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The details of paper 63 from the university of cambridge's mathematical tripos part iii exam held on june 7, 2005. The paper covers topics in differential geometry, lie algebras, and their applications in physics. Students were required to answer questions on fibre bundles, principal bundles, frame bundles, pseudo-orthonormal frame bundles, lie groups, lie algebras, symplectic manifolds, poisson manifolds, and their connections to physics. The document also includes instructions for the exam, stationery requirements, and special requirements.
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Tuesday 7 June, 2005 1.30 to 4.
Attempt FOUR questions.
There are SEVEN questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Define a fibre bundle and a principal fibre bundle. What is a section of a bundle? Show that a principal bundle is trivial if and only if it admits a global section.
What is the frame bundle of a manifold? What is a pseudo-orthonormal frame bundle? Give a group-theoretic description of the orthonormal frame bundles of Minkowski spacetime and of De-Sitter spacetime.
Show that the frame bundle of any connected Lie Group is trivial. What can you say about the tangent bundle of a Lie group?
2 In five-dimensional supergravity, whose bosonic fields comprise the spacetime metric and a vector field, one adds to the usual action functional for an exact two-form F = dA,
a so-called Chern-Simons term of the form
c
where c is a constant.
Obtain the field equation for F. Explain why the field equation is gauge-invariant despite the the fact that the Chern-Simons integrand is not invariant under gauge transformations. The energy momentum tensor Tμν for any field theory in a spacetime with metric gμν is given by the functional derivative
Tμν = −
−g
δS δgμν^
Obtain the contribution to the energy momentum tensor from the two-form action, including any which arises from the Chern-Simons term.
Paper 63
5 Define a symplectic manifold and a Poisson manifold. Give examples, in particular, examples of manifolds which are Poisson but not symplectic.
By showing that every symplectic manifold has an everywhere non-vanishing volume form, prove that every symplectic manifold is orientable.
Show that the cotangent bundle T ?(M ) of a manifold M is a symplectic manifold. Give a condition on a Poisson manifold that the Jacobi identity holds for the Poisson bracket defined on functions. Show that this condition is satisfied on a symplectic manifold.
6 A group G acts on a symplectic manifold {P, ω} preserving the symplectic form ω. Show how one obtains a moment map μ : P → g?. Give a necessary condition that the Poisson algebra of the moment maps coincides with the Lie algebra g of the group G. Show that this condition is satisfied if the Killing form, or Killing metric, of g is non-degenerate. For what groups is this latter property true?
Illustrate your answer by reference to the isotropic simple harmonic oscillator in three spatial dimensions with Hamiltonian
p^2 +
x^2.
Show in particular that there is such an action of U (3) on T ?(R^3 ), and that the moment map for the U (1) factor is the Hamiltonian. Exhibit some other moment maps. Which ones arise from the obvious geometric action of SO(3) on R^3? Which ones do not?
7 Write an essay, either on one of the following topics
Geometric Quantization, in which case, your essay should give the aims and motivations behind the idea, describe the prequantization construction and discuss the problem of finding a suitable polarization.
or
Applications of Stokes’s Theorem, in which case you should indicate briefly how Stokes’s theorem works and show how it is used to construct gauge-invariant equations of motion for p-branes and how, using the Brouwer degree construction, it may be used to obtain topological conservation laws, giving as an example the Skyrmion, i.e. a theory based on an SU (2) target space.
Paper 63
