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REVIEW 2 major objections 5 minor 184 references

Four-dimensional Chern-Simons theory generates time-dependent integrable field theories whose allowed time dependence is exactly the one-loop renormalization-group flow.

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2026-07-12 07:57 UTC pith:GNAG3HBY

load-bearing objection Clean geometric construction that forces time-dependent integrable models from 4d CS to track one-loop RG, with real extension to ultralocal theories and dilaton gravity. the 2 major comments →

arxiv 2607.02648 v1 pith:GNAG3HBY submitted 2026-07-02 hep-th cond-mat.stat-mechcond-mat.str-elmath-phmath.MP

Time-Dependent Integrability from Gauge Theory, I

Shota Komatsu , Jun-ichi Sakamoto , Anders Wallberg , Masahito Yamazaki This is my paper
classification hep-th cond-mat.stat-mechcond-mat.str-elmath-phmath.MP
keywords four-dimensional Chern-Simonstime-dependent integrabilityLax connectionone-loop RG flowultralocalnon-ultralocaldilaton gravityisomonodromy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Exact solutions for systems whose couplings change with time are rare, yet they are needed both for non-equilibrium physics and as benchmarks for quantum simulators. This paper shows that a mild, geometrically natural generalization of four-dimensional Chern-Simons theory supplies a systematic construction of such systems. The usual holomorphic one-form that encodes the spectral data is allowed to depend on space-time; consistency of the four-dimensional theory then forces the time dependence of every coupling to coincide with the one-loop renormalization-group flow of the corresponding time-independent model. Despite the explicit time dependence the resulting two-dimensional theories still possess a Lax connection, so they remain solvable by inverse-scattering methods. The same construction covers both ultralocal models (Gross-Neveu type) and non-ultralocal sigma models, and it can be rewritten as two-dimensional dilaton gravity coupled to integrable matter. The result therefore unifies previously scattered examples and produces large new families of classically solvable time-dependent systems.

Core claim

For every integrable field theory obtained from the spacetime-dependent four-dimensional Chern-Simons construction, the allowed time dependence of the couplings is identical to the one-loop renormalization-group flow; the theories nevertheless retain a Lax pair and remain solvable by inverse scattering.

What carries the argument

The spacetime-dependent meromorphic one-form ω̂=φ̂ dz+Ψ̂+dx++Ψ̂−dx− whose components are constrained by the requirements that dω̂ have no bulk support away from poles and that the pole orders of the Lax connection remain compatible with the zeros of φ̂; these geometric conditions automatically enforce both the modified flatness equation and the one-loop RG flow.

Load-bearing premise

The only allowed spacetime dependence of the one-form is the one forced by preservation of shift gauge symmetry and vanishing of bulk dω̂; if other consistent deformations exist outside this geometric constraint, the claimed universality fails.

What would settle it

Construct any classically integrable, purely time-dependent deformation of a known model (Gross-Neveu, principal chiral model, or their deformations) whose coupling flow is not proportional to the one-loop beta function, yet still admits a Lax pair of the form required by the four-dimensional construction.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper generalizes four-dimensional Chern–Simons theory by replacing the holomorphic one-form with a spacetime-dependent meromorphic one-form. Consistency of the resulting variational problem (preservation of shift gauge symmetry, vanishing of bulk dω away from poles, and compatible pole orders for edge modes) yields a systematic construction of classically integrable field theories with explicit spacetime dependence. Specializing to purely time-dependent couplings forces the allowed time evolution to coincide with the one-loop renormalization-group flow of the corresponding time-independent theory. The resulting models retain a (modified) Lax connection and are therefore solvable by inverse-scattering methods. The construction covers both ultralocal (order-defect) and non-ultralocal (disorder-defect) theories, recovers and extends known time-dependent sigma models, admits a reformulation as two-dimensional dilaton gravity coupled to integrable matter, and is illustrated by a range of explicit examples (Gross–Neveu, Faddeev–Reshetikhin, PCM with WZ term, η- and λ-deformations, coupled models).

