Harmonic Superspaces from Superstrings

We derive harmonic superspaces for N=2,3,4 SYM theory in four dimensions from superstring theory. The pure spinors in ten dimensions are dimensionally reduced and yield the harmonic coordinates. Two anticommuting BRST charges implement Grassmann analyticity and harmonic analyticity. The string field theory action produces the action and field equations for N=3 SYM theory in harmonic superspace.


Introduction
Pure spinors [1] in ten dimensions are complex commuting chiral spinorial ghosts λα withα = 1, . . . , 16 satisfying the ten nonlinear constraints λαγm αβ λβ = 0 , (1.1) (hats denote 10-dimensional indices). They form the starting point for a new approach to the quantization of the superstring with coordinates xm, θα and λα [2]. Due to these constraints on λ, the troublesome second class constraints of the superstring become effectively first class. One can relax these constraints and obtain a covariant formulation by introducing more ghosts as Lagrange multipliers [3]. The result is an N = 2 WZNW model [4]. The pure spinors in this covariant approach are real and the BRST charge maps θα into λα. In this letter, though, we use complex constrained λα. Pure spinors also exist in other dimensions [1].
Harmonic superspace (see [5] for a complete review of the subject and references 3 ) was constructed to circumvent the no-go theorems for a full-fledged superspace description of N-extended supersymmetries (susy). The main idea is to let the R-symmetry group U (N ) (or SU (N ) for N=4), which acts on the susy generators, become part of a coset approach. , (1.2) respectively, although other choices are also possible [6].
In this letter we present a derivation of four-dimensional harmonic superspaces from ten-dimensional pure spinors by using ordinary dimensional reduction in which we set the extra six coordinates to zero by hand. The spinors λα decompose into λ α I andλα I where I = 1, . . . , 4 is an SU (4) ∼ SO(6) index. The main idea is to factorize the pure spinors 3 Two useful accounts of the subject can be found in [6] and in [7]. Projective harmonic superspace has been introduced in [8]. The application to the AdS/CFT correspondence is studied in [9], and some developments of N = 4 harmonic superspace for SYM can be found in [10] and in [11]. λα into auxiliary variables λ α a andλα a with a = 1, 2, and harmonic variables u a I andv aI . In this way we factorize the Lorenz group and the internal symmetry group SU (4). Using this factorization, the pure spinor constraints turn into constraints on λ α a andλα a , and on u a I andv aI . Contracting the operator d zα in the BRST charge [2] with the harmonic coordinates leads to eight spinorial covariant derivatives which satisfy the constraints as a consequence of the constraints on u andv, and in terms of which G(Grassman)analyticity (dependence on half the θ's) of superfields is defined. If one does not provide the information that d a α andd ȧ α are linear in u a I andv aI , one looses information. We therefore construct a second BRST charge which only anticommutes with Q H if d a α andd ȧ α are factorized as in (1.4). It is constructed from the generators of U (N ) represented by the following differential operators 4 Requiring that the vertex operators are annihilated by these BRST charges should yield the field equations of N=4 harmonic superspace. In this letter we work out the case of N=3 and obtain by truncation the field equations of N=3 SYM theory in harmonic superspace.
We end by deducing an action for N=3 SYM theory in harmonic superspace from the Chern-Simons action for string field theory [12].
The present analysis might provide a link between string theory with pure spinors and recent developments in twistor theory [13]. Another interesting aspect not covered in the present letter is deformed harmonic superspace [14]. It would be interesting to discover which kind of harmonic superspace one obtains for suitable Ramond-Ramond backgrond fields [15].
In a future article we intend to extend these results to the N=4 case and construct an action for N=4 SYM theory [16]. In particular, this should give a conceptually simple derivation of the rather complicated measure. This suggests that the second BRST charge might be obtained by dimensional reduction of the BRST charge in ten dimensions, extended to include the ten dimensional Lorentz generators. 5 A similar analysis is pursued in [17].
2. The coordinates of N=4, N=3, and N=2 harmonic superspace from pure spinors We substitute the decomposition λα = (λ α I ,λα I ) into the pure spinor constraints, and use the representation of the matrices γm αβ given in [18]. In this representation the Dirac matrices with m = 0, 1, 2, 3 are labelled by γ αβ and those for m = 4, . . . , 9 are labelled by γ IJ = −γ JI , and all matrix elements are expressed in terms of Kronecker delta's and the epsilon symbols ǫ αβ , ǫαβ and ǫ IJKL . The pure spinor constraints decompose then into the following six plus four constraints The first relation corresponds to m = 4, . . . , 9 while the second one corresponds to m = 0, 1, 2, 3. To solve these constraints we adopt the following ansatz where a = 1, 2. The new variables u a I andv aJ are complex and commuting.They carry GL(2, C) and SU (4) indices. The spinors λ α a ,λα a are also complex and commuting, and carry a representation of SL(2, C) and GL(2, C). In this way, we separate the Lorentz group from the internal symmetry group SU (4).

