Non-renormalization theorem in a lattice supersymmetric theory and the cyclic Leibniz rule

N=4 supersymmetric quantum mechanical model is formulated on the lattice. Two supercharges, among four, are exactly conserved with the help of the cyclic Leibniz rule without spoiling the locality. In use of the cohomological argument, any possible local terms of the effective action are classified into two categories which we call type-I and type-II, analogous to the D- and F-terms in the supersymmetric field theories. We prove non-renormalization theorem on the type-II terms which include mass and interaction terms with keeping a lattice constant finite, while type-I terms such as the kinetic terms have nontrivial quantum corrections.


Introduction
Supersymmetry is not only a fascinating idea for solving the gauge hierarchy problem, but also has many interesting properties worth investigating in its own light, one of which is nonrenormalization of F -terms [1,2] in the Wess-Zumino model. In order to investigate such non-trivial properties by non-perturbative means, a lattice formulation has been desired for a long time [3,4]. It is, however, difficult due to the lack of the Leibniz rule of finite difference operators on the lattice [5,6].
In our previous paper, we proposed a novel approach where we use a finite difference operator and field products which satisfy cyclic Leibniz rule (CLR) instead of ordinary Leibniz rule (LR) [7,8,9]. Both rules are coincide with each other in the continuum limit, i.e. they reduce to the Leibniz rule of differential operator. With the CLR approach we realized a lattice quantum mechanical model in which kinetic and interaction terms are supersymmetric invariant separately, so that we succeeded to apply localization technique to obtain some exact results of the model. An important point we should emphasize here is that there indeed exist concrete sets of finite difference operator and field product which satisfy locality, translational invariance and the CLR. A systematic method for getting a general solution of the CLR is recently found by Kadoh and Ukita [10].
In the present paper, we proceed further to realize more supersymmetries by our approach. We consider N = 4 supersymmetric quantum mechanical model which is obtained by dimensional reduction of N = 2 Wess-Zumino model in two dimensions. We will construct its lattice version which exactly preserve two supersymmetry among four thanks to the CLR. As far as our knowledge is concerned, this is the first example which exactly realize two independent supersymmetries in the lattice model 1 . Two exactly realized supersymmetric charges satisfy maximal nilpotent subalgebra, and surprisingly are sufficient to lead us to the non-renormalization theorem in the lattice model.
Let us note that if we use Wilson term to avoid doubling problem, the term must also keep the same number of supersymmetries. In our approach Wilson term as well as ordinary mass term is constructed in use of the CLR, so that they keep all the necessary supersymmetries and are protected by the non-renormalization theorem from quantum corrections. 1 It is known that a Nicolai map leads to an exact nilpotent supercharge (see Ref. [4]). It is easy to construct a single Nicolai map but it is hard to find a second one. In fact, other approaches without the CLR have not succeeded in constructing two Nicolai maps or two exact supercharges for the model considered in the manuscript. On the other hand, the CLR is nothing but a necessary condition for two exact nilpotent supercharges to exist in the model. This is a reason why we require the CLR for the model. (If we abandon the locality, we could realize a full supersymmetry on lattice. We have however restricted our considerations to a class of lattice models keeping the locality because non-local lattice models might cause trouble in continuum limit.) The paper is organized as follows. In the next section, we first construct a lattice quantum mechanical model with two exact supersymmetries with the help of the CLR. There we introduce a kind of superfields which are slightly different from conventional ones, but useful for the subsequent discussions such as non-renormalization theorem. We then discuss on the one-particle irreducible effective action including quantum correction. Since our model contains all the necessary auxiliary fields, supersymmetries are realized off-shell and the fields are transformed linearly. Thus supersymmetry transformations of the field variables in the effective action have the same form as the elementary field variables in the tree action.
This fact means that the difference operator itself appeared in the transformation has no quantum correction. Since the realized symmetry is a maximal nilpotent subalgebra of the N = 4 supersymmetry, we utilize cohomological argument for the classification of the supersymmetric invariant terms which is discussed in section 3. There are two categories which we call type-I and type-II each of which is an analogue of D-and F-term in the Wess-Zumino model. In section 4, we will see that this model is not a trivial one by showing one-loop quantum corrections for the type-I terms such as the kinetic term. In section 5, we prove non-renormalization theorem for the type-II terms without taking continuum limit. Section 6 is devoted to the summary and discussion. Some useful definitions and formulae are given in appendices.

