Quantum Spectral Curve (QSC) Overview

• The Quantum Spectral Curve (QSC) is an advanced integrability framework that encodes the exact spectral problem in planar and fishnet CFTs.
• It employs Baxter Q-functions, analytic constraints, and gluing conditions to determine scaling dimensions and operator correlation data non-perturbatively.
• QSC’s formulation reveals deep connections to integrable spin chains, Yangian symmetry, and Bethe ansatz methods with broad applications in theoretical physics.

The quantum spectral curve (QSC) framework is an advanced integrability-based formalism encoding the exact spectral problem and related observables in planar, typically supersymmetric, quantum field theories and their special limits, such as the bi-scalar or fishnet conformal field theories (CFTs). QSC provides a powerful set of functional and analytic constraints on Baxter Q-functions, which, together with analyticity and gluing (sewing) conditions, determine scaling dimensions and OPE data of local operators at all values of the coupling. Within the context of fishnet CFTs—emerging as double-scaling, strongly twisted limits of $\gamma$-deformed $\mathcal N=4$ SYM—the QSC formalism degenerates into a system tightly related to non-compact conformal spin chains, making the integrable structure manifest and allowing full non-perturbative solution of spectral problems.

1. Historical Development and Origin in Planar Integrability

The QSC originated as a systematization of the spectral problem in @@@@2@@@@, first for planar $\mathcal N=4$ SYM, where the integrability of the spectrum was encoded in a closed set of functional equations for Q-functions organized on a Hasse hypercube, subject to Wronskian and analytic conditions (Kazakov, 2018). In these models, the underlying symmetry is usually that of $su(2,2|4)$. With the advent of $\gamma$- and $\beta$-deformations, and their subsequent double scaling limits, the fishnet CFTs provided a drastically simplified context. In these highly chiral, non-unitary yet strictly conformal models, the QSC reduces to its minimal "bosonic core," capturing the spin chain structure of the fishnet graphs (Gromov et al., 2017, Ekhammar et al., 28 Dec 2025). This setting, both in four and two dimensions, exposes integrability in its purest form, isolating the role of conformal symmetry and the Yang–Baxter equation.

The fishnet limit further motivates the study of QSC since integrable observables in this sector are governed by difference equations for Q-functions reminiscent of the Baxter TQ equation, with physical quantities (e.g., anomalous dimensions) encoded in their exact analytic structure.

2. Algebraic Structure of the Quantum Spectral Curve

At its heart, the QSC formalism in fishnet models is captured by a distinguished set of Baxter Q-functions, which both parametrize eigenstates of the underlying non-compact spin chain and organize the analytic data of the QFT. For the 4D bi-scalar theory, the QSC is associated with the $SU(2,2)$ (or, in general, conformal) spin chain; in 2D reductions, the relevant algebra is $SL(2)$ or $SL(2,\mathbb C)$ (Ekhammar et al., 28 Dec 2025, Kazakov, 2018).

The basic objects are:

• Q-functions: Solutions $q(u)$ of finite-difference Baxter equations of fixed order (e.g., second-order for $SL(2)$; fourth-order for $SU(2,2)$, see Eq. (2.22) (Ekhammar et al., 28 Dec 2025) and Eq. (3) (Kazakov, 2018)).
• Analyticity constraints: $q(u)$ must possess specified asymptotic power-law behavior as $u\to\infty$ (e.g., $u^{\Delta/2-S/2}$), have prescribed singularities (poles/cuts), and be analytic in prescribed half-planes ("upper" and "lower" half-plane analyticity—UHPA and LHPA—bases).
• Gluing (sewing) matrix: Periodicity and analyticity impose that the Q-bases for UHPA and LHPA are related by an $i$-periodic gluing matrix $\Omega_{ij}(u)$, which must satisfy algebraic conditions (e.g., $\det \Omega = 1$, $\Omega_{11} = \Omega_{22}$, see (2.26) (Ekhammar et al., 28 Dec 2025)).
• Quantization (sewing) conditions: The spectrum is determined by the requirement that the two Q-bases glue into a single well-defined Q-system, translating into quantization constraints on operator dimensions and coupling dependence.

In the 4D bi-scalar fishnet limit, the QSC emphasizes the degeneration from the full $su(2,2|4)$ Q-system to a bosonic $SU(2,2)$ core, with the associated Baxter equations and analytic constraints being the minimal closed structure sufficient to determine the physical spectrum (Kazakov, 2018, Gromov et al., 2017).

