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ODE/IM Correspondence in Toda Field Theories and Fermionic Basis in sin(h)-Gordon Model
High Energy Physics - Theory · Submitted 22 Feb 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.
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The abstract, sentence by sentence
The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Ka\v{c}-Moody algebra $\hat{\mathfrak g}$. In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. These isomorphisms are used to construct a set of quadratic functional relations, called $\psi$-system, for the solutions to complex differential equations associated to $\hat{\mathfrak g}$. Finally, from the $\psi$-system for the algebras $\mathfrak a$ and $\mathfrak d$, we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. | contribution 0.99 | 0.85 | 0.05 | asserted 0.96 |
| s1 | It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Ka\v{c}-Moody algebra $\hat{\mathfrak g}$. | future work 0.37 | 0.21 | 0.06 | asserted 0.97 |
| s2 | In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. | method 0.81 | 0.93 | 0.05 | asserted 1.00 |
| s3 | These isomorphisms are used to construct a set of quadratic functional relations, called $\psi$-system, for the solutions to complex differential equations associated to $\hat{\mathfrak g}$. | contribution 0.45 | 0.93 | 0.10 | asserted 1.00 |
| s4 | Finally, from the $\psi$-system for the algebras $\mathfrak a$ and $\mathfrak d$, we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. | result 0.93 | 0.95 | 0.20 | asserted 1.00 |
| s5 | The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. | contribution 0.94 | 0.63 | 0.03 | asserted 0.97 |
| s6 | The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. | method 0.84 | 0.16 | 0.04 | asserted 0.98 |
| s7 | We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. | contribution 0.75 | 0.94 | 0.14 | asserted 1.00 |
| s8 | In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. | interpretation 0.90 | 0.92 | 0.10 | asserted 1.00 |
| s9 | This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding. | result 0.85 | 0.92 | 0.21 | asserted 0.95 |
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The paper
Original abstract
The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories. It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Ka\v{c}-Moody algebra $\hat{\mathfrak g}$. In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms. These isomorphisms are used to construct a set of quadratic functional relations, called $\psi$-system, for the solutions to complex differential equations associated to $\hat{\mathfrak g}$. Finally, from the $\psi$-system for the algebras $\mathfrak a$ and $\mathfrak d$, we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.
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How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it · 1 sentence without a confident label
Cites it as background 3 papers
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Our analysis in this work rests on the fact that the ODE/IM correspondence[For reviews of the ODE/IM correspondence, we refer the reader to [ref] .] provides a link between the spectral theory of ODEs [ref] and certain integrable quantum field theories, and which relates classical and quantum integrals of motion [ref] .
Integrable Structure of Higher Spin CFT and the ODE/IM Correspondence · Introduction and Summary · 0.82
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Off-critical generalisations of the ODE/IM correspondence associated to higher-rank algebras have been discussed in [ref] , although without specific analysis of the connection with surface embedding.
Geometric aspects of the ODE/IM correspondence · Surfaces embedded in AdS$_n+1$ · 0.90
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From the ODEs, one obtains the massless integrable models or conformal field theories [ref] .
Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations · Introduction · 0.86
What it cites · 105 sentences without a confident label
Builds on 3 papers
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Zamolodchikov [ref] to a general affine Lie-Kač-Moody algebra $𝔤̂$.
1003.53331 · Introduction · 0.99
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This point of view gives the fermionic basis a more “physical" interpretation with respect to the formal mathematical introduction of [ref] and, in addition, allows to extend its application to the sinh-Gordon model for which a rigorous definition of the fermions was not available, given the complicated nature of the lattice regularisation; the only “weak spot" in this interpretation is the fact that the formula ([ref]) has to be introduced as a conjecture.