Significance. If correct, the result supplies a unifying geometric origin for time-dependent integrability that simultaneously explains the previously observed coincidence with one-loop RG flow and generates a large new family of solvable non-autonomous systems. The framework extends well beyond the special coset models arising from dimensional reduction of gravity, encompasses ultralocal theories, and produces classically integrable dilaton-gravity models. The appendices contain new, self-contained computations of one-loop RG flows for general rational, trigonometric and elliptic order defects, including a proof of linear evolution of defect positions in the rational case and a detailed treatment of the Belavin elliptic r-matrix. These technical contributions stand independently of the main claim and strengthen the paper’s value as a reference.

major comments (2)
  1. The central derivation in §5 (especially the consistency conditions (5.10), (5.22), (5.35) and (5.38)) is presented as exhausting all admissible spacetime dependences compatible with a well-defined 4d variational problem. While the conditions are necessary, the manuscript does not contain an explicit argument that they are also sufficient for the existence of a global solution of the axial-gauge equations of motion on a compact Σ. A short remark clarifying the global versus local character of the construction would remove any residual ambiguity about the claimed universality.
  2. In §7 the map between fixed-spectral and variable-spectral Lax connections is derived under the assumption that the algebraic equation (7.21) admits a globally single-valued branch on the relevant covering of the w-plane. For the coupled PCM_k example the resulting multi-sheeted structure is left largely unanalyzed; a brief statement of the monodromy data that remain invariant would make the isomonodromic interpretation (promised for the sequel) more concrete already in the present paper.
minor comments (5)
  1. Abstract and Introduction: the phrase “the allowed time dependence coincides with the one-loop renormalization group flow” is repeated almost verbatim; a single concise statement would improve readability.
  2. Equation (3.8) and surrounding text: the coordinate redefinition that removes the finite-time singularity is given only for the GN model; a one-sentence remark that the same holomorphic change works for all models of the paper would be helpful.
  3. Table 1: the boundary conditions for the λ-model are written with a slightly inconsistent placement of parentheses relative to the η-model; aligning the notation would aid comparison.
  4. Appendix A.3: the claim that the linear flow (A.27) holds for trigonometric and elliptic cases is labelled a conjecture based on examples; stating the precise range of r-matrices for which the conjecture has been verified would prevent over-interpretation.
  5. References: several arXiv preprints cited as “to appear” (e.g. [35], [81]) should be updated with arXiv numbers if available before final publication.

Circularity Check

1 steps flagged

No significant circularity: time-dependence constraints are derived independently from 4d CS consistency and merely coincide with previously computed one-loop RG data.

specific steps
  1. self citation load bearing [§5.1 and §5.3, eqs. (5.2)–(5.3) vs. (5.40)]
    "the same structural data that control the one-loop RG flow in the 4d CS description [36–40] reappear here as the consistency conditions governing time dependence. … This is precisely the same structure that appeared in the one-loop RG flow (5.2)."

    The comparison target (the explicit form of the RG vector field Ψ) is taken from prior papers that share co-authors with the present work. While the time-dependence conditions are derived independently, the claim that they “coincide with the one-loop RG flow” relies on those self-citations for the precise expression of the RG data. The circularity is minor: the self-cited results are independently checkable calculations, not an unverified uniqueness theorem, and the matching itself is a new observation rather than a definitional identity.

full rationale

The central claim (allowed pure time dependence of couplings equals one-loop RG flow for any theory obtained from the spacetime-dependent 4d CS construction) is obtained by imposing well-definedness of the generalized action: preservation of shift gauge symmetry (5.9)–(5.10) for order defects, vanishing of bulk dω̂ away from poles together with pole-order compatibility (5.21)–(5.22), (5.35), (5.38) for disorder defects. Specializing to pure time dependence then forces the meromorphic data (Ψ̂, periods) to obey exactly the same algebraic conditions that govern the ordinary one-loop RG flow of the same 4d CS theories (5.2)–(5.3) and (5.40). The match is therefore a comparison of two independently derived sets of equations, not a definitional identity or a fitted input renamed as prediction. Self-citations to the prior RG analyses of disorder defects ([36–40], some of which share co-authors) supply the comparison target; they are not used to justify the consistency conditions themselves. Order-defect RG is recomputed from scratch in the appendices and again yields the linear flow. No ansatz is smuggled, no uniqueness theorem is imported as an external fact that forbids alternatives, and no parameter is fitted to data and then “predicted.” The derivation is therefore self-contained within the stated geometric class; the only residual limitation is the scope of the 4d CS framework, which the paper already acknowledges.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The construction rests on the established 4d Chern-Simons framework for ordinary integrable field theories, the classical definition of Lax integrability, and the geometric characterization of one-loop RG flows already derived for disorder defects. No free parameters are fitted; the overall time-scale a is an arbitrary normalization. No new physical entities are postulated.

