Notes on String Theory, Study notes of Relativity Theory

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String Theory

Gang Xu

of Cornell University

a set of notes on string theory based on the string theory class by Liam McAllister

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Declaration

These notes are based on various of resources. The credit will be given in due time.

Gang Xu

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Acknowledgements

Of the many people who deserve thanks, some are particularly prominent:

My supervisor...

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Contents

• 1 Introduction to String Theory
  • 1.1 What does String Theory Describe?
  • 1.2 Why would we Want a Theory of Quantum Gravity?
• 2 Classical Relativistic String Theory
  • 2.1 Adventure of Finding an Action
    • 2.1.1 First Guess: an Action with a Square Root
    • 2.1.2 A Tougher Task: Looking for an Action without Square Root
  • 2.2 Symmetries and Boundary Conditions
    • 2.2.1 Symmetries of the Action and Equation of Motion
    • 2.2.2 Boundary Conditions
    • 2.2.3 Energy Momentum Tensor and Relation with Noether Theorem
  • 2.3 Summary
• 3 Light Cone Quantization of Bosonic String Theory
  • 3.1 The Options of Quantization
  • 3.2 solve the classical theory
• List of Figures
• List of Tables

“My plants died, one of you guys killed them.” — Enrico Pajer

4 Introduction to String Theory

• that the spacetime metric obeys the Einstein equations Gμν = 8πTμν [needs a picture of mapping from worldsheet to spacetime] Open string theories readily contain nonabelian gauge fields and chiral fermions. Furthermore, open string theories always contain closed strings.[needs a picture of "pant" demonstration]

Hence, except in exotic cases, we find that string theory is a theory for quantum gravity. It’s actually a finite one.

Moreover, string theory naturally exists in D > 4 and readily contains nonabelian gauge fields+chiral fermions(+SUSY).

1.2 Why would we Want a Theory of Quantum Gravity?

Recall in QFT, eg λφ^4 , we can write the Lagranrian as

L =^12 (∂φ)^2 − 12 m^2 φ^2 − λ 4 φ^4 − ( (^) Mλ^6 2 )φ^6 + · · · (1.1)

where the higher order terms will generically exist except for symmetry reasons. As we are very familiar in QFT, the couplings with positive, vanishing, and negative mass dimensions are termed superrenormalizable, renormalizable, nonrenormalizable respec- tively.

In gravity, the coupling constant is GN ∼ M −^2 (can be easily seen from Newtonian Gravity formula for the force F = GN^ m r 21 m^2 with mass dimension 2), which is, from the previous naive argument, highly nonrenormalizable.

There is a heuristic argument: point-particle scattering at sufficiently high ener- gies leads to black hole formation(unlikely in LHC, though), while string scattering at high energies is much softer(as we will see quantitatively): [needs a picture of scattering of particles and strings]

So is that all? A finite theory of Quantum Gravity with some prospect of model- building?

NO! The extremely rich structure of string theory has led to important insights into

Introduction to String Theory 5

• nonperturbative dualities
• gauge theories (eg at strong coupling)
• mathematics (eg in algebraic geometry)
• black holes (eg microscopic counting of states giving Bekenstein-Hawking entropy.
• SYM holography (AdS/CFT correspondence)
• theories on branes

Chapter 2

Classical Relativistic String Theory

“There is a hierachy between graduate students.” — Csaba Csaki, 2009 Feb 25th

2.1 Adventure of Finding an Action

2.1.1 First Guess: an Action with a Square Root

So how to formulate a classical relativistic string theory? [need a picture of embedding]

We can view the string worldsheet as an embedding Φ : Σ → M , where Σ is a (1 + 1)-dimensional worldsheet with coordinates ξα α = 1, 2. A point on the world sheet ξα is mapped by Φ to a point in the spacetime with embedding functions Xμ(ξα).

We define the pullback by the embedding Φ of spacetime metric by

Φ∗g : ∂Xμ ∂ξα

∂Xν ∂ξβ^ gμν^ ≡^ hαβ^ (2.1)

This leads to a natural measure of the size of an embedded submanifold ∫ (^) √ − det hαβ d^2 ξ (2.2)

8 Classical Relativistic String Theory

We can check if this gives the correct action in case of a point particle: ξα^ ↔ τ , where τ is the worldline parameter. [need a picture of worldline] Then we have h = dX dτμdX dτμ. Using (2.2), we have

Spoint =

dτ

dXμ dτ

dXμ dτ (2.3)

which agrees with the action from special relativity.

More generally, for a theory of relativistic membranes, dimension p + 1, we would have again,

hαβ = ∂X

μ ∂ξα

∂Xν ∂ξβ^ gμν^ (2.4)

as the ”induced metric”, where α, β = 1...p + 1, and the action is given by

S =

dp+1ξ

− det h (2.5)

We’ll come back to this action when we talk about D-branes.

2.1.2 A Tougher Task: Looking for an Action without Square Root

The action we just obtained in (2.2) is not easy to quantize. How can we get a simple- looking but equivalent action, so we can quantize that instead?

First, we go back to the point particle case to do a warmup. Remind ourselves that the action of a relativistic point particle is written as,

S = m

X^ ˙μ^ X˙μdτ (2.6)

where · : (^) dτd , pay attention that τ is a parameter of the embedding, especiall τ 6 = X^0. The equation of motion we find from this action (2.6) is

√^ m X˙μ X˙ν^ X˙ν^ =^ const.^ (2.7)

Note that this invariant end the conformal change of parameter τ → λτ.