New Results on N = 4 SuperYang–Mills Theory

The N = 4 SuperYang–Mills theory is covariantly determined by a U (1) × SU (2) ⊂ SL (2 , R ) × SU (2) internal symmetry and two scalar and one vector BRST topological symmetry operators. This determines an oﬀ-shell closed sector of N = 4 SuperYang–Mills, with 6 generators, which is big enough to fully determine the theory, in a Lorentz covariant way. This reduced algebra derives from horizontality conditions in four dimensions. The horizontality conditions only depend on the geometry of the Yang–Mills ﬁelds. They also descend from a genuine horizontality condition in eight dimensions. In fact, the SL (2 , R ) symmetry is induced by a dimensional reduction from eight to seven dimensions, which establishes a ghost-antighost symmetry, while the SU (2) symmetry occurs by dimensional reduction from seven to four dimensions. When the four dimensional manifold is hyperK¨ahler, one can perform a twist operation that deﬁnes the N = 4 supersymmetry and its SL (2 , H ) ∼ SU (4) R-symmetry in ﬂat space. (For deﬁning a TQFT on a more general four manifold, one can use the internal SU (2)-symmetry and redeﬁne a Lorentz SO (4) invariance). These results extend in a covariant way the light cone property that the N = 4 SuperYang–Mills theory is actually determined by only 8 independent generators, instead of the 16 generators that occur in the physical representation of the superPoincar´e algebra. The topological construction disentangles the oﬀ-shell closed sector of the (twisted) maximally supersymmetric theory from the (irrelevant) sector that closes only modulo equations of motion. It allows one to escape the question of auxiliary ﬁelds in N


Introduction
Recently, we have constructed the genuine N = 2 supersymmetric algebra in four and eight dimensions in their twisted form, directly from extended horizontality conditions [1]. The new features were the geometrical construction of both scalar and vector topological BRST symmetries. A remarkable property is that the supersymmetry Yang-Mills algebra, both with 8 and 16 generators, contain an off-shell closed sector, with 5 and 9 generators, respectively, which is big enough to completely determine the theory. The key of the geometrical construction is the understanding that one must determine the scalar topological BRST symmetry in a way that is explicitly consistent with reparametrization symmetry. This yields the vector topological BRST symmetry in a purely geometrical way.
Here we will extend the result to the case of the maximally supersymmetric N = 4 algebra in four dimensions, with its 16 supersymmetric spinorial generators and SL(2, H) internal symmetry 1 . The most determining phenomenon occurs when one computes the dimensional reduction from eight to seven dimensions, which provides an SL(2, R) symmetry. The subsequent dimensional reductions to six and four dimensions are then straightforward, and add a further Z 2 and SU (2) internal symmetry, respectively. This explains the organization of the paper and the eventual obtaining of four-dimensional horizontality conditions, which determine the N = 4 algebra in a twisted form. We will also discuss the possible other twists of the supersymmetric theory.
As for the physical application of our construction, having obtained an off-shell closed algebra allows one to escape the question of auxiliary fields in N = 4 SuperYang-Mills theory. In a separate publication, using this algebra and its consequences, we will give an improved demonstration of the renormalization and finiteness of the N = 4 superYang-Mills theory [2].
2 From the D = 8 to the D = 7 topological Yang-Mills theory 2