The decomposition in (2.2) is left invariant by the gauge transformations
where M andM are independent GL(2, C) matrices. The factorization (2.2) plus the gauge invariance (2.3) yields 16 complex parameters. To reduce to the usual 11 independent complex parameters of pure spinors, we further impose the following two covariant constraints u a Iv bI = 0 , λ α a ǫ αβ ǫ ab λ β b +λα a ǫαβǫ abλβ b = 0 . S(U(2)×U(2)) used in [7](see also [11] and [9]). The restriction of U (2) × U (2) to the subgroup S(U (2) × U (2)) is due to second constraint of (2.4). The latter is preserved by the transformations M andM only after the identification detM = detM .
To identify the SU (4) of the coset space, we introduce new coordinates u a,ḃ and is a U (4) matrix because the harmonic variables u a I andv aI satisfy the constraints (2.4) and they can be normalized as follows, using the gauge This gauge choice is preserved by S(U (2) × U (2)).
The normalizations ( (2)) . Let us turn to N=3 harmonic superspace. If we decompose the λ α I 's and theλα I 's into N=3 vectors and N=3 scalars we have λ α I = (λ α i , ψ α ) andλα I = (λα i ,ψα). In that basis, the pure spinor constraints in (2.1) become The reduction to the N=3 case is obtained by setting ψ α =ψα = 0. Inserting this ansatz into the first two equations of (2.8), we obtain

9)
6 Denoting this relation by N IJ = 0, it is clear that N IJv aJ = 0 and ǫ IJ KL N KL u a J = 0 due to (2.4). This leaves the phase of det u aḃ I undetermined. The gauge in (2.5) sets this phase to zero.
which is equivalent to requiring that all determinants of order 2 of the matrices λ α i and λα i vanish. 7 This means that the pure spinors can be factorized into and the equations (2.8) are solved by So for the N=3 case no constraint is needed for λ α andλα. Notice that the two complex vectors u i andv i are defined up to a gauge transformation where ρ, σ ∈ C. The two real parameters |ρ| and |σ| are used to impose the normalizations u iū i = 1 and v iv i = 1. If one also gauges away the overall phases of u i andv i , the space of harmonic coordinates u i andv i is parametrized by six real parameters. This coincides with the number of free parameters of the coset SU (3)/U (1) × U (1). Indeed, we can construct (2.14) For later use we also list the components of the inverse matrix u i I : Finally, we consider a further reduction to N=2. We decompose the N=3 pure spinors λ α i andλα i into a vector of N=2 and a singlet, λ α i = (λ α I , λ α 3 ) andλα i = (λα I ,λα 3 ) where I = 1, 2. We set λ α 3 andλα 3 to zero. The pure spinor equations (2.8) reduce then to The first two equations imply that λ α I andλα I are factorized into λ α I = λ α u I andλα J = λαv J where u Iv I = 0. The vectorv I is proportional to ǫ IJ u J . Hence without loss of generality one may write With this parametrization of the N=2 case there are neither constraints on the λ's nor on the u's.
The vector u I yields the usual parametrization of N=2 harmonic superspace [5].
Namely, one introduces the SU (2) matrix (u + I , u − I ) where u + I = u I and u − I = (u +I ) * with u + J = ǫ J K u +K . The coset SU (2)/U (1) is obtained by dividing by the subgroup U (1) which generates the phases u ± I → e ±iα u ± I . In fact, eqs. (2.17) are defined up to a rescaling of λ α ,λα and of u I given by u I → ρu I , for ρ = 0. This yields the compact space CP 1 .

