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Finite size effects for giant magnons on physical strings

J. A. Minahan, O. Ohlsson Sax

TL;DR

The paper develops a finite-gap framework to compute leading finite-size corrections for an arbitrary number of giant magnons on physical strings with total momentum $P=0\mod 2\pi$. Using a reduction to single-cut condensates, the authors derive explicit expressions for the energy shifts: for $M$ fundamental magnons the shift is a simple product over magnon momenta with an exponential suppression $\exp(-2(J+\sum_k E_k)/E_j)$, while dyons introduce a phase and a modified exponential scale; these results smoothly connect to the Heisenberg spin-chain in the large-$Q$ limit and are corroborated by TBA analyses. The work demonstrates that finite-size effects on giant magnons can be captured in closed form (especially when all $Q=0$) and provides a bridge between string, spin-chain, and TBA approaches in the AdS/CFT integrability framework.

Abstract

Using finite gap methods, we find the leading order finite size corrections for an arbitrary number of giant magnons on physical strings, where the sum of the momenta is a multiple of 2π. Our results are valid for the Hofman-Maldacena fundamental giant magnons as well as their dyonic generalizations. The energy corrections turn out to be surprisingly simple, especially if all the magnons are fundamental, and at leading order are independent of the magnon flavors. We also show how to use the Bethe ansatz to find finite size corrections for dyonic giant magnons with large R-charges.

Finite size effects for giant magnons on physical strings

TL;DR

The paper develops a finite-gap framework to compute leading finite-size corrections for an arbitrary number of giant magnons on physical strings with total momentum . Using a reduction to single-cut condensates, the authors derive explicit expressions for the energy shifts: for fundamental magnons the shift is a simple product over magnon momenta with an exponential suppression , while dyons introduce a phase and a modified exponential scale; these results smoothly connect to the Heisenberg spin-chain in the large- limit and are corroborated by TBA analyses. The work demonstrates that finite-size effects on giant magnons can be captured in closed form (especially when all ) and provides a bridge between string, spin-chain, and TBA approaches in the AdS/CFT integrability framework.

Abstract

Using finite gap methods, we find the leading order finite size corrections for an arbitrary number of giant magnons on physical strings, where the sum of the momenta is a multiple of 2π. Our results are valid for the Hofman-Maldacena fundamental giant magnons as well as their dyonic generalizations. The energy corrections turn out to be surprisingly simple, especially if all the magnons are fundamental, and at leading order are independent of the magnon flavors. We also show how to use the Bethe ansatz to find finite size corrections for dyonic giant magnons with large R-charges.

Paper Structure

This paper contains 10 sections, 87 equations, 7 figures.

Table of Contents

  1. Introduction
  2. From Heisenberg to finite gap
  3. Finite size corrections for giant magnons
  4. Single fundamental magnon ($Q=0$)
  5. Single dyonic magnon ($Q \ne 0$)
  6. Several fundamental magnons (all with $Q=0$)
  7. $M$ magnons with the same momentum and $Q$
  8. Checking results --- back to Heisenberg
  9. Results from TBA
  10. Discussion

Figures (7)

  • Figure 1: A solution with 4 branch cuts. The different cuts correspond to different log branches and are symmetric under complex conjugation.
  • Figure 2: An A-cycle circles around a single branch cut, while a B-cycle traverses two cuts in going from the top sheet to the bottom sheet and back again.
  • Figure 3: Recutting the hyperelliptic surface. The condensate is now added to preserve the cycles.
  • Figure 4: Two cuts joined by a condensate with density $\rho(x)=-i$.
  • Figure 5: Cuts for several finite size giant magnons
  • ...and 2 more figures