.1 Determination of both topological scalar symmetries in seven dimensions
The D = 8 topological Yang-Mills theory relies on the following horizontality equation, completed with its Bianchi identity [1]: One has the closure relations: 1 The internal symmetry group SU (4) of N = 4 superYang-Mills, defined on a Minkowski space, must be replaced on a Euclidean one by the Minkowski equivalent SL(2, H) ∼ SO(5, 1) of SU (4). This is implied by the fact that N = 4 superYang-Mills is the dimensional reduction of the ten-dimensional N = 1 superYang-Mills theory, which is only defined on Minkowski space.
We refer to [1] for a detailed explanation of these formula and the twisted fields that they involve. Ψ is a 1-form topological ghost and χ is an antiselfdual 2-form in eight dimensions, with 7 independent components. Selfduality exists in eight dimensions when the manifold has a holonomy group included in Spin (7). The octonionic invariant 4-form of such a manifold allows one to define selfduality, by the decomposition of a 2-form as 28 = 7 ⊕ 21. κ is a covariantly constant vector, which exists if the holonomy group is included in G 2 ⊂ Spin(7). Φ andΦ are respectively a topological scalar ghost of ghost and an antighost for antighost. c andc can be interpreted as the Faddeev-Popov ghost and antighost of the eight-dimensional Yang-Mills field A. s and δ are the scalar and vector topological BRST operators. In flat space, one can write δ + |κ|δ gauge (c) = κ µ Q µ , and we understand that Q = s + δ gauge (c) and Q µ count for 9 independent generators, giving an off-shell closed sector of N = 2, D = 8 twisted supersymmetry, which fully determines the theory [1].
To determine the topological symmetry in seven dimensions, we start from a manifold in eight dimensions that is reducible, We can chose κ as a tangent vector to S 1 . Thus, the 8-dimensional vector symmetry along the circle reduces to a scalar one in seven dimensions,s, and the reduction of the antiselfdual 2-form χ gives a seven-dimensional 1-formΨ. So, the dimensional reduction of the horizontality condition (1) is Indeed, with our choice of κ, L = i κ A = A 8 in eight dimensions. Eq. (3) can be given a different interpretation than Eq. (1). It contains no vector symmetry and looks like a standard 7-dimensional BRST-antiBRST equation. The transformation of L, the origin of which is A 8 , is now given by the Bianchi identity of Eq. (3). In fact, after dimensional reduction, L is understood as a curvature. Using the convenient pyramidal diagrammatic description of ghostantighost structures, we can rewrite the field description, as follows: As compared to the asymmetrical diagram on the left hand-side, the one on the right hand-side exhibits an SL(2, R) symmetry, which counts the ghost antighost numbers. Indeed, each line of this diagram corresponds to an irreducible representation of SL(2, R), namely a completely symmetrical SL(2, R)-spinorial tensor with components φ g,G−g , where g and G − g are respectively the (positive) ghost and antighost numbers of φ, and 2g − G is the effective ghost number of φ. This SL(2, R) symmetry actually applies to the covariant ghost-antighost spectrum of a p-form gauge field φ p ,φ p = 0≤G≤p 0≤g≤G φ g,G−g p−G . In fact, the fields (Ψ,Ψ) and (η,η) can be identified as SL(2, R) doublets, Ψ α and η α , α = 1, 2, and the three scalar fields (Φ, L,Φ) as a SL(2, R) triplet, Φ i , i = 1, 2, 3. The index α and i are respectively raised and lowered by the volume form ε αβ of SL(2, R) and the Minkowski metric η ij of signature (2, 1). Both BRST and antiBRST operators can be assembled into a SL(2, R) doublet s α = (s,s).
The horizontality condition (3) can be solved, with the introduction of three 0-form Lagrange multipliers, η,η, b and a 1-form T .
The transformation laws of the b field cannot be made however SL(2, R) covariant. Its introduction allows one to raise the degeneracy arising from the equation sc +sc + [c,c] = L, which basically selects a ghost direction, and freezes the SL(2, R) symmetry. In fact, going from a Cartan to a nilpotent algebra breaks this SL(2, R), and in this way, defines the net ghost number symmetry from the residual unbroken U (1) ⊂ SL(2, R). The Cartan algebra (that we will denote by the subindex (c)) is on the other hand fully SL(2, R) covariant. It is obtained by adding gauge transformations with parameters c andc, from s ands, respectively. It reads as: The equivariant (Cartan) algebra is the one that will match by twist with the relevant part of the twisted supersymmetry algebra. Its closure is only modulo gauge transformations, with parameters that are ghosts of ghosts.

