Superfishnet Theory in CFT

Updated 4 September 2025

Superfishnet theory is a conformal field theory derived from double-scaling limits in deformed gauge and supersymmetric models, resulting in integrable fishnet lattices.
Refined Lagrangians with imposed flavor symmetries preserve single-trace interactions, ensuring perturbative consistency and RG stability across various dimensions.
Integrability in these models manifests through mappings to spin chains and sigma models, revealing exact computations of anomalous dimensions and holographic dualities.

Superfishnet Theory refers to a class of planar, non-supersymmetric or supersymmetric conformal field theories in various spacetime dimensions whose perturbative expansions are dominated by highly constrained, integrable families of Feynman diagrams. These configurations—regular “fishnet” lattices—arise from special double-scaling limits of deformed gauge or supersymmetric models (notably γ- or β-deformed $\mathcal{N}=4$ SYM, ABJM, or similar) and exhibit enhanced integrability, allowing for exact, all-loop calculations of anomalous dimensions, correlation functions, and thermodynamical quantities. The distinctive feature is that only a subset of single-trace (chiral) interactions survive, leading to lattice-like diagrammatics, emergent dualities (including AdS/CFT correspondences), and nontrivial scaling limits governed by integrable sigma models.

1. Construction and Lagrangians of Fishnet and Superfishnet Theories

The origins of fishnet and superfishnet theories lie in the double-scaling limits of highly symmetric quantum field theories:

Bi-scalar and Tri-scalar fishnets are obtained from γ-deformed $\mathcal{N}=4$ SYM or ABJM theory by sending the gauge coupling to zero while scaling deformation parameters to infinity, keeping certain effective couplings fixed. The resulting Lagrangian includes only specific chiral quartic (or sextic) scalar interactions, e.g. in four dimensions:

$\mathcal{L}_{4d} = \operatorname{Tr}\left(-\partial_\mu \bar X \partial^\mu X - \partial_\mu \bar Z \partial^\mu Z + \xi^2 \bar X \bar Z X Z\right)$

Superfishnet theories emerge by analogous double-scaling limits of β- or γ-deformed supersymmetric models, often formulated in superspace. For example, the 3d $\mathcal{N}=2$ superfishnet theory arises from such a limit of $\beta$ -deformed ABJM (Kade, 23 Oct 2024) with the action:

$S_{\mathrm{SFN}} = N \int d^3x \, d^2\theta \, d^2\bar\theta \, \mathrm{tr}\left[ - \sum_{i=1}^4 \Phi_i^{\dagger} \Phi_i + \xi ( \bar\theta^2 \Phi_1 \Phi_2 \Phi_3 \Phi_4 + \theta^2 \Phi_1^{\dagger} \Phi_2^{\dagger} \Phi_3^{\dagger} \Phi_4^{\dagger}) \right]$

Nonlocal deformations can be introduced by replacing $\Box$ (the Laplacian) in kinetic terms with fractional powers, leading to fields with generalized propagator scaling dimensions.

These models are nonunitary due to the chiral and nonhermitian nature of their interactions, but are rendered ultraviolet finite and superconformal at their respective fixed points. The Feynman graphs in the planar limit are characterized by regular “square”, “hexagonal”, or “triangular” fishnet lattices, with all vertices and propagators fixed by the underlying symmetry and deformation constraints (Mamroud et al., 2017, Kade, 23 Oct 2024).

Fishnet and superfishnet theories display distinct renormalization behaviors depending on the spacetime dimension and field content:

3d $\phi^6$ (triangular fishnet): In the large $N$ limit, planar diagrams only arise with at least eight external legs per loop, eliminating superficial divergences. No double-trace counterterms are radiatively generated, ensuring perturbative consistency and RG stability (Mamroud et al., 2017).
4d $\phi^4$ (square fishnet): Here, planar contractions generate double-trace terms such as $(\operatorname{tr} \phi^2)^2$ via RG flow. These cannot be absorbed in the original lagrangian. To prevent such instabilities, a “refinement” is applied: additional flavor indices and cyclic selection rules are imposed at interaction vertices (e.g., using a Kronecker delta $\delta_{i, j\mathrm{\ mod}\ n+1}$ ) such that only contraction patterns compatible with this extended flavor symmetry survive in planar diagrams, thereby protecting the single-trace structure of the theory (Mamroud et al., 2017).
6d $\phi^3$ (hexagonal fishnet): Similar to 3d, dangerous divergences and radiative double-trace terms are absent due to the structure of the vertex and the nullification of single-field traces (by constraints akin to Gauss’s law).

Refined versions thus allow the formulation of perturbatively stable and integrable fishnet models in arbitrary dimensions.

3. Integrability: Lattice Regularization and Spin Chains

A defining property of superfishnet theories is the integrability visible both at the level of Feynman diagrams and in the regularized continuum limits:

Spin chains and transfer matrices: The “graph-building” operators that generate fishnet diagrams are realized as transfer matrices of integrable (often noncompact) $SO(1,5)$ spin chains. Their eigenfunctions are related to separated variable wave functions that diagonalize the mirror channels of the correlators (Olivucci, 2021, Olivucci, 2023).
Hexagon formalism: Originally developed for $\mathcal{N}=4$ SYM, the hexagon approach is extended to fishnet theories. Here, Feynman diagrams are “cut” along appropriate channels to factorize the computation into hexagon building blocks, which are then glued via mirror magnon overlaps (with factorized $R$ -matrices and S-matrix constraints) to assemble full correlators or structure constants (Basso et al., 2018, Olivucci, 2023).
Star–triangle and chain relations: Fishnet integrals satisfy star–triangle (uniqueness) identities, even in the presence of spin or superspace, which underlie the Yang–Baxter equation for the corresponding lattice models (Derkachov et al., 2021, Kade, 3 Sep 2025, Kade, 23 Oct 2024).
Thermodynamical and double-scaling limits: In the thermodynamical limit of large fishnet graphs, the summation yields continuum partition functions of integrable two-dimensional sigma models on AdS target spaces (e.g. AdS $_5$ ), providing a bridge to holography (Basso et al., 2018).