free parameters (1)
  • overall time-scale a (and additive constants b_i)
    Appears as an arbitrary constant rescaling of the time coordinate that matches the dual Coxeter number in the RG equation; not fitted to data.
axioms (4)
  • domain assumption Classical Lax integrability (flatness of a spectral-parameter family of connections) is the appropriate definition of integrability for time-dependent systems.
    Stated in §3.1; replaces the usual infinite set of conserved charges which cannot exist once energy is non-conserved.
  • domain assumption The 4d Chern-Simons action with meromorphic one-form ω and surface defects generates all (or a sufficiently large class of) classical 2d integrable field theories.
    Taken from Costello-Yamazaki et al.; used throughout §§4-5 as the starting point.
  • standard math One-loop RG flow of disorder-defect theories is given by dφ/dt_RG = dΨ/dz with Ψ fixed by the zeros of φ (eq. 5.2).
    Previously derived in the 4d CS literature (Delduc et al., Levine, Lacroix-Wallberg); re-used as the geometric data that reappear for time dependence.
  • ad hoc to paper Shift gauge symmetry A → A + χ dz must be preserved after the one-form is made spacetime-dependent.
    Imposed in §5.2-5.3 to constrain ∂±ẑ∓ = 0 and the flow equations for Ψ±; the key consistency requirement that selects the RG vector field.

pith-pipeline@v1.1.0-grok45 · 71294 in / 2611 out tokens · 31712 ms · 2026-07-12T07:57:31.419016+00:00 · methodology

0 comments
read the original abstract

Solvable time-dependent systems provide important settings for studying non-equilibrium physics, where exact results are rare. They are also useful for benchmarking quantum simulations, which can directly probe real-time dynamics beyond the reach of conventional numerical approaches. In this paper, we show that the four-dimensional Chern-Simons theory offers a natural and unifying framework for constructing such systems. Focusing on classically integrable field theories, we consider a generalization of the four-dimensional Chern-Simons theory in which the usual holomorphic one-form is replaced by a more general, spacetime-dependent one-form. This yields a systematic procedure for generating time-dependent integrable field theories and establishes a universal relation: for every theory obtained in this way, the allowed time dependence coincides with the one-loop renormalization group flow. Despite the explicit time dependence, these theories retain Lax integrability and remain solvable by inverse scattering methods. Our construction applies to both ultralocal and non-ultralocal theories and extends previously known time-dependent sigma models to a much broader class of integrable systems. It also admits rewriting as dilaton gravity coupled to matter, producing a large family of classically integrable dilaton gravity theories in two dimensions. We also comment on connections to time-dependent integrable models studied recently in condensed matter physics and non-autonomous integrable systems arising from dimensionally-reduced Einstein gravity.

discussion (0)

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Reference graph

Works this paper leans on

184 extracted references · 2 canonical work pages

  1. [1]

    Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition,

    L. Onsager, “Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition,”Phys. Rev.65(1944) 117–149

    1944

  2. [2]

    Exact solution of a 1D asymmetric exclusion model using a matrix formulation,

    B. Derrida, M. R. Evans, V. Hakim, and V. Pasquier, “Exact solution of a 1D asymmetric exclusion model using a matrix formulation,”J. Phys. A26(1993) 1493–1517

    1993

  3. [3]

    An exactly soluble non-equilibrium system: The asymmetric simple exclusion process,

    B. Derrida, “An exactly soluble non-equilibrium system: The asymmetric simple exclusion process,”Phys. Rept.301no. 1-3, (1998) 65–83

    1998

  4. [4]

    The Kardar-Parisi-Zhang equation and universality class,

    I. Corwin, “The Kardar-Parisi-Zhang equation and universality class,”Random Matrices: Theory Appl.1no. 1, (2012) 1130001,arXiv:1106.1596 [math.PR]

    Pith/arXiv arXiv 2012

  5. [5]

    The exclusion process: A paradigm for non-equilibrium behaviour,

    K. Mallick, “The exclusion process: A paradigm for non-equilibrium behaviour,”Physica A 418(2015) 17–48

    2015

  6. [6]

    Dynamic Scaling of Growing Interfaces,

    M. Kardar, G. Parisi, and Y.-C. Zhang, “Dynamic Scaling of Growing Interfaces,”Phys. Rev. Lett.56(1986) 889–892

    1986

  7. [7]

    One-Dimensional Kardar-Parisi-Zhang Equation: An Exact Solution and Its Universality,

    T. Sasamoto and H. Spohn, “One-Dimensional Kardar-Parisi-Zhang Equation: An Exact Solution and Its Universality,”Phys. Rev. Lett.104(2010) 230602,arXiv:1002.1883 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2010

  8. [8]

    Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals,

    K. A. Takeuchi and M. Sano, “Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals,”Phys. Rev. Lett.104(2010) 230601

    2010

  9. [9]

