Anti-symmetric rank-2 Matter Field on Superspace for N_{T}=2

In this work, we discuss the interaction between anti-symmetric rank-two tensor matter and topological Yang-Mills fields. The matter field considered here is the rank-2 Avdeev-Chizhov tensor matter field in a suitably extended $N_{T}=2$ SUSY. We start off from the $N_{T}=2$, D=4 superspace formulation and we go over to Riemannian manifolds. The matter field is coupled to the topological Yang-Mills field. We show that both actions are obtained as $Q-$exact forms, which allows us to write the energy-momentum tensor as $Q-$exact observables.


Introduction
Topological field theories such as Chern-Simons and BF-type gauge theories probe space-time in its global structure, and this aspect has a significative relevance in quantum field theories. On the other hand, there is great deal of interest in anti-symmetric rank-2 tensor fields that can be put into two categories: gauge fields or matter fields. In recent years, Avdeev Chizhov [1,2,3] proposed a model where the antisymmetric tensor behaves as a matter field.
In a recent work [4], Geyer-Mülsch presented a formulation until then unknown in the literature, which is a construction of the Avdeev-Chizhov action described in the topological formalism [5]. This was built for N T = 1 and generalized for N T = 2. Known the properties of the anti-symmetric rank-two tensor matter field theory, also called Avdeev-Chizhov field [6], the supersymmetric properties and characteristics are presented also in ref. [7]; following this formalism, we shall write this action in the superfield formalism, as presented by Horne [8] in topological theories as a Donaldson-Witten topological theories [9,5].
Our goal in this work is to discuss the interaction between matter and topological Yang-Mills fields as presented by Geyer-Mülsch [4] for N T = 1 and N T = 2. The matter field considered here is the rank-2 tensor matter field as a complex self-duality condition [6]. Thus, we write this field now as an anti-symmetric rank-two tensor matter superfield in N T = 2 SUSY in the superspace formalism, founded also in [7]. The matter field is coupled to the topological Yang-Mills connection by means of the Blau-Thompson action. We write the Yang-Mills superconnection as a 2−superform in a superspace with four bosonic dimensions spacetime described by Grassmannodd coordinates and two fermionic dimensions described by Grassmann-even coordinates, and them construct the action in a superfield formalism following the definitions by Horne [8]. Then, we go over to Riemannian manifolds duely described in terms of the vierbein and the spin connection, where we take the gravitation as a background. We introduce and discuss the Wess-Zumino gauge condition induced by the shift supersymmetry better detailed in [10]. Then, we arrive at a topological invariant action as the sum of the Avdeev-Chizhov's action coupled to the topological super-Yang-Mills action; both actions are obtained as Q−exact forms, and the energy-momentum tensor is shown to be Q−exact.
2 The N T = 2 Super-conection, Super-curvature and Shift Algebra Let us now consider the Donaldson-Witten theory, whose space of solutions is the space of self-dual instantons, F = * F . To follow our superfield formulation, we shall proceed with the definition of the action of Horne [8] and Blau-Thompson [13,14]. The N T = 2 superfield conventions are the ones of [10]. The superfields superconnection and its associated superghosts are given as below: A =Â a T a ,Ĉ =Ĉ a T a , (2.1) whose the generators belonging the Lie algebra: Expanding the superforms (2.1) in component superfields, we havê with I = 1, 2; in component fields, it comes out as below: The associated supercurvature is defined aŝ which can also be expressed as:F = F + Ψ I dθ I + Φ IJ dθ I dθ J , whose components read as follows: where f = da + a 2 and the covariant derivatives in a being given by D a (·) = d(·) + [a, (·)]; the symbol (·) represents any field which the derivative act upon. This formalism with N T = 2, it can be found as an example in the work [11].
The SUSY number, s, is defined by attributing −1 to θ. Thus, the supersymmetry generators, Q, have s = 1. The BRST tranformation of the superconnection (2.3) is sÂ = −dĈ − [Â,Ĉ] = −DÂĈ and component superfields, is given by which in components take the form: (2.12) and the super-covariant derivative is decomposed as: The supersymmetry transformations or shift symmetry transformations are defined as: in components, they read as follows: (2.13) Next, we believe it is interesting to introduce and discuss a sort of Wess-Zumino gauge choice associated to the shift symmetry above, which is the topological BRST transformation. The Wess-Zumino 2 gauge seen in [12,10], is here defined by the condition due to the linear shift in the transformations (2.12) for scalar fields χ I and φ IJ respectively, with parameters given by the ghost fields, c I and c F . There exists now, only the symmetric field φ (IJ) , that we write from now on simply as φ IJ . This condition is not SUSY-invariant under Q I , and it can be defined in terms of the infinitesimal fermionic parameter ǫ I as This operator leaves the conditions (2.14) invariant, and it is built up by the combinations of Q with the BRST transformations in the Wess-Zumino gauge, such that The results in terms of component fields are displayed below: (2. 16) in agreement with the transformation found in the works of [15,14]; the nilpotence reads as that is an infinitesimal transformation of φ IJ . With the result of the previous section, we are ready to write down the Blau-Thompson action, which is the invariant Yang-Mills action for the topological theory.

The Blau-Thompson action
The associated action for N T = 2, D = 4 is the Witten action [8,15,16], described in N T = 2 by the Blau-Thompson action [13,14], with gauge completely fixed in terms of the superfield. For the construction of this action, we wish a Lagrange multiplier that couples to the topological super-Yang-Mills so as to manifest its self-duality: F = * F . We then define a 2-form-superfield Lagrange multiplier, with the property of anti-self-duality and super-gauge covariant: We still wish a quadratic term in the last component field of K. Still, we need a 0-form-superfield to complete the gauge-fixing for Ψ I , which is defined as: To fix the super-Yang-Mills gauge, we define an anti-ghost superfield for C, being a 0-formsuperfield of fermionic nature we define a 0-form-superfield Lagrange mulptiplier Therefore the complete Blau-Thompson action in superspace takes the form with ζ being constant. In components, we have where g is the beckground metric of the Riemannian manifold.
In the next section, we shall discuss the Avdeev-Chizhov action in a general Riemannian manifold with the same background metric.

