A.Sako S.Kuroki T.Ishikawa

Graduate school of Mathematics, Hiroshima UniversityGraduate school of Science, Hiroshima University Higashi-Hiroshima 739-8526 ,Japan

Noncommutative Shift Invariant Quantum Field Theory

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.

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

srescaling operator

(1)

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.

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.

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)

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 modelBRS operator

B

H

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 spaceRescale:

Commutative limit θ→0Bosonic Leading Term

Fermionic Leading Term

Integrate out without Zero modeIntegration 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

The stationary field configuration is decided by the field equation

Integrating over the fields

, , , and

the action is

H H B

In the noncommutative space, there are specific stationary solution, that is, GMS-soliton.

where Pi is the projection. The coefficient is determined by i

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 .

nASA ,,2,1

AA Si iS PP

For the sets SA and SB are orthogonal each other

nBABA ,,2,1,

Bosonic Lagrangian expanded around the specific GMS soliton

The second derivative of the potential is

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 ）

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)

nn

nnn

nn

n

n

Potential is given as

Then

nn

: n is even number: n is odd number

Morse function :

5. Noncommutative Morse Theory

Noncommutative cohomological field theoryis understood as Noncommutative Morse theory.5-1 In the commutative limit

zero mode of is just a real constant number

Critical Point :

Hessian :

np : the number of negative igenvalue of the Hessian

From the basic theorem of the Morse theory,From the basic theorem of the Morse theory,this is a Euler number of the isolate points this is a Euler number of the isolate points {p}.{p}.

In this commutative limit np 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.

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 Mn :

Mn Number of GMS soliton whose Hessian has np negative igenvalue Number of GMS soliton that include np projections combined to p_

Number of choice of N-np 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 np projection combined to p_ : (a)

(b)

n

nN

nNn M

p

p

1lim

N

N

nn

22

1lim

n

n

nn

This is consistent result with the Mathai-Quillen formalism.

: n is even number: n is odd number

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 R4.We can change the base manifold to noncommutative torus.