Twisted Superspace

We formulate the ten-dimensional super-Yang-Mills theory in a twisted superspace with 8+1 supercharges. Its constraints do not imply the equations of motion and we solve them. As a preliminary step for a complete formulation in a twisted superspace, we give a superspace path-integral formulation of the N=2, d=4 super-Yang-Mills theory without matter. The action is the sum of a Chern--Simons term depending on a super-connection plus a BF-like term. The integration over the superfield B implements the twisted superspace constraints on the super-gauge field, and the Chern-Simons action reduces to the known action in components.


Introduction
Superspace formulations of supersymmetric theories are often very efficient for practical computations and proofs of non-renormalization theorems. A complete superspace path-integral formulation requires that the supersymmetry algebra admits a functional representation on the fields, but the latter is believed not to exist for maximal supersymmetry. This has lead to several proposals for restricting the whole super-Poincaré algebra to subalgebras with such a representation. For instance, the maximally supersymmetric Yang-Mills theory has been formulated within N = 3 harmonic superspace [1]. There are severe restrictions on such off-shell closed representations. In six dimensions, to maintain the full Poincaré invariance, one must reduce the N = 2 super-Poincaré symmetry to the N = 1 one. From dimension seven and above, no non-trivial subalgebra includes the whole Poincaré algebra. In fact, a superspace path-integral formulation of maximally supersymmetric Yang-Mills theories in higher dimensions must presumably give up manifest Lorentz invariance.
In [2] we have shown that SO(1, 1) × Spin(7) ⊂ SO (1,9) is the biggest subgroup of the ten-dimensional Lorentz group that can be preserved for obtaining an off-shell closed supersymmetric algebra of the N = 1, d = 10 supersymmetric theory. We introduced for this theory SO(1, 1) × Spin(7) ⊂ SO (1,9) invariant constraints for the curvatures of superfields depending of 1 + 8 fermionic coordinates, as a hint for a possible off-shell superspace description. Part of this Letter is devoted to solve explicitly these constraints, in function of the fields of the component formalism, in dimension up to d = 10.
The maximally supersymmetric theory in ten dimensions is a chiral model with a gauge anomaly that spoils its quantization. A consistent approach implies in fact its coupling to supergravity at the quantum level. However, its chiral anomaly often disappears after dimensional reduction to lower dimensions. It is thus a relevant question to investigate a possible superspace off-shell formulation of the pure ten-dimensional supersymmetric Yang-Mills theory.
The path integral quantization procedure in superspace usually requires to solve the superspace constraints by introduction of an unconstrained prepotential. This procedure is difficult within the considered twisted superspace, mainly because some constraints are cubic in the gauge superfields. This justifies that we first consider the quantization of a simpler model in four dimensions. As a matter of fact, the twisted SU(2) × SU(2) invariant formulation of the N = 2 super-Yang-Mills theory without matter in four dimensions is formally very similar to the discussed formulation of the ten-dimensional theory. Instead of introducing a prepotential, we quantize this theory by implementing the constraints by mean of Lagrange multipliers. The path-integral is then formulated in term of the unconstrained potential superfields themselves.
The four-dimensional action is written as an integral over the full twisted superspace of three different parts. The superspace constraints are implemented by Lagrange multipliers through a BF -like action. where F stands for the components of the super-curvature that define the constraints. Because of Bianchi identities, the auxiliary superfields B possess a set of zero modes that must be taken into account in the super-Feynman rules. The complete gauge-fixing of the BF component of the action requires the introduction of an infinite tower of ghosts and ghosts for ghosts. This problem is reminiscent of the infinite set of auxiliary fields required in the harmonic-superspace formulation of the theory [3]. The classical part is a Chern-Simons-like action for the superspace connection along the scalar odd coordinate. Finally, the gauge-fixing part is a generalization in superspace of the usual Landau gauge-fixing action. Its decomposition in components turns out to be equivalent to a supersymmetric gauge-fixing involving shadow fields, which generalize those introduced in [4].
We will ignore through the Letter the problems associated with unitarity and the doubling of fermions in four and eight-dimensional euclidean space. This is justified within the context of describing the ten-dimensional structure.
The Letter is organized as follows. In the first section we define the N = 2 twisted superspace, and its generalization in higher dimensions. Then we define the twisted super-Yang-Mills constraints and solve them in term of the component fields in four dimensions and generalize the results in ten dimensions, with an obvious application in eight dimensions. In the last section, we construct the action for the N = 2, d = 4 theory. We explain the problem associated with the gauge invariance of the Lagrange multipliers that enforce the covariant constraints on the supercurvature, but postpone to a forthcoming publication the definition of the corresponding gauge-fixing action.
There are earlier references for the idea of twisted superspace for the N = 2, d = 4 twisted super-Yang-Mills vector multiplet [5,6,7]. The general superspace methodology for super-connections relies on [8,9]. No path-integral formulation in twisted superspace had been proposed so far.
2 Twisted superspace set-up 2.1 The N = 2, d = 4 case Let us recall the basic features of the twisted formulation of N = 2 super-Yang-Mills theory [10]. It is defined in a four-dimensional euclidean space with the manifest invariance reduced to L ′ = SU(2) ′ × SU(2) R , where SU(2) ′ is the diagonal subgroup of SU(2) L ×SU(2) I , and SU(2) I is the internal symmetry group associated to N = 2 supersymmetry. The vector multiplet in representations of L ′ is made of the gauge field A µ , two commuting scalar fields Φ andΦ, an anticommuting vector Ψ µ , an anticommuting antiselfdual 2-form χ µν − , an anticommuting scalar η, and a commuting auxiliary field G µν − . These fields transform under the scalar and vector anticommuting generators δ ≡ ǫ αı Q αı and δ µ ≡ iσα ı µ Qα ı . The invariance under the action of these 5 generators completely determines the classical action of the theory, which is nothing but the super-Yang-Mills action, in twisted form [4]. In order to recover the complete super-Poincaré symmetry with 8 generators, one must introduce the anti-selfdual generator δ µν − ≡ σ αı µν Q αı . The δ µν − invariance is an additional symmetry of the action, which is obtained for free from the requirement of δ and δ µ symmetry. Moreover the absence of trivial anomalies for the tensor symmetry shows that forgetting about the tensor symmetry does not introduce ambiguities in the renormalization program [11]. Therefore, as long as we only consider correlation functions of the fields, the scalar and vector supersymmetry generators unambiguously determine the theory to be invariant by the action of all the supersymmetry generators, including the tensor generator δ µν − .
To express the scalar and vector supersymmetry in terms of superspace derivatives, we complete the four-dimensional space by five anticommuting coordinates, a scalar one θ and a vector one ϑ µ (µ = 1 · · · 4). We define as follows the superspace differential operators Q and Q µ , whose action on superfields provide component by component a linear realization of the scalar δ and vector δ µ supersymmetry generators Here the index A stands for the L ′ representation of the superfield and S A carries ♯(A)×2 5 components, where ♯(A) is the dimension of the corresponding L ′ representation.
The covariant superspace derivatives and their anticommuting relations are They anticommute with the supersymmetry generators. A connection superfield (C, Γ µ , A µ ) valued in the adjoint of the gauge group of the theory can be defined in correspondence with the set of the superspace derivatives (∇, ∇ µ , ∂ µ ). This provides the following gauge covariant superderivativeŝ The corresponding covariant superspace curvatures are so that These different objects can be assembled into an extended exterior differential and the extended connection The Bianchi identity implies the following constraints on the components of F The super-gauge transformations of the extended connection A and curvature F are where the gauge superparameter α can be any given general superfield valued in the Lie algebra of the gauge group. The "infinitesimal" gauge transformation is δA = ∆α+[A, α].

