The basic cohomology of the twisted N=16, D=2 super Maxwell theory

We consider a recently proposed two-dimensional Abelian model for a Hodge theory, which is neither a Witten type nor a Schwarz type topological theory. It is argued that this model is not a good candidate for a Hodge theory since, on-shell, the BRST Laplacian vanishes. We show, that this model allows for a natural extension such that the resulting topological theory is of Witten type and can be identified with the twisted N=16, D=2 super Maxwell theory. Furthermore, the underlying basic cohomology preserves the Hodge-type structure and, on-shell, the BRST Laplacian does not vanish.


Introduction
It has been known for some time that two-dimensional Yang-Mills theory without matter is exactly soluble [1]. Moreover, it has been shown that Yang-Mills theory with matter on an arbitrary orientable two-dimensional manifold M of genus G and area A is equivalent to a closed, orientable, string theory with target space M [2]. Two-dimensional Yang-Mills theory was revisited using a non-Abelian version of the Duistermaat-Heckman formula in [3] and a simple mapping to topological Yang-Mills theory with an underlying N T = 1 equivariantly nilpotent shift symmetry was given. For a previous work on quantum gauge theories in two dimensions see, e.g., [4]. In this paper we consider a Hodge type cohomological gauge theory with an underlying N T = 8 strictly nilpotent shift and co-shift symmetry (i.e., even prior to the introduction of the gauge ghost and anti-ghost fields), which leads to other topological observables.
The study of such theory was motivated by a recently series of papers [5] where a class of topological gauge theories in two dimensions was presented which is neither of Witten nor of Schwarz type. Rather, they show some of the characteristic features of both types of topological quantum field theories (TQFT), namely, the form of the action turns out to be of Witten type whereas the underlying supersymmetries are reminiscent of Schwarz type (for a review of TQFT, see, e.g., [6]).
The aim of this Letter is to reveal these rather unusual properties of a TQFT. Thereby, we focus on the simplest of the models considered in [5], namely the Euclidean Maxwell theory in two dimensions in the Feynman gauge. Their action is given by where A a is the Abelian gauge field and C,C are the gauge (anti)ghost fields, respectively.
BRST operator Ω, differential d, co-BRST operator ⋆ Ω = ±JΩJ, co-differential δ = ± ⋆ d⋆, Hence, the action (1) provides a Hodge type field theoretical model in two dimensions. However, owing to the absence of a shift and co-shift symmetry, the action is not of Witten type and, because the Maxwell action in a gravitational background is not metric independent, it is also not of Schwarz type. Differently, the topological nature of that model is a consequence of the fact that in two dimensions there are no propagating degrees of freedom associated with the gauge field. On the other hand, as there is no topological supersymmetry, on-shell, the BRST Laplacian vanishes. This is, in fact, an unsatisfactory property of a Hodge type theory because their physical states should lie entirely in the set of harmonic states, i.e., the set of the zero modes of the BRST Laplacian. Therefore, in order to incorporate a topological supersymmetry into that model, it will be shown that it can be regarded as part of a more complex topological model of Witten type, namely the twisted N = 16, D = 2 super Maxwell theory with global symmetry group SU (4), whose basic cohomology [10] possesses actually a Hodge type structure.
The Letter is organized as follows: In Sect. 2, as a first step, we substitute in (1) the Maxwell action by the cohomological action of twisted N = 16, D = 2 super Maxwell theory with global symmetry group SU (4). And we show that the BRST complex of the 8 twisted scalar supercharges, i.e., the generators of the shift and co-shift symmetries, is really of Hodge type. In Sect. 3, as a second step, we complete the cohomological action by introducing the ordinary gauge fixing terms and verify that the basic cohomology, i.e., the BRST complex including also the ordinary gauge symmetry, preserves the underlying Hodge type structure. In Sect. 4 we construct topological observables for that theory.
