Off-shell Invariant D=N=2 Twisted Super Yang-Mills Theory with a Gauged Central Charge without Constraints

We formulate N=2 twisted super Yang-Mills theory with a gauged central charge by superconnection formalism in two dimensions. We obtain off-shell invariant supermultiplets and actions with and without constraints, which is in contrast with the off-shell invariant D=N=4 super Yang-Mills formulation with unavoidable constraints.


Introduction
Supersymmetry (SUSY) is one of the most important guiding principle in contemporary particle physics. In particular it has been recognized that D=N=4 super Yang-Mills (SYM) theory plays a crucial role in string motivated gauge theory formulations [1]. It has also been recognized that D=N=2 and D=N=4 SYM formulation play a special role for Dirac-Kähler twisting procedure [2], which gives links to quantization and supersymmetry [2,3], and to lattice supersymmetry [4]. There is a long-standing question if one can find a superspace formulation for D=N=4 SYM to obtain off-shell invariant formulation. It is known that D=N=4 SYM with SU(4) R-symmetry can be formulated only on-shell level [5] while one can find off-shell invariant SYM formulation if we introduce a central charge and change the R-symmetry from SU (4) to USp(4) [6,7]. In this case we, however, need a constraint equation which can be seen as remnant of an equation of motion of higher dimensions. The corresponding superspace formulation has been developed in [8,9,10]. Harmonic superspace has also been developed in the similar context [11].
One may thus ask a question whether or not such a constraint is unavoidable for gauged central charge SYM formulation. In this paper we investigate D=N=2 off-shell invariant twisted SYM with gauged central charge by suerconnection formalism [12,13]. This kind of gauged central charge formulation is called vector-tensor multiplet and has been investigated intensively [14]. It turns out that for A model ansatz we obtain off-shell invariant formulation with a constraint. Although this constraint can be solved in the case of Abelian gauge group, it is not possible to solve the constraint at least locally in the case of non-Abelian gauge group [15]. For B model ansatz we obtain off-shell invariant formulation without any constraints. This paper is organized as follows: In Section 2 we first discuss twisted superalgebra with central charges. In Section 3 we consider three models for three different supercurvature ansatz. Then we obtain supermultiplets and actions in Section 4. We then summarize the result and give some discussions in the last section.

D=N=Twisted Superalgebra with Central Charge
In this section we introduce central charge to N=2 superalgebra in two dimensions and perform Dirac Kähler twisting. We concentrate on Euclidean spacetime in this paper according to the twisting procedure [2].

Superalgebra with central charge
In two dimensional Euclidean spacetime γ-matrices satisfying {γ µ , γ ν } = 2δ µν and charge conjugation matrix C can be chosen to satisfy [16] Cγ µT C −1 = γ µ , C T = C. (2.1) Thus we can choose the representation of these matrices as Note that γ µ and γ 5 are symmetric and antisymmetric matrices, respectively.
The general extended superalgebra is given by where Q αi is supercharge and R 1 , R 2 are generators of R-symmetry. Here we consider D=N=2 case. Majorana condition is given by Q αi = Q αi * . Jacobi identities w.r.t. Q, Q, R 1 and Q, Q, R 2 lead, respectively, S ij = −S ji and S ′ ij = −S ′ ji , which means R 1 and R 2 generate U (1) symmetry.
We now introduce possible extra terms as follows: where U ij = U ji , V ij = −V ji to be consistent with simultaneous replacements of α ↔ β and i ↔ j. U ij and V ij get the following restrictions according to a Jacobi identity w.r.t. Q, Q, R 1 : We can then solve the constraints up to an over all constant as On the other hand Jacobi identity w.r.t. Q, Q, R 2 leads to the relation: which can be solved as 8) where u, u ′ are real parameters. The solutions (2.6) and (2.8) are not compatible. In other words we cannot keep both of R 1 and R 2 symmetries. In case we choose R 1 (≡ R) symmetry we obtain the following algebra: where we identify U 0 and V 0 as central charges. On the other hand if we choose R 2 symmetry, we cannot carry out Dirac-Kähler twisting procedure. We thus not choose this case.