Higher dimensions
The formalism for the scalar and vector supersymmetry generalizes directly to the euclidean eight-dimensional case, by extending the eight-dimensional space-time with nine fermionic coordinates and considering a reduction of the Wick rotated Lorentz group SO (8) to Spin (7), with all previous equations remaining formally identical. One can further "oxidise" the eight-dimensional theory into the N = 1, d = 10 theory. This has already been described in [2], and we shall only summarise the equations that are relevant for the following. (One can go from four to six dimensions in an analogous way). The N = 1, d = 10 superspace is made of ten bosonic coordinates x m and nine fermionic ones θ and ϑ i . The x m (m = 0, · · · 9) split into euclidean eight-dimensional coordinates x i and light-cone coordinates x + and x − , so that a general ten-dimensional form splits as A m dx m = A i dx i + A + dx + + A − dx − . The Grassmann coordinates θ and ϑ i are scalar and vector, the latter being identified with the spinorial representation 8 of Spin (7). The covariant superspace derivatives are defined as ∇ ≡ ∂ ∂θ − θ∂ + and Super-curvatures are defined by the analogue of Eq.(9) for ten dimensions 1 We have analogous notations Φ andΦ αβ for the curvatures of the different N = 1 and N = 2 cases, in six (respectively four) dimensions (α, β=µ, ν), and ten (respectively eight) dimensions (α, β=i, j). However, after dimensional reduction and once the constraints Φ N =1 =Φ N =1 ij = 0 are imposed, we have the correspondence A + → Φ N =2 and A − →Φ N =2 .
3 Constraints and their resolution 3.1 The N = 2, d = 4 case To eliminate superfluous degrees of freedom and to make contact with the component formulation, we must impose superspace gauge covariant constraints, as follows The super-gauge symmetry defined in Eq.(11) allows us to simplify the resolution of the constraints. We partially fix super-gauge invariance by setting to zero all antisymmetric as well as the first component C| 0 of C. 2 In this gauge, the remaining gauge invariance reduces to that of the component formalism (α = α| 0 ). The details of the procedure will be found in [11]. After solving the constraints in this particular super-gauge, we will reintroduce the super-gauge invariance by a general gauge transformation depending on new fields that stand for the longitudinal components.
We start with Γ µ . The constraint Eq.(15) onΦ µν and its Bianchi identity leave its ϑ µ independent trace components unconstrained. We define them asΦ| 0 ≡Φ and ( ∂ ∂θΦ )| 0 ≡ η. Using the definition ofΦ µν in terms of Γ µ and its Bianchi identity, we then obtain The constraint L µ = 0 allows us to express A µ in terms of Γ µ and C. It is convenient to parametrize the superfield C as whereΦ andÃ are general functions in ϑ variables, except thatÃ| 0 = 0 as it is required by our special gauge choice. Moreover, we define ( ∂ ∂ϑ µÃ )| 0 ≡ A µ andΦ| 0 ≡ Φ. We can then determine A µ as The explicit content ofΦ andÃ is determined through the resolution of the anti-selfdual constraint on the χ µν curvature. We first observe that the Bianchi identities and the constraint L µ = 0 imply This allows one to express η and χ [µν] in terms ofΦ,Ã andΦ and η, The component ( ∂ ∂ϑ µΦ )| 0 is not constrained. We define ( ∂ ∂ϑ µΦ )| 0 ≡ −Ψ µ and we solve the constraint χ [µν] + = 0, component by component. From the θ-independent part, we get and the part proportional on θ gives us that where χ, G are anti-selfdual 2-forms and F = dA + A 2 . As a result, the general solution of the constrained superfields in the chosen Wess-Zumino-like gauge can be written in term of the known component fields of the theory, with the auxiliary field required for the functional representation of the supersymmetry algebra. The general solution to the constraints (15) can now be obtained by application of a general super-gauge transformation, which we parametrize as follows 3 e α = e θϑ µ ∂µ eγe θc = eγ 1 + θ(c + e −γ ϑ µ ∂ µ eγ) whereγ andc are respectively commuting and anticommuting functions of ϑ µ and the coordinates x µ , with the conditionγ| 0 = 0. The superfield connections C, Γ and their curvatures then have the following expressions and One can check that the supersymmetry transformations of the connection superfields reduce in components to the known twisted transformation laws of the N =2 super-Yang-Mills theory in the Wess-Zumino gauge. This is obtained forγ =c = 0 and redefining the supersymmetry transformations by adding appropriated field-dependent super-gauge transformations such that these fields are left invariant.

