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Partial Breaking of N = 2 Supersymmetry and
Decoupling Limit of Nambu-Goldstone Fermion
in U(N) Gauge Model
K.F. hep-th/0609039
Kazuhito Fujiwara(Osaka City Univ.)
motivation
Gauge-Gravity(Matrix) duality
[Dijkgraaf-Vafa ’02] [Cachazo-Douglas-Seiberg-Witten ’02]
N = 1 U(N)
・・・
N=2
The vector superfields
The fermionic field
( index 0 refers to the overall U(1) part of U(N) ) is decoupled .
in
is a free fermion.
low energy (SU(N) is confined )
fermionic shift symmetry
free energy F (Matrix model)
N=2?
Find the relation between fermionic shift symmetry and
spontaneous partial breaking of N = 2 supersymmetry.
( Derive the N = 1 action
from spontaneous
partial breaking of N = 2 supersymmetry. )
Table of Contents
next
N = 2 U(N) gauge model with
spontaneous partial breaking of rigid SUSY(review)
supercharge algebra
resulting N = 1 U(N) action
decoupling limit of Nambu-Goldstone fermion
summary & future plans
N = 2 U(N) gauge model
[Antoniadis-Partouche-Taylor]; hep-th/9512006 Phys.Lett. B372 (1996) 83-87
N = 2 U(1) gauge model equipped with electric and magnetic FI terms.
N= 2
N= 1 (on the vacuum)
contribution from
FI D-term and superpotential.
[K.F-Itoyama-Sakaguchi]; hep-th/0409060 Prog.Theor.Phys. 113 (2005) 429-455
N = 2 U(N) gauge model described by
N = 1 chiral superfields
and N = 1 vector superfields
cf.
.
[K.F-Itoyama-Sakaguchi]; hep-th/0503113 Nucl.Phys. B723 (2005) 33-52
(on vacua)
[K.F-Itoyama-Sakaguchi]; hep-th/0510255 Nucl.Phys. B740 (2006) 58-78
Harmonic superspace formalism
[Itoyama-Maruyoshi] hep-th/0603180
Local supersymmetry
ingredients
U(N) adjoint N = 1 chiral superfields
overall U(1)
U(N) adjoint N = 1 vector superfields
Vector superfield strengths
K hler potential
where
is an analytic function of
(prepotential) , and
K hler metric：
Killing vector
Killing potential
: structure constant of
U(N) gauge symmetry.
.
total action ( N =2 )
K hler term
Counterterm for the U(N) gauging of
Gauged kinetic term for the vector superfield
Superpotential term
condition for the
supersymmetry
FI D-term
These parameters
play a key role of partial breaking of N = 2 supersymmetry.
forms the real part of an ``electric" FI term and
forms the real part
of a ``magnetic" FI term in [K.F.-Itoyama-Sakaguchi ’0510]
We impose
for later convenience.
where
analysis of vacua
Let us examine the case with
and gauge group U(N) is unbroken on vacua.
scalar potential
vacuum condition
where
denote
reduces to
(
evaluated at
: index for non-Cartan generators)
Nambu-Goldstone fermion
：vacuum expectation value of
.
N=2→1!
Table of Contents
N = 2 U(N) gauge model with
spontaneous partial breaking of rigid SUSY(review)
next
supercharge algebra
resulting N = 1 U(N) action
decoupling limit of Nambu-Goldstone fermion
summary & future plans
general supercurrent algebra
central extension
(constant)
The N = 2 supersymmetric transformation rule is given as
where
We can obtain the 1st supercurrent
where
and
from the action
satisfy following relations,
.
After some algebra, we obtain
The 2nd supercurrent
of
with
.
is given by the discrete
transformation
N = 2 Supercharge algebra is derived by
It may be irrelevant to denote supercharges as
because the supercharge
corresponding to the broken supersymmetry is ill-defined. For convenience, we ignore
this point here and write the divergent part explicitly.
We obtain the central charge
To get the resulting N = 1 supercharge algebra, we define
. Anti-commutators of
and
and
itself are given as
∞
central extension (divergent)
is the unbroken generator
is the broken generator.
Table of Contents
N = 2 U(N) gauge model with
spontaneous partial breaking of rigid SUSY(review)
supercharge algebra
next
resulting N = 1 U(N) action
decoupling limit of Nambu-Goldstone fermion
summary & future plans
spinor fields mixing
The spinor fields
and
are to be mixed as well as supercharges.
Rewrite the N = 2 action in
and
.
Following terms keep the same form as before
with
The F.I. term parameter
so that we get
the same form as before
is absorbed into the superpotential terms
shifted scalar fields
The scalar fields
The prepotential
are to be sifted from its vacuum expectation value,
is expanded in the shifted fields
Similarly, derivatives of the prepotential are written as
We can rewrite the action in the shifted scalar fields
K hler metric
K hler potential
Killing potential
with
as
Resulting N = 1 action
where
superpotential
complex
(×N = 2 supersymmetry.)
We can write the off-shell N = 1 action by introducing auxillialy fields
and
form massive chiral multiplets
form massless vector multilets
The Nambu-Goldstone fermion
is coupled with other fields.
Table of Contents
N = 2 U(N) gauge model with
spontaneous partial breaking of rigid SUSY(review)
supercharge algebra
resulting N = 1 U(N) action
next
decoupling limit of Nambu-Goldstone fermion
summary & future plans
If the prepotential
is a second order polynomial, there are no
Yukawa couplings in the action and the Nambu-Goldstone fermion
will be a free fermion.
But derivatives of the superpotential become zero,
This means that the superpotential does not contribute to the
action and it recovers the N = 2 supersymmetry.
(partial breaking does not occur!)
This problem can be solved by a large limit of the parameters
i.e. large limit of the electric and magnetic FI terms.
reparametrization
We reparametrize
and
The prepotential is written as
Derivatives of the prepotential
where
and
are
and they are vanished at
K hler metric and Killing potential
structure constant
Derivatives of the superpotential are
cancellation !
The superpotential keeps non-trivial form at
.
decoupling limit
Take a limit
The superpotential
, and the action
is converted into
is given as
where
binomial coefficient
The Nambu-Goldstone fermion
is decoupled from the other fields.
We can rewrite the action
where
in superfield formalism as
is the field strength of
decoupling limit
summary
・We obtain general N = 1 U(N) action, which include a free fermion, from
spontaneously broken N = 2 supersymmetry.
・We conclude that the fermionic shift symmetry is identified with
2nd, spontaneously broken supersymmetry in the decoupling limit of the
Nambu-Goldstone fermion in our model.
future plans
・Relation between geometric transition and spontaneous partial breaking.
・Relation between domain wall solution and spontaneous partial breaking.
・Realization of our model in string theory.