Strings in Ramond-Ramond Backgrounds from the Neveu-Schwarz-Ramond Formalism

TL;DR

This work develops a closed NSR superstring field theory approach to RR backgrounds and applies it to two nontrivial settings: a pp-wave with RR 5-form flux and AdS$_3\times$S$^3\times M_4$ with mixed flux. By perturbatively solving the string field equations around RR-deformed backgrounds and analyzing linearized fluctuations, it extracts dispersion-mass corrections to order $\mu^2$ for carefully chosen string states. In the pp-wave case the results reproduce known Green–Schwarz light-cone spectrum, while in AdS$_3\times$S$^3\times M_4$ they yield finite-radius mass corrections that agree with semiclassical and BMN limits, including BPS and non-BPS sectors. The study validates NSR-based background deformations as a viable, finite-radius framework for RR backgrounds and provides new quantitative data on string spectra in mixed-flux AdS backgrounds.

Abstract

We treat RR flux backgrounds of type II string theory in the framework of closed superstring field theory based on the NSR formalism, focusing on two examples: (1) the pp-wave background supported by 5-form flux, and (2) $AdS_3\times S^3\times M_4$ supported by mixed 3-form fluxes. In both cases, we analyze the classical string field solution perturbatively, and compute the correction to the dispersion relation of string states to quadratic order in the RR flux. In the first example, our result is in a delicate way consistent with that obtained from lightcone quantization of the Green-Schwarz string. In the second example, we will obtain numerically the mass corrections to pulsating type IIB strings in $AdS_3\times S^3\times M_4$. Our results, valid at finite $AdS$ radius, agree with previously known answers in the semiclassical limit and in the BMN limit respectively.

Figures (3)

• Figure 1: The leading anomalous weight $\delta h$ in units of $\mu^2$ as a function of the oscillator number $n$ (with $h_{\rm int}=0$). In the first three plots, the blue dashed curve represents the anomalous weight inferred from the semiclassical quantization of the pulsating string solution (\ref{['semiclassic']}), while the black dots denote the quantum result. In the bottom-right plot, we have plotted the quantum anomalous weights (colored dots) for various $S^3$ angular momentum quantum numbers $j'$.
• Figure 2: RR correction to the spacetime energy ${\delta}{\Delta}$ in units of $k\mu^2$, numerically extrapolated to infinite level $k$ at fixed ${n\over k}$ (black dots), compared to the semiclassical result (\ref{['semiclassic']}) (dashed blue curve).
• Figure 3: The order $\mu^2$ anomalous weight $\delta h$ extrapolated to infinite level $k$ at fixed $\gamma={j'\over k}$ (black dots), compared to that of the known pp-wave string spectrum (\ref{['eq:PPExpectation']}) (dashed blue curve), at two sample values of the oscillator level $n$.