Fermion zero modes for the mixed-flux AdS3 giant magnon

Abstract

We explicitly construct the four and two fermion zero modes for the mixed-flux generalization of the Hofman-Maldacena giant magnon on two of the AdS 3 backgrounds with maximal amount of supersymmetry, AdS 3 ×S 3 ×T 4 and AdS 3 ×S 3 ×S 3 ×S 1 . We also show how to get the psu(1|1) 4 and su(1|1) 2 superalgebras from the semiclassically quantized fermion zero modes.

Semiclassical quantization of the mixed-flux AdS3 giant magnon

“…Therefore, the mixed-flux magnon on AdS 3 × S 3 × T 4 and AdS 3 × S 3 × S 3 × S 1 should have 4 and 2 fermion zero modes. We found these zero modes in [63], and showed how they can be used to construct the odd generators of the residual algebras. Our objective here is to find the complete spectrum of fluctuations around the AdS 3 giant magnon.…”

“…Our objective here is to find the complete spectrum of fluctuations around the AdS 3 giant magnon. Throughout, we will only consider the stationary magnon, a subclass of solutions we identified as the mixed-flux generalisation of the HM giant magnon in [63]. The rest of this paper is structured as follows.…”

“…Although the perturbation equations are rather complicated, one can construct explicit solutions algebraically using the dressing method, which we adapt to be more suited to the fluctuation analysis. In section 3 we find the fermionic fluctuations, closely following the methods developed in [28,63] extended to non-zero angular frequencies. Using the symmetries of the system and an explicit kappa-fixed ansatz, the full 2 × 32 component spinor equations are reduced to a 4 dimensional system, which we can solve explicitly.…”

“…This is a special case of the dyonic mixed-flux magnon, which was first constructed in [54] for the AdS 3 × S 3 × T 4 background. The stationary magnon was identifed in [63] as the mixed-flux equivalent of the Hofman-Maldacena magnon [24], as compared to the more general AdS 5 dyonic magnon of [26]. The dispersion relation 3…”

“…In this section we solve for the complete fermion fluctuation spectrum around the mixedflux stationary magnon (2.8). Our approach will be mostly based on [63], but rather than normalizable zero modes, we will be looking for solutions with plane-wave asymptotes. The leading order (quadratic) action for fermion fluctuations around a general bosonic string solution X µ (t, x) is given by [70]…”