This integrability constrains the spectrum of scaling dimensions to the solutions of Bethe ansatz-type or transfer-matrix eigenvalue equations, even in the presence of nontrivial dynamical (e.g., mass or flavor) deformations (Mamroud et al., 2017, Olivucci, 2021).

4. Exact Results: Anomalous Dimensions, Free Energy, and Structure Constants

Superfishnet theories enable the computation of physically significant observables to all orders:

Anomalous dimensions: All-loop anomalous dimensions of local operators (e.g. $\operatorname{tr} \phi_i \phi_j$ ) are computed via closed integral equations stemming from Schwinger-Dyson resummation and spin chain transfer matrices. Typical forms include (Mamroud et al., 2017):

$\gamma_{11} ( \gamma_{11} + 4 ) = -4 \alpha^2 I_{2,2,1-\gamma_{11}/2}$

where $I_{2,2,1-\gamma_{11}/2}$ is a computable Feynman-parameter integral.

Scaling dimensions in the continuum: For long operators (e.g. BMN vacuum), the scaling dimension is expressible in terms of solutions of thermodynamic Bethe ansatz (TBA) integral equations (Basso et al., 2018):

$\Delta = L - \int_{-B}^B \frac{du}{\pi} \chi(u)$

Vacuum free energy and critical coupling: The zero-mode fixed free energy in the thermodynamic limit is computable, and its radius of convergence yields the critical coupling, e.g. for the superfishnet model (Kade, 23 Oct 2024):

$\xi_{\mathrm{cr}} = \left( \mathbb{K} \right)^{-1/2}$

where $\mathbb{K}$ is the thermodynamic free energy density.

Structure constants and correlators: Leclair–Mussardo resummation formulas yield exact expressions for structure constants; wrapping effects are accounted for by subtracting decoupling pole residues and point-splitting in the hexagon expansion (Basso et al., 2018).

Notably, these calculations are valid at finite coupling (not just perturbatively), and the analytic structure of the observables reveals a complex pattern of branch cuts and spurious pole cancellation (Kazakov et al., 2018).

5. Generalizations: Twistor, Massive, and Supersymmetric Extensions

Twistor space formulation: The fishnet (and more generally, superfishnet) theory admits a reformulation in twistor space (Adamo et al., 2019). This allows manifest realization of conformal invariance and an abelian gauge symmetry absent in spacetime, leading to cohomological formulas for scattering amplitudes and efficient treatment of divergences and cancellation at criticality.
Massive fishnet theories: Mass terms can be introduced either by spontaneous symmetry breaking (leading to product-mass propagators and new three-point couplings) or via a double-scaling limit of the Coulomb branch of $\mathcal{N}=4$ SYM, resulting in difference-mass propagators (Loebbert et al., 2020, Loebbert et al., 2021). These massive theories retain integrability, with amplitudes obeying a generalized massive Yangian invariance.
Supersymmetric fishnets: Superfishnet models in three and four dimensions are constructed using superspace formalism; nonlocal deformations can be implemented by fractional Laplacians in the Kähler potential (subject to certain constraints for superconformality) (Kade, 23 Oct 2024, Kade, 3 Sep 2025). Superspace chain and inversion relations (super--x--unity and superconformal triangle decompositions) generalize the bosonic integrability to supersymmetric settings, enabling the computation of critical couplings and scaling dimensions at all loops.

6. Holography, Dualities, and Future Directions

Holographic correspondences: In the continuum (thermodynamic or double-scaling) limit, the sum over fishnet diagrams is mapped to sigma models on AdS target spaces (e.g. AdS $_5$ ), elucidating the emergence of extra (fifth) dimensions and realizing a discrete-string or fishchain worldsheet description (Basso et al., 2018, Gromov et al., 2019, Huang, 2022).
Dualities: The fishnet/fishchain mapping provides a concrete weak/strong coupling duality, with classical fishchain dynamics reproducing the strong-coupling scaling of CFT correlators. This plays an instructive role in simulating holography without supersymmetry and suggests that supersymmetric extensions (superfishnet) may allow analogous constructions in settings with integrable S-matrices and discrete worldsurface models.
Mathematical structures: Antipodal self-duality of m×m square fishnet graphs demonstrates invariance under a Hopf algebra antipode combined with kinematic transformations on multiple polylogarithms (Dixon et al., 2 Feb 2025), indicating deep connections between the number-theoretic structure of ladder integrals and the algebraic structures controlling scattering amplitudes or form factors in superfishnet theory.

Superfishnet theory thus serves as a key testing ground for advances in the classification of integrable QFTs, holographic dualities, and exact computational methodologies, with ongoing research extending to boundary integrability, triangulations of higher-point or supersymmetric fishnet diagrams, and rigorous exploration of observer-independent operator dynamics in both bosonic and supersymmetric settings.