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This is the Past Paper of Mathematical Tripos which includes Turbulence, Burgers’ Vortex, Vortex Tube with Vorticity, Irrotational Straining Flow, Unsteady Vorticity Equation, Kinematic Viscosity, Azimuthal Velocity, Rate-Of-Strain Tensor etc. Key important points are: String Theory, Closed Bosonic String Theory, Scattering Amplitude, Minkowski Spacetime, Spacetime Co-Ordinates, World-Volume Metric, Equations of Motion, Virasoro Operator, Orthonormal Gauge, Majorana Fermions
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Thursday, 7 June, 2012 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.
In closed bosonic string theory explain how, starting from the action, one constructs the four-point scattering amplitude for tachyons.
The action for a bosonic p-brane propagating in d-dimensional Minkowski spacetime is I =
Σ
dp+1ξ γ^1 /^2 (γμν^ ∂μXa∂ν Xbηab − (p − 1)) ,
where Σ is the world-volume of the p-brane, ξμ^ labels points on the world-volume, Xa(ξ) are the spacetime co-ordinates of the point ξμ, ηab is the Minkowski spacetime metric, γμν is the world-volume metric and γ = − det γμν.
Derive the equations of motion for Xa^ and γμν , taking care to explain how and why the case p = 1 differs from p 6 = 1 and what boundary conditions are needed.
For the case p = 1, in other words the string, and assuming that we are dealing with the open string with NN boundary conditions, derive the operator version of the Virasoro constraints assuming we can choose the orthonormal gauge for the world-sheet metric.
Find the commutators of the Virasoro operator, paying particular attention to the central term.
Part III, Paper 51
The action for the closed bosonic string in a curved spacetime with metric gab and with vanishing dilaton in the orthonormal gauge is
d^2 ξ
∂μXa∂ν Xbημν^ gab(X) ,
where the world-sheet metric is flat ημν , ξμ^ are the string world-sheet coordinates and Xa labels where in spacetime the points ξμ^ on the string are located.
Explain how to derive the background field equation
Rab = 0 ,
where Rab is the Ricci tensor of the metric gab, from requiring that the classical symmetries of the string are preserved quantum mechanically.
Suppose that the background metric has vanishing Ricci tensor and admits a (spacelike) Killing vector
ka^
∂xa^
∂z and that the d-dimensional background metric is of the form
gzz = V, gzI = 0, I = 1, 2 ,... (d − 1) ,
with all metric coefficients independent of z. By making the appropriate field transforma- tions on the string worldsheet, show that the T -dual metric with gzz replaced by V −^1 but gzI and gIJ unchanged is equivalent to the original string theory.
Briefly explain why the fact that Rab 6 = 0 for the new metric does not imply an inconsistency for bosonic string theory.
Part III, Paper 51