Twisted Superalgebra
Dirac-Kähler twisting procedure includes two steps: Expansion of supercharge by complete set of γ-matrices and redefinition of Lorentz rotation generator [2,9,13].
We identify the representations of R-symmetry as that of spinor, and treat the extended SUSY suffix and spinor suffix of supercharge Q αi in the same manner. We thus expand the charge as Q αi = (1s + γ µ s µ − iγ 5s ) αi , (2.12) where s, s µ ands are called twisted supercharges. Note that these supercharges can be expressed by the original charge as The charges may be looked strange because s µ has, for example, vector suffix although it is fermionic charge. This Dirac-Kähler mechanism can be understood in the following. In two dimensions Lorentz generator is represented by one component generator J satisfying On the other hand we can rewrite (2.10) in the same form as (2.15) because S ij and γ 5 ij are both antisymmetric and thus can be chosen to be proportional to each other, from (2.13), (2.15) and (2.16). These relations mean that twisted supercharges s, s µ ands transform as scalar, vector and (pseudo-)scalar under J ′ , respectively. As can be seen above the equivalence between Lorentz group and R-symmetry group is required to realize Dirac-Kähler twist. R-symmetry group is inevitably a compact group and thus Lorentz group need to be also compact. There is a natural reason that Euclidean spacetime is chosen.
The algebra among the twisted supercharges is derived from (2.9) and (2.13) as follows: This is the N=2 twisted superalgebra with central charges in two dimensions.

Ansatz on Supercurvature
In this section we consider so-called superconnection formalism and find appropriate ansatz on supercurvatures based on the algebra derived in the previous section [9,13]. We introduce superfields in the superspace parametrized by (x µ , θ A , z) where θ A represents θ, θ µ andθ which are Grassmann coordinates, and z is a real parameter associated with a central charge. Using the supercharge differential operator Q A generating a parameter shift in the superspace, we define the supertransformations of the component fields φ, φ A , · · · , as follows: One can find the supercovariant derivative D A which anticommutes with Q A and then introduce fermionic gauge supercovariant derivative as where Γ A is a fermionic superfield called superconnection. Similarly bosonic gauged supercovariant derivatives are introduced as where Γ µ and Γ z are bosonic superfields. The gauge transformation of where K is a gauge parameter superfield. The zeroth order terms of Γ A and Γ z w.r.t. θ A can be taken to be 0 by choosing Wess-Zumino gauge while that of Γ µ is defined as a usual gauge connection where | represents the zeroth order term w.r.t. θ A . We can thus define standard gauge covariant derivative as We then define the supercurvatures by (anti-)commutation relations of all pair of ∇ I . The supercurvatures transform gauge covariantly under (3.5). Then some suitable ansatz on supercurvatures leads to an irreducible supermultiplet. To find such ansatz it is useful to introduce supercurvatures X, X ′ and X µ defined as where ∇ αi are gauged supercovariant derivative corresponding to Q αi in (2.9). Right hand side in (3.8) is the most general terms for the consistency with simultaneous replacement of α ↔ β and i ↔ j. As can be seen from (2.9), X and X ′ can be identified as gauged central charge of U 0 and V 5 , respectively. It may be further possible to identify X or X ′ as ∇ z . Table 1 shows the relations between gauged supercovariant derivative and supercurvaturs given in (3.8) in the twisted space. It is in principle possible to find different types of supercurvature ansatz. We eventually find three types of ansatz. The first ansatz is shown in Table 2. Here one of X and X ′ is identified as ∇ z . We call A model ansatz when bosonic scalar supercurvature (W in the case) is placed in diagonal positions. It is also possible to include X µ as supercurvatures. In this case the bosonic vector superurvatures are placed in off-diagonal positions, which we call B model ansatz. One naive candidate for B model ansatz is given in Table 3. Jacobi identities, however, lead to G I = G µ = 0, which coincides with a model without central charge. On the Table 1: Twisted version of supercurvature ansatz of (3.8).     Table 4 and Table 5 we show the two kinds of ansatz which we name B (0,0,Z) model ansatz and B (Z,Z,0) model ansatz, respectively. Once suitable ansatz on supercurvatures is obtained, a set of relations between the supercurvatures can be derived from Jacobi identities w.r.t. ∇ I , by which degrees of freedom of the component fields are reduced. We use a notaion ∇W ≡ [∇, W ] . For example the component fields in an irreducible supermultiplet can be defined as where W represents a supercurvature while A and ρ are bosonic and fermionic component field, respectively. The supertransformations and central charge transformations are obtained by The third equality holds at the zeroth order of θ A while the fourth equality holds due to the first relation of (3.6). More complicated supertransformations can be defined by more sophisticated Jacobi identities. One thus obtain all supertransformations of each component field in an irreducible supermultiplet.

Supermultiplets and Actions
We derive the supermultiplets and the actions for each model with supercurvature ansatz found in the previous section.