    Conserved charges in the quantum simulation of integrable spin chains,

    K. Maruyoshi, T. Okuda, J. W. Pedersen, R. Suzuki, M. Yamazaki, and Y. Yoshida, “Conserved charges in the quantum simulation of integrable spin chains,” arXiv:2208.00576 [quant-ph]

    Pith/arXiv arXiv

  10. [10]

    Integration of the Einstein Equations by the Inverse Scattering Problem Technique and the Calculation of the Exact Soliton Solutions,

    V. A. Belinsky and V. E. Zakharov, “Integration of the Einstein Equations by the Inverse Scattering Problem Technique and the Calculation of the Exact Soliton Solutions,”Sov. Phys. JETP48(1978) 985–994

    1978

  11. [11]

    The integration of Einstein’s equations, by the method of inverse scattering, and exact soliton solutions,

    V. Belinsky and V. Zakharov, “The integration of Einstein’s equations, by the method of inverse scattering, and exact soliton solutions,” inWorkshop on Sources of Gravitational Radiation, pp. 161–172. 1979

    1979

  12. [12]

    Sigma models with local couplings: a new integrability – RG flow connection,

    B. Hoare, N. Levine, and A. A. Tseytlin, “Sigma models with local couplings: a new integrability – RG flow connection,”JHEP11(2020) 020,arXiv:2008.01112 [hep-th]

    Pith/arXiv arXiv 2020

  13. [13]

    Integrable deformations of dimensionally reduced gravity,

    M. Ces` aro and D. Osten, “Integrable deformations of dimensionally reduced gravity,” JHEP06(2025) 064,arXiv:2502.01750 [hep-th]

    Pith/arXiv arXiv 2025

  14. [14]

    Exact out-of-equilibrium central spin dynamics from integrability,

    D. Fioretto, J.-S. Caux, and V. Gritsev, “Exact out-of-equilibrium central spin dynamics from integrability,”New J. Phys.16(2014) 043024,arXiv:1211.5905 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2014

  15. [15]

    Integrable Time-Dependent Quantum Hamiltonians,

    N. A. Sinitsyn, E. A. Yuzbashyan, V. Y. Chernyak, A. Patra, and C. Sun, “Integrable Time-Dependent Quantum Hamiltonians,”Phys. Rev. Lett.120no. 19, (2018) 190402, arXiv:1711.09945 [quant-ph]

    Pith/arXiv arXiv 2018

  16. [16]

    Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets,

    E. A. Yuzbashyan, “Integrable time-dependent Hamiltonians, solvable Landau-Zener models and Gaudin magnets,”Annals Phys.392(2018) 323–339,arXiv:1802.01571 [cond-mat.stat-mech]

    Pith/arXiv arXiv 2018

  17. [17]

    Integrable Floquet dynamics,

    V. Gritsev and A. Polkovnikov, “Integrable Floquet dynamics,”SciPost Phys.2no. 3, (2017) 021,arXiv:1701.05276 [cond-mat.stat-mech]. – 77 –

    Pith/arXiv arXiv 2017

  18. [18]

    A map between time-dependent and time-independent quantum many-body Hamiltonians,

    O. Gamayun and O. Lychkovskiy, “A map between time-dependent and time-independent quantum many-body Hamiltonians,”Proc. Steklov Inst. Math.313(2021) 41–57, arXiv:2009.13873 [quant-ph]

    Pith/arXiv arXiv 2021

  19. [19]

    Integrability of the Kondo model with time-dependent interaction strength,

    P. R. Pasnoori, “Integrability of the Kondo model with time-dependent interaction strength,”Phys. Rev. B112no. 6, (2025) L060409,arXiv:2505.20125 [cond-mat.str-el]

    Pith/arXiv arXiv 2025

  20. [20]

    Exact many-body wavefunction of the Kondo model with time-dependent interaction strength,

    P. R. Pasnoori and E. A. Yuzbashyan, “Exact many-body wavefunction of the Kondo model with time-dependent interaction strength,”arXiv:2509.05640 [cond-mat.str-el]

    Pith/arXiv arXiv

  21. [21]

    Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-dependent Interaction Strength,

    P. R. Pasnoori, “Quantum Knizhnik-Zamolodchikov Equations and Integrability of Quantum Field Theories with Time-dependent Interaction Strength,”Phys. Rev. B113 (2026) 094514,arXiv:2510.14764 [hep-th]

    arXiv 2026

  22. [22]

    Quantum Integrability of Hamiltonians with Time-Dependent Interaction Strengths and the Renormalization Group Flow,

