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The questions and instructions for paper 56 of the mathematical tripos part iii exam held on 12 june 2007. The topics covered include solitons and instantons in scalar field theory, sigma models, yang-mills-higgs fields, and holomorphic line bundles on complex projective space.
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Tuesday 12 June 2007 9.00 to 11.
Attempt THREE questions.
There are FOUR questions in total.
The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
1 Let φ : RD+1^ −→ R be a smooth function. Define the term soliton in the context of scalar field theory with the Lagrangian
RD
(∂tφ)^2 −
|∇φ|^2 − U (φ)
dD^ x,
and use the Derrick scaling arguments to show that solitons can only exist if D = 1.
Derive the first order Bogomolny equations and use these equations to find the static kink if U (φ) = φ^6 − 8 φ^4 + 16φ^2.
2 Write an essay on sigma model lumps, paying particular attention to the role of topological degree.
3 Consider a Yang–Mills–Higgs field (Ai, Φ) : R^3 −→ su(2). Define the covariant derivative Di and the curvature Fij of the potential Ai.
Consider the Bogomolny equations
1 2
εijkFjk = DiΦ (1)
on R^3 and discuss their gauge invariance. State the boundary condition needed to interpret (Ai, Φ) as a non-abelian magnetic monopole.
Show that the one–form A = Aidxi^ + Φdτ
satisfies the anti–self–dual Yang–Mills (ASDYM) equations on R^4 = R^3 ×R, where (Ai, Φ) is a solution to (1) which does not depend on τ ∈ R. Calculate the four–dimensional Euclidean action density in terms of the three–dimensional field Fjk.
Define the term instanton in the context of a pure Yang–Mills theory in R^4 , and show that A is not an instanton. What can you deduce about translational symmetries of ASDYM instantons?
4 Define the holomorphic line bundles O(n) −→ CP^1 and show that if n > 0 the space of holomorphic sections is H^0 (CP^1 , O(n)) = Cn+1.
Define H^1 (CP^1 , O(n)) with respect of an open cover of your choice, and show that
dim H^1 (CP^1 , O(n)) = 0, if n > − 2.
Consider the vector bundle O(1) ⊕ O(1) −→ CP^1 and characterise those holomorphic sections of this bundle which vanish at a single point of CP^1.
Paper 56