.1 Supersymmetric transformations and superfields
We consider a lattice version of complex quantum mechanical model with N = 4 supersymmetry. Our model has four sets of complex fundamental lattice fields 2 (i.e. eight real degrees of freedom): where φ ± , F ± (χ ± ,χ ± ) are bosonic (fermionic) variables, and n stands for a lattice site. The lattice constant is set to be unity for brevity. An inner product between these fields is defined These lattice fields are transformed under two supersymmetries δ ± as follows: where ∇ is a local difference operator on a field such as (∇φ) m ≡ n ∇ mn φ n . These transformations are diagrammatically depicted in figure 1. As we will see later, (2) is exactly realized in our lattice theory. The remaining transformations in full N = 4 SUSY are The SUSY transformations δ ± ,δ ± are realized by the following differential operators, From (4), we find the following anti-commutation relations as a part of the N = 4 SUSY algebra, which could be realized on lattice. The last piece of the algebra is given by The problem is that P cannot be realized as an exact symmetry on the lattice, because it does not satisfy the relation P X n = −i(∇X) n for general composite fields X n . This comes from the fact that the Leibniz rule of a finite difference operator cannot be realized on the lattice due to the no-go theorem [5,6]. Therefore we must try to realize only a part of nilpotent subalgebra (5). A possible maximal set of supercharges is either (Q + , Q − ) or For concreteness we take as a maximal nilpotent subalgebra to be realized in our model. From here on we call this nilpotent-SUSY. Our task is now to find functionals consisting of fundamental fields which satisfy To this end, it is helpful to examine cohomology of the nilpotent supercharges as will be seen shortly.
For later convenience, we assign a U(1) charge ±1 to the fields with ± index respectively.
In addition to this , we assign another U(1) charge (we call it U(1) R ) which is defined by the eigenvalue of each field to the operator In the holomorphy-like argument in later section, we will assign these U(1) and U(1) R charges also to the parameters in the action such as coupling constants as well as mass parameter and the Wilson coefficient (a coefficient of the Wilson term). The quantum numbers (fermion number N F , U(1) and U(1) R charges) for fields are summarized in Table 1.
We define following additional operators The U(1) charge defined before is equivalent to the eigenvalue of the operator N + − N − .
These operators and supercharges Q ± satisfy the following algebra This algebra (12) is useful to analyze a cohomology of nilpotent SUSY, because K ± plays an analogous role of homotopy operator.
Let us consider a monomial O k consists of the fundamental fields, where k stands for a collection of lattice site of each field in the monomial, e.g. O n,m,l = χ n φ m φ l . And also consider a linear sum of a monomial over its lattice sites O = k C k O k . If the coefficient C k satisfies the following two conditions, then we say the C k is translationally invariant local coefficient or simply TILC.

1.
Translational invariance: C k is invariant under discrete translation of lattice site, i.e.
C m 0 +1,m 1 +1,··· ,m I +1 = C m 0 ,m 1 ,··· ,m I . Then it is a function only of the site differences, 2. Locality: For each index (site difference) k ℓ , if k ℓ is large enough, then there exist M, L > 0 such that If C is TILC, then we can define holomorphic functioñ which is holomorphic in an I-dimensional complex domain, with ǫ i < M (i = 1, 2, · · · , I). We callC holomorphic-representation or simply H-representation of C.
As for the fields, we introduce "superfields" which are slightly different from conventional ones in the continuum theory, but correspond to a certain rearrangement of the original fields.
With the Grassmann variables θ ± , we define eight superfields Φ ± , Ψ ± , Υ ± and S ± : Then nilpotent SUSY transformations δ ± have simple expressions on the superfields, namely derivatives with respect to θ ± : Furthermore, any functional O of original fields (1) can be expressed in terms of that of where we used the following integral formulae for the Grassmann variables θ ± , Note that our superfields are merely a change of variables from the original fields; the number of superfields is the same as that of the original fields, and the expression of any quantity in terms of original fields can be recovered from that of superfields by putting θ ± = 0.
Nonetheless, because of the simplicity of the expression (18) of Q ± , the relations (19) and (20) will largely simplify the classification of supersymmetric invariant local functionals in the fermion number zero sector in later section. Therefore, we shall use these eight superfields instead of eight fundamental fields for the discussion of the Q ± -cohomology and for a proof of nonrenormalization theorem. Quantum numbers for these superfields and θ ± are assigned as listed in Table 2.