Higher dimensions
We now consider the N = 1, d = 10 theory, which also encodes the case N = 2, d = 8. The constraints Eq.(15) become where Ω ijkl is the octonionic eight-dimensional Spin (7)-invariant 4-form [2]. Proceeding along the same line as for the resolution of the constraints in four dimensions, we get the gauge-fixed solution (once again we refer the reader to [11] for more details) which gives the solution to ∇ dϑ Γ + Γ 2 = −|dϑ| 2 A − as Γ i = −ϑ i A − . Then, by introducing the functionsÃ andÃ + of ϑ i to parametrize C, and by using the constraints Φ = L i = 0 and the Bianchi identities, one can write A + , A i and χ ij in terms of C and Γ i . Eventually, the anti-selfdual constraint on χ [ij] permits one to completely determine the component field content of each superfield. The expansion ofÃ andÃ + is in fact By introducing the fieldsc andγ, one can reinforce the super-gauge invariance and get the following expression for the ten-dimensional superfield C The supersymmetry transformation laws of the ten-dimensional super-Yang-Mills in components in the Wess-Zumino gauge [2] are then recovered in an analogous way as in the four dimensional case.

Action in superspace 4.1 The gauge invariant part
We observe from the Bianchi identity ∇Φ + [C, Φ] = 0 that the gauge invariant function Tr Φ 2 is θ independent. Therefore, its components of highest order in ϑ µ can be used to write the equivariant part of the action. The latter can be expressed as a full superspace integral of a Chern-Simons-like term One can check that this action reproduces the known action for super-Yang-Mills in components. Notice that the superfield C has a positive canonical dimension, which is an interesting point for its renormalization properties. Unfortunately this formula does not generalize to higher dimensions. However, the one-loop invariant counter-terms involved in the eight-dimensional theory can be expressed as simple integrals over superspace The constraints can be covariantly implemented by the following superspace integral depending on auxiliary Lagrange multipliers superfields where B (µν) is symmetric traceless andΨ [µν] + is antisymmetric selfdual. The superfields K µ and A µ can be trivially integrated, giving rise to a simple substitution of A µ by minus ∇Γ µ +∇ µ C+{Γ µ , C}. The resolution of the constraints is such that the formal integration over the auxiliary superfields B (µν) andΨ [µν] + leads to the non-manifestly supersymmetric formulation of the theory in components, without introducing any determinant contribution in the path-integral. However, B (µν) andΨ [µν] + admit a large class of zero modes that must be considered in the manifestly supersymmetric superspace Feynman rules. They can be summarized by the following invariance of the action where λ (σµν) is a superfield in the rank three symmetric traceless representation and

