1. A.Sako S.Kuroki T.Ishikawa Graduate school of Mathematics, Hiroshima University Graduate school of Science, Hiroshima University Higashi-Hiroshima 739-8526,Japan Noncommutative Shift Invariant Quantum Field Theory

2. 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 This is an example of independent partition function. To calculate the Euler number of the GMS-soliton space 1.Introduction We will understand the relation between the GMS soliton and commutative cohomological field theory, soon.

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

4. Note that this transformation is just rescaling of the coordinates so any action and partition function are not changed under this transformation. This shift change the action and the partition function in general. Contrary, our purpose is to construct the invariant field theory under this shift. The next step, we change the noncommutative parameter.

5. BRS operator Action Partition function The partition function give us a representation of Euler number of space The lagrangian of cohomological field theory is BRS-exact form. Cohomological field theory is possible to be nominated for the invariant field theory.

6. This partition function is invariant under any infinitesimal transformation which commute BRS transformation. 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. The VEV of any BRS-exact observable is zero. (2)

7. 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 The stationary field configuration is decided by the field equation Integrating over the fields,,, and the action is

12. In the noncommutative space, there are specific stationary solution, that is, GMS-soliton. where P i 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 P i belongs to S A, the projection P i takes the coefficient ν A. And we defines. For the sets S A and S B are orthogonal each other

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

14. the coefficients of the crossed index term Formula (A ＞ B) 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 n A is the number of the indices in a set S A, and we call this number as degree of S A :n A =deg(S A ). The total partition function (includes all the GMS soliton) n n n n n n n n n

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

17. Morse function : 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 Critical Point : Hessian : n p : the number of negative igenvalue of the Hessian

18. From the basic theorem of the Morse theory, this is a Euler number of the isolate points {p}. In this commutative limit n p is 0 or 1, so the partition function is written by 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 : p_ : p + : These are operators and we have to pay attention for their order. = Morse index M n : MnMn Number of GMS soliton whose Hessian has n p negative igenvalue Number of GMS soliton that include n p projections combined to p_

20. Number of choice of N-n p projection combined to p + : When the total number of projection is fixed N, We can define the Euler number of the isolate with Basic theory of the Morse theory, From (a) and (b), the Morse index is given by Number of p_ : [(n+1)/2] Number of p + : [n/2] Number of combination to chose n p projection combined to p_ : (a) (b) n n nn

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

22. We have studied noncommutative cohomological field theory. Especially, balanced scalar model is examined carefully. Couple of theorems is provided. 6. Conclusion and Discussion 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 R 4. We can change the base manifold to noncommutative torus.