Nilpotent-SUSY invariant action and cyclic Leibniz rule
Now, we write down a kinetic term S 0 of the nilpotent-SUSY invariant tree action, Due to the translational invariance and locality, the difference operator ∇ in the above is actually a function of the site difference, i.e. ∇ mn = ∇(m − n), and also is bounded for large |k| as |∇(k)| < L exp(−M|k|) with positive real numbers L, M; in other words, it is TILC.
In general, this kinetic term may have a doubling problem. In order to avoid it we introduce Wilson terms as well as ordinary mass terms, Here G ± expresses mass and Wilson terms for ± fields. As a typical example, in the case of (∇) mn = (δ m+1,n − δ m−1,n )/2, we can take the term (G + ) mn = m + δ m,n + r + (δ m+1,n − 2δ m,n + δ m−1,n )/2, where m ± are mass parameters with m * + = m − and r ± are Wilson parameters with r * + = r − . In the H-representation, they are expressed as Note that, as far as site indices are symmetric, any choice of (G ± ) mn = (G ± ) nm makes S m invariant under nilpotent SUSY (2) with the symmetric difference operator (∇) mn = −(∇) nm . 4 Let us turn to the interaction terms. We begin with the simplest case, namely three point interaction which can be written in the form where we denote a two-field product as i.e.
where the last k indices of the coefficient M are totally symmetric. The above * -product corresponds to the k = 2 case: Again if this multi-field product satisfies the following CLR for any k + 1 bosonic fields B (i) i.e.

Effective action on lattice and Ward-Takahashi identities
We will discuss quantum corrections for our model in the subsequent sections. In order to Here we use an index i for distinguishing various fields and n for lattice sites. The argument field φ i n of Γ(φ) is defined as an expectation value of ϕ i n under the source J: We use left-derivative convention for fermionic fields. So we have with grassmann parity |A| = 0 or 1 for grassmann even or odd field A, respectively.
Let us make the change of integration fields from ϕ to ϕ + δϕ in eq. (35). If we choose δϕ as an infinitesimal symmetry transformation and the action S(ϕ) and the measure Dϕ are invariant under the transformation, then we obtain the identity that is, Since our supersymmetry transformations Q ± are linearly realized in the typical form of we find Thus defining the transformation for φ i n as δφ i n = ǫ j,m A ij nm φ j m , we obtain the Ward-Takahashi identity for the effective action This shows that the supersymmetry transformation for the effective action has the same form of a tree level action. This comes from the fact that we retain auxiliary fields and the transformation is kept linear. Also notice that the coefficients A ij nm in the transformation for the original field ϕ is inherited to those of φ, which means the difference operator in the coefficients is the same both for ϕ and φ. In other words, we can just replace original fields by its expectation values in supersymmetry transformations Q ± and the Ward-Takahashi identity becomes In the following sections we use the same symbol for the original field and its expectation value in the discussion of the effective action for simplicity.

Cohomology of Q ± and the classification of Q ± -invariant functionals of fields
In this section, we are going to analyze cohomology of the nilpotent-SUSY transformations Q ± and classify all Q ± -invariant functionals of fields with TILC. As will be seen shortly, there are two categories which we call type-I and -II. Typical examples of the former are kinetic terms, while those of the latter are mass terms and interaction terms appeared in the tree action.
Since Q ± do not change the number of '+'-fields nor that of '−'-fields, we can safely restrict our discussion in the subsector with definite number of '±'-fields without loss of generality. If we write O(n + , n − ) for the functional consisting of n + '+'-fields and n − '−'fields, then for the operator defined in (11).
Immediate consequence at this point is the following. In use of the algebra {Q ± , K ± } = N ± in (12), Q ± -closed functional O(n + , n − ) can be written in the form Thus Q + and Q − cohomologies in the space of functionals {O(n + , n − )} are trivial for n + = 0 and n − = 0 respectively. Our next task is to determine Q − cohomology in the subspace {O(n + , 0)} and Q + cohomology in the subspace {O(0, n − )}. Before stating the result, let us define the CLR terms in the N F = 1 sector as where the coefficients C are TILC and satisfy the CLR relation (32). Note that CLR terms with N F = 1 have the maximum U(1) R charge (eigenvalue of the operator R) in the space {O(k + 1, 0)} and the minimum in {O(0, k + 1)}.
Then we have the following theorem for N F = 1 sector:

Theorem 1. (Fundamental theorem on the cohomology of nilpotent SUSY)
If P + is a local functional in {O(n + , 0)|n + > 0} with N F = 1 and satisfies then P + can be written in the form where Q + is a local functional in {O(n + , 0)|n Similarly, if P − is a local functional in {O(0, n − )|n − > 0} with N F = 1 and satisfies then P − can be written in the form The proof for this theorem is shown in Appendix C with some preparations in Appendix B.
We turn to discuss a nilpotent-SUSY invariant local functional S with N F = 0. The quantity shall be useful in discussing an effective action and proving a nonrenormalization theorem. where Proof. Since nilpotent SUSY-transformations Q ± do not change the number of +-fields nor −-fields, we can separately argue each term with definite numbers of ±-fields. For a term with n + = 0 and n − = 0 (let us denote T ), from (45) and (46) any nilpotent-SUSY invariant local functional takes the following form, Note that both K + and K − map from local functionals to local functionals.
For a term with n − = 0, n + = 0 and N F = 0 (let us denote U + ), any nilpotent-SUSY invariant functional can be written as From Q − U + = 0 and {Q − , K + } = 0 in (12), it follows that P + ≡ K + U + /n + is a Q − -invariant local functional with N F = 1. Thus we can apply Theorem 1 to P + , so that we have where Q + is a local functional with n + > 0, n − = 0 and N F = 2. Thus From similar discussions, for a nilpotent-SUSY invariant local functional V − with n + = 0, n − = 0 and N F = 0, we can get where Q − is a local functional with n + = 0, n − > 0 and N F = 2.
Thus, putting all terms together, we have This is indeed a form (52). 5 The result (52) of Proposition 1 means that nilpotent-SUSY invariant local functionals thing is that functionals of type-II are only CLR terms.
Let us consider a superfield counterpart of Proposition 1. From (19) and (20), we can translate a nilpotent SUSY transformed functional Q ± O into a functional of superfields, And also, By utilizing (59), (60) and (52), the next proposition follows: where the "±type CLR terms with N F = 1" consisting of k + 1 superfields (k = 0, 1, 2, · · · ) are given by whose coefficient C is a TILC satisfying the CLR.
In Section 5, we will use a trick in which we lift constant parameters in the model, such as coupling constants, mass parameters and Wilson-term coefficients, to constant superfields by introducing constant super-partner for each constant parameter. Therefore we need an extended version of Proposition 2 including constant superfields. Let us take N CF constant parameters ρ i ± (i = 1, · · · , N CF ) as constant fields, and their super-partners ζ i ρ± (i = 1, · · · , N CF ) as well. Then Q ± , K ± , N ± are modified to include constant fields: and their algebra is the same as before: We assign N F = 0 for ρ i ± and −1 for ζ i ρ± . Then, ±type local functionals are defined by new operators as We should notice that the constant fields do not depend on lattice site, so the coefficient appeared in functionals only depend on the site index of ordinary fields to which the notion of locality refers.
From (63), we can easily show an extended version of the fundamental theorem as follows, Let P ′ ± be any ±type local functional with N F = 1. If then P ′ ± = (±type CLR extended terms with where ±type CLR extended terms with N F = 1 of k + 1-th order are expressed as and the coefficient C ′ is a TILC satisfying the CLR condition and may depend on constant fields ρ ± . Q ′ ± are N F = 2 local functionals which may depend on ρ i ± , ζ i ρ± .
The proof goes almost same way as that of the previous Theorem 1 with a care of the fact that ρ i ± , ζ i ρ± are invariant under δ ∓ . We also note that ζ i ρ± does not contribute to nontrivial CLR terms with N F = 1, likeχ ± .
From this theorem, we can prove two additional propositions: where T ′ (+, −) is a local functional with N F = 2.
where the above ±type CLR extended terms with N F = 1 of k + 1-th order is defined as and the coefficient C ′ is a TILC satisfying the CLR condition and may depend on constant superfields ρ i ± (θ ± ).
We could think the mass term as one of the interaction terms and might pursuit our perturbative calculation with massless propagator. In order to avoid infrared divergences, instead, we use perturbation with massive propagator. We denote a symbol · · · 0 as an expectation value in the tree-level. By defining D mn = D nm ≡ (∇ T ∇ + G − G + ) −1 mn , the tree-level propagators can be written as φ ∓m φ ±n 0 = D mn , φ ±m φ ±n 0 = 0, where (∇ T ) nm = (∇) mn . Note that our mass operators G + , G − = (G + ) † include Wilson terms, thus have site-dependence in general. In terms of superfields, relevant 2-point func- where δ-functions for the Grassmann variables are defined as δ(θ ± ) ≡ θ ± , δ 2 (θ) ≡ θ + θ − .
For the interaction term, we only consider three-point interaction for simplicity: where λ ∓ = λ * ± and the coefficient M in the definition of * product (27) is a TILC satisfying the CLR condition,