N=3 Harmonic Superspace for SYM Theory from Superstrings
The field equation for D = 4, N = 3 SYM-theory in ordinary (not harmonic) superspace are given by [19] The coordinates for this N=3 superspace, (x m , θ α i ,θα i ), are obtained by imposing the constraint θ α 4 =θα 4 = 0. Since θ's transform into λ's under BRST transformations we also impose for consistency λ α 4 =λα 4 = 0. Using the decomposition of the N=3 spinors λ α i andλα i given in (2.10), and contracting the harmonic variables with the operators d zα in (1.3) yields two new spinorial operators The operator d 1 α corresponds to ξ i D i α andd 3α to η iDα i in [5]. Due to the constraints on the u's the operators d 1 α andd 3α satisfy the commutation relations To derive these relations one may use the dimensionally reduced relations Hence Q G (where G stands for Grassmann) is nilpotent for any λ α andλα.
The BRST operator Q G implements naturally the G-analyticity on the space of superfields Φ(x, θ,θ, λ,λ, u). A superfield with ghost number zero is given by Φ(x, θ,θ, u) and G-analyticity means θ,θ, λ,λ, u) and similarly ford 3α ). Such a superfield is called a G-analytic superfield in [5]. A generic superfield Φ(x, θ,θ, λ,λ, u) with ghost number one can be parametrized in terms of two u-dependent spinorial superfields A α ,Āα as follows Φ (1) (x, θ,θ, λ,λ, u) = λ α A α +λαĀα , (3.4) and {Q G , Φ (1) } = 0 implies the following constraints on these superfields Assuming that A α and Aα factorize in the same way as D 1 α = u i D i α andD 3α =v iDα i , so A α = u i A i α and Aα =v i Aα i , the equations (3.5) reproduce (3.1). We stress that (3.5), unlike (3.1), do not put the theory on-shell; only the extra assumption of the factorization of A α and Aα puts the theory on-shell. To determine on which harmonic variables superfields depend, we construct a second BRST operator Q H which is constructed from the SU (3) generators where p i b can be represented by ∂/∂u b i and similarly for p b i . These generators split into three raising operators d 1 2 = d (2,−1) , d 2 3 = d (−1,2) , d 1 3 = d (1,1) , three lowering operators −1) , and two Cartan generators d 1 1 and d 2 2 . The raising operators operators commute with Q G and form an algebra, in particular [d (2,−1) , d (−1,2) ] = d (1,1) . This suggests to construct a new nilpotent BRST operator Q H where we introduced new pairs of anticommuting (anti)ghosts (ξ 3 1 , β 1 3 ), (ξ 2 1 , β 1 2 ), (ξ 3 2 , β 2 3 ) with canonical anticommutation relations. It is convenient to use a notation in which the . Since Q H and Q G anticommute their sum Q tot is obviously nilpotent. A generic superfield Φ (1) with ghost number one can be decomposed into into the following pieces where A

The equations of motion for N=3 SYM follow from the BRST-cohomology equations
H denotes the terms with ξghosts and Φ (1) G the terms with λ-ghosts, the Maurer-Cartan equations in (3.10) decompose as follows This system of equations is invariant under the infinitesimal gauge transformation where Ω is a generic harmonic superfield with ghost number zero. According to the above decomposition of Φ (1) , one obtains δΦ is solved by a pure gauge superfield Φ (1) where ∆ is a ghost-number zero superfield known in the literature as the bridge (see for example [5]). Also the BRST cohomology of Q H vanishes on the unconstrained superspace and therefore one can also solve the system (3.11)-(3.13) starting from the last equation.
In the harmonic superspace framework, one usually employs the bridge superfield ∆(x, θ,θ, u) to bring the spinorial covariant derivatives to the 'pure gauge' form (3.15) Here the bridge is seen as the most general solution of (3.11). By making a finite gauge transformation which sets Φ (1) H is given by where the superfield ω satisfies

The Action and Measure for N = 3 SYM theory
We start from the observation that the field equations (3.10) are of Chern-Simons form and can be derived from an action of the form where ⋆ denotes conventional matrix multiplication. The measure dµ has to be determined.
Instead of dimensionally reducing (4.1) we follow a different path. We have to define the integration measure for all zero modes in the theory. Since we are dealing with worldline models, the only contribution comes from the zero modes of x µ , θ α i ,θα i ,λ α i ,λα i , u I i and ξ 1 3 , ξ 2 1 , ξ 1 2 . The set of ghosts λ α i ,λα i pertains to the BRST charge Q G which implements the G-analyticity. Therefore, they implement kinematical constraints on the theory expressed by the equations: where S N=3 is the off-shell N = 3 action and dµ H is the invariant measure in the space of the zero modes of x µ , θ α i ,θα i , u I i and ξ 1 3 , ξ 2 1 , ξ 1 2 . In addition, S N=3 has zero ghost number, while dµ H has ghost number three. Form [2] and [20]it is known that dµ H ∈ H 3 (Q H ). This implies that dµ H = dξ 1 3 dξ 2 1 dξ 1 2 dµ ′ where the measure dµ ′ = dµ ′ (x µ , θ α i ,θα i , u I i ) has to be fixed by the G-analyticity (4.2).

Acknowledgemnets
We thank N. Berkovits, M. Porrati, G. Policastro, M. Roček and W. Siegel for useful discussions. This work was partly funded by NSF Grants PHY-0098527. PAG thanks L.
Castellani and A. Lerda for discussions and financial support.