Determination of the equivariant part of both topological vector symmetries in seven dimensions
We have produced by dimensional reduction a new scalar (antiBRST) topological symmetry operator. However, we have apparently lost the rest of the vector symmetry in eight dimensions, since we have chosen κ along the circle of dimensional reduction. The freedom of choosing this circle generates an automorphism of the seven dimensional symmetry. It allows one to obtain two vector topological symmetries, in seven dimensions, in a SL(2, R) symmetric way.
In the genuine theory in eight dimensions, the definition of a chosen Spin (7) structure is actually arbitrary. The different choices of a Spin(7) structure on the eight dimensional Riemann manifold can be parametrized by the transformations of SO(8) \ Spin (7). However, such transformations cannot be represented on eight dimensional fields such as the antiselfdual 2-form χ µν . They are generated by antiselfdual 2-form infinitesimal parameters, which can be parametrized by seven-dimensional vectors, after dimensional reduction to 7 dimensions. One can thus define the following (commuting) 7-dimensional derivation γ, which depends on a covariantly constant seven-dimensional vector κ µ , (the indices µ, ν... are seven dimensional), and acts as follows on the 7-dimensional fermion fields: The action of γ is zero on the bosonic field of the equivariant BRST algebra, but on the Lagrange multiplier field T . (We will shortly define γT , by consistency). C µνσ is the seven dimensional G 2 invariant octonionic 3-form and its dual is the 4-form C ⋆ µνσρ . We will use the notation i κ ⋆ C ⋆ w 1 = −C µν σ κ ν w σ dx µ . The derivation γ expresses the arbitrariness in the choice of an eighth component, in order to perform the dimensional reduction. To each constant vector on N , one can assign a U (1) group, which is subset of SO(8) \ Spin (7). Since , one can verify on all fields: One defines the vector operator: To ensure the validity of this formula on T , one defines the transformation of the auxiliary field T as follows: Computing the commutators of s α (c) and γ, one gets the action of δ α (c) : is a pair of two vector symmetries in seven dimensions, which transform as an SL(2, R)doublet. This completes the scalar doublet s α (c) . Then, one can verify that the anticommutation relations for the vector operator δ α (c) are: Reciprocally, these closure relations uniquely determine δ α (c) , from the knowledge of s α (c) . In the next section, we will re-derive these transformation laws, from horizontality equations. Moreover, we will extend them as nilpotent transformations, by including gauge transformations.
It is instructive to check the expression we have just obtained for δ α (c) , by starting from Eq. (1) in 8 dimensions (with the notational changec → γ), and computing the dimensional reduction with a vector κ along N , instead of along the circle. This gives: One can then verify: with the modified definition that The difference between this expression of δ + |κ|δ gauge (γ) and that of a component of the SL(2, R) covariant vector symmetry operators δ α (c) is just a field redefinition.

The complete Faddeev-Popov ghost dependent vector and scalar topological symmetries in seven dimensions
We now directly construct the scalar and vector BRST topological operators s α and δ α , the equivariant analogs of which are s α (c) and δ α (c) . One needs scalar Faddeev-Popov ghosts, c,c, γ,γ, which are associated to the equivariant BRST operators s (c) ,s (c) , δ (c) ,δ (c) , respectively.
Dimensional reduction therefore transforms the N T = 1 eight-dimensional theory into a N T = 2 theory, with an SL(2, R) internal symmetry, and a G 2 ⊂ Spin(7) Lorentz symmetry.
As a matter of fact, this algebra gives the SL(2, R) invariant twisted supersymmetry transformations [5] in the limit of flat manifold.

Seven-dimensional invariant action
The most general gauge invariant topological gauge function Ψ, which yields a δ invariant action S = sΨ − 1 2 M C ∧ Tr F ∧ F , is: This function turns out to bes-exact and thuss invariant. Moreover, S isδ invariant. By using the SL(2, R) covariant form of the algebra, we can compute the gauge function Ψ in a manifestly invariant way, as a component of a doublet Ψ α . One has σ i αβ δ (c) α Ψ β = 0 and , and, with our conventions, Ψ ∝ Ψ 1 . The automorphism generated by γ leaves invariant neither the gauge function, nor the action, since its action breaks the Spin(7)-structure of the 8-dimensional theory. These properties will remain analogous in 4 dimensions, and we will give more details in this case.

