https://cds.cern.ch/record/549532/files/0204253.pdf

Dilaton gravity in two-dimensions

We prove that no local diffeomorphism invariant two-dimensional theory of the metric and the dilaton without higher derivatives can describe the exact string black hole solution found a decade ago by Dijkgraaf, Verlinde and Verlinde. One of the key points in this proof is the concept of dilaton-shift invariance. We present and solve (classically) all dilaton-shift invariant theories of two-dimensional dilaton gravity. Two such models, resembling the exact string black hole and generalizing the CGHS model, are discussed explicitly.

/pdf/non-existence-of-a-dilaton-gravity-action-for-the-exact-k9mlghhpus.pdf

Non-existence of a dilaton gravity action for the exact string black hole

Envelope solitons of the Nonlinear Schrodinger equation (NLS) under quantum potential's influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s 1, to the reaction–diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1+1)-dimensional Jackiw–Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space–time metric. The space–time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.

Resonance NLS solitons as black holes in Madelung fluid

The recently proposed SO(2,1)-invariant quantization of the Liouville theory is elaborated. We develop a renormalized perturbation expansion which preserves this symmetry to all orders, but spontaneously breaks Poincare invariance. Some Green's functions and scattering amplitudes are calculated in low perturbative order, and it is established that the S matrix is trivial in the tree approximation. Whether this is also true of the complete S matrix remains an open question.

SO(2,1) Invariant Quantization of the Liouville Theory

The classical solutions of the dimensionally reduced gravitational Chern–Simons theory

Using the methods of differential geometry it is shown that the Born-Infeld scalar field in two-dimensional space-time and the relativistic string in three dimensions are described by the same non-linear Liouville equation utt-uxx=R eu. This equation admits soliton solutions which may be stable or unstable, and there are periodical solutions among the stable ones. In the quantum case the solitons can be interpreted as massive particles either stable or unstable with respect to the stability of the corresponding classical solution. The periodical soliton generates a series of resonances which have the equidistant mass spectrum. This result appears to be well suited to the theory of the closed relativistic string. In four dimensions the relativistic string is described by the same Liouville equation, but for the complex-valued function u.

https://iopscience.iop.org/article/10.1088/0305-4470/13/1/030/pdf

The solitons in some geometrical field theories

We present a model of a one-dimensional extended relativistic object, whose motion is defined by the requirement that its time track in Minkowski space is a surface of the constant mean curvature H. The world surface of the relativistic string is a particular case of such surfaces, namely, a minimal surface with H=0. By differential-geometry methods the theory of the proposed object moving in three-dimensional space-time is reduced to one nonlinear equation ϕττ − ϕσσ = Hshϕ. In the theory under consideration, there naturally arises the pair of Lax's operators needed to solve this nonlinear equation by the inverse scattering method.

Generalization of the relativistic string model in the geometrical approach

Exact classical solutions of the equations of motion for the infinite relativistic string loaded with a point-like mass are obtained in terms of the Fourier integrals. The gauge when the time evolution parameter τ is proportional to the proper time of a point-like particle is used. The gauge conditions in this case are the second-class constraints. So in the quantum case the commutators of the dynamical variables have to be defined by the Dirac brackets.

A. M. Chervjakov

Papers

The solitons in some geometrical field theories

Generalization of the relativistic string model in the geometrical approach

Infinite relativistic string with a point-like mass