Ph230a Homework 2 Solutions
Nick Hunter-Jones
Fall 2013
Abelian Higgs Model
Consider the static action for the (2+1)-dimensional Abelian Higgs model, a gauge eld coupled to
a complex scalar,
1
S = d2 x |D |2 F F (|2 a2 )2 ,
4
2
where the indices run from
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 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 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 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 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
Ph230a Homework 3 Solutions
Nick Hunter-Jones
Fall 2013
t Hooft - Polyakov Monopole
We want to evaluate the surface integral,
dS i
a j b k c
ijk abc x x x ,
2
S
where spatial indices, i, j, k , and the color indices, a, b, c, both take values 1, 2, 3, and
Ph230a Homework 7 Solutions
Nick Hunter-Jones
Fall 2013
ADHM Construction
In nding explicit instanton solutions in the ADHM construction, we introduced the (N + 2k ) 2k
complex-valued matrix ai
T
,
(1)
(x) =
+ x
X
which had the nice factorization prope
Ph230a Homework 5 Solutions
Nick Hunter-Jones
Fall 2013
Abelian Duality
The Wilson line operator, WR ( ) = TrR Hol (A), is the trace in a representation R of the gauge
group G, of the holonomy of the connection A around a curve . In a U (1) gauge theory w
Ph230a Homework 6 Solutions
Nick Hunter-Jones
Fall 2013
BPST Instanton
a
Using the ansatz, Aa = x f (r), we calculate the eld strength to be
a
F = Aa Aa +
abc
Ab Ac
a
a
a
= 2 f (r) + ( x x x x )
Noting that a =
a 4
f (r )
+
r
abc b c
x x f (r)2 .
+ a 4 a
Ph230a Homework 4 Solutions
Nick Hunter-Jones
Fall 2013
Monopole Scattering
We want to consider monopoles moving through space very slowly and claim that the moving
solution is given by the static solution with time dependent collective coordinates varyin
Ph230a Homework 1 Solutions
Nick Hunter-Jones
Fall 2013
Kinks in Sine-Gordon Theory
We consider a real scalar eld theory with the potential, V () = A 1 cos 2 . The action is
a
given by,
1
2
S = d2 x
()2 A 1 cos
.
2
a
which gives the equation of motion,
2
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 .