Large-spin and large-winding expansions of giant magnons and single spikes

Abstract

We generalize the method of our recent paper on the large-spin expansions of Gubser-Klebanov-Polyakov (GKP) strings to the large-spin and large-winding expansions of finitesize giant magnons and finite-size single spikes. By expressing the energies of long open strings in R × S 2 in terms of Lambert's W-function, we compute the leading, subleading and next-to-subleading series of classical exponential corrections to the dispersion relations of Hofman-Maldacena giant magnons and infinite-winding single spikes. We also compute the corresponding expansions in the doubled regions of giant magnons and single spikes that are respectively obtained when their angular and linear velocities become smaller or greater than unity. * E-mails: mflorato@phys.uoa.gr, glinard@inp.demokritos.gr. arXiv:1406.0796v4 [hep-th] 11 Nov 2015 E Lambert's W-Function 36 F Elliptic Integrals and Jacobian Elliptic Functions 381 We shall employ the following convention in our paper: E, J, p = ∞ and v, ω = 1 will denote infinite size (as obtained by computing the lim J/p→∞ , lim v/ω→1 ) and E, J, p → ∞, v, ω → 1 will denote large but still finite size.2 The compact su (2) sector of N = 4 super Yang-Mills consists of the single-trace operators Tr Z J X M , where X , Y, Z are the three complex scalar fields of N = 4 SYM, composed out of the six real scalars Φ of the theory. The su (2) sector is dual to (closed) strings that rotate in R × S 3 ⊂ AdS5 × S 5 and its one-loop dilatation operator is given by the Hamiltonian of the ferromagnetic XXX 1/2 Heisenberg spin chain.

Finite size effect from classical strings in deformed AdS3× S3

“…There are several examples where finite size correction has been computed in string theory side and was compared with their gauge theory counter part, for example [32][33][34][35][36][37]. Some of recent works in this direction can be found in [38][39][40][41][42].…”

The omega-infinity limit of single spikes

“…Finite-size giant magnons were introduced in [18,19]. A study of their classical finite-size corrections can be found in [20]. 7 The direct analogs of giant magnons for the AF ground state are single spikes (SSs) [23,24].…”

“…In fact, (1.7) maps the elementary region of single spikes to the elementary region of giant magnons and the doubled region of single spikes to the doubled region of giant magnons. In [20], two more symmetries between the equations of motion of giant magnons and single spikes were found. 9…”

“…When v → 1 ± , single spikes have finite but still large size/winding 10 and their dispersion relation assumes the following general form [20] (at strong coupling, λ = ∞):…”

“…These either transform between the elementary regions of GMs and SSs, or map their elementary regions to their doubled regions. For more, refer to the appendix A.5 of [20]. 10 As a reminder, a string of infinite size is defined as having infinite worldsheet size r and therefore infinite energy E (c.f.…”