A model
We now consider the following algebra: {s, s µ } = P µ , {s, s µ } = −ǫ µν P ν , {s,s} = 0 , where + represents the case Z = U 0 and − represents the case Z = −V 5 of (2.18) in the double sign. The corresponding supercharge differential operators for the superspace are given by where {Q A , D B } = 0 . It should be noted that D A satisfies the same algebraic relations as (4.1) with the identification of D A → s A , while Q A satisfies the similar relations with the replacements; Q A → s A , P µ → −P µ and Z → −Z in (4.1). We now consider supercurvature ansatz in Table 2. The following relations can be derived by Jacobi identities: (4.4) In addition to them we need to impose the following relation: This relation is not derived by Jacobi identity but interpreted as a constraint on supercurvatures to kill the reducibility of representation. The component fields are then defined as (4.6) where φ, D and g µ are bosonic fields, and ρ,ρ and λ µ are fermionic fields. Table 6 shows the supertransformations of each component field. Note that F µν | ≡ F µν = i[D µ , D ν ] is a curvature in usual gauge theory. Off-shell closure of the supertransformations up to gauge transformations is shown with the following constraint on the component fields: One can regard this constraint as the same type of constraint found for D=N=4 SYM case [7]. Because of the constraint the degrees of freedom of g µ can be regarded as one. Thus the bosonic degrees of freedom at the off-shell level is four (φ, A µ , D, g µ ). Note that the gauge field A µ has one bosonic degree of freedom at the off-shell level.
For Abelian gauge group the constraint (4.7) becomes simply as (4.8) which can be solved as The degrees of freedom is in fact one. The explicit form of an action which includes field B is where e is a parameter with mass dimension 1 . In this case invariance of the action and closure of superalgebra are satisfied without constraints. It is interesting to note that topological BF term is included in the action. For non-Abelian gauge group the constraint cannot be solved locally [15]. Finally one can find an action for non-Abelian gauge group as It is worth to mention that this action cannot be derived by superspace. In this subsection we found a SYM formulation with a constraint. In the next subsections we can find SYM formulations without constraints.

B (0,0,Z) model
The following algebra is considered: where Z = 2U 0 = 2V 5 in (2.18). The corresponding supercharge and supercovariant derivative differential operators are given by B model (0,0,Z) ansatz is shown in Table 4. The following relations are derived by Jacobi identities: ∇G =∇G = ∇G +∇G = 0 , (4.17) The component fields are defined as where φ µ and D are bosonic fields and ρ,ρ and λ µ are fermionic fields. The supertransformations of each component field are derived straightforwardly. In contrast with the previous model where G A is related to W , G andG should satisfy (4.17) and seem to be independent from F µ as far as Jacobi identities are concerned. Superalgebra is closed up to gauge transformations without constraints at the off-shell level.
To obtain the action we introduce linear combination of s µ We define λ ± ≡ λ 1 ± iλ 2 similarly, and introduce the notation ∇ ± µ ≡ ∇ µ ± F µ and D ± µ ≡ D µ ± φ µ for convenience. Then one can derive action by using the nilpotency of s,s and s ± . In fact we can find S 1 , S 2 and S 3 satisfying sS 1 =sS 1 = s + S 2 = s − S 3 = 0 where S 1 , S 2 and S 3 are not generally identical. However, in the case of ∇G =∇G = 0 together with (4.17), we find where (4.25) which corresponds to the action without a central charge and to the twisted versions of the action in [17]. Here it is important to recognize that we can find solutions satisfying ∇G =∇G = 0 and (4.17), where a is a parameter with mass dimension −1. Moreover the above choice of G andG makes S 2 and S 3 identical We can then find the following action satisfying sS =sS = s µ S = 0, where the supertransformations are given in Table 7, (4.28) In Table 7 the following expressions are used: whereρ ≡ ρ − 1 2 G| andρ ≡ρ − 1 2G | .
where Z = 2U 0 = −2V 5 . The model is completely similar in construction to the previous model, we thus show mainly results. B (Z,Z,0) supercurvature ansatz is shown in Table 5. The component fields are defined as in (4.19).
The key relation derived by Jacobi identity is Similar to the previous model G µ is not directly related to F µ by Jacobi identity, it is then necessary to solve (4.31). As far as the above relation holds one can derive the supertransformations of each component field and show off-shell closure up to gauge transformations without constraints.

Conclusion and Discussions
We have constructed off-shell invariant N=2 twisted SYM theory with a gauged central charge in two dimensions. Depending on the supercurvature ansatz we have introduced A and B models. In A model superalgebra is closed at off-shell level with an extra constraint (4.7). This model has a similarity with D=N=4 SYM theory with gauged central charge with unavoidable extra constraint. For Abelian gauge group we can explicitly solve the constraint (4.7), we can thus construct the off-shell supertransformations and action without any other constraints. It is interesting to note that the action has two dimensional topological BF term. We cannot, however, solve the constraint for non-Abelian case.
On the other hand we have found two types of B model whose superalgebra is closed at the off-shell level without any constraints. This gives us a hope that we may use the similar ansatz of B model in four dimensions to get off-shell invariant N=4 formulation without constraints [18].