3. Operatorial Realization: From Integrable Spin Chains to Baxter/QSC System

The QSC formalism is deeply intertwined with the integrable spin chain picture justified by the mapping between planar fishnet Feynman graphs and the transfer matrices of non-compact conformal spin chains (Kade, 3 Sep 2025, Ekhammar et al., 28 Dec 2025, Gromov et al., 2017, Kazakov et al., 2022). In 4D, each closed single-trace operator $\mathcal O_L = \operatorname{Tr}(\phi_1^L)$ is mapped to a periodic chain of $L$ sites, each carrying a non-compact principal series representation of $SU(2,2)$ (scaling dimension $\Delta=1$).

• Transfer matrices: Built from R-matrices (satisfying the Yang–Baxter equation) associated with local star–triangle relations, the transfer matrix encodes the eigenvalue problem, with the Baxter operator and Q-functions emerging from its spectral decomposition (Kazakov, 2018, Gromov et al., 2017).
• Graph-building operator: In position space, this is an integral operator re-summing all wheel-like ("wrapping") Feynman graphs, acting as a Hamiltonian for the spin chain. Its spectral analysis reduces the computation of correlators and anomalous dimensions to the solution of the QSC (Gromov et al., 2018, Kazakov et al., 2022).
• Baxter equation: The transfer matrix commutes with itself for different spectral parameters, leading to finite-difference equations for Q-functions, whose analytic and asymptotic properties are entirely captured by the QSC system (Gromov et al., 2017).

In 2D bi-scalar models, the essential structure survives with $SL(2)$-spin chains and second-order Baxter equations (Ekhammar et al., 28 Dec 2025, Derkachov et al., 2018).

4. Analytic Structure, Gluing, and Quantization Conditions

The determination of the physical spectrum and the operator content relies on a detailed analysis of the analytic properties of $Q$-functions and their sewing (gluing) conditions.

• Asymptotics: For large $u$, independent solutions $q_1(u) \sim u^{\Delta/2-S/2}$, $q_2(u)\sim u^{1-\Delta/2-S/2}$ (and their antiholomorphic analogues) are chosen (Ekhammar et al., 28 Dec 2025).
• Upper and lower half-plane analyticity: The physical basis is constructed by requiring analyticity except for prescribed singularities in either the upper or lower half-plane.
• Gluing matrix: The two sets of solutions are related by an $i$-periodic, algebraically constrained matrix $\Omega_{ij}(u)$. Physical single-trace states are those for which the gluing yields a unique QSC solution—a nontrivial constraint leading to quantization of scaling dimensions (see Eq. (2.26) (Ekhammar et al., 28 Dec 2025) and the quantization condition (5) (Kazakov, 2018, Gromov et al., 2017)).
• Cyclicity condition: Projecting to physical (single-trace) states requires enforcing a cyclicity (level-matching) constraint on the spin-chain eigenstates, implemented via the shift/translation operator (eigenvalue 1).

In summary, solving the coupled functional and analytic system—Baxter equations, asymptotics, gluing/quantization, and cyclicity—produces the complete non-perturbative spectrum of scaling dimensions and quantum numbers (Ekhammar et al., 28 Dec 2025, Kazakov, 2018).

5. Solution Techniques and Extracted Spectral Data

Exact and numerical solution of the QSC system yields the full non-perturbative spectrum for local operators at arbitrary coupling:

• Algorithm: For each operator (specified by quantum numbers length $L$, number of magnons $M$, spin $S$), construct a power-series ansatz for large-$u$ $q_i(u)$, recursively solve the Baxter equations, build the gluing matrix, enforce quantization, coupling, and cyclicity conditions, and solve the resulting nonlinear system (Ekhammar et al., 28 Dec 2025).
• Weak/strong coupling expansions: The QSC accommodates perturbative (weak-coupling) expansions—e.g., for $J=3$, $M=0$, $S=0$,

$$\Delta(\xi) = 2 - \frac{1}{6}\xi^2 - \frac{5}{27}\xi^4 - \frac{991}{15552}\xi^6 + \mathcal{O}(\xi^8)$$

and systematic strong-coupling semiclassical analyses (Ekhammar et al., 28 Dec 2025, Gromov et al., 2017).