2. The BRST complex of the twisted N = 16, D = 2 super Maxwell theory with global symmetry group SU(4) As mentioned above, in order to avoid the vanishing of the BRST Laplacian of the topological model proposed in [5] we view the Maxwell action in (1) as the classical part of the twisted action of N = 16, D = 2 super Maxwell theory with global symmetry group SU (4) (see, e.g., [11] where the non-Abelian extension of that action has been constructed), Here, V a is a co-vector field which is combined with A a to forms the complexified gauge fields A a ± iV a , i.e., the action (3) localizes onto the moduli space of complexified flat connections. Furthermore, we have introduced a SU (4)-quartet of Grassmann-odd vector fields ψ α a , two SU (4)-quartets of Grassmann-odd scalar fields, η α and ζ α , which transform as the fundamental and its complex conjugate representation of SU (4), respectively, and a SU (4)-sextet of Grassmanneven complex scalar fields M αβ = 1 2 ǫ αβγδ M γδ , which transform as the second-rank complex selfdual representation of SU (4), where α = 1, 2, 3, 4 denotes the internal group index of SU (4). ǫ ab is the antisymmetric Levi-Civita tensor in D = 2.
Let us notice that the action (3) can be obtained from the Euclidean N = 16, D = 2 super Maxwell theory with R-symmetry group SU (4) ⊗ U (1) by twisting the Euclidean rotation group SO E (2) ∼ U E (1) in D = 2 by the U (1) of the R-symmetry group (by simply putting together both U (1) charges), thereby leaving the group SU (4) intact [12].
The action (3) is invariant under the following discrete Hodge type ⋆-symmetry, defined by the replacements with the property ⋆(⋆ϕ) = ±ϕ.
Let us now describe the full set of twisted supersymmetry transformations which leave the action (3) invariant. The transformation rules for the on-shell shift symmetries Q α are From combining Q α with the above displayed Hodge type ⋆-symmetry one gets the corresponding transformation rules for the on-shell co-shift symmetries ⋆ Q α , i.e., ⋆ Q α ϕ = ± ⋆ Q α ⋆ ϕ, where the signs are the same as in the relation ⋆(⋆ϕ) = ±ϕ. Furthermore, by making use of the identity ǫ ab ∂ c + ǫ bc ∂ a + ǫ ca ∂ b = 0, one simply verifies that (3) is also invariant under the following on-shell vector supersymmetries Q aα , In principle, owing to the Hodge type ⋆-symmetry, one can also introduce on-shell co-vector supersymmetries ⋆ Q aα , namely, similar as before, according to ⋆ Q aα ϕ = ± ⋆ Q aα ⋆ ϕ. However, they become i times the on-shell vector supersymmetries Q aα , i.e., ⋆ Q aα = iQ aα . Hence, it holds = 0, and the total number of (real) supercharges is actually N = 16. By an explicit calculation one establishes that the 8 scalar supercharges Q α and ⋆ Q α , being interrelated by the ⋆-operation, together with the 8 vector supercharges Q µα , obey the following topological superalgebra on-shell, , where the field-dependent gauge transformations δ G (ϕ), with ϕ = (M αβ , A a ± iV a ), are defined by δ G (ϕ)A a = ∂ a ϕ, and zero otherwise. (The symbol . = means that the corresponding relation is satisfied only on-shell.) Finally, let us show that the twisted action (3) is actually of Witten type. For that purpose we break down the global symmetry group SU (4) in such way that the resulting action splits into a topological term (Q-cocyle) and a Q-exact term, where Q is a certain linear combination of Q α and ⋆ Q α , and that it possesses a discrete Hodge type ⋆-symmetry. This is precisely what we need in order to put the theory on a two-dimensional compact Riemannian manifold.
To begin with, let us consider the following N T = 1 topological action in five dimensions, which is build up from the Abelian gauge field A A , the co-vector field V A and the Grassmannodd scalar, vector and antisymmetric tensor fields η, ψ A and χ AB , respectively. Here, the space index A runs from 1 to 5 and ǫ ABCDE is the complete antisymmetric Levi-Civita tensor in D = 5. This action can be obtained from the Euclidean N = 2, D = 5 super Maxwell theory with R-symmetry group SO(5) by twisting the Euclidean rotation group SO E (5) in D = 5 by the R-symmetry group (for details, we refer to [11], where the non-Abelian extension of that action was given).
The full set of twisted on-shell supersymmetry transformations, generated by the scalar, vector and antisymmetric tensor supercharges Q, Q A and Q AB , respectively, are given by and By the help of the identity it is straightforward, but tedious to prove that the action (7) is really left invariant under the above transformations.
