Restoring Lorentz invariance in classical N=2 string

Abstract

We study classical N=2 string within the framework of the N=4 topological formalism by Berkovits and Vafa. Special emphasis is put on the demonstration of a classical equivalence of the theories and the construction of an action for the N=4 topological string. The SO(2, 2) Lorentz invariance missing in the conventional Brink-Schwarz action for the N = 2 string is restored in the N = 4 topological action.

Geometry of anN=4twisted string

“…[10] we choose to get rid of d = 2 γ-matrices and work directly in terms of irreducible components on the world-sheet (for example ψ (24), which we rewrite here in terms of the irreducible components…”

“…Just like we proceeded in the former case, in order to gauge two remaining twisted global supersymmetries in the N = 2 gauged nonlinear sigma model it suffices to mimic the structure of the terms entering the N = 4 twisted string. To this end, on the world-sheet there must be introduced two new real vectors C α and D α , and an extra complex fermion µ Aα [10], these complementing an N = 2, d = 2 supergravity multiplet to an N = 4, d = 2 one and playing the role of the gauge fields for the extra local symmetry transformations. An amendment composed of the new fields reads…”

“…According to the N = 4 topological prescription, one adds to the theory two more fermionic currents (of conformal spin 3/2) and two more bosonic ones (of conformal spin 1) which, on the one hand extend the N = 2 SCA to a small N = 4 SCA, but on the other hand do not change the physical content of the model as they prove to be functionally dependent. The key point, however, is that this extension brings an extra U(1, 1) outer symmetry to the formalism (see [10,11] 4 N = 3 automatically implies N = 4, as a product of two complex structures yields a third one [5].…”

“…[12]. Based on the symmetry argument, the N = 4 topological string action has been constructed in [10] just by installing the U(1, 1) outer into the action of the N = 2 string. Curiously enough, to a great extent the situation resembles what happens for the Green-Schwarz superstring, where extracting an independent set of fermionic first class constraints is known to be in a conflict with manifest Lorentz covariance 6 .…”

“…At the classical level the former drawback has been overcome recently [10,11] based on an earlier N = 4 topological formalism by Berkovits and Vafa [12]. According to the N = 4 topological prescription, one adds to the theory two more fermionic currents (of conformal spin 3/2) and two more bosonic ones (of conformal spin 1) which, on the one hand extend the N = 2 SCA to a small N = 4 SCA, but on the other hand do not change the physical content of the model as they prove to be functionally dependent.…”