Physics 683 Homework 4 Solutions by Matt Reece Problem 1 Consider N = 1 SU(N) gauge theory with N + 2 Nf < 3N flavors (Qf of SU(N) with tree-level superpotential ~ ~ Wtree = (Qf Qg )(Qg Qf ), ~ and Qg (1)
~ ~ where (Qf Qg ) and (Qg Qf ) are gauge invarian
Physics 683 Homework 3 Solutions by Matt Reece Problem 1 (a) Given N = 1 SUSY transformations = 2 and = i 2 , find L where
L = + i . (1) Answer: I seem to need a different sign, = -i 2 . The variation of the first term is: Lscalar = ( ) + ( ) = 2 + . For
Physics 683 Homework 2 Solutions by Matt Reece Problem 1 W = 1 P M is the Pauli-Lubanski spin vector. 2 (a) Show that W 2 = -m2 J 2 , where m is mass and J is angular momentum. The vector J is defined by Ji = 1 ijk Mjk , where i, j, k are all spatial indi
Physics 683 Homework 1 Solutions by Matt Reece (I'll try to type up some solutions to each homework, but they might be considerably less thorough at times.) Problem 1 (a) Show that we can define Weyl spinors in all even dimensions. Answer: much of the sol
1 Physics 683 Spring 2007 Homework 5 Due on April 19 Problem 1. The quantum moduli space of pure N = 2 U (N ) gauge theory can be parameterized by the hyperelliptic curve y 2 = PN (x)2 - 42N , where y and x are complex coordinates such that
N
(1)
PN (x) =
1
Physics 683 Spring 2007
5.3
N = 1 U (1) with Nf = 2
The rst important step in the Klebanov-Witten construction involves identifying the
moduli space of N = 1 supersymmetric U (1) gauge theory with two avors with the base
of the conifold. In this case, t
1
Physics 683 Spring 2007
Matrix models
Random matrix models were introduced in physics by Wigner in studying heavy nuclie in
the 1950s. The quantum energy levels in heavy nuclei are very dense. Wigner suggested
that the energy levels of heavy nuclie coul
1
Physics 683 Spring 2007
3.11
Seiberg-Witten elliptic curve
Now the main feature of the general solution must be clear. We started with showing that
the quantum N = 2 SU (2) gauge theory was invariant under S and T duality transformations which generate
1
Physics 683 Spring 2007
3.10
Monodromies
The basic idea now is simple. We know the asymptotic value of the coupling coecient or
the dual scalar eld, since we have already determined the exact perturbative prepotential
2
and () = F . Moreover, the coupli
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Physics 683 Spring 2007
3.7
S-duality
We have determined the perturbative part of the prepotential. The gauge coupling is
renormalized only at one loop in N = 2 and we used the chiral anomaly (the anomaly
is always exact at one loop) to calculate the pe
1
Physics 683 Spring 2007
3.1
N = 2 supersymmetric Lagrangian
Our interest here will be restricted to pure N = 2 supersymmetric gauge theories. The
massless vector multiplet in N = 2 contains states with helicity |1 , 2|1/2 , |0 and their
CPT conjugates |
Physics 683 Spring 2007
2.23
1
Seiberg duality
We have made some observations in order for N = 1 SU (N ) gauge theory with Nf avors
to ow to a nontrivial conformal xed point.
Seiberg put forward the following conjectures:
(1). N = 1 SU (N ) (electric) gau
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Physics 683 Spring 2007
2.21
More on Nf = N
We would like to emphasize some important features of N = 1 supersymmetric SU (N )
gauge theory with Nf = N avors. First let us show that the dynamical superpotential
vanishes and calculate the quantum moduli
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Physics 683 Spring 2007
2.15
Gaugino condensation
Consider pure N = 1 SU (N ) gauge theory. The Lagrangian is given by
L=
where
1
Im tr
8
d2 W W ,
a
W = ta ia + Da ( ) F + 2 ( D a ) .
(2.15-1)
(2.15-2)
In component elds,
L=
1 a a
i
1
FF
+
F a F a 2 a D
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Physics 683 Spring 2007
2.10
Wilsonian renormalization group ow
There is important dierence between the running of holomorphic couplings and the running of physical couplings. The running of physical couplings depends on the nonholomorphic Kahler kineti
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Physics 683 Spring 2007
2.5
Classical global avor symmetries
The supersymmetric QCD Lagrangian with Nf avors with no tree level superpotential has U (Nf ) U (Nf ) global symmetry, where one U (Nf ) rotates Qf and the other
U (Nf ) rotates Qg . Let us fa
1
Physics 683 Spring 2007
2.2
Lagrangian
Our interest will be SU (N ) gauge theory with Nf avors. It consists of Nf left
handed fermions that transform in the (N + N ) representation of the gauge group.
The Lagrangian that describes this system is
d4 e2V
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Physics 683 Spring 2007
1.3
Poincare symmetry in four dimensions
Can we combine Poincare spacetime symmetry with internal symmetries and embed them
in a larger symmetry group? The motivation for this question in the 1960s was the
discovery of internal s
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Physics 683 Spring 2007
1.2
Cliord algebra and spinor representations in D-dimensions1
The supercharge operator which changes the helicity of a state by a 1/2 unit in 1 + (D 1)
dimensional Minkowski spacetime transforms as a spinor under the spinor repr
Physics 683 Spring 2007
1.1
1
Motivation for supersymmetry
Our interest in this section is to make a qualitative overview of some of the important motivations for supersymmetry before we get started with details and formal developments.
What is supersymme