Corrections for kinetic terms
One-loop corrections to the kinetic term F − F + , for instance, are given by Figure 3. Here internal lines with filled circles symbolize massive propagators.
At the one-loop level, we can write down the F − F + propagator with correction in the form and the self-energies at the one-loop are explicitly given by and where we have used the symmetric property of D mn = D nm in (75) and the CLR relation  The χ −χ+ propagator is evaluated at the one-loop level as where In this calculation the CLR (78) is important. Similarly, Forχ − χ + at the one-loop level, the propagator is evaluated as where and In use of the CLR (78), self-energy parts (80), (86) and (87)  Finally, the φ − φ + propagator at the one-loop level becomes In summary, we have evaluated one-loop contributions to the part of self-energy which mixes ±-fields and thus to the type-I functionals, i.e. kinetic terms. Relevant diagrams are shown in Figures 3, 4, 5 and 6. Note that Also we can see each self-energy is related to each other by using the CLR (78) repeatedly, These equations are exact relations thanks to the CLR and are nothing but a part of the SUSY Ward-Takahashi identities.

Quantum effects on type-II functionals
Let us turn to the type-II functionals, the simplest of which is the mass term. So we have to evaluate self-energy parts which do not mix ±-fields. Immediate consequences we obtain are the vanishing of Σ ++ F φ and Σ ++ χχ due to the vanishing of φ + φ + 0 from (75). (We can easily see from Figure 7 that these self-energy parts are proportional to φ + φ + 0 .) We emphasize here that not only pure mass term but also Wilson term have no quantum correction at one-loop (and actually any loop order as will be shown in next section), thus the Wilson parameter is unchanged.
The situation becomes a bit nontrivial in two-loop order. Actually, for the diagrams  in Figure 8, the correction for F + φ + is proportional to λ 4 + φ + φ + 2 0 and thus it vanishes as before. On the other hand, diagrams in Figure 9 with λ 2 + λ 2 − can be shown to be proportional to the difference operator ∇ by the CLR (78), where D ±F φ ij ≡ F ±i φ ±j 0 . Since n is the site index of external φ + , the correction to the effective action is proportional to ∇φ + . Therefore, this effect can be shown to be a 2-loop contribution to type-I functional.
5 Non-renormalization theorem on the lattice