space.
To fix the super-gauge invariance, one first introduces a Fadeev-Popov ghost superfield Ω and a BRST differential s that anticommutes with ∆. As indicated by the super-gauge transformations (11) and their infinitesimal version, the BRST symmetry is defined as One also needs a Fadeev-Popov antighost superfieldΩ and its Lagrange multiplier superfield B. In fact, the BRST transformation laws of the super-connection, super-ghost and super-antighost follow from the following generalization of the horizontality equation Eq.(9), which involves both the anti-BRST operators and the BRST operator s A fully invariant gauge-fixing action can then be written as One has also to write a gauge-fixing action for the action of constraints. The gauge invariance (33) can be written in terms of the BRST operator, thanks to the introduction of the ghostsΨ (1,0)µν,σ and B (1,0)µνσ . As discussed in the previous section, the BRST transformations are themselves subject to a gauge invariance and one has to introduce an infinite tower of ghosts for ghosts to correctly gauge-fix the theory. We define the commuting ghostsΨ (n,0)µν,··· in the rank n + 2 irreducible representation obtained by applying the symmetric traceless projector on the n + 1 last indices and then the antisymmetric selfdual projector to the two first indices, as well as the anticommuting ghost B (n,0)µν··· in the rank n + 2 symmetric traceless representation. The BRST transformations are the following whereΨ (0,0)µν and B (0,0)µν are simplyΨ µν and B µν . The BRST operator is nilpotent modulo the constraints, that is modulo the equations of motion of the fieldsΨ µν , B µν and K µ . The Batalin-Vilkovisky formalism permits one to solve this problem, by introducing antifields as sources for the BRST transformations. 4 We have not yet worked out the gauge-fixing of this BF system. Even if it shares similarities with a standard bosonic BF model, the choice of gauge-functions cannot be defined by naively replacing the space derivative of the bosonic case by the anticommuting vector covariant derivative ∇ µ . It seems that the free case can be worked out, by introducing transverse projectors for the auxiliary fields, but more work is yet required for a complete procedure. It will be described in the forthcoming publication [11], as well as a practical way for doing computations that takes into account the existence of the infinite tower of ghosts in loops.
Despite our present ignorance of the gauge-fixing of the BF system that enforces the covariant constraints, we thus propose as a defining superspace action the following integral over the twisted superspace The four-dimensional expressions (32) and (37) of S C and S GF can be extended to eight and ten dimensions. It is not clear however if these expressions are relevant in higher dimensions, where the introduction of a prepotential is required in order to write the equivariant part of the action.

Conclusion
By using twisted variables, one can reexpress the N = 2, d = 4 supersymmetry algebra in such a way that the pure super-Yang-Mills theory is determined by a subalgebra of the super-Poincaré algebra. We have seen the existence of a corresponding twisted superspace, with coordinates (x µ , θ, ϑ µ ). The result generalizes in higher dimensions. Quite interestingly, the constraints on the super-curvatures are such that they do not imply the equations of motion. This general property makes it plausible that one can obtain a superspace path-integral formulation of maximally supersymmetric theories.
Moreover, we have shown in this publication that a twisted superspace path-integral formulation of the N = 2 super-Yang-Mills theory does exist in four dimensions. This theory is formulated as a Chern-Simons term for the classical action plus a BF term for expressing the covariant constraints in superspace. Despite the fact that the gauge-fixing of the BF part requires the introduction of an infinite tower of ghosts and ghosts for ghosts, we hope that it will exhibit a general structure for a compact resumation of the ghost contributions. We have solved explicitly the constraints in component formalism and verified that the theory reduces to the usual Yang-Mills theory in components, after integration of the superspace longitudinal components of the super-gauge fields and their corresponding Faddeev-Popov ghosts.
Finally, it must be understood that the construction of a twisted superspace for the N = 2 supersymmetric theory is not an attempt for an alternative to its harmonic superspace formulation. Rather, it is a preliminary construction, as an example of a non-manifestly Lorentz invariant superspace-path-integral that can be generalized in ten dimensions, but must be completed within an harmonic superspace path-integral formulation for a complete description of the ten-dimensional super-Yang-Mills theory. Eventually, one expects the full Lorentz invariance to be recovered for the on-shell amplitudes.