Integrability and the spectrum of two-dimensional fishnet CFT

TL;DR

This work constructs a non-perturbative spectral framework for the 2D bi-scalar fishnet CFT by deriving a Quantum Spectral Curve–like system from an sl(2) spin chain. The framework combines closed Baxter equations for holomorphic/anti-holomorphic Q-functions with a gluing (quantisation) condition and a coupling–QSC relation that encodes the graph-building operator, enabling finite-coupling numerics and an analytic ABA at weak coupling. The authors validate the approach through an exact J=2 solution, extensive numerical spectra for J=3 and magnons, Lüscher corrections, and a Twisted QSC extension, while outlining a clear path to Separation of Variables and potential AdS3/CFT2 connections. The results establish 2D fishnet theory as a tractable laboratory for operatorial QSCs, SoV, and non-perturbative spectral analysis with implications for higher-dimensional integrable systems and correlation function studies.

Abstract

We formulate a closed set of equations for the spectrum of two-dimensional bi-scalar fishnet conformal field theory, comprising Baxter equations and quantisation conditions, which we derive operatorially from the underlying sl(2) spin chain. These equations are reminiscent of the Quantum Spectral Curve (QSC) framework found in other holographic CFTs and are expected to provide a complete non-perturbative description of the spectrum at arbitrary coupling. We solve the QSC numerically at finite coupling and uncover a rich analytic structure, including state collisions and complex energy levels. Analytically, we introduce a new method to derive the Asymptotic Bethe Ansatz equations, which control the spectrum up to wrapping order and incorporate spinning states. We further extend our results to the twisted case, which may be particularly useful for future separation of variables analyses of correlation functions in this theory.

Figures (8)

• Figure 1: Graphical illustration of the action of the graph-building operator $\hat{B}$ on the fishnet graphs. Panel (a) shows the vacuum configuration (no magnons), while panel (b) shows the case of two adjacent magnons.
• Figure 2: Action of $\hat{\mathbb Q}_+(-3i/4-i\epsilon)$ on a CFT wave function with a magnon at site $k\!+\!1$ adjacent to an empty site $k$. Dashed lines denote $\delta$-functions. In the first step we use that the $$-32$$-propagator, viewed as an integration kernel, annihilates the standard propagator. In the second step the remaining standard propagator cancels against the $\tfrac{1}{2}$-propagator. After the action, the magnons remain at the same sites, while the integration variables on the vertical lines are shifted by one index. This operation can therefore be viewed as a generalised shift operator. The highlighted node carries the extra factor $4\pi\xi^2$ from the Feynman rules.
• Figure 3: We plot the low-lying spectrum for 2D bi-scalar fishnet theory from weak to strong coupling for operators with $J=3$, $M=0$, and $S=0$. As expected, we find collisions among states and complex energy levels.
• Figure 4: The strong coupling behaviour of the imaginary part of $\Delta$ for the states that become complex at $\Delta=1$; see figure \ref{['fig:LowLyingSpectrum']}. We find numerically that $\Im[\Delta] \sim \xi^{3}$ for all these states.
• Figure 5: The strong coupling behaviour of the imaginary part of $c_{-1}$; see \ref{['eq:BaxterJ3']}. We show the states that become complex at $\Delta=1$; see figure \ref{['fig:LowLyingSpectrum']}. We find numerically that $\Im[c_{-1}] \sim \xi^{6}$ for all these states.