1. 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 ,Japan

2. 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.

3. 2.Noncommutative Cohomological Field Theory
Let us make a theory that is invariant under the shift of noncommutative parameter
rescaling operator
(1)
Inverse matrix of transformation (1)
Integral measure, differential operator and　
Moyal product is shifted as

4. The next step, we change the noncommutative parameter.
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.

5. 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

6. 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.

7. Generally, it is possible to define to commute with the BRS operator.
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.

8. 3．Noncommutative parameter deformation
3‐1　Balanced Scalar model
BRS operator
The Action
For simplicity, we take a form of the potential as
Bosonic Action
Fermionic Action

9. 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

10. Potential
Result
This result is not changed even
in the　θ→∞ limit as seen bellow.

11. 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
, , , and
the action is
The stationary field configuration is decided by the field equation

12. In the noncommutative space, there are specific stationary solution, that is, GMS-soliton.
where Pi is the projection.
The coefficient is determined by
The general GMS solution is the linear combination of projections.
where is the set of the projection indices, if the projection Pi belongs to SA, the projection Pi takes the coefficient νA. And we defines
For the sets SA and SB are orthogonal each other

13. Bosonic Lagrangian expanded around the specific GMS soliton
The second derivative of the potential is

14. 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）

15. The partition function is
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

16. Potential is given as n n
Then
: n is even number
: n is odd number

17. 5. Noncommutative Morse Theory
Noncommutative cohomological field theory
is understood as Noncommutative Morse theory.
5-1 In the commutative limit
zero mode of is just a real constant number
Morse function :
Critical Point :
Hessian :
np : the number of negative igenvalue of the
Hessian

18. 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.

19. 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 :
Number of GMS soliton whose Hessian has np negative igenvalue Mn
Number of GMS soliton that include np projections combined to p_ =
p_ :
p+ :

20. Number of combination to chose np projection combined to p_ :
Number of p_ : [(n+1)/2]
Number of p+ : [n/2]
Number of combination to chose np projection combined to p_ :
(a) n
When the total number of projection is fixed N,
Number of choice of N-np projection combined to p+ :
(b) n
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,

21. with the Mathai-Quillen formalism.
: n is even number
: n is odd number
This is consistent result
with the Mathai-Quillen formalism.

22. 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.