• Spectrum features: The spectrum displays phenomena such as state collisions (real branches meeting and developing complex conjugate branches) and transitions from real to complex scaling dimensions, reflecting non-unitarity and integrability features (Ekhammar et al., 28 Dec 2025, Gromov et al., 2018).
• Bethe Ansatz reduction: In the asymptotic regime (weak coupling, below wrapping), the QSC reduces to algebraic Bethe ansatz equations for integrable spin chains, allowing systematic weak-coupling expansion and identification of Bethe roots (Ekhammar et al., 28 Dec 2025, Gromov et al., 2017).
• Twisted boundary conditions: Twists modify asymptotic and cyclicity properties by introducing phases in the Q-functions and shift operators, yielding quasi-periodic spectra and selection rules for global quantum numbers (Ekhammar et al., 28 Dec 2025).

These techniques provide not only anomalous dimensions but also the structure constants, OPE coefficients, and full correlation functions via the operatorial approach (separation of variables) (Ekhammar et al., 28 Dec 2025, Derkachov et al., 2018).

6. Connections to Yangian Symmetry and Quantum Integrability

The QSC and associated Baxter equations are intimately linked to hidden infinite-dimensional Yangian symmetry present in planar correlators of the fishnet CFT (Chicherin et al., 2017, Corcoran et al., 2021, Beisert et al., 24 Nov 2025). This symmetry underlies the construction:

• Lax formalism: Monodromy matrices built from local Lax operators act as generating functions for Yangian charges, enforcing the integrability constraints that are algebraically encoded in the QSC system (Chicherin et al., 2017, Corcoran et al., 2021).
• Ward identities: In special cases (fishnet graphs of planar, unique, square-lattice topology), Yangian Ward identities ensure the invariance of correlators under both level-zero (conformal) and level-one generators, contingent on precise evaluation parameters (Beisert et al., 24 Nov 2025).
• Breakdown of exact symmetry: For more general graphs, especially those with nonzero dual Coxeter number ($h^*\neq 0$, e.g. in $SU(2,2)$), or in the presence of non-unique planar contributions/non-square loops, quantum-level Yangian invariance can break down, constraining the range of applicability of integrability-based exact methods (Beisert et al., 24 Nov 2025).
• Generalization: In maximally supersymmetric models (e.g., $\mathcal N=4$ SYM with $h^*=0$), QSC integrability and Yangian symmetry are further enhanced, supporting full QSC closure and exact spectral equations for all sectors (Kazakov, 2018).

In fishnet models, the QSC therefore directly realizes the integrable sector determined by the underlying Yangian symmetry.

7. Broader Implications and Current Research Frontiers

The QSC framework, through its minimal embodiment in fishnet CFTs, serves as a unique laboratory for the study of AdS/CFT integrability, integrable QFTs, and the interplay between conformal and Yangian symmetry. Current directions include:

• Exact correlation functions and SoV (Separation of Variables): QSC allows construction of SoV bases, determinant representations for structure constants, and closed-form expressions for correlators even at finite coupling (Ekhammar et al., 28 Dec 2025, Derkachov et al., 2018).
• Extension to general fishnet and chiral models: QSC/Baxter formalisms have proven flexible in encoding the spectral problem in anisotropic, supersymmetric, or higher-rank fishnets, as well as in their D-dimensional generalizations (Ekhammar et al., 28 Dec 2025, Kazakov et al., 2022, Kazakov et al., 2018).
• Connection to geometry and higher-loop dynamics: The operatorial QSC approach exposes links to Calabi–Yau geometry (in 2D), structure of Feynman integrals, and the emergence of Riemann-Hilbert and algebraic-geometric constraints (Ekhammar et al., 28 Dec 2025).
• Integrability in non-supersymmetric settings: The fishnet QSC paradigm provides a controlled setting for addressing questions of non-unitary integrable CFTs, spontaneous conformal symmetry breaking, and the fate of exact integrability in the absence of supersymmetry (Karananas et al., 2019, Loebbert et al., 2020).
• Open problems: The full extension of QSC/SoV techniques to arbitrary correlation functions, twist deformations, and other integrable deformations remains an active domain of research (Ekhammar et al., 28 Dec 2025).

The QSC remains the central algebraic machinery enabling exact, non-perturbative control of the spectrum and integrable structure in a broad class of planar conformal field theories. Its further refinement promises to shed light on the origin of integrability in QFT and the non-perturbative structure of quantum spectra (Ekhammar et al., 28 Dec 2025, Kazakov, 2018, Beisert et al., 24 Nov 2025).