On String S-matrix, Bound States and TBA

TL;DR

The paper develops a coherent framework for the AdS5×S5 light-cone string by constructing and analyzing the mirror model obtained via a double Wick rotation. It establishes a meromorphic elliptic S-matrix on the rapidity torus, enforces unitarity and crossing, and carefully relates the string and mirror theories through a consistent Hopf-algebraic structure. It then derives Bethe Ansatz equations for both the string and mirror S-matrices, and analyzes the spectrum of bound states, including finite-size effects and their implications for the validity of the asymptotic Bethe Ansatz. The work highlights how physical regions, degeneracies, and finite-J corrections shape the bound-state structure and motivates a TBA program for finite-volume spectra in this integrable AdS/CFT context.

Abstract

The study of finite J effects for the light-cone AdS superstring by means of the Thermodynamic Bethe Ansatz requires an understanding of a companion 2d theory which we call the mirror model. It is obtained from the original string model by the double Wick rotation. The S-matrices describing the scattering of physical excitations in the string and mirror models are related to each other by an analytic continuation. We show that the unitarity requirement for the mirror S-matrix fixes the S-matrices of both theories essentially uniquely. The resulting string S-matrix S(z_1,z_2) satisfies the generalized unitarity condition and, up to a scalar factor, is a meromorphic function on the elliptic curve associated to each variable z. The double Wick rotation is then accomplished by shifting the variables z by quarter of the imaginary period of the torus. We discuss the apparent bound states of the string and mirror models, and show that depending on a choice of the physical region there are one, two or 2^{M-1} solutions of the M-particle bound state equations sharing the same conserved charges. For very large but finite values of J, most of these solutions, however, exhibit various signs of pathological behavior. In particular, they might receive a finite J correction to their energy which is complex, or the energy correction might exceed corrections arising due to finite J modifications of the Bethe equations thus making the asymptotic Bethe ansatz inapplicable.

Figures (6)

• Figure 1: On the left figure the torus is divided by the curves $|x^+|=1$ and $|x^-|=1$ into four non-intersecting regions. The middle figure represents the torus divided by the curves ${\rm Im}(x^+)=1$ and ${\rm Im}(x^-)=1$, also in four regions. The right figure contains all the curves of interest.
• Figure 2: Divisions of the torus by the curves $|x^\pm|=1$ (upper figures) and by the curves ${\rm Im}\, x^{\pm}=0$ (lower figures) for $g=1/2$, $g=1$ and $g=50$. The red curves are $|x^-|=1$, and the pink ones are $|x^+|=1$. The coordinates $x$ and $y$ are the rescaled real and imaginary parts of $z$: $x = {\rm Re}( {2\over\omega_1}z)$, $y = {\rm Re}( {4\over\omega_2}z)$. In the limit $g\to\infty$ the curves $|x^{\pm}|=1$ and ${\rm Im}\, x^{\pm}=0$ are related by the shift $z\to z+{\frac{\omega_2}{2}}$.
• Figure 3: On the left figure the upper and lower curves correspond to $|x^+|=1+0$ and $|x^-|=1+0$, respectively. The map $z\to u(z)$ folds each of these curves onto the corresponding cut on the $u$-plane.
• Figure 4: Four copies of the $u$-plane (the Riemann sphere) glued together through the cuts to produce the torus of the kinematical variable $z$. We indicated four branch points ${\rm \bf B}_{1,2}$ and ${\rm \bf C}_{1,2}$ which are images of those on Fig.3.
• Figure 5: Two-particle bound states of string theory. Figure a) describes the first BPS family corresponding to $p<p_{\rm cr}$. The green curves are ${\rm Im}(x^-)=0$ for ${\rm Im}(z) <0$ and ${\rm Im}(x^+)=0$ for ${\rm Im}(z) >0$. For any $p<p_{\rm cr}$ there are two solutions: the first one is represented by the continuous curves ${\rm\bf B}_1{\bf O}{\rm\bf C}_1$ (1st particle) and ${\rm\bf B}_2{\bf O}{\rm\bf C}_2$ (2nd particle), the second one corresponds to the dashed curves ${\rm\bf A}_1{\rm\bf B}_1 \cup {\rm\bf C}_1{\rm\bf D}_1$ (1st particle) and ${\rm\bf A}_2{\rm\bf B}_2 \cup {\rm\bf C}_2{\rm\bf D}_2$ (2nd particle). Figure b) describes the second BPS family corresponding to $p>p_{\rm cr}$. Again, for any $p>p_{\rm cr}$ there are two solutions ${\rm\bf B}_2{\rm\bf C}_2 \cup {\rm\bf A}_1{\rm\bf B}_1\cup {\rm\bf C}_1{\rm\bf D}_1$ and ${\rm\bf B}_1{\rm\bf C}_1 \cup {\rm\bf A}_2{\rm\bf B}_2\cup {\rm\bf C}_2{\rm\bf D}_2$. Figure c) represents one of the four possibilities to smoothly connect solutions from the first and the second BPS families. Here the variable $z_1$ of the 1st particle is on the curve ${\rm\bf A}_1{\rm\bf B}_1{\bf O}{\rm\bf C}_1{\rm\bf D}_1$. When $z_1$ runs along the curve from ${\rm \bf A}_1$ to ${\rm \bf D}_1$ the real part of the momentum ${\rm Re}(p_1)$ increases monotonically from $-\pi$ to $\pi$. At the same time, the variable $z_2$ corresponding to the 2nd particle encloses the curve ${\rm\bf A}_2{\rm\bf B}_2{\bf O}{\rm\bf C}_2{\rm\bf D}_2$.