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Register a new userThe Moutard transformation of two-dimensional Dirac operators and conformal geometry of surfaces in the four-space
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Added by
Iskander A. Taimanov
Publication date
2016
fields
Physics
and research's language is
English
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The Moutard transformation for a two-dimensional Dirac operator with a complex-valued potential is constructed. It is showed that this transformation relates the potentials of Weierstrass representations of surfaces related by a composition of the inversion and a reflection with respect to an axis. It is given an analytical description of an explicit example of such a transformation which results in a creation of double points on the spectral curve of a Dirac operator with a double-periodic potential.
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We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.
We investigate second order quasilinear equations of the form f_{ij} u_{x_ix_j}=0 where u is a function of n independent variables x_1, ..., x_n, and the coefficients f_{ij} are functions of the first order derivatives p^1=u_{x_1}, >..., p^n=u_{x_n} only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n+1, R), which acts by projective transformations on the space P^n with coordinates p^1, ..., p^n. The coefficient matrix f_{ij} defines on P^n a conformal structure f_{ij} dp^idp^j. In this paper we concentrate on the case n=3, although some results hold in any dimension. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived. These conditions constitute a complicated over-determined system of PDEs for the coefficients f_{ij}, which is in involution. We prove that the moduli space of integrable equations is 20-dimensional. Based on these results, we show that any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. Reformulated in differential-geometric terms, the integrability conditions imply that the conformal structure f_{ij} dp^idp^j is conformally flat, and possesses an infinity of 3-conjugate null coordinate systems. Integrable equations provide an abundance of explicit examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.
In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the square of these operators, which play an important role in several topics of modern Mathematics, in particular in the study of the geometry of moduli spaces of a class of black holes, the geometry of NS-5 brane solutions of type II supergravity theories and BPS solitons in some string theories. We obtain a generalized Lichnerowicz formula, decompositions of the Dirac and spin-Dirac operators and their squares in terms of the standard Dirac and spin-Dirac operators and using the fact that spinor fields (sections of a spin-Clifford bundle) have representatives in the Clifford bundle we present also a noticeable relation involving the spin-Dirac and the Dirac operators.
By using the Moutard transformation of two-dimensional Schroedinger operators we derive a procedure for constructing explicit examples of such operators with rational fast decaying potentials and degenerate $L_2$-kernels (this construction was sketched in arXiv:0706.3595) and show that if we take some of these potentials as the Cauchy data for the Novikov-Veselov equation (a two-dimensional version of the Korteweg-de Vries equation), then the corresponding solutions blow up in a finite time
We show how to assign to any immersed torus in $R^3$ or $S^3$ a Riemann surface such that the immersion is described by functions defined on this surface. We call this surface the spectrum or the spectral curve of the torus. The spectrum contains important information about conformally invariant properties of the torus and, in particular, relates to the Willmore functional. We propose a simple proof that for isothermic tori in $R^3$ (this class includes constant mean curvature tori and tori of revolution) the spectrum is invariant with respect to conformal transformations of $R^3$. We show that the spectral curves of minimal tori in $S^3$ introduced by Hitchin and of constant mean curvature tori in $R^3$ introduced by Pinkall and Sterling are particular cases of this general spectrum. The construction is based on the Weierstrass representation of closed surfaces in $R^3$ and the construction of the Floquet--Bloch varieties of periodic differential operators.
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