    P. R. Pasnoori, “Quantum Integrability of Hamiltonians with Time-Dependent Interaction Strengths and the Renormalization Group Flow,”Phys. Rev. B113no. 20, (2026) L201405, arXiv:2512.13625 [cond-mat.str-el]

    arXiv 2026

  23. [23]

    Time-Dependent Dynamical Dimensional Transmutation in the SU(2) Gross-Neveu Model with Time-Dependent Interaction Strength,

    P. R. Pasnoori, “Time-Dependent Dynamical Dimensional Transmutation in the SU(2) Gross-Neveu Model with Time-Dependent Interaction Strength,”arXiv:2605.05111 [math-ph]

    Pith/arXiv arXiv

  24. [24]

    Integrable multistate Landau-Zener models with parallel energy levels,

    V. Y. Chernyak, F. Li, C. Sun, and N. A. Sinitsyn, “Integrable multistate Landau-Zener models with parallel energy levels,”J. Phys. A53no. 29, (2020) 295201, arXiv:2001.10091

    Pith/arXiv arXiv 2020

  25. [25]

    Integrability in the multistate Landau-Zener model with time-quadratic commuting operators,

    V. Y. Chernyak and N. A. Sinitsyn, “Integrability in the multistate Landau-Zener model with time-quadratic commuting operators,”J. Phys. A54no. 11, (2021) 115204, arXiv:2006.15144

    Pith/arXiv arXiv 2021

  26. [26]

    Higher spin Richardson-Gaudin model with time-dependent coupling: Exact dynamics,

    S. Barik, L. Bakker, V. Gritsev, J. Minar, and E. A. Yuzbashyan, “Higher spin Richardson-Gaudin model with time-dependent coupling: Exact dynamics,”Phys. Rev. B 113no. 20, (2026) 195147,arXiv:2507.10856 [quant-ph]

    Pith/arXiv arXiv 2026

  27. [27]

    Supersymmetric gauge theory and the Yangian,

    K. Costello, “Supersymmetric gauge theory and the Yangian,”arXiv:1303.2632 [hep-th]

    Pith/arXiv arXiv

  28. [28]

    Gauge Theory and Integrability, I,

    K. Costello, E. Witten, and M. Yamazaki, “Gauge Theory and Integrability, I,”ICCM Not. 06no. 1, (2018) 46–119,arXiv:1709.09993 [hep-th]

    Pith/arXiv arXiv 2018

  29. [29]

    Gauge Theory and Integrability, II,

    K. Costello, E. Witten, and M. Yamazaki, “Gauge Theory and Integrability, II,”ICCM Not.06no. 1, (2018) 120–146,arXiv:1802.01579 [hep-th]

    Pith/arXiv arXiv 2018

  30. [30]

    Gauge Theory And Integrability, III,

    K. Costello and M. Yamazaki, “Gauge Theory And Integrability, III,”arXiv:1908.02289 [hep-th]

    Pith/arXiv arXiv 1908

  31. [31]

    Four Dimensional Holomorphic Theories,

    N. Nekrasov, “Four Dimensional Holomorphic Theories,”. http://media.scgp.stonybrook.edu/papers/prdiss96.pdf

  32. [32]

    Elliptic Calogero-Moser system from two-dimensional current algebra,

    A. Gorsky and N. Nekrasov, “Elliptic Calogero-Moser system from two-dimensional current algebra,”arXiv:hep-th/9401021

    Pith/arXiv arXiv

  33. [33]

    A Kahler-Chern-Simons Theory and Quantization of Instanton Moduli Spaces,

    V. P. Nair and J. Schiff, “A Kahler-Chern-Simons Theory and Quantization of Instanton Moduli Spaces,”Phys. Lett. B246(1990) 423–429

    1990

  34. [34]

    Four-dimensional avatars of – 78 – two-dimensional RCFT,

    A. Losev, G. W. Moore, N. Nekrasov, and S. Shatashvili, “Four-dimensional avatars of – 78 – two-dimensional RCFT,” inStrings ’95: Future Perspectives in String Theory, vol. 46, pp. 130–145. 1996.arXiv:hep-th/9509151

    Pith/arXiv arXiv 1996

  35. [35]

    Time-Dependent Integrability from Gauge Theory, II,

    S. Komatsu, J. Sakamoto, A. Wallberg, and M. Yamazaki, “Time-Dependent Integrability from Gauge Theory, II,”. To appear

  36. [36]

