Ph230b Homework 13 Solutions
Nick Hunter-Jones
Winter 2014
Deformations of 2d Yang-Mills Theory
The tools with which we will construct our 2d Yang-Mills theory are
=
1
,
f ()
= ,
= f ()
where , , and run over irreps of G.
a) We rst want to compute the par
Homework 9
Bose-Fermi correspondence in 0+1 dimensions:
a) For the eigenvalue En of the bosonic Hamiltonian
HB =
N
mb bm
m
m=1
calculate the degeneracy D(n) of the corresponding eigenspace. Calculate the
large-n asymptotics of D(n) for n N and for n N .
Homework 10
2d Sine-Gordon theory (revisited):
a) The following time-dependent periodic solution in the Sine-Gordon theory
)
(
4
sin(t)
1
(x, t) = tan
cosh(x)
is called a breather. Here = m and m. By evaluating the energy of the
breather solution at a x
Homework 13
Deformations of 2d Yang-Mills theory: In this homework, we are going to study
several variants of 2d Yang-Mills theory dened by the cap, cylinder and pants
amplitudes:
=
1
f ( )
=
,
=
C = f () ,
where , , and run over irreducible representati
Homework 11
The generalized Schwinger model: Consider a two-dimensional U (1) gauge theory
coupled to NL (resp. NR ) left (resp. right) charged Weyl fermions. Assume that the
fermion charges, q and qr , satisfy the anomaly cancelation condition
NL
=1
2
q
Homework 12
The chiral Gross-Neveu model: Consider the IR physics of the model
N2
/
L = a i a
( + 2 ) + ( + i5 )
2g0
around the vacuum = 0 and 0 = 0, where 0 is a dynamically generated mass
scale. (As usual, and are scalar elds, and a , i = 1, . . . , N
Homework 8
Massless 2D Fermions with Axial Coupling: Show that in a system of massless
2D fermions with axial coupling to a background gauge eld,
L = i ( i 5 A ) ,
the total chirality is conserved whereas the fermion number is not:
Q5 = 0
,
QF = 0