = T the pullback of the metric is simply,

So that the volume of the worldline is embedded into the target space,

We are thus inspired to use Vol(ξ₂) as an action for the relativistic string. This is called Nambu-goto action.

S_NG = -T vol (ξ₂), Equation- 6

Equation of motion for the to-be quantized field's X :

δ/δ_l (MX^μ/√X^μX_μ) =0

      = MXμ/√XμXμ = Const,	Equation- 7.

As quantum field theories, we could write down

Swrong = ∫ δ l(XμXμ + m 2), Equation- 8

Equation 5, obeys, T → λT , Equation- 9, 

But equation 8 does not obey this action, to fix up this action we introduce a compensator field e:

S`l = ∫ δ T (e-1XμXμ+em2) , Equation-10

One can think of this compensator as a lagrange multiplier, or more usefully as a metric on the wordline. If we assume that under reparameterization we have,

T → λT, Equation- 11
e → λ-1e , Equation- 12

The equation of motion for the compensator will now be:

-1/e2Xμ Xμ+m2= 0
e = 1/m √XμXμ, Equation-13

Similarly, The EOM for the X^μ field is:

δ/δt(e-1Xμ) = 0, Equation- 14

We would now like to find the analogous quantization friendly action for the string. This is called the Polyakov action.

Sp=-T/2 ∫ δ2ξ√-γ γαβ δαXμδβXδνζν, Equation- 15

According to sigma model,

Sσ= ∫ δ4η√g gμ ν δμφmδν φN Gmn,Equation- 16

Where G_mn, is a metric on the field space. To go back and forth from the polyakov action for string theory and our F model analogy we note,

Xμ ↔ φM,Equation- 17
ημ ν ↔ Gmn, Equation- 18
γαβ ↔ gμ μ, Equation- 19

GR in (1 +1)D, The Einstein Hilbert Action gives,

S_EH = 1/2k²∫ &delta²2 ξ √ -γ R

R_αβ - 1/2 g_αβ R = T_αβ, Equation- 20

Which vanishes in Vacuum. However, let us note that the Riemann Tensor obeys,

R_αβγδ = -R_βγδ = -R_αβγδ = R_αβγδ , Equation- 21

So that since α,β,γ,δ, ∈ {0,1}, we can write,

Rαβγδ= fεαβγδ, Equation- 22

Contracting the Indices to get the ricci tensor, we find:

R = fεαβγεγδ gαγ gβδ
=  Rαβγδ f gαγ ( εδα εγδ)
= f  εγδ  εγδ =	2F, Equation- 23

We have used ε^δ_α ε_γδ = g_αγ

plugging this back into the Riemann tensor, we obtain:

Rαβγδ = R/2  εδα εγδ, Equation- 24

Which can again plug back into Ricci tensor to find (Vacuum):

Rαγ  =  Rαβγδ gβδ = 1/2 Rg αβ, Equation- 25

We want to take the variation of the action with respect to the worksheet metric γ_αβ. In particular, we would like to write out the tricky term δ√-r/δ γ_αβ , recalling that r = δeδ γ_ζσ We can involve a well known result for a general metric g_ab,

δg = ggab δ gab, Equation- 26

Which can prove by using det m = expTrLog m.

By swapping the height of the indices, we can see that we pick up a negative sign since,

δ (g^ab g_ab) = 0 δg = -g g_ab g^ab, Equation- 27

We thus have,

δ√-g = 1/2 √-g  gab δ gab
= 1/2 √-g δ gab  gab , Equation- 28

The variation of our action with respect to γ^ρσ is then,

δS = -1/2 ∫ δ2 ξ {-1/2 √-γ γγρ δ γγρ hαβ + √-γ hσρ δ γσρ}, Equation- 29

δS/ δγσρ = -1/2 ∫ δ2 ξ √-γ { hσρ -1/2 γσρ &gammaαβ hαβ}, Equation- 30

With equation of motion,

h_σρ = 1/2 γ^σρ γ^αβ h_αβ, Equation- 31

When that EOM are satisfied, one can take a determinant to prove.

γαβ  hαβ = 2 √-h/ √-γ, Equation- 32

SP/EOM = -1 ∫ δ2 ξ √-h	
= SNG, Equation- 33

Pick a background spacetime M endowed with a metric, which in some local co-ordinates looks like,

δS2 = gμν (η) δxμ δxν

There are some other fields to specify, too, but let's ignore them for now. Consider the set of map's

Xμ: Worldsheet → Target spacetime.
Σ ↔ M
(σ,T) ↔ Xμ(σ,T)

Where, σ,T, are local coordinates on the world sheets.

Now try to compute the following path integrals,

I = ∫ [Dx (σ,T)] exp(iSw3[X...])