Perturbative proof of the nonrenormalization theorem for type-II functionals
In the previous section, we explicitly saw that there is no quantum correction in mass and Wilson terms at the one loop level, while the kinetic terms have nontrivial quantum corrections. In this section, we show the non-renormalization theorem that the type-II terms are not suffered from quantum corrections at all.
Our starting action in the tree level is where Here we have a little bit generalized from the preceding sections by adding a linear term v ± Ψ ± in the action as a type-II functional. The inclusion of the linear term is not a necessary thing but the argument below goes well either with or without this term. Also we have slightly changed our notation as G ± → m ± G ± from (24), and so this is accompanied by the change of the Wilson coefficient r ± → r ± /m ± .
Type-II part of the effective action with full quantum corrections can be separated into the sum of tree action S type-II and quantum correction ∆Γ eff type-II Γ eff type-II = S type-II + ∆Γ eff type-II .
First of all, we are going to show that the latter should be of the form Here emphasis is on that W eff , and this holomorphic property (even in the parameter dependence) will turn out to be crucial in the proof of the non-renormalization theorem.
To this end, we replace the parameters v ± , m ± and λ ± by constant superfields such a way as [2] where the lowest components of v ± (θ ± ), m ± (θ ± ) and λ ± (θ ± ) correspond to the original parameters v ± , m ± and λ ± . Note that the tree action (96) is still supersymmetric under the replacement (102) with the supersymmetry transformations: Then, it is straightforward application of Proposition 4 to show that ∆Γ eff type-II can be super- As a next step, we further restrict the form of the functions W eff ± (Φ ± ; v ± , m ± , λ ± ) by assigning the fermion number N F , the U(1) and U(1) R charges to the fields, as listed in Table 3. Then, ∆Γ eff type-II is found to be generally expressed as where f ± (z ± , w ± ) are some holomorphic functions of the complex variables In this perturbative calculation, we may expand the functions f ± (z ± , w ± ) in powers of z ± and w ± as where a ± kl are some constant coefficients. The SUSY invariance then requires the CLR relations as they should be. Since we are considering perturbation theory, in which the weak coupling limits of v ± , λ ± → 0 are assumed to exist, the powers of k and l should be restricted to in order for ∆Γ eff type-II to be non-singular. Note that the functions f ± in (104) are non-singular in the weak coupling limits of v ± , λ ± → 0 with (107).
Let us next clarify what kind of diagrams lead to the terms given in (105). To this end, we use the topological relation for Feynman diagrams where L, I, V 1 and V 3 denote the numbers of loops, internal lines, v ± -vertices and λ ± -vertices of a Feynman diagram, respectively. Further, we have the relation because we have one-point vertices v ± and three-point vertices λ ± in the tree action. Here, E denotes the number of external lines. From (108) and (109), we obtain Since the values of V 1 , V 3 and E for each term in (105) are given by V 1 = k, V 3 = k + l − 1 and E = l + 1, corresponding diagrams turn out to be tree ones without loops, i.e.
Therefore, all the terms given in (105) have to be excluded from ∆Γ eff type-II because the effective potential ∆Γ eff type-II consists of 1PI diagrams only. This implies that there is no quantum correction to the tree type-II action S type-II , i.e. Γ eff type-II = S type-II .
This completes the perturbative proof of the nonrenormalization theorem on the lattice.
In the essential point of our proof, any nilpotent-SUSY invariant local functional is classified into type-I or type-II. In the addition, possible type-II local functionals are only terms so called as CLR-type. 7

Consideration of nonperturbative nonrenormalization property for type-II functional
We briefly consider a non-perturbative justification beyond a perturbative proof. Let us assume that no massless mode appears even non-perturbatively in the massive theory.
The assumption leads us that our theory is sigularity-free at the origin in the coupling constant space with weak fields. In our quantum mechanical model, the assumption seems natural.
But for higher dimensional cases, it we may need more careful treatment.
Any way, from the assumption, the complex analysis in the previous subsection tells us that the holomorphic functions f ± (z ± , w ± ) in (104) equals to zero in a neighborhood of the origin, since the perturbative proof in the previous subsection holds there. Then f ± (z ± , w ± ) vanish identically on the whole complex plane because of the analytical continuation or the identity theorem in complex analysis. This suggests our nonrenormalization theorem holds nonperturbatively.

