[Paper Review] Lightcone Quantization of String Theory Duals of Free Field Theories

Andreas Karch|ArXiv.org|Dec 4, 2002

Black Holes and Theoretical Physics16 references15 citations

TL;DR

This paper demonstrates that in the zero curvature radius limit of string theory on AdS space, the bosonic string spectrum reduces to a theory of free partons in light-cone quantization, exactly matching the spectrum of a free U(N) scalar field theory in light-front formalism. The momentum in the extra dimension maps to the light-cone momentum fraction of partons, confirming a strong-weak duality in the weak-coupling regime of the AdS/CFT correspondence.

ABSTRACT

We quantize in light cone gauge the bosonic sector of string theory on Anti-de Sitter space in the zero curvature radius limit. We find that the worldsheet falls apart into a theory of free partons and map the Hilbert space of the string theory to the Hilbert space of a free scalar in light-front description. We outline how the string worldsheet reproduces the field theory at weak coupling.

Motivation & Objective

• To investigate the behavior of string theory on Anti-de Sitter (AdS) space in the limit of vanishing curvature radius, where it should reduce to a free field theory.
• To establish a precise mapping between the Hilbert space of the string theory and that of a free scalar field theory in light-front quantization.
• To demonstrate that the string worldsheet in this limit decomposes into independent free partons, with momentum in the extra dimension encoding parton momentum fractions.
• To explore the implications for the AdS/CFT correspondence in the weak-coupling (small 't Hooft coupling) regime.
• To lay groundwork for extending the analysis to the full superstring and interacting field theories via inclusion of fermions.

Proposed method

• Quantize the bosonic sector of string theory on AdS space using light-cone gauge, fixing $X^+ = \tau$ and $\gamma_{01} = 0$ to derive the light-cone Hamiltonian.
• Use the metric $ds^2 = \frac{R^2}{Y^2}(2dX^+dX^- + dX^idX^j\delta_{ij} + dY^2)$ to describe the warped AdS background.
• Take the $\lambda = R^4/l_s^4 \to 0$ limit, which removes all $\sigma$-derivative terms from the Hamiltonian, leaving only kinetic terms in transverse and longitudinal momenta.
• Map the string's Fourier modes $p_n^\perp$ and $p_n^y$ to the momentum modes $k_a^\perp$ and $k_a^+$ of $N$ partons in the gauge theory via discrete Fourier transforms.
• Derive the correspondence between $p_n^y$ and $k_a^+$ using the constraint $p_n^- = \sum_{m,M} p^M_m p^M_{n-m}$ and the field theory relation $k_a^- = (k_a^\perp)^2 / k_a^+$.
• Confirm consistency by showing that the total $P_+$ and $P_\perp$ on the string side match the sum of $k_a^+$ and $k_a^\perp$ on the field theory side.

Experimental results

Research questions

• RQ1 How does the spectrum of the bosonic string on AdS space behave in the limit of vanishing curvature radius ($\lambda \to 0$)?
• RQ2 Can the string theory in this limit be mapped exactly to a free field theory in light-front quantization?
• RQ3 What is the physical interpretation of the momentum in the extra dimension ($Y$) in terms of parton momentum fractions in the dual field theory?
• RQ4 How do the Fourier modes of the string worldsheet correspond to the momentum modes of individual partons in the gauge theory?
• RQ5 What is the role of the $\sigma$-dependent modes in the string Hamiltonian, and why do they decouple in the $\lambda \to 0$ limit?

Key findings

• In the $\lambda \to 0$ limit, the string Hamiltonian reduces to $H = \frac{1}{2}\int d\sigma \left( P_X P_X + P_Y P_Y \right)$, indicating decoupled free particles at each $\sigma$-mode.
• The string worldsheet decomposes into an infinite set of independent free partons, each labeled by $\sigma$, with no interactions due to the absence of $\sigma$-derivatives.
• The momentum in the fifth dimension ($p_n^y$) maps to the light-cone momentum fraction $x_a = k_a^+ / P^+$ via $p_n^y = \frac{1}{\sqrt{N}} \sum_a e^{2\pi i a n / N} |k_a^\perp| \left( \frac{1}{x_a} - 1 \right)^{1/2}$, with $x_a \in (0,1)$.
• The total $P_+$ and $P_\perp$ on the string side match the sum of $k_a^+$ and $k_a^\perp$ on the field theory side, confirming consistency of the mapping.
• The string's closed-string spectrum is reproduced via translation-invariant Fourier modes, satisfying the zero-momentum constraint $\sum_n p_n^y = 0$ through cyclicity of the trace.
• The result confirms that the weak-coupling limit of the AdS/CFT correspondence maps a string theory on a highly curved AdS space to a free field theory with $U(N)$ gauge invariance in light-front formalism.