Reduction to four dimensions and N = 4 theory
The process of dimensional reduction can be further done, from 7 dimensions. The ghost antighost symmetry that has appeared when going down from 8 to 7 dimensions will continue to hold true, and therefore, one remains in the framework of an N T = 2 theory, with SL(2, R) invariance.
One is concerned by going down from seven to four dimensions. The SO(7)-symmetry is decomposed into SO(4) × SO(3) ∼ Spin(4) × SU (2), and insides this decomposition, the G 2 -symmetry is decomposed into SU (2) × SU (2). So the (twisted) 4-dimensional theory is expected to have SU (2) ⊂ Spin(4) Lorentz symmetry, with an SL(2, R) × SU (2) internal symmetry. Thus, starting from a manifold of holonomy included in G 2 suggests that the theory will be defined on a 4-dimensional hyperKähler manifold.
In fact, by identifying the SU (2) part of the internal symmetry group as the missing chiral SU (2) part of the Spin(4)-invariance, one can eventually restore the full Spin(4) invariance, and define the topological theory on more general manifolds. It is useful to use a hyperKähler structure to simplify the form of the equations. For instance, given the 3 constant hyperKähler 2-forms J I µν , the antiselfdual 2-form χ µν can be written as χ µν = χ I J I µν , where χ I is a SU (2) triplet made of scalars. (Capital indices as I are devoted to the adjoint representation of the SU (2) symmetry, the scalars h I correspond to A 7 , A 6 , A 5 , etc...). This allows simplified expressions for scalars, such as, for instance, one is interested in obtaining by twist the N = 4 superYang-Mills theory. This is a further justification of the restricted choice of a hyperKähler manifold, since two constant spinors are needed to perform the twist operation and eventually to map the topological ghosts on spinors.

Equivariant scalar and vector algebra in four dimensions
By dimensional reduction of the seven-dimensional equations of section 2, one can compute the Cartan SL(2, R) doublet of scalar topological BRST operators for the topological symmetry in four dimensions: One has the closure relation s The Cartan vector algebra is: One has, δ {α (c) δ β} (c) = |κ| 2 σ i αβ δ gauge (Φ i ), and δ α (c) anticommute with s α (c) , as follows 2 : The four dimensional vector operators are s α -exact, as in seven dimensions, δ α (c) = [s α (c) , γ], where the non zero component of γ are As in seven dimensions, one has a U (1) automorphism of the algebra, which is not a symmetry of the theory, with e tγ s α (c) e −tγ = cos t s α (c) − sin t δ α (c) and e tγ δ α (c) e −tγ = cos t δ α (c) + sin t s α (c) .

Invariant action
There are two gauge functions which fit in a fundamental multiplet of SL(2, R), and satisfy: The action is defined as: Eq. (27) completely constrains the gauge function (up to a global scale factor), as follows: The action (28) is δ α and s α invariant. Indeed, one can check that it verifies: These facts remind us that we are in the context of a N T = 2 theory. The critical points of the Morse function F in the field space are given by the equations Eqs. (33) are the dimensional reduction of selfduality equations in 7 dimensions. [6,7,8] display analogous equations, corresponding to the dimensional reduction of the selfduality equation in 8 dimensions. [4] indicates that the moduli problems defined by both equations are equivalent 3 . Expanding the action S, and integrating out T and H I , reproduces the N = 4 action in its twisted form [6,7] In fact, it was not necessary to ask SL(2, R)-invariance from the beginning. Rather, looking for a δ, s ands invariant action, with internal symmetry SU (2)×U (1), (U(1) is the ghost number symmetry), determines a unique action, Eq. (28), with additional SL(2, R) andδ invariances. This shows that the N = 4 supersymmetric action is determined by the invariance under the action of only 6 generators s,s, δ, with a much smaller internal symmetry than the SL(2, H) Rsymmetry, namely SU (2) × U (1). After untwisting, this determines the relevant closed off-shell sector of N = 4 supersymmetry.

