Bi-Scalar Fishnet CFT in 2D

• Bi-scalar fishnet CFT is a two-dimensional conformal field theory defined by two complex scalar fields with non-unitary integrability and exact spectral quantization.
• Its structure is encoded by Baxter equations and a graph-building operator, mapping fishnet Feynman diagrams to an integrable SL(2) spin-chain framework.
• Coupling normalization and gluing conditions discretize the spectrum, providing non-perturbative insights analogous to quantum spectral curves in AdS/CFT.

The bi-scalar fishnet conformal field theory (CFT) is a class of exactly solvable, non-unitary, planar, conformally invariant quantum field theories involving two complex matrix-valued scalar fields. Originally formulated as a double-scaling limit of strongly γ-twisted $\mathcal{N}=4$ SYM, the fishnet CFT exhibits remarkable integrable and algebraic characteristics, especially in two dimensions, where the full non-perturbative spectrum is governed by quantum spectral curve (QSC) and spin-chain machinery. This article provides a comprehensive technical overview of the bi-scalar fishnet CFT in two dimensions, emphasizing its algebraic structure, integrable dynamics, spectral data, and analytic solution space (Ekhammar et al., 28 Dec 2025).

1. Definition, Action, and Symmetries

The two-dimensional bi-scalar fishnet CFT employs two complex $N \times N$ matrix scalars, $\phi_1$ and $\phi_2$, each with canonical scaling dimension $\Delta_{\phi_1} = \Delta_{\phi_2} = \frac{1}{2}$. The Euclidean action is

$$S = \int d^2x \left[ \operatorname{Tr} \left( \phi_1^\dagger (-\Box)^{\alpha} \phi_1 + \phi_2^\dagger (-\Box)^{\alpha} \phi_2 \right) + \frac{(4\pi)^\beta}{N} \xi \operatorname{Tr}(\phi_1^\dagger \phi_2^\dagger \phi_1 \phi_2) \right]$$

where $\alpha = 1-\frac{2}{d}$, so for $d=2$ one finds $\alpha = \frac{1}{2}$, yielding a fractional Laplacian kinetic operator. The single-trace interaction parameter $\xi$ governs the strength of the chiral quartic vertex. The fundamental propagators are

$$\langle \phi_i(z) \phi_i^\dagger(w) \rangle = [z-w]^{-\alpha} \equiv (z-w)^{-\alpha} (\bar{z} - \bar{w})^{-\alpha}$$

in terms of complex coordinates $z = x_1 + i x_2$ and $\bar{z} = x_1 - i x_2$.

The theory is invariant under the global conformal group $\mathrm{SO}(2,2) \cong SL(2) \times SL(2)$. Each sector acts holomorphically ($z$) or antiholomorphically ($\bar{z}$) on coordinate wavefunctions. Local operators carry left/right spins $(h,\bar{h})$, total scaling dimension $\Delta = h + \bar{h}$, and spin $S = h - \bar{h}$.

2. Spin-Chain Interpretation and Graph-Building Operator

Single-trace operators of the form

$$\mathcal O(z) = \operatorname{Tr}\left[(\partial^a \bar{\partial}^{\bar{a}} \phi_1)^{J-M} (\partial^a \bar{\partial}^{\bar{a}} \phi_2)^{M}\right]$$

are identified as wavefunctions in an integrable $\mathfrak{sl}(2)$ spin chain of length $J$, where each site hosts local spins $s_k = \frac{1}{2}$. Magnon excitations correspond to insertions of $\phi_2$.

The central object for integrability is the graph-building operator $\mathbb B$, adding a "wheel" of fishnet propagators to planar diagrams. Its eigenfunctions diagonalize the dilatation operator, giving access to the operator spectrum.

3. Integrability and Baxter Equations

The spin-chain Hamiltonian is derived via an infinite-dimensional transfer matrix $\mathbb T_s(u)$ built from $R$-operators acting on $SL(2)$ Verma modules. Specializing the auxiliary spin and spectral parameter recovers the graph-building operator as: $$\mathbb B = \mathbb Q_+\left(u = \frac{i}{4}\right), \quad \mathbb Q_+(u) = \left[\mathbb T_s(u + \tfrac{i}{2}, u - \tfrac{i}{2})\right]_{s = 1/2}$$ where $\mathbb Q_+(u)$ is a Baxter-style $Q$-operator relevant for spectral analysis.

The eigenfunctions $q(u)$ of the $Q$-operator obey a second-order finite-difference Baxter equation: $(u+2) q(u+i) - t(u) q(u) + (u-2) q(u-i) = 0$ and similarly for the antiholomorphic sector $\bar{q}(\bar{u})$, with $t(u)$ a degree-$J$ transfer-matrix polynomial. The large-$u$ asymptotics distinguish two independent solutions $q_1(u) \sim u^{h - J/2}$, $q_2(u) \sim u^{1 - h - J/2}$, which form the analytic basis.

4. Quantization and Gluing Condition

The spectrum is discretized via quantization (gluing) conditions on the analytic continuations between upper and lower $u$-half planes. The gluing matrix $M(u)$ connects the two analytic bases: $$\begin{pmatrix} q_1(u) \ q_2(u) \end{pmatrix}_{\rm upper} = M(u) \begin{pmatrix} q_1(u) \ q_2(u) \end{pmatrix}_{\rm lower}, \quad M(u+i) = M(u)$$ with the non-trivial requirement that $M(u)$ is anti-diagonal up to a fixed constant $c$: $$M(u) = \begin{pmatrix} 1 & 0 \ c & 1 \end{pmatrix}$$ This fixes allowed $(h, \bar{h})$ and polynomial coefficients in $t(u)$ to a discrete spectrum. The coupling $\xi$ is introduced via normalization at $u = i/4$: $$\lim_{u \to i/4} \frac{\mathbb Q_+(u)}{(u - i/4)^{-J}} \propto \xi^J$$ Baxter equations, analytic gluing, and coupling normalization collectively determine the complete non-perturbative spectrum $\Delta = h + \bar{h}$.

