Sigma Models with N=8 Supersymmetries in 2+1 and 1+1 Dimensions

We introduce an N=8 supersymmetric extension of the Bogomolny-type model for Yang-Mills-Higgs fields in 2+1 dimensions related with twistor string theory. It is shown that this model is equivalent to an N=8 supersymmetric U(n) chiral model in 2+1 dimensions with a Wess-Zumino-Witten-type term. Further reduction to 1+1 dimensions yields N=(8,8) supersymmetric extensions of the standard U(n) chiral model and Grassmannian sigma models.


Introduction and Summary
Nonlinear sigma models in k dimensions describe mappings of a k-dimensional manifold X into a manifold Y (target space). In particular, as target spaces one can consider Lie groups G (chiral models) and homogeneous spaces G/H for closed subgroups H ⊂ G. Sigma models and their Nextended supersymmetric generalizations play an important role both in physics and mathematics (see e.g. [1,2]). For instance, two-dimensional sigma models serve as a theoretical laboratory for the study of more complicated (quantum) super Yang-Mills theory since they share many of its features such as asymptotic freedom, nontrivial topological structure, the existence of instantons, ultraviolet finiteness for the N =4 supersymmetric case etc. [3]. Moreover, supersymmetric two-dimensional sigma models are the building blocks for superstring theories [3,4].
Recall that for two-dimensional nonlinear sigma models admitting a Lagrangian formulation the number of supersymmetries is intimately related to the geometry of the target space. Namely, it was argued that Lagrangian N =1 models can be defined for any target space Y , for N =2 the target space must be Kähler, for N =4 it must be hyper-Kähler, and no Lagrangian models were introduced for N >4 [5,6]. Similar results hold for sigma models in three dimensions. In particular, this means that a target space Y admits no more than N =1 supersymmetry in the case of (non-Kähler) group manifolds G and N ≤2 supersymmetries for homogeneous Kähler spaces G/H.
The field equations of the standard G and G/H sigma models in 1+1 and 2+0 dimensions can be obtained by dimensional reduction of the self-dual Yang-Mills (SDYM) equations in 2+2 dimensions, with a gauge group G [7]. Concretely, the SDYM model reduced to two dimensions is equivalent to the sigma model with G-valued scalar fields, while the G/H sigma model arises after imposing additional algebraic constraints. Similar reduction to 2+1 dimensions yields a modified integrable chiral model [8]. Recall that the SDYM model in 2+2 dimensions can be endowed with up to four supersymmetries [9,10]. Reducing the N -extended supersymmetric SDYM equations in 2+2 dimensions to 2+1 and 1+1 dimensions yields models which have twice as many supersymmetries (cf. [11] for reductions from 3+1 dimensions). We will show that for G=U(n) and N =4 these models are equivalent to U(n) chiral models with N =8 supersymmetries. These new supersymmetric sigma models in 2+1 and 1+1 dimensions are well defined on the level of equations of motion, but their Lagrangian formulation is not known yet.
In this note we concentrate on the reduction of the N =4 SDYM equations (instead of arbitrary N ≤4) in 2+2 dimensions since for this case a Lagrangian can be written down at least in terms of the component fields of a reduced Yang-Mills-type supermultiplet. Moreover, it was shown by Witten [12] that the N =4 SDYM model appears in twistor string theory, which is a B-type topological string with the supertwistor space CP 3|4 as a target space 1 . This fact gives additional arguments in favour of introducing N = 8 supersymmetric sigma models in 2+1 and 1+1 dimensions related with twistor string theory and of studying their properties.
N =4 SDYM in superfields. The field content of N =4 supersymmetric SDYM is given by a supermultiplet (A αα , χ iα , φ ij ,χα i , Gαβ) of fields on R 2,2 of helicities (+1, + 1 2 , 0, − 1 2 , −1). Here A αα are the components of a gauge potential with the field strength Note that the scalars φ ij are antisymmetric in ij and all the fields, including the fermionic ones χ iα andχα i , live in the adjoint representation of the gauge group U(n).
The N = 4 SDYM equations [15,9] can be written in terms of superfields on antichiral superspace R 4|8 [9,16]. Namely, all fields from the above N = 4 supermultiplet can be combined into superfields A αα and A iα on R 4|8 in terms of which the N = 4 SDYM equations read where we have introduced the covariant derivatives ∇ αα := ∂ αα + A αα and ∇ iα := ∂ iα + A iα . (2.12) Note that (2.11) can be combined into the manifestly supersymmetric equations where A αα and A iα depend only on x αα and ηα i .
N = 8 supersymmetric Bogomolny-type equations on R 2,1 . After imposing the condition of t-independence on all fields in the linear system (2.15), we obtain the equations andD i α given in (3.5). Here A i α , A (αβ) andφ are superfields depending only on y αβ and η β i . The compatibility conditions for the linear system (3.7) read As usual, these manifestly N = 8 supersymmetric equations are equivalent to equations in component fields, Obviously, these equations are ∂ 4 -reduction of (2.16).
Supersymmetric sigma models. Note that matrices ψ ± in (3.7) are defined up to a gauge transformation generated by a matrix which does not depend on λ ± and therefore one can choose a gauge such that where Φ is a U(n)-valued superfield and Υ is a u(n)-valued superfield both depending only on y αβ and η α i . For this gauge, from (3.7) we obtain and from (3.8) we have Substituting (3.12) into (3.9), we obtain equationŝ which after using (3.5) and (3.13) read Note that the last two terms in (3.15) are the Wess-Zumino-Witten terms which spoil the standard Lorentz invariance but yield an integrable U(n) chiral model in 2+1 dimensions. For reduction to 1+1 dimensions one should simply put ∂ y Φ = 0 in (3.15)- (3.17) obtaining an N =8 supersymmetric extensions of the standard U(n) chiral model in two dimensions with field equations There is not yet a Lagrangian description of equations (3.15)-(3.17) or (3.18). However, using the equivalence of equations (3.10) to (3.14), one can write explicitly a Lagrangian in terms of fields (A (αβ) , χ iα , ϕ, φ ij ,χ α i , G αβ ). The proper Lagrangians follow from (2.17) by reduction to 2+1 and 1+1 dimensions. It is a challenging task to find Lagrangians in terms of the U(n)-valued superfield Φ. where ǫ iα and ǫ α i are 16 Grassmann parameters. In particular, for coordinates y αβ and η β i on the antichiral superspace R 3|8 we haveδy αβ = −2ǫ i(α η β) i andδη α i = ǫ α i . It is obvious that the sigma model field equations (3.14) are invariant under the supersymmetry transformations (3.20) because the operatorsD i α as well asD iα anticommute with the supersymmetry generatorsQ iα andQ j β . Note that these N = 8 supersymmetric extensions of the U(n) and Gr(k, n)=U(n)/U(k)×U(n−k) sigma models in 2+1 and 1+1 dimensions are not the standard ones defined only for N ≤ 1 and N ≤ 2, respectively. It will be interesting to study this new kind of sigma models in more detail.