Light-cone Gauge for N=2 Strings *

Covariant quantization of self-dual strings in 2+2 flat dimensions reduces them to their zero modes, a consequence of extended world-sheet supersymmetry. We demonstrate how to arrive at the same result more directly by employing a ‘double’ light-cone gauge. An unconventional feature of this gauge is the removal of anticommuting degrees of freedom by commuting symmetries and vice versa. The reducibility of the N=4 string and its equivalence with the N=2 string become apparent. ∗ Contribution to a special issue of the Russian Physics Journal + On sabbatical leave from Institut für Theoretische Physik, Universität Hannover, Germany 1. N=2 and N=4 strings in superconformal gauge. String theories with more than one world-sheet supersymmetry ‘suffer’ from the absence of higher dimensions [1, 2]. Indeed, covariant quantization and naive BRST ghost-counting for the N=2 and N=4 superconformal algebras of constraints yield critical dimensions of four and minus eight(!), respectively. However, as was realized by Siegel [3], the N=4 constraints are reducible, and the N=4 string turns out to be the same as the N=2 string. Yet, in contrast to the N=2 formulation, the N=4 description is manifestly Lorentz covariant. Since the required signature of the four-dimensional target is (++−−), by ‘Lorentz group’ we mean SO(2, 2) ≃ SL(2,R) ⊗ SL(2,R). These global symmetries are not to be confused with the local R symmetry of N=4 supersymmetry, denoted by SL(2,R). In this letter we shall employ (Majorana) spinor notation for all Lorentz and internal indices. We distinguish the different groups by using α ↔ SL(2,R) , α̇ ↔ SL(2,R) , α̈ ↔ SL(2,R) , where α ∈ {0, 1} . (1) In particular, fundamental spinors are taken to be real.1 In the NSR formulation, both N=2 and N=4 strings are parametrized by the four coordinatesX plus four anticommuting NSR fields ψ. From the world-sheet point of view, the former are scalars while the latter form two-component Majorana spinors, ψ = (ψ + , ψ α̈α̇ − ) in a Weyl basis. With ‘±’ we generally indicate light-cone components of world-sheet tensors, i.e. ∂± = 1 2(∂τ ± ∂σ). Sharing a world-sheet supersymmetry multiplet with X, the anticommuting coordinates ψ should also carry an undotted SL(2,R) index. However, its value is coupled to that of the SL(2,R) index and we suppress it. The action defines a (target-space) light-cone pairing, (X,X) , (ψ, ψ) and (X,X) , (ψ, ψ) , (2) which decomposes the variables into two N=1 light-cone sets. Starting from the formulation with auxiliary world-sheet N=2 or N=4 supergravity [4, 5], we advance to the superconformal gauge. In the critical dimension, all supergravity remnants then disappear thanks to super Weyl invariance. The residual (superconformal) freedom in this gauge does not prevent quantization but implies that physical states are subject to constraints and gauge identifications. The N=2 constraints (T,G0̈1, G1̈0, J 0̈1̈) represent a non-degenerate subset of the (reducible) N=4 constraints (T,G, J (α̈β̈)), which entails the selection of a one-parameter subgroup of the SL(2,R) R-symmetry group generated by the spin-one constraints J (α̈β̈). Because we work with real spinors it is preferable to choose a noncompact subgroup GL(1,R) ⊂ SL(2,R). As the R-symmetry index of ψ is tied to the space-time SL(2,R) index, the choice of J 0̈1̈ incidentally also breaks the Lorentz group, SO(2, 2) ≃ SL(2,R) ⊗ SL(2,R) −→ GL(1,R)⊗ SL(2,R) . (3) Being non-degenerate, each (commuting or anticommuting) N=2 constraint essentially removes one timelike and one spacelike degree of freedom (of matching statistics). Hence, we expect (T,G0̈1, G1̈0, J 0̈1̈) to eliminate all string coordinates X and their partners ψ in the 2+2 dimensional space-time. Indeed, this expectation is confirmed by direct analysis [6] as well as by amplitude computations which reveal that the N=2 string is just a point particle encoding the dynamics of self-dual Yang-Mills and gravity [7], at least at tree-level.2 1 Hence, v and v are not related by complex conjugation, as in 3+1 dimensions. 2 For a review see [8, 9]. Quantization is detailed in [10]. The loop structure is subject of [11, 12].