Summary and discussion
In this article, we have constructed a supersymmetric complex quantum mechanics model on lattice. The action is invariant under two nilpotent-SUSY transformations (Q ± ) which form a maximal nilpotent subalgebra of full N = 4 SUSY and they keep a holomorphic property.
Furthermore, they enable us to study Q ± cohomology exactly. As the result of the analysis, we can classify all local Q ± -invariant functionals with N F = 0 into type-I (such as kinetic terms) and type-II (such as a mass term including the Wilson term and interaction terms.) The local functionals in a nilpotent-SUSY invariant effective action is also classified into the two types and we have proved nonrenormalization theorem for the type-II local functionals at any order of perturbative expansion and without taking continuum limit, namely with finite lattice constant. This means that we are able to realize more than one nilpotent-SUSY and the holomorphy even in a regularized theory and this is extremely nontrivial result. For nonperturbative justification of the nonrenormalization property of type-II functionals, we gave reasonable arguments.
The reasons why we succeeded to prove the theorem are (1)  Finally, we would like to emphasize notable features of our lattice models. Note that supersymmetry in continuum theories is not the Lagrangian symmetry but the action one.
That is, supersymmetric Lagrangians are invariant under supersymmetry transformations up to total divergences, in general. Our lattice model holds this property. Under the supersymmetry transformations (2), our Lagrangian on lattice is not invariant but the action becomes invariant with the CLR by taking the summation over the lattice sites. Thus, our lattice model mimics an important feature of continuum supersymmetric theories. We would like to notice that other lattice supersymmetric models without the CLR do not possess this property and fail in inheriting a property with respect to translations in supersymmetry.
Another notable feature of our lattice model is as follows. Our lattice model has been shown to be cohomologically non-trivial. This property turns out to be crucial to prove the non-renormalization theorem for F-terms, as discussed in the manuscript.
The locality implies thatÃ(z 1 , z 2 , z 3 ) is holomorphic in a domain D = {1 − ǫ < |z i | < 1 + ǫ | ǫ > 0, i = 1, 2, 3}. Although we refer [5] and [7] for the detailed arguments on locality, we note that the meaning of locality include not only ultralocality but also exponential damping. In this article, a terminology, local functional or functional with locality, used as a collection of fields with a local coefficient. A local difference operator ∇ mn in this paper is also a TILC From this hermiticity, we can show the reality of our action with (22), (23), (26) , We note that any two-point functions with translational invariance and locality commute with each other in the sense of matrices.
for M ≤ N. Note that ∇ has two site-indices and the coefficient has N + 1 site-indices in lattice site representation.
The solution of (1) can be generally written as whereÃ {ab} for a, b = 1, 2, · · · M are holomorphic functions in the domain D N .
As an example, we explain the above result in the case of m = 2 below, although (13) generally holds. For m = 2, (12) becomes Note that the two zeros z (0) k 2 are not necessarily the same. We substitute this to the following identitỹ Here the Lagrange's interpolation functions are used. The result is Since z 1 = z (0) k 1 and z 2 = z (0) k 2 are removable singularities on the right hand side of (17) and the multiplication of f k (z) for holomorphic functions does not change the holomorphy on the annulus, we obtaiñ whereB (3) a are holomorphic functions in D 3 . From (13), (11) can be regarded as an inhomogeneous linear equation forÃ where the 1st term on the right-hand-side gives a special solution to (11) and the 2nd term corresponds to the general solution for a homogeneous equation m a=1Ã (m+1) a (z 1 , · · · , z N )∇(z a ) = 0 (see (10)). By definingÃ we obtain the general solution (2) for M = m + 1.
By repeating the above argument, the following corollary is immediately obtained. If M a 1 =1Ã for M ≤ N, theñ Here allÃ are holomorphic functions in a domain D N .
In applying these results to the proof of the fundamental theorem of Q − -cohomology, we where C's are some TILC. Although the index k seems redundant, at least one extra index is necessary for writing down corresponding lattice site representation, since the number of independent indices are one less than the total number of indices for the translationally invariant coefficient.

C Proof of fundamental theorem on the cohomology of nilpotent SUSY
Since the analysis of −type fields is similar to that of +type fields, we concentrate on only + fields (+type functionals) and Q − in this appendix. Thus, we omit the subscript of fields as where n + and K are the number of +type fields and the combined number of F andχ, respectively. When K = 0, namely R = R max ≡ n + − 1, the functional O = Cχφ · · · φ is called as a CLR term with N F = 1. The N is defined as the total number of φ and χ. K ×F k 1 · · · F k K−pχ ℓ 1 · · ·χ ℓp χ m 1 · · · χ m p+1 φ n 1 · · · φ n N−p .
There remains an extra condition (6). For K > N and p = N, the condition (6) reads By directly solving (4), we also have for N = 0. It is consistent with N = 0 case of (16).