Resonances in sinh- and sine-Gordon models and higher equations of motion in Liouville theory
Michael Lashkevich
备注：
22 pages, no figures; v2: no changes, a misprint in metadata corrected; v3: a new appendix added, numerous misprints corrected; v4: minor changes; v5: eq. (4.8) corrected
The notion of operator resonances was introduced earlier by Al. Zamolodchikov within the framework of the conformal perturbation theory. The resonances are related to logarithmic divergences of integrals in the perturbation expansion, and manifest themselves in poles of the correlation functions and form factors of local operators considered as functions of conformal dimensions. The residues of the poles can be computed by means of some operator identities. Here we study the resonances in the Liouville, sinh- and sine-Gordon models, considered as perturbations of a massless free boson. We show that the well-known higher equations of motion discovered by Al. Zamolodchikov in the Liouville field theory are nothing but resonance identities for some descendant operators. The resonance expansion in the vicinity of a resonance point provides a regularized version of the corresponding operators. We try to construct the corresponding resonance identities and resonance expansions in the sinh- and sine-Gordon theories. In some cases it can be done explicitly, but in most cases we are only able to obtain a general form so far. We show nevertheless that the resonances are perturbatively exact, which means that each of them only appears in a single term of the perturbation theory.
Using the fermionic basis we conjecture exact expressions for diagonal finite volume matrix elements of exponential operators and their descendants in the sinh-Gordon theory. Our expressions sum up the LeClair-Mussardo type infinite series generalized by Pozsgay for excited state expectation values. We checked our formulae against the Liouville three-point functions for small, while against Pozsgay's expansion for large volumes and found complete agreement.
The diagonal matrix elements $\langleθ_1,θ_2|T_{2n}T_{-2n}|θ_1,θ_2\rangle$ between two-particle states in the sinh-Gordon model are computed analytically for all integers $n>0$. This confirms the proposal by F. Smirnov and A. Zamolodchikov for these matrix elements and demonstrates effectiveness of the algebraic approach to form factors.
In this paper we derive the type-II integrable defect for the N = 2 supersymmetric sinh-Gordon (sshG) model by using the fusing procedure. In particular, we show explicitly the conservation of the modified energy, momentum and supercharges.
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We construct the generalised Eigenfunctions of the entries of the monodromy matrix of the $N$-site modular XXZ magnet and show, in each case, that these form a complete orthogonal system in $L^2(\mathbb{R}^N)$. In particular, we develop a new and simple technique, allowing one to prove the completeness of such systems. As a corollary of out analysis, we prove the Bystko-Teschner conjecture relative to the structure of the spectrum of the $\boldsymbol{ \texttt{B} }(\la)$-operator for the odd length lattice Sinh-Gordon model.
We study quasilocal operators in the quantum complex sinh-Gordon theory in the form factor approach. The free field procedure for descendant operators is developed by introducing the algebra of screening currents and related algebraic objects. We work out null vector equations in the space of operators. Further we apply the proposed algebraic structures to constructing form factors of the conserved currents $T_k$ and $Θ_k$. We propose also form factors of current-current operators of the form $T_kT_{-l}$. Explicit computations of the four-particle form factors allow us to verify the recent con…
We study 1+1 dimensional dilaton gravity models which take into account backreaction of the sinh-Gordon matter field. We found a wide class of exact solutions which generalizes black hole solutions of the Jackiw-Teitelboim gravity model and its hyperbolic deformation.
This paper is concerned with a class of approximate non-linear transformations that compress solutions of the (generalized) sinh-Gordon equation into parametrically small regions in two-dimensional spacetime. Given the sinh-Gordon field near a time-slice, a long Nambu-Goto string can be constructed in three-dimensional anti-de Sitter space. The string is then approximated to arbitrary accuracy by a slightly smoothed piecewise linear string of N segments. The corresponding sinh-Gordon field has a comb-like structure and its size is controlled by the amount of smoothing applied to the segmented …
We derive a characterization of the spectrum of the Sinh-Gordon model in terms of certain nonlinear integral equations. There exists a large class of solutions to these equations which allows a continuation between the infrared and the ultraviolet limits, respectively. We present nontrivial evidence for the claim that the class of solutions in question describes the spectrum of the Sinh-Gordon model completely in both of these limits. The evidence includes some nontrivial relations to Liouville theory.
We analyze the Thermodynamic Bethe Ansatz (TBA) for various integrable S-matrices in the context of generalized $T\bar T$ deformations. We focus on the sinh-Gordon model and its elliptic deformation in both its fermionic and bosonic realizations. We confirm that the determining factor for a turning point in the TBA, interpreted as a finite Hagedorn temperature, is the difference between the number of bound states and resonances in the theory. Implementing the numerical pseudo-arclength continuation method, we are able to follow the solutions to the TBA equations past the turning point all the …
Using generalized hydrodynamics (GHD), we develop the Euler hydrodynamics of classical integrable field theory. Classical field GHD is based on a known formalism for Gibbs ensembles of classical fields, that resembles the thermodynamic Bethe ansatz of quantum models, which we extend to generalized Gibbs ensembles (GGEs). In general, GHD must take into account both solitonic and radiative modes of classical fields. We observe that the quasi-particle formulation of GHD remains valid for radiative modes, even though these do not display particle-like properties in their precise dynamics. We point…
We examine the question of the integrability of the recently defined $\mathbb{Z}_2\times \mathbb{Z}_2$-graded sine-Gordon model, which is a natural generalisation of the supersymmetric sine-Gordon equation. We do this via appropriate auto-Bäcklund transformations, construction of conserved spinor-valued currents and a pair of infinite sets of conservation laws.
