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Finite-size corrections for quantum strings on AdS_4 x CP^3

Davide Astolfi, Valentina Giangreco M. Puletti, Gianluca Grignani, Troels Harmark, Marta Orselli

TL;DR

The work computes curvature-induced finite-size corrections to string energies on $ ext{AdS}_4 imes ext{CP}^3$ by expanding the near-pp-wave Hamiltonian to order $1/R^2$, showing that cubic and quartic interactions introduce divergent sums that cancel when a heavy/light mode regularization and Weyl normal ordering are properly applied. The resulting energy shifts for two-oscillator and more general states in the $ ext{SU}(2) imes ext{SU}(2)$ sector reveal a universal, convergent term $ ext{S}(n)$ that acts as a finite-size correction to the single-magnon dispersion, in addition to explicit exponential (wrapping) corrections captured by sums of Bessel functions. Crucially, the regularization implied by the cubic Hamiltonian enforces a heavy/light cutoff relation, ensuring that the strong-weak coupling interpolating function $h(oldsymbol{ extlambda})$ receives no one-loop correction, in agreement with the algebraic-curve spectrum. The results reinforce the integrability of the $ ext{AdS}_4/ ext{CFT}_3$ duality (ABJM), provide a finite-worldsheet confirmation of wrapping effects, and connect spectrum calculations to Lüscher-type corrections in the gauge theory.

Abstract

We revisit the calculation of curvature corrections to the pp-wave energy of type IIA string states on AdS_4 x CP^3 initiated in arXiv:0807.1527. Using the near pp-wave Hamiltonian found in arXiv:0912.2257, we compute the first non-vanishing correction to the energy of a set of bosonic string states at order 1/R^2, where R is the curvature radius of the background. The leading curvature corrections give rise to cubic, order 1/R, and quartic, order 1/R^2, terms in the Hamiltonian, for which we implement the appropriate normal ordering prescription. Including the contributions from all possible fermionic and bosonic string states, we find that there exist logarithmic divergences in the sums over mode numbers which cancel between the cubic and quartic Hamiltonian. We show that the cubic Hamiltonian naturally requires that the cutoff for summing over heavy modes must be twice the one for light modes. With this prescription the strong-weak coupling interpolating function h(λ), entering the magnon dispersion relation, does not receive a one-loop correction, in agreement with the algebraic curve spectrum. However, the single magnon dispersion relation exhibits finite-size exponential corrections.

Finite-size corrections for quantum strings on AdS_4 x CP^3

TL;DR

The work computes curvature-induced finite-size corrections to string energies on by expanding the near-pp-wave Hamiltonian to order , showing that cubic and quartic interactions introduce divergent sums that cancel when a heavy/light mode regularization and Weyl normal ordering are properly applied. The resulting energy shifts for two-oscillator and more general states in the sector reveal a universal, convergent term that acts as a finite-size correction to the single-magnon dispersion, in addition to explicit exponential (wrapping) corrections captured by sums of Bessel functions. Crucially, the regularization implied by the cubic Hamiltonian enforces a heavy/light cutoff relation, ensuring that the strong-weak coupling interpolating function receives no one-loop correction, in agreement with the algebraic-curve spectrum. The results reinforce the integrability of the duality (ABJM), provide a finite-worldsheet confirmation of wrapping effects, and connect spectrum calculations to Lüscher-type corrections in the gauge theory.

Abstract

We revisit the calculation of curvature corrections to the pp-wave energy of type IIA string states on AdS_4 x CP^3 initiated in arXiv:0807.1527. Using the near pp-wave Hamiltonian found in arXiv:0912.2257, we compute the first non-vanishing correction to the energy of a set of bosonic string states at order 1/R^2, where R is the curvature radius of the background. The leading curvature corrections give rise to cubic, order 1/R, and quartic, order 1/R^2, terms in the Hamiltonian, for which we implement the appropriate normal ordering prescription. Including the contributions from all possible fermionic and bosonic string states, we find that there exist logarithmic divergences in the sums over mode numbers which cancel between the cubic and quartic Hamiltonian. We show that the cubic Hamiltonian naturally requires that the cutoff for summing over heavy modes must be twice the one for light modes. With this prescription the strong-weak coupling interpolating function h(λ), entering the magnon dispersion relation, does not receive a one-loop correction, in agreement with the algebraic curve spectrum. However, the single magnon dispersion relation exhibits finite-size exponential corrections.

Paper Structure

This paper contains 22 sections, 170 equations.

Table of Contents

  1. Introduction and summary
  2. Preliminaries
  3. The procedure
  4. Mode expansions and plane-wave spectrum
  5. The near plane-wave spectrum in the $\mathrm{SU}(2)\times \mathrm{SU}(2)$ sector
  6. The term $\sum_{|i\rangle \neq |s\rangle}\frac{\left|\langle i|H_{3}|s\rangle\right|^2}{E^{(0)}_{|s\rangle}-E^{(0)}_{|i\rangle}}$
  7. The term $\langle s |H_{4}|s\rangle$
  8. String energy spectrum at order $1/R^2$
  9. States with arbitrary number of $SU(2)\times SU(2)$ oscillators.
  10. States with one $\mathrm{SU}(2)$ oscillator.
  11. Regularization prescription
  12. Normal ordering prescription
  13. Normal ordering and plane-wave algebra
  14. Cancelation of divergences
  15. Finite size corrections for states outside the $\mathrm{SU}(2)\times \mathrm{SU}(2)$ subsector
  16. ...and 7 more sections