1007.05561 · Figures and Tables · 0.86
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By adding parameters, it is possible to generalise further the Bessis-Zinn-Justin Hamiltonian; we might add an angular-momentum-like term $l(l+1)x^-2$, which gives [ref]
Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations · $𝒫𝒯$ symmetric Hamiltonians · 0.80
Uses the method or tool of 7 papers
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Takeyama [ref] , which allow the computation of any one-point function of sine-Gordon model[The method was originally developed on the XXZ model and later extended to the sine-Gordon model by means of an appropriate scaling limit.] exploiting the existence of a particular fermionic basis in the space of states of the theory.
0911.37310 · The $n=r$ case · 0.99
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Takeyama [ref] , which allow the computation of any one-point function of sine-Gordon model[The method was originally developed on the XXZ model and later extended to the sine-Gordon model by means of an appropriate scaling limit.] exploiting the existence of a particular fermionic basis in the space of states of the theory.
1105.62091 · The $n=r$ case · 0.99
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Takeyama [ref] , which allow the computation of any one-point function of sine-Gordon model[The method was originally developed on the XXZ model and later extended to the sine-Gordon model by means of an appropriate scaling limit.] exploiting the existence of a particular fermionic basis in the space of states of the theory.
1007.05561 · The $n=r$ case · 0.99
Compares with 7 papers
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A first clue of this fact comes by specialising ([ref]) to the various algebras as in Table [ref] and by comparing the forms obtained with those of the $ψ$-systems in [ref] .
Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras · Product representations and weak isomorphisms · 0.81
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In this section we investigate the behaviour of the function $F(α)$ introduced above in the classical limit and compare it against a result presented by Lukyanov in [ref] .
Finite temperature expectation values of local fields in the sinh-Gordon model · Classical limit · 1.00
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In this chapter we wish to check the results obtained from the fermionic basis framework against a result found by LeClair and Mussardo in [ref] .
Finite Temperature Correlation Functions in Integrable QFT · LeClair-Mussardo formula · 0.99
Cites as support 3 papers
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A verification of the correctness of these definition was carried out in [ref] , where the behaviour of $T(ζ)$ in the ultraviolet region $R→0$ is investigated numerically, showing how the asymptotics of $T(ζ)$ for $ζ→0$ and for $ζ→∞$ correctly reproduce the eigenvalues of CFT integrals of motion and, moreover, that their normalisation is the same as in the sine-Gordon case [ref] ; this is an extremely convincing argument.
Integrable Structure of Conformal Field Theory III. The Yang-Baxter Relation · One-point functions in sinh-Gordon model · 0.97
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A verification of the correctness of these definition was carried out in [ref] , where the behaviour of $T(ζ)$ in the ultraviolet region $R→0$ is investigated numerically, showing how the asymptotics of $T(ζ)$ for $ζ→0$ and for $ζ→∞$ correctly reproduce the eigenvalues of CFT integrals of motion and, moreover, that their normalisation is the same as in the sine-Gordon case [ref] ; this is an extremely convincing argument.
Finite temperature expectation values of local fields in the sinh-Gordon model · One-point functions in sinh-Gordon model · 0.97
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Again, as proved in Appendix B of [ref] , the spectrum of ([ref]) is real and positive for $M≥1$ and $|2l+1|< M+1$.
Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics · $𝒫𝒯$ symmetric Hamiltonians · 0.97
Cites as background 27 papers
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This means that, for sin(h)-Gordon model, we have two equivalent descriptions of the space of local fields; writing one description as a function of the other give rise to the reflection relations [ref] , which will be discussed later in chap.[ref].
Expectation values of boundary fields in the boundary sine-Gordon model · The $n=r$ case · 0.90
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This means that, for sin(h)-Gordon model, we have two equivalent descriptions of the space of local fields; writing one description as a function of the other give rise to the reflection relations [ref] , which will be discussed later in chap.[ref].
Expectation values of descendent fields in the sine-Gordon model · The $n=r$ case · 0.90
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This means that, for sin(h)-Gordon model, we have two equivalent descriptions of the space of local fields; writing one description as a function of the other give rise to the reflection relations [ref] , which will be discussed later in chap.[ref].
Expectation values of local fields in Bullough-Dodd model and integrable perturbed conformal field theories · The $n=r$ case · 0.90
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