Tensorial Matter in a General Riemannian Manifold
To couple the theory above to the Avdeev-Chizhov model, we start describing the Avdeev-Chizhov action through the complex self-dual field ϕ [6], initially written in the 4-dimensional Minkowskian manifold, whose indices are: m, n, ... . We write this action, according to the work of [6], as S matter = d 4 x{(D m ϕ mn ) † (D p ϕ pn ) + q(ϕ † mn ϕ pn ϕ †mq ϕ pq )}. (4.1) Here q is a coupling constant for the self-interaction, and the covariant derivative D m a ϕ mn = ∂ m ϕ mn − [a m , ϕ mn ]; a m is the Lie-algebra-valued gauge potential and we assume ϕ mn to belong a given representating of the gauge group G. This action is invariant under the folowing transformations: with ϕ given by which exhibit the properties ϕ mn = i ϕ mn , ϕ mn = −ϕ mn , where the duality is defined by ϕ mn = 1 2 ε mnpq ϕ pq . To treat this theory, in a general Riemannian manifold as a topological theory, Geyer-Mülsch [4] rewrite the field in a four-dimensional Riemannian manifold, endowed of the vierbein e m µ and a spin-connection ω mn µ , i.e., the tensorial matter read as ϕ µν = e m µ e n ν ϕ mn , where the action (4.1) is given by In this 4-dimensional Riemannian manifold, we find the folowing properties: √ gε µνρλ ε mnpq = e m [µ e n ν e p ρ e q λ] , (4.5) e m µ e n ν g µν = η mn , e m µ e n ν η mn = g µν . (4.6) The covariant derivative in the Riemannian manifold is now written in terms of the spin-connection: where ω µ = 1 2 ω mn µ σ mn , being σ mn the generator of the holonomy Euclidean group SO(4), also we have: D µ = (D a ) µ , where, a, is the Yang-Mills connection.

Supersymmetrization of the Avdeev-Chizhov Action
From now on, we can write the action (4.4) in terms of superfields, mentioning the conventions of the works [10,8]. The superfield that accommodates the rank-two anti-symmetric tensorial matter field, is similar to the one defined in [7], being now expressed as a linear fermionic. This is defined as a rank-two anti-symmetric tensor in the 4-dimensional Riemannian manifold, and with the topological fermionic index I referring to the topological SUSY index: where ϕ µν (x) is the Avdeev-Chizhov field. The super-manifold is composed by Riemannian manifold and the N T = 2 topological manifold.
The superfield is defined under the SUSY transformations and in components: Based on the work of ref. [6], we rewrite the BRST transformations, referring the non-Abelian Avdeev-Chizhov model, in terms of the transformations: where (2.2) is the Lie algebra. We wish to write the BRST−transformation for a supergauge transformation, generalizing the transformations for the Avdeev-Chizhov fields, according to in components, we get: By now performing BRST−transformations on the components that survive in the N T = 2 Wess-Zumino gauge (2.15), we find: in agreement to (2.17).
We build up rank-two anti-symmetric tensorial matter field in a superspace formulation, leaving the superfield with the same properties as shown in [7]; this is invariant under gauge transformations (5.6) and SUSY transformations. The kinetic term is proposed as In components, we get: The interaction term has the peculiarity of presenting two derivatives of the Grassmann coordinates; it should also be invariant under the gauge transformations (5.6) and supersymmetry. We write it as where q is a quartic coupling constant. In components, we have the Avdeev-Chizhov action plus its partness: It is invariant under conformal transformations. Therefore, the total gauge invariant action can be written as: S AC + S BT . We could also have replace S BT by the super−BF action described in the work of ref. [11].
The Q−exactness of the total action above is also true for N T = 2 SUSY as in [4]; this is so because the fermionic volume element Q 2 ∝ Q 1 Q 2 , which means the exactness in the charge Q 1 , Q 2 of this action. This proof for N T = 1 and general N T , is given in the works [10], where the total action is also s−exact. According to Blau-Thompson in their review [17], the energy-momentum tensor Θ µν is also Q−exact, (5.14) ensuring the topological nature of the theory, where we shall just use the Avdeev-Chizhov kinetic term, because the interaction term carries the coupling constant q, which is irrelevant for the attainment of the observables of the theory [4].

Concluding Remarks
The main goal of this paper is the settlement of a topological superspace formulation for the investigation of the coupling between the rank-two Avdeev-Chizhov matter field and Yang-Mills fields. It comes out that the stress tensor is Q−exact. This opens us the way for the identification of a whole class of obsevables that we are trying to classify [19].
It is worthwhile to draw the attention here to the shift symmetry that allows us to detect the ghost caracter of the Avdeev-Chizhov field. On the other hand, it is known that there appears a ghost mode in the spectrum of excitations of our tensor matter field [1]. The connection between these two observations remain to be clarified. The fact that the Avdeec-Chizhov field manifest itself as a ghost guide future developments in the quest for a consistent mechanism to systematically decouple the unphysical mode mentioned above.
We are also trying to embed the tensor field in the framework of a gauge theory with Lorentz symmetry breaking [18]. We expect that this breaking may identify the right ghost mode present among the two spin 1 components of the Avdeev-Chizhov field. This result is applied to a superfunction f (x, θ), so that the volume element is therefore, the square of the supersymmetric charge operator (shift operator) is defined by: which is a volume element too.