This is meant to be a proxy for a physical quantity like a scattering amplitude for two strings to go to two strings. The data about the external states are hidden in measures, Hence the subscript 'ws' stands for worldsheet.

The Feynman-Kac formula says that a transition amplitude takes the form:

 x(T1)∫x(T2)eSwρ[x(T)]

= (η2, T2/x1, T1)

Where,

Xμ : WorldLine → Target spacetime

(T) → x^μ(T)

Can be written as a sum over trajectories that interpolate between specified initial and final states. For example; for a massless charged scalar particle, propagating in a background spacetime with metric g_μν and background (Abelian) gauge field A_μ, The worldline action takes the form,

S_wρ[η] = ∫ δ₁ g_{μν (η) x^μ x^ν + ∫ δ1 x^μ Aμ}

Where, x = δ₁x. Note that minimal coupling to the gauge field can be written as,

∫ δ1 xμ Aμ = ∫wρ A

Taylor expands around some points in the target space, which let us call O, for example:

Aμ(η) = Aμ(0) + xρ δρ Aμ (0) + .....

And the coupling to the gauge field becomes,

∫ δT ( xμ Aμ(0) +  xμ  xρ  δρ Aμ(0) + ...)

The Crucial point:

Near any point in the worldsheet, it is just free propagation. This implies a certain uniqueness( Good thing Number four): Once the free theory is specified, the interactions are determined but only for some choices( M,*), and can be even defined as I. To see why, let us consider the simplest example where, M = IR^D-1,1, with g_μν = η_μν = diag(-1,1,1,1...1) then the worldsheet is

Sws[X] = 1/4παl ∫ δ2 σ δμ Xμ δα Xν ζμν

Here α =1,2 is a worldsheet index and μ = 0,...D-1 indexes the spacetime coordinates.

Crudely rendered perturbative interactions of closed strings. Time goes upward. Slicing this picture by horizontal planes shows that these are diagrams which contribute to the path integral for 2 to 2 scattering, Tilting the horizontal plane corresponds to boosting your frame of reference and moving the point ,where you think the splitting and joining happened, alluded to twice so far:

Ms2 = 1/4παl

People will disagree about the factor of 2 is π, Here, (2) is acting for free Bosons, we can compute:

Ms2 = 1/4παl

The equation of motion, O :

O = δSws/δX

Here, the equation of motion, "O" gives the massless 2d wave equation:

Θ = ((-δT)2 + (δρ)2)Xμ
=  δ+ δ- Xμ

Where, σ^± = T, ± σ ,

This is solved by superposition of right movers and left movers.

Xμ (σ, T ) = Xμ (σ+) + (XR)μ (σ-)

To study closed strings, we can treat the spatial direction of the worldsheet as a circle,

 σ = σ + 2π, 

In which case we can expand in modes,

Xμ (σ, T) = Ση ∈Z  αμ einσ± + Ση ∈Z  α`μ einσ-

For open strings, the boundary conditions will relate the right moving and left moving modes: α^l ∝ α , we can quantize this canonically and find;

[(αn)μ, (αm)ν] = n δn+m + mημν,  [(`αn)μ, (`αm)ν] =  n δn+m + ημν,  [(α, `α] = 0

This says that we can think of ( α_n < 0) as creation of operators, (α_n >0)as annihilation operators. Let |0 > be annihilated by all the (α_n > 0), Thus,

`αρ2 + Σn>0 (α-n)μ αnμ-a

Where, "a" is a normal ordering constant, which turns out to be 1. Here comes the problem: Consider the state,

BAD> = (α_-n) ^μ=0 | 0 >,

For n>0, because of commutator [(α_n)⁰ , (α_n)⁰] = - n

We face, the choice ||| BAD > ||² < 0, or E_BAD < 0 Well, Neither is okay, especially since E_BAD becomes arbitrarily negative as n gets larger. Now, We can build physical states that results look like.

Where [..] is meant to indicate antisymmetric indices, whereas, {..} is meant to indicate symmetric indices. The scalar is called Dilation. For open string, we would have, among other states, string state : -1|0>, Field Mass² (α_-1)^μ | 0 > Vector A_α m² = 0. The world sheet gauge symmetry leads directly to the spacetime gauge symmetries under which, For example:

A → A + δA(0) 
B  = Bμν δxαA δxγ 
→ B + δA(1)

There is a general theorem that massless symmetric tensor like g _μν, which interacts must couple to stress energy, and hence is a gravitation using the machinery built from the above picture. This mode can be seen to participate in nontrivial scattering amplitudes, which indeed reduces, those computed from GR, to lower energies. So, string theory, however unwieldy, this description can be, is a Quantum theory of gravity.

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