Let us now group the components of the 5-dimensional fields A A , V A , χ AB , ψ A , η into those of the 2-dimensional fields A a , V a , M αβ , η α , ζ α , ψ α a according to Then, it is easily seen that by dimensional reducing the action (7) and the supersymmetry transformations (8)-(10) onto two dimensions, one arrives precisely at the N T = 8 topological action (3) and the supersymmetry transformations (5) and (6).
On the other hand, on-shell, upon using the equation of motion for η, the action (7) can be written as a sum of a topological term (Q-cocycle) and a Q-exact term, with the gauge fermion Hence, by reducing also (11) in the same way as above into two dimensions we get exactly the decomposition of the action (3) we are looking for, consisting of a topological term (Q-cocycle) and of a Q-exact term. More precisely, if we group the components A M , V M , ψ M , χ aM , χ M N , (M, N = 3, 4, 5) into SU (2) triplets M ij , N ij , ρ ij , χ ij a , λ ij (i, j = 1, 2) and identify 1 2 ǫ ab χ ab with the SU (2) singlet ζ, we obtain the following N T = 8 topological action with a residual global symmetry group SU (2) ⊗ U (1), The transformation rules for the on-shell shift symmetry Q are given by Moreover, the action (12) is also invariant under the following duality ⋆-operation, which maps Hence, as anticipated, after breaking down the SU (4)-symmetry to SU (2) ⊗ U (1) the Hodge type structure of the theory is still preserved. Obviously, because the maximal number of scalar supercharges of twisted N = 16, D = 2 super Maxwell theory is N T = 8, the generator Q must be a certain linear combination of Q α and ⋆ Q α . Up to now we have shown that the BRST complex of the twisted N = 16, D = 2 super Maxwell theory possesses actually a Hodge type structure. Moreover, we have verified that the cohomological action is of Witten type and that it corresponds to the case with the maximal number N T = 8 of scalar supercharges and the largest possible global symmetry group SU (4). On the other hand, we could also perform a topological twist of N = 8, D = 2 super Maxwell theory, whose underlying BRST complex possesses a Hodge type structure, too. In that case we would get the minimal number of N T = 4 scalar supercharges and the lowest possible global symmetry group SU (2) [13].

The Hodge type structure of the basic cohomology
The twisted action (3) has still an ordinary gauge symmetry, which we have not considered yet. Now, we want to show that by adding to (3) the usual gauge fixing and (anti)ghost dependent terms the Hodge type structure of the underlying basic cohomology, i.e., the BRST complex including also the ordinary gauge symmetry, is preserved. To this end we introduce two SU (4)quartets of Grassmann-odd scalar ghost and antighost fields, C α andC α , which transform as the fundamental and its complex conjugate representation of SU (4), respectively.
Then, the complete gauge-fixed action reads This action, in spite of the fact that the gauge symmetry is fixed, exhibits still an invariance under the following bosonic symmetry, where we have written down only the non-vanishing transformation rules. However, below it will be shown that the generator W αβ of that symmetry is just the BRST Laplacian. Furthermore, in order to ensure the nilpotency of the BRST and co-BRST operators, we introduce a set of auxiliary fields, namely, the SU (4)-singlets of Grassmann-even scalar fields B,B and G,Ḡ. By the help of these additional fields the action (13) can be rewritten as and the Hodge type ⋆-symmetry (4) must be supplemented by the following replacements, where, again, two successive ⋆-operations on ϕ yield ⋆(⋆ϕ) = ±ϕ.
The transformation rules for the off-shell BRST symmetries Ω α are , From combining Ω α with the Hodge-type ⋆-symmetry (4) and (15) one gets the corresponding transformation rules for the off-shell co-BRST symmetries ⋆ Ω α . Then, for the non-vanishing transformations generated by the BRST Laplacian W αβ one obtains which, after elimination of G andḠ through their equations of motion, agree, as promised, with (14). Let us emphasize, that the BRST Laplacian W αβ , in contrast to [1], does not vanish on-shell, due to the presence of scalar fields M αβ . This is a consequence of the fact that the topological nature of our model is actually encoded in the shift and co-shift symmetries Q α and ⋆ Q α , and not in the vanishing of the BRST Laplacian as in [5]. Furthermore, by a straightforward calculation it can be verified that Ω α and ⋆ Ω α actually leave the action (15) invariant, i.e., it holds (Ω α , ⋆ Ω α )S (N T =8) = 0. Both operators, together with W αβ , satisfy the following BRST complex, Obviously, this basic cohomology is analogous to the de Rham cohomology: The both nilpotent BRST and co-BRST operators, Ω α and ⋆ Ω α = ± ⋆ Ω α ⋆, being interrelated by the duality ⋆-operation, correspond to the exterior and the co-exterior derivatives, d and δ = ± ⋆ d⋆, respectively, and the BRST Laplacian W αβ = {Ω α , ⋆ Ω α } is the analogue of ∆ = {d, δ}, so that we have indeed a perfect example of a Hodge type cohomological theory in two dimensions.