    RG flows of integrableσ-models and the twist function,

    F. Delduc, S. Lacroix, K. Sfetsos, and K. Siampos, “RG flows of integrableσ-models and the twist function,”JHEP02(2021) 065,arXiv:2010.07879 [hep-th]

    Pith/arXiv arXiv 2021

  37. [37]

    Universal 1-loop divergences for integrable sigma models,

    N. Levine, “Universal 1-loop divergences for integrable sigma models,”JHEP03(2023) 3, arXiv:2209.05502 [hep-th]

    Pith/arXiv arXiv 2023

  38. [38]

    Equivalence of 1-loop RG flows in 4d Chern-Simons and integrable 2d sigma-models,

    N. Levine, “Equivalence of 1-loop RG flows in 4d Chern-Simons and integrable 2d sigma-models,”Phys. Lett. B862(2025) 139325,arXiv:2309.16753 [hep-th]

    Pith/arXiv arXiv 2025

  39. [39]

    Geometry of the spectral parameter and renormalisation of integrable sigma-models,

    S. Lacroix and A. Wallberg, “Geometry of the spectral parameter and renormalisation of integrable sigma-models,”JHEP05(2024) 108,arXiv:2401.13741 [hep-th]

    Pith/arXiv arXiv 2024

  40. [40]

    1-loop renormalisability of integrable sigma-models from 4d Chern-Simons theory,

    S. Lacroix, N. Levine, and A. Wallberg, “1-loop renormalisability of integrable sigma-models from 4d Chern-Simons theory,”JHEP09(2025) 153,arXiv:2505.23890 [hep-th]

    arXiv 2025

  41. [41]

    Inverse scattering method with variable spectral parameter,

    V. Burtsev, S.P. Zakharov and A. Mikhailov, “Inverse scattering method with variable spectral parameter,”Theor. Math. Phys.70(1987) 227––240

    1987

  42. [42]

    On the Geroch Group,

    P. Breitenlohner and D. Maison, “On the Geroch Group,”Ann. Inst. H. Poincare Phys. Theor.46(1987) 215

    1987

  43. [43]

    Twistor Actions for Integrable Systems,

    R. F. Penna, “Twistor Actions for Integrable Systems,”JHEP09(2021) 140, arXiv:2011.05831 [hep-th]

    Pith/arXiv arXiv 2021

  44. [44]

    Integrability in gravity from Chern-Simons theory,

    L. T. Cole and P. Weck, “Integrability in gravity from Chern-Simons theory,”JHEP10 (2024) 080,arXiv:2407.08782 [hep-th]

    Pith/arXiv arXiv 2024

  45. [45]

    Integrable Deformations of the Breitenlohner-Maison Model from 4d Chern-Simons Theory,

    M. Ashwinkumar and M. Blau, “Integrable Deformations of the Breitenlohner-Maison Model from 4d Chern-Simons Theory,”arXiv:2604.26452 [hep-th]

    Pith/arXiv arXiv

  46. [46]

    The application of monodromy preserving deformation to the gravitational field equation,

    K. Ueno, “The application of monodromy preserving deformation to the gravitational field equation,”Proc. Japan Acad. Ser. A Math. Sci.57no. 4, (1981) 210–215

    1981

  47. [47]

    Separation of variables and Hamiltonian formulation for the Ernst equation,

    D. Korotkin and H. Nicolai, “Separation of variables and Hamiltonian formulation for the Ernst equation,”Phys. Rev. Lett.74(1995) 1272–1275,arXiv:hep-th/9412072

    Pith/arXiv arXiv 1995

  48. [48]

    An integrable model of quantum gravity,

    D. Korotkin and H. Nicolai, “An integrable model of quantum gravity,”Phys. Lett. B356 (1995) 211–216,arXiv:hep-th/9504088

    Pith/arXiv arXiv 1995

  49. [50]

    On 2-D quantum gravity coupled to a sigma model,

    D. Korotkin, H. Nicolai, and H. Samtleben, “On 2-D quantum gravity coupled to a sigma model,”Nucl. Phys. B Proc. Suppl.49(1996) 1–9,arXiv:gr-qc/9602027

    Pith/arXiv arXiv 1996

  50. [51]

    Integrable classical and quantum gravity,

    H. Nicolai, D. Korotkin, and H. Samtleben, “Integrable classical and quantum gravity,” in Quantum Fields and Quantum Space Time, pp. 203–243. 1997.arXiv:hep-th/9612065

    Pith/arXiv arXiv 1997

  51. [52]

