Homework 4
Monopole Scattering:
By studying geodesics on the Atiyah-Hitchin manifold, nd the result of the classical scattering in a head-on collision of two BPS monopoles in SU (2) gauge theory.
Do the same problem for two monopoles of dierent type in
Homework 3
Evaluate the surface integral
ijk abc xa j xb k xc dS i
(a xa = 1)
x
2
S
that enters the calculation of the magnetic charge of the t Hooft - Polyakov monopole
solution.
Homework 1
Kinks in sine-Gordon theory: consider a theory of a single scalar eld in 1+1 dimensions with the canonical kinetic term and a potential:
V () = A 1 cos
2
a
The vacua of this theory are at = na with n Z. Find the explicit form of the kink
soluti
Homework 5
Duality: Consider a Wilson line operator Wq () in four-dimensional U (1) gauge theory
with = D,
Wq () = exp iq
A
= exp iq
F
D
Show that Wq () is dual to t Hooft line operator (disorder operator) Tq () dened by
the following prescription: remove
Homework 6
BPST instanton: Using the ansatz
a
A = x a f (r)
nd an instanton solution, that is a function f (r) such that A obeys the self-duality
equation,
F = F
Homework 7
ADHM Construction: Prove that the ADHM construction gives a solution to the
self-duality equation,
F = F
Start with the vector potential A = iU U expressed in terms of a (N + 2k) N
matrix U .