Topological observables
As already pointed out earlier, when the gauge-fixed action (15) is formulated on a compact two-dimensional Riemannian manifold we break down the global symmetry group SU (4) in such a way that the resulting action splits into a topological term (Ω-cocycle) and a Ω-exact term, and that the discrete Hodge type ⋆-symmetry is preserved, mapping Ω to ⋆ Ω. Alternatively, the same action can be also obtained from the cohomological action (12) by adding the usual gauge-fixing and (anti)ghost dependent terms, see Eq. (3), and by introducing an appropriate set of auxiliary fields. With regard to this, let us note two unusual features of the action (12) which are relevant for the construction of two-dimensional observables of that topological model. Its most striking property is that the both, shift and co-shift symmetry Q and ⋆ Q, are not equivariantly nilpotent (due to the absence of the usual ghost for ghost field φ) but, on-shell, rather they are strictly nilpotent even prior to the introduction of the ghost and anti-ghost fields C andC.
Another remarkable property is that A a − iV a and A a + iV a are invariant under one of the supercharges, namely Q in the former and ⋆ Q in the latter case. Thus, for the both BRST and co-BRST operators Ω and ⋆ Ω one should expect the existence of two different sets of observables, depending either on A a − iV a in the former case or on A a + iV a in the latter case. In fact, these observables can be constructed in a similar way as in the case of the topological sigma models [14]. Therefore, we shall omit any details and simply quote the results.
To begin with, we first associate the zero-forms W 0 and ⋆ W 0 to the BRST and co-BRST transforms ΩC and ⋆ ΩC (which, on-shell, correspond to the gauge-fixing function ∂ a A a ), respectively, via the BRST and co-BRST invariant ghost field C. These zero-forms can be used as building blocks for constructing the following both sets of k-forms, W k and ⋆ W k , being interrelated by the Hodge type ⋆-operation, W 0 = (ΩC)C, and ⋆ W 0 = ( ⋆ ΩC)C, a dx a and δx µ = ǫ µν e νa dx a , with e µ a being the two-bein on a smooth connected, oriented Riemannian manifold M endowed with metric g µν . Here, d = dx µ ∂ µ and δ = dx µ ǫ µν ∂ ν are the exterior and co-exterior derivative, respectively.
These k-forms obey the following recursion relations, which are typical for any topological gauge theory, 0 = ΩW 0 , dW 0 = ΩW 1 , dW 1 = ΩW 2 , dW 2 = 0, Now, if γ is a k-dimensional homology cycle, ∂γ = 0, on M then the integrated k-forms I k (γ) = Moreover, if β = ∂α is the boundary of a (k + 1)-dimensional surface, k < 2, so that β is trivial in homology, then I k (γ) depends only upon the homology class of γ up to a Ω-exact term, Finally, following [14], one can introduce gauge invariant correlation functions of arbitrary products of the I k (γ), Z(γ 1 , . . . , γ r ) = Dϕ exp(−S(ϕ)) which, by construction, both are Ω-invariant and invariant under metric deformations which preserve the holonomy structure. The same constructions hold for the k-forms ⋆ W k . Summarizing, we have shown that, on-shell, the vanishing of the BRST Laplacian of the Hodge theory proposed in [5] can be avoided, if we view the Maxwell action as the classical part of a more involved cohomological action, which is obtained by a N T = 8 topological twist of N = 16, D = 2 super Maxwell theory with global symmetry group SU (4). Then, the complete gauge-fixed cohomological action is of Witten type and the underlying basic cohomology is really of Hodge type. The non-Abelian case will be presented elsewhere.