    Quantization of coset space sigma models coupled to two-dimensional gravity,

    D. Korotkin and H. Samtleben, “Quantization of coset space sigma models coupled to two-dimensional gravity,”Commun. Math. Phys.190(1997) 411–457, arXiv:hep-th/9607095

    Pith/arXiv arXiv 1997

  52. [53]

    Solutions of Schlesinger system and Ernst equation in terms of theta-functions,

    D. Korotkin and V. Matveev, “Solutions of Schlesinger system and Ernst equation in terms of theta-functions,”arXiv:gr-qc/9810041. – 79 –

    Pith/arXiv arXiv

  53. [54]

    Note sur l’int´ egration des ´ equations diff´ erentielles de la Dynamique, pr´ esent´ ee au Bureau des Longitudes le 29 juin 1853,

    J. Liouville, “Note sur l’int´ egration des ´ equations diff´ erentielles de la Dynamique, pr´ esent´ ee au Bureau des Longitudes le 29 juin 1853,”Journal de Math´ ematiques Pures et Appliqu´ ees 1e s´ erie, 20(1855) 137–138.https://www.numdam.org/item/JMPA_1855_1_20__137_0/

  54. [55]

    Babelon, D

    O. Babelon, D. Bernard, and M. Talon,Introduction to Classical Integrable Systems. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2003

    2003

  55. [56]

    V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin,Quantum Inverse Scattering Method and Correlation Functions. Cambridge Monographs on Mathematical Physics. Cambridge University Press, 1993

    1993

  56. [57]

    Solutions of the classical Yang - Baxter equation for simple Lie algebras,

    A. A. Belavin and V. G. Drinfel’d, “Solutions of the classical Yang - Baxter equation for simple Lie algebras,”Funct. Anal. Appl.16no. 3, (1982) 159–180

    1982

  57. [58]

    Some exact results for the many body problems in one dimension with repulsive delta function interaction,

    C.-N. Yang, “Some exact results for the many body problems in one dimension with repulsive delta function interaction,”Phys. Rev. Lett.19(1967) 1312–1314

    1967

  58. [59]

    Complex structures and zero-curvature equations for sigma-models,

    D. Bykov, “Complex structures and zero-curvature equations for sigma-models,”Physics Letters B760(Sept., 2016) 341–344,arXiv:1605.01093 [hep-th]. http://arxiv.org/abs/1605.01093

    Pith/arXiv arXiv 2016

  59. [60]

    Flag manifold sigma models: Spin chains and integrable theories,

    I. Affleck, D. Bykov, and K. Wamer, “Flag manifold sigma models: Spin chains and integrable theories,”Phys. Rept.953(2022) 1–93,arXiv:2101.11638 [hep-th]

    Pith/arXiv arXiv 2022

  60. [61]

    String Theory and the Nonabelian Thirring Model,

    D. Kutasov, “String Theory and the Nonabelian Thirring Model,”Phys. Lett. B227(1989) 68–72

    1989

  61. [62]

    On the beta function for anisotropic current interactions in 2-D,

    B. Gerganov, A. LeClair, and M. Moriconi, “On the beta function for anisotropic current interactions in 2-D,”Phys. Rev. Lett.86(2001) 4753,arXiv:hep-th/0011189

    Pith/arXiv arXiv 2001

  62. [63]

    Renormalization of the nonlinear sigma model in 2 + epsilon dimensions. Application to the Heisenberg ferromagnets,

    E. Brezin and J. Zinn-Justin, “Renormalization of the nonlinear sigma model in 2 + epsilon dimensions. Application to the Heisenberg ferromagnets,”Phys. Rev. Lett.36(1976) 691–694

    1976

  63. [64]

    Four-dimensional Chern–Simons theory and integrable field theories,

    S. Lacroix, “Four-dimensional Chern–Simons theory and integrable field theories,”J. Phys. A55no. 8, (2022) 083001,arXiv:2109.14278 [hep-th]

    Pith/arXiv arXiv 2022

  64. [65]

    Gauge Theory and Integrability: An Overview,

    M. Yamazaki, “Gauge Theory and Integrability: An Overview,”arXiv:2509.07628 [hep-th]

    Pith/arXiv arXiv

  65. [66]

    Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories,

    M. Benini, A. Schenkel, and B. Vicedo, “Homotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories,”Commun. Math. Phys.389no. 3, (2022) 1417–1443, arXiv:2008.01829 [hep-th]

    Pith/arXiv arXiv 2022

  66. [67]

    Deformed integrableσ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles,

    B. Vicedo, “Deformed integrableσ-models, classical R-matrices and classical exchange algebra on Drinfel’d doubles,”J. Phys. A48no. 35, (2015) 355203,arXiv:1504.06303 [hep-th]

