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Noncommutative Shift
Invariant
Quantum Field Theory
A.Sako S.Kuroki T.Ishikawa
Graduate school of Mathematics, Hiroshima University
Graduate school of Science, Hiroshima University
Higashi-Hiroshima 739-8526 ,Japan
1.Introduction
Aim 1
To construct quantum field theories that
are invariant under transformation of
noncommutative parameter  
Partition functions of these theories is
independent on  
These theories are constructed as
cohomological field theory on the
noncommutative space.
Aim 2
To calculate the Euler number of the
GMS-soliton space
This is an example of   independent
partition function.
We will understand the relation between the
GMS soliton and commutative cohomological
field theory, soon.
2.Noncommutative
Cohomological Field Theory
Let us make a theory that is invariant under
the shift of noncommutative parameter  
rescaling operator
s
(1)
Inverse matrix of transformation (1)
Integral measure, differential operator and
Moyal product is shifted as
Note that this transformation is just rescaling of
the coordinates so any action and partition
function are not changed under this
transformation.
The next step, we change the noncommutative
parameter.
This shift change the action and the partition
function in general.
Contrary, our purpose is to construct the
invariant field theory under this shift.
Cohomological field theory is possible to be
nominated for the invariant field theory.
The lagrangian of cohomological field theory is
BRS-exact form.
BRS operator
Action
Partition function
The partition function give us a representation
of Euler number of space
This partition function is invariant under any
infinitesimal transformation which commute
BRS transformation .
(2)
The VEV of any BRS-exact observable is zero.
Note that the path integral measure is invariant
under   transformation since every field has
only one supersymmetric partner and the
Jacobian is cancelled each other.
θ shift operator as
Generally, it is possible to define  to commute
with the BRS operator.
Following (ref (2)), the partition function is
invariant under this θ shift.
The Euler number of the space
is independent of the
noncommutative parameter θ.
3．Noncommutative parameter deformation
3‐1 Balanced Scalar model
BRS operator








H
B




H

The Action
For simplicity, we take a form of the potential as
Bosonic Action
Fermionic Action
3-2 Commutative limit θ→0
2-dimensional flat noncommutative space
Rescale:
Commutative limit θ→0
Bosonic Leading Term
Fermionic Leading Term
Integrate out without Zero mode
Integration of Zero mode
Bosonic part
Fermionic part
Potential
Result
This result is not changed even
in the θ→∞ limit as seen bellow.
4．Strong noncommutative limit θ→∞
In the strong noncommutative limit θ→∞, the
terms that has derivative is effectively dropped
out.
Action
Integrating over the fields


H , H  , B  ,  and 
the action is
The stationary field configuration is decided by
the field equation
In the noncommutative space, there are specific
stationary solution, that is, GMS-soliton.
where Pi is the projection.
The coefficient  i is determined by
The general GMS solution is the linear
combination of projections.
where S A  A 1,2,, n is the set of the projection
indices, if the projection Pi belongs to SA, the
projection Pi takes the coefficient νA. And we
defines PS  iS Pi .
A
A
For A  B A, B  1,2,, n the sets SA and SB are
orthogonal each other
Bosonic Lagrangian expanded around the
specific GMS soliton
The second derivative of the potential is
Formula
(A＞B)
the coefficients of the crossed index term
are always vanished.
The bozon part of the action is written as
Fermionic part
（the calculation is same as the bosonic part）
The partition function is
n
n
Here nA is the number of the indices in a set SA,
and we call this number as degree of
SA:nA=deg(SA).
The total partition function
(includes all the GMS soliton)
n
n
n
n
n
n
n
Potential is given as
n
n
Then
: n is even number
: n is odd number
5. Noncommutative Morse Theory
Noncommutative cohomological field theory
is understood as Noncommutative Morse
theory.
5-1 In the commutative limit
zero mode of
number
is just a real constant
Morse function :
Critical Point :
Hessian :
np : the number of negative igenvalue of the
Hessian
In this commutative limit np is 0 or 1,
so the partition function is written by
From the basic theorem of the Morse theory,
this is a Euler number of the isolate points {p}.
This result is consistent with regarding the
cohomological field theory as Mathai-Quillen
formalism.
5-2 Large θ limit
Critical points :GMS solitons
Hessian :
These are operators and we have to pay
attention for their order.
Morse index Mn :
Mn 
=
Number of GMS soliton whose
Hessian has np negative igenvalue
Number of GMS soliton that
include np projections combined
to p_
p_ :
p+ :
Number of p_ : [(n+1)/2]
Number of p+ : [n/2]
Number of combination to chose np projection
combined to p_ : n
(a)
When the total number of projection is fixed N,
Number of choice of N-np projection combined
to p+ :
n
(b)
From (a) and (b), the Morse index is given by
n
n
We can define the Euler number of the isolate
with Basic theory of the Morse theory,
 n  lim
N
  1
N  n p
 lim
N 
np
Mn
  n  1  n  
 
   

 2  2
N
: n is even number
: n is odd number
This is consistent result
with the Mathai-Quillen formalism.
6. Conclusion and Discussion
We have studied noncommutative
cohomological field theory.
Especially, balanced scalar model is examined
carefully.
Couple of theorems is provided.
The partition function is invariant under the
shift of the noncommutative parameter.
The Euler number of the GMS soliton space on
Moyal plane is calculated and it is 1 for even
degree of the scalar potential and 0 for odd
degree.
It is possible to extend our method to more
complex model.
We can estimate the Euler number of moduli
space of instanton on noncommutative R4.
We can change the base manifold to
noncommutative torus.