Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Final Exam of Mathematical Tripos which includes Wave Scattering in Inhomogenous Media, Wiener-Hopf Technique, Boundary Conditions, Infinite Line, Dirichlet Boundary Conditions, Neumann Boundary Conditions, Incident Field, Diffraction Problem etc. Key important points are: String Theory, Mode Expansion, Circular Closed String, Equations of Motion, Minkowski Space, Expression for Classical Mass, Non-Interacting Scalars, Operator Product Expansion, Stress-Energy Tensor
Typology: Exams
1 / 5
This page cannot be seen from the preview
Don't miss anything!
Thursday, 3 June, 2010 9:00 am to 12:00 pm
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
The worldsheet of a classical relativistic closed string is parameterized by coordinates τ and σ ∈ [0, 2 π). The embedding of the string into Minkowski space R^3 ,^1 is described by the functions Xμ(σ, τ ), with μ = 0, 1 , 2 , 3. Write down the equations of motion and constraints obeyed by Xμ.
Explain why there are no solutions describing a circular closed string at constant radius, R.
The most general solution to the equations of motion is given by
Xμ(σ, τ ) = XLμ(σ+) + XRμ(σ−) where σ±^ = τ ± σ.
The mode expansion for these fields can be written as
XLμ(σ+) =
xμ^ +
α′p μ^ σ+^ + i
α′ 2
n 6 = 0
n
α˜ (^) nμ e−in σ
,
XRμ(σ−) =
xμ^ +
α′p μ^ σ−^ + i
α′ 2
n 6 = 0
n
α (^) nμ e−in σ − .
Write down the constraints in terms of the modes pμ^ and ανn. Use this to provide an expression for the classical mass of the string and explain what is meant by the term “level matching”
Consider now a string moving on R^2 ,^1 × S^1 , where the circle S^1 is parameterised by X^3 ≡ X^3 + 2πR. Explain how the mode expansion changes. Construct a classical solution obeying the constraints which, when projected onto R^2 ,^1 , looks like a static string of constant radius R.
Part III, Paper 44
Under an infinitesimal Weyl rescaling and reparametrization, the worldsheet metric gαβ transforms as δ gαβ = 2 ωgαβ + ∇αvβ + ∇β vα.
Use this to provide a path integral expression for the inverse Faddeev-Popov determinant, ∆− F P^1 [g].
Assuming that ∆F P [g] is gauge invariant, explain the Faddeev-Popov procedure that results in the ghost action,
Sghost =
d^2 z
bzz ∂z¯ cz^ + b¯z ¯z ∂z c¯z^
Sketch in outline how the presence of the ghosts results in the critical dimension D of the string.
The Veneziano amplitude describing two-to-two tachyon scattering in the open string is given by ratios of Gamma functions
A ∼ gs
B(−α′s − 1 , −α′t − 1) + B(−α′s − 1 , −α′u − 1)
where B(a, b) = Γ(a)Γ(b)/Γ(a + b). The gamma function Γ(x) has simple poles at x = 0, − 1 , − 2 ,.... Determine the open string spectrum from the position of these poles. What information would we obtain by computing the residues of the poles?
The Veneziano amplitude is exponentially suppressed for fixed angle, high-energy scattering. Explain the relevance of this observation.
Part III, Paper 44
Write down the worldsheet action describing a string coupled to a background spacetime metric Gμν (X), an anti-symmetric tensor Bμν (X) and the dilaton Φ(X).
What are the local symmetries of the various terms in the action?
Describe how the couplings to Gμν (X) and Bμν (X) are related to the states of the string that arise from quantization in flat space.
Show that the Bμν coupling has a spacetime gauge symmetry. Explain how string perturbation theory arises from the dilaton coupling and the sum over worldsheets of different topologies.
Describe in outline how quantization of the two dimensional worldsheet theory leads to equations of motion for fields in D = 26 dimensional spacetime.
Part III, Paper 44