    Pith/arXiv arXiv 2015

  67. [68]

    ηandλdeformations as E -models,

    C. Klimcik, “ηandλdeformations as E -models,”Nucl. Phys. B900(2015) 259–272, arXiv:1508.05832 [hep-th]

    Pith/arXiv arXiv 2015

  68. [69]

    On integrable deformations of superstring sigma models related toAdS n ×S n supercosets,

    B. Hoare and A. A. Tseytlin, “On integrable deformations of superstring sigma models related toAdS n ×S n supercosets,”Nucl. Phys. B897(2015) 448–478,arXiv:1504.07213 [hep-th]

    Pith/arXiv arXiv 2015

  69. [70]

    Generalised integrableλ- and η-deformations and their relation,

    K. Sfetsos, K. Siampos, and D. C. Thompson, “Generalised integrableλ- and η-deformations and their relation,”Nucl. Phys. B899(2015) 489–512,arXiv:1506.05784 [hep-th]. – 80 –

    Pith/arXiv arXiv 2015

  70. [71]

    IntegrableE-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects,

    S. Lacroix and B. Vicedo, “IntegrableE-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects,”SIGMA17(2021) 058,arXiv:2011.13809 [hep-th]

    Pith/arXiv arXiv 2021

  71. [72]

    On space of integrable quantum field theories,

    F. A. Smirnov and A. B. Zamolodchikov, “On space of integrable quantum field theories,” Nucl. Phys. B915(2017) 363–383,arXiv:1608.05499 [hep-th]

    Pith/arXiv arXiv 2017

  72. [73]

    T¯T-deformed 2D Quantum Field Theories,

    A. Cavagli` a, S. Negro, I. M. Sz´ ecs´ enyi, and R. Tateo, “T¯T-deformed 2D Quantum Field Theories,”JHEP10(2016) 112,arXiv:1608.05534 [hep-th]

    Pith/arXiv arXiv 2016

  73. [74]

    On the Classical Integrability of Root-T TFlows,

    R. Borsato, C. Ferko, and A. Sfondrini, “On the Classical Integrability of Root-T TFlows,” arXiv:2209.14274 [hep-th]

    Pith/arXiv arXiv

  74. [75]

    Infinite Family of Integrable Sigma Models Using Auxiliary Fields,

    C. Ferko and L. Smith, “Infinite Family of Integrable Sigma Models Using Auxiliary Fields,”Phys. Rev. Lett.133no. 13, (2024) 131602,arXiv:2405.05899 [hep-th]

    Pith/arXiv arXiv 2024

  75. [76]

    4D Chern-Simons theory with auxiliary fields,

    O. Fukushima and K. Yoshida, “4D Chern-Simons theory with auxiliary fields,”JHEP09 (2025) 001,arXiv:2407.02204 [hep-th]

    Pith/arXiv arXiv 2025

  76. [77]

    T ¯Tand root-T ¯Tdeformations in four-dimensional Chern-Simons theory,

    J. Sakamoto, R. Tateo, and M. Yamazaki, “T ¯Tand root-T ¯Tdeformations in four-dimensional Chern-Simons theory,”arXiv:2509.12303 [hep-th]

    arXiv

  77. [78]

    The Courant-Hilbert construction in 4D Chern-Simons theory,

    O. Fukushima, T. Matsumoto, and K. Yoshida, “The Courant-Hilbert construction in 4D Chern-Simons theory,”arXiv:2509.22080 [hep-th]

    arXiv

  78. [79]

    Integrable DegenerateE-Models from 4d Chern–Simons Theory,

    J. Liniado and B. Vicedo, “Integrable DegenerateE-Models from 4d Chern–Simons Theory,”Annales Henri Poincare24no. 10, (2023) 3421–3459,arXiv:2301.09583 [hep-th]

    Pith/arXiv arXiv 2023

  79. [80]

    A unifying 2d action for integrable σ-models from 4d Chern-Simons theory,

    F. Delduc, S. Lacroix, M. Magro, and B. Vicedo, “A unifying 2d action for integrable σ-models from 4d Chern-Simons theory,”Lett. Math. Phys.110(2020) 1645–1687, arXiv:1909.13824 [hep-th]

    Pith/arXiv arXiv 2020

  80. [81]

    Renormalization Group Flow from 4d Chern-Simons Theory

    M. Ashwinkumar and M. Yamazaki, “Renormalization Group Flow from 4d Chern-Simons Theory.” To appear

Showing first 80 references.