Pohlmeyer reduction of superstring sigma model

Abstract

Motivated by a desire to find a useful 2d Lorentz-invariant reformulation of the AdS 5 × S 5 superstring world-sheet theory in terms of physical degrees of freedom we construct the "Pohlmeyerreduced" version of the corresponding sigma model. The Pohlmeyer reduction procedure involves several steps. Starting with a coset space string sigma model in the conformal gauge and writing the classical equations in terms of currents one can fix the residual conformal diffeomorphism symmetry and kappa-symmetry and introduce a new set of variables (related locally to currents but non-locally to the original string coordinate fields) so that the Virasoro constraints are automatically satisfied. The resulting equations can be obtained from a Lagrangian of a non-abelian Toda type: a gauged WZW model with an integrable potential coupled also to a set of 2d fermionic fields. A gauge-fixed form of the Pohlmeyer-reduced theory can be found by integrating out the 2d gauge field of the gauged WZW model. The small-fluctuation spectrum near the trivial vacuum contains 8 bosonic and 8 fermionic degrees of freedom with equal mass. We conjecture that the reduced model has world-sheet supersymmetry and is ultraviolet-finite. We show that in the special case of the AdS 2 × S 2 superstring model the reduced theory is indeed supersymmetric: it is equivalent to the N=2 supersymmetric extension of the sine-Gordon model.

Alleviating the non-ultralocality of the AdS5 × S5 superstring

“…We then show that these Hamiltonian equations of motion coincide with those associated with the fermionic extension of the G/H gauged WZW model with an integrable potential as given in [18]. This canonical analysis is presented in section 4.…”

“…After summarizing the situation in section 3.3, we discuss the reduced theory in section 3.4. First of all, the resulting reduced equations of motion are exactly as in [18] and exhibit a H L × H R -gauge invariance where H L,R ≃ [SU(2)] 4 . However, they are not Hamiltonian with respect to the modified Poisson bracket but this is remedied by partially fixing the H L × H R -gauge invariance to the diagonal subgroup.…”

“…In view of the possible generalization of the results of [10] to semi-symmetric space σ-models, in [17] we already investigated directly the canonical structure of the semisymmetric space sine-Gordon model obtained by Pohlmeyer reduction of the AdS 5 × S 5 superstring [18,19]. We have shown that the corresponding non-ultralocality is only mild and have given the corresponding lattice Poisson algebra for the discretized Lax matrix.…”

“…Even after setting all of the constraints to zero, the modified Poisson bracket is still degenerate. All fields take values in f but describing the remaining Casimirs requires lifting one field to G. Remarkably, it turns out that these Casimirs correspond to gauge fixing conditions used in the Pohlmeyer reduction of the AdS 5 × S 5 superstring [18]. We thus set their values accordingly.…”

“…But in order for the Virasoro constraint Str(A (2) + A (2) + ) = 0 to be satisfied, T has to be taken such that Str T 2 = 0. We shall choose 2 the same T as in [18]. Its definition and the fact that it induces a Z 2 -grading of f, denoted f [0] ⊕ f [1] , are recalled in appendix B, along with the definitions of some other matrices used below.…”