Horizontality condition in four dimensions
The algebra is not contained in a single horizontality condition, as in the seven-dimensional case. In fact, one has splitted conditions for the Yang-Mills field, and for the scalar fields h I . (This gives the possibility of building matter multiplets, by relaxing the condition that h I is in the adjoint representation of the gauge group). They are : (d + s +s + δ +δ) A + c +c + |κ|γ + |κ|γ + A + c +c + |κ|γ + |κ|γ 2 = F + Ψ +Ψ + g(κ) η +η + g(J I κ) χ I +χ I + (1 + |κ| 2 ) Φ + L +Φ These equations and their Bianchi identities fix the action of s,s, δ andδ, by expansion in ghost number, up to the introduction of auxiliary fields that are needed for solving the indeterminacies of the form "s antighost +s ghost" . These indeterminacies introduce auxiliary fields in the equivariant part of the algebra, T and H I , as well as in the Faddeev-Popov sector, with several auxiliary fields that induce a breaking of the SL(2, R) invariance. The latter does not affect the equivariant, that is, gauge invariant, sector.
To be more precise about the number of auxiliary fields, all the actions given by a symmetrized product of operators on the four Faddeev-Popov ghosts are determined by the closure relations of the algebra. There is one indeterminacy for each antisymmetrized product of operators. To close the algebra in the Faddeev-Popov sector, 11 = 6 + 4 + 1 Lagrange multiplier fields must be introduced, with the standard technique. We do not give here the complete algebra for these fields, which we postpone for a further paper, devoted to a new demonstration of the finiteness of the N = 4, D = 4 theory. We can gauge-fix the action in a s ands invariant way and/or in a s and δ invariant way. In the former case, one uses an ss-exact term which gauge-fixes the connection A.. This ss-exact term eliminates all fields of the Faddeev-Popov sector, but c,c, and b ≡ sc. In the former case, one uses an sδ-exact term. In this case,c is replaced by γ.

Different twists of N = superYang-Mills
As noted in [7,10] there are three non equivalent twists of N = 4 superYang-Mills, corresponding to the different possible choices of an SU (2) in the R-symmetry group SL(2, H). When one defines the symmetry by horizontality conditions, these 3 different possibilities correspond to different representations of the matter fields. These matter fields are respectively organized in an SL(2, R)-Majorana-Weyl spinor, a vector field and a quaternion. The latter is the one studied in the previous section 5 , and, as a matter of fact, the most studied in the literature [4,6,8,12]. It is the only case that can be understood as a dimensional reduction of the eight-dimensional topological theory. We will shortly see that the two other cases have scalar and vector symmetries that are not big enough for a determination of the action.

Spinor representation
The first twist gives an N T = 1 theory. It is obtained by breaking Spin(4) ⊗ SL(2, H) into Spin(4)⊗ SU (2)⊗ SL(2, R)⊗ U (1), and then taking the diagonal of the chiral SU (2) of Spin(4) with the SU (2) of the previous decomposition of SL (2, H). The bosonic matter field h α is then a chiral SL(2, R)-Majorana-Weyl spinor. The ghost λ α + and antighost λ α − of the matter field are respectively chiral and antichiral SL(2, R)-Majorana-Weyl spinors. The horizontality condition reads: Introducing the auxiliary antichiral SL(2, R)-Majorana-Weyl spinors D α , one gets : With this definition of the twist, there is no other scalar or vector charge, which leaves us with a N T = 1 theory. The action is not completely determined by these two symmetries.

Vector representation
One breaks SL(2, H) into Spin(4) ⊗ SO(1, 1) and then takes the diagonal of Spin(4) ⊗ Spin(4) [11]. The matter horizontality condition involves a vector field V µ ≡ h µ , its vector ghostΨ and antighosts scalarη and selfdual 2-formχ, This give the following transformations of the fields This corresponds to a N T = 2 theory. However, the mirror operatorss (c) andδ (c) have the same ghost number as the primary ones. The internal symmetry in this case is Z 2 instead of SL(2, R) [7]. As a matter of fact the four symmetries are not enough to fix the action and do not give an algebra which can be closed off-shell without the introduction of an infinite set of fields.