5. Quantum Spectral Curve (QSC) Analogy

The 2D fishnet QSC comprises two Baxter equations, power-law asymptotics, quantization matrix $M(u)$, and coupling normalization—rendering it a rank-1 analog of the multidimensional QSC encountered in AdS/CFT or higher-dimensional fishnet theories (Gromov et al., 2017, Kazakov, 2018). The 2D structure captures:

• finite-difference Baxter equations as QSC functional relations
• quantum numbers from large-$u$ expansions
• gluing matrix as the $\mu$-function in QSC nomenclature
• coupling normalization akin to $\mathbf{P}_a(u)$ expansions in QSC

This simplified $2 \times 2$ QSC lacks the full Hirota/T-system structure but retains the analytic bootstrap mechanisms for spectrum determination.

6. Solution Structure and Analytic Properties

The numerical solution strategy includes:

1. Asymptotic expansion of $q_{1,2}(u)$ at large $u$, parametrized by coefficients in $t(u)$.
2. Baxter recursion evaluation of $q_{1,2}(u)$ on a discrete imaginary lattice.
3. Enforcement of gluing conditions to fix $M(u)$.
4. Application of coupling normalization at $u = i/4$.
5. Newton iteration to converge on discrete spectrum $\Delta(\xi)$.

The analytic structure in the $\xi$-plane exhibits:

• Collisions of real $\Delta(\xi)$ branches at finite $\xi$ leading to complexification and branch cuts.
• Operator/shadow solutions merging at $\Delta = 1$, post-collision forming complex conjugate pairs.
• Level collisions and monodromies analogous to BFKL and higher-dimensional fishnet scenarios.

7. Asymptotic Bethe Ansatz and Twisted Extensions

At weak coupling (up to $O(\xi^{2J})$), the QSC reduces to algebraic Bethe Ansatz (ABA) equations. Bethe roots $u_k$ and auxiliary roots $v_a, v_{\dot{a}}$ satisfy: $$\left[\frac{u_k + i}{u_k - i}\right]^J = \prod_a \frac{u_k - v_a + i}{u_k - v_a - i} \frac{u_k - v_a - i}{u_k - v_a + i}$$ and

$$\prod_k \frac{v_a - u_k + i}{v_a - u_k - i} = \prod_{b \ne a} \frac{v_a - v_b + 2i}{v_a - v_b - 2i}$$

with dispersion relation

$$\Delta = J + M + \sum_{k=1}^M \left( \frac{i}{u_k + i/2} - \frac{i}{u_k - i/2} \right) + O(\xi^{2J})$$

and cyclicity constraint

$\prod_{k=1}^M \frac{u_k + i/2}{u_k - i/2} = 1$

The twisted model incorporates a phase $\phi$ implementing quasi-periodic boundary conditions; Q-functions acquire exponential twists in asymptotics: $$q_1(u) \sim e^{+i\phi u} u^{h - J/2}, \quad q_2(u) \sim e^{-i\phi u} u^{1 - h - J/2}$$ Twisted cyclicity and Bethe equations further modify phase factors, e.g.,

$$\left[ \frac{u_k + i}{u_k - i} \right]^J e^{-2i\phi} = \prod_{\rm aux} \frac{u_k - v + i}{u_k - v - i}\cdots$$

and

$$\prod_k \frac{u_k + i/2}{u_k - i/2} = e^{2iS\phi/J}$$

where $S$ is total spin. This facilitates future separation-of-variables analysis of correlators.

8. Broader Context and Analytical Applications

The two-dimensional bi-scalar fishnet CFT connects deeply to exactly solvable lattice models (with integrable $SL(2)$-spin chains as the continuum limit of vertex models), the star-triangle relation, and determinant representations for multi-point correlators (Derkachov et al., 2018). Its structure exemplifies a solvable sector of planar QFT dominated by square-lattice fishnet Feynman diagrams, which are themselves direct realizations of transfer matrices.

The full spectrum and OPE data for CFT primaries, including spinning and twisted cases, are computable via operatorial QSC techniques and provide analytic control over non-perturbative dynamics at arbitrary coupling.

9. Summary Table: Bi-Scalar Fishnet CFT—Core Structures

Feature	Mathematical Object	Role in Theory
Fields & Action	$\phi_1$, $\phi_2$, $S$	Fundamental model definition
Graph-building operator	$\mathbb{B}$, $\mathbb{Q}_+$	Generates fishnet Feynman graphs
Integrability engine	Transfer matrix, Baxter eqn	Spin-chain/QSC formalism
Spectrum quantization	Gluing matrix $M(u)$	Discretizes scaling dimensions
Coupling normalization	$u = i/4$ point	Sets interaction strength
ABA equations (weak coupling)	Bethe roots, polynomial $Q$	Determines pre-wrapping spectrum
Twisted extension	Phase $\phi$, asymptotics	Enables SoV, lifts degeneracies

All constructs and workflow steps above are implemented precisely as formulated in (Ekhammar et al., 28 Dec 2025), and the analytic/numerical solution strategy is directly extensible to generalized and twisted fishnet sectors. This framework, with the underlying $SL(2) \times SL(2)$ symmetry and rank-1 QSC structure, sets the technical foundation for the study and computation of operator spectra and correlation functions in the two-dimensional bi-scalar fishnet CFT.