Zeta-function method in field theory at finite temperature

Engineering-Physics Institute, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 78, No. 3, pp. 444–457, March, 1989.     

Solving linear equations in a vector space over a finite field

We study the maximum possible size of a subset in a vector space over a finite field which contains no solution of a given linear equation (or a system of linear equations). This is a finite field version of Ruzsa's work [7].     

Boundary conditions and confinement in the gauge field theory at finite temperature

A Hamiltonian formulation of a non-Abelian gauge theory confined in a finite domain is constructed in a generalized three-dimensional Fock-Schwinger gauge in the presence of surface terms. The dependence of the partition function on the boundary value of the longitudinal electric-field component, which because of the Gauss law, coincides with the electric-field flow through an infinitesimal boundary-surface element in this gauge, is investigated. This dependence is related to the confinement—deconfinement phase transition. In the confinement phase, the chromoelectric current through any boundary element is zero, because all observable quantities are singlet w.r.t. the remaining gauge-transformation group, i.e., color objects are unobservable at spatial infinity. In addition to the non-Abelian theory, a simpler example of quantum electrodynamics with an external-charge density in a spherical domain is considered in which the effective partition function is exactly calculable. With a feeling of deep sadness and irrecoverable loss, I had to complete this paper on my own. In N. A. Sveshnikov, theoretical physics has lost an extremely talented scientist, a great educator, and a very nice person. No matter how big an influence Nikita Alexeevich's ideas had on his students, which I am privileged to be one of, and on colleagues who knew him well, it would be only fair to say that future generations will judge their true merit. This paper is a logical conclusion to our years of joint research on the confinement problem. Although, in my view, it sheds light on one of the most difficult problems in elementary particle physics, it is really only a small step forward. It was Sveshnikov's gift to see the hidden mathematical beauty of the physical world, the beauty that expresses the essence of all things. Deceased Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 2, pp. 206–220, November, 1998.     

Three-dimensional Gross-Neveu model at nonzero temperature and in an external magnetic field

The phase structure of the (2+1) Gross-Neveu model at nonzero temperatureT and in an external fieldH is studied in the leading order of the 1/N expansion. It is shown that for any fixed value ofT there exists a critical valueH _c of the magnetic field such that whenH>H _c the chiral invariance of the model is spontaneously broken. For fixedH, there exists a critical value of the temperatureT _c (H) such that forT>T _c (H) the original symmetry of the model is restored. The phase portrait of the model is constructed in theH-T plane.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 1, pp. 3–11, January, 1992.     

Supersymmetric p-branes at finite temperature

The regularized free energy of open and closed supersymmetric p-branes in (D – 1)-dimensional flat space is calculated by the method of quasiclassical quantization. The critical temperature, indicating a phase transition, is obtained. It is shown that the quantum properties of bosonic p-branes (open and closed) with p = D – 1 are identical to the properties of a massless scalar particle.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 36–40, 1990.     

Thermo-magneto-dynamic stresses and perturbation of magnetic field vector in a non-homogeneous hollow cylinder

An analytical method is presented to investigate thermo-magneto-elastic stresses and perturbation of the magnetic field vector in a conducting non-homogeneous hollow cylinder under thermal shock. The interaction between the deformation and the magnetic field vector in a non-homogeneous hollow cylinder is considered by adding a Lorentz’s electro-magneto-force into the equation of thermo-elastic motion of the non-homogeneous hollow cylinder in an axial magnetic field. The exact solution for magneto-thermo-dynamic stresses and perturbation responses of an axial magnetic field vector in a conducting non-homogeneous hollow cylinder was obtained by using finite integral transforms. From numerical calculations, the dynamic characteristics on both thermo-magneto-stresses and perturbation of the axial magnetic field vector in the conducting non-homogeneous hollow cylinder is revealed and discussed.     

The effects of strong magnetic fields and rotation on soliton stars at finite temperature

We study the effects of strong magnetic fields and uniform rotation on the properties of soliton stars in Lee-Wick model when a temperature dependence is introduced into this model. We first recall the properties of the Lee-Wick model and study the properties of soliton solutions, in particular, the stability condition, in terms of the parameters of the model and in terms of the number of fermions N inside the soliton (for very large N) in the presence of strong magnetic fields and uniform rotation. We also calculate the effects of gravity on the stability properties of the soliton stars in the simple approximation of coupling the Newtonian gravitational field to the energy density inside the soliton, treating this as constant throughout. Following Cottingham and Vinh Mau, we also make an analysis at finite temperature and show the possibility of a phase transition which leads to a model with parameters similar to those considered by Lee and his colleagues but in the presence of magnetic fields and rotation. More specifically, the effects of magnetic fields and rotation on the soliton mass and transition temperature are computed explicitly. We finally study the evolution on these magnetized and rotating soliton stars with the temperature from the early universe to the present time.     

On distribution by rank of bases for vector spaces of matrices over a finite field

Let F^m×n_q denote the vector space of all m×n matrices over the finite field F_q of order q, and let $\text{B}\text{=(A}{}_1\text{,A}{}_2\text{,…,A}{}_{\mathrm{mn}}\text{)}$ denote an ordered basis for F^m×n_q. If the rank of A_i is r_i,i=1,2,…,mn, then $\text{B}$ is said to have rank (r₁,r₂,…,r_mn), and the number of ordered bases of F^mxn_q with rank (r₁,r₂,…,r_mn is denoted by N_q(r₁, r₂,…,r_mn). This paper determines formulas for the numbers N_q(r₁,r₂,…,r_mn) for the case m=n=2, q arbitrary, and while some of the techniques of the paper extend to arbitrary m and n, the general formulas for the numbers N_q(r₁,r₂,…,r_mn) seem quite complicated and remain unknown. An idea on a possible computer attack which may be feasible for low values of m and n is also discussed.     

Contact metric manifolds whose characteristic vector field is a harmonic vector field

In this paper, contact metric manifolds whose characteristic vector field ξ is a harmonic vector field are called H-contact manifolds. We show that a (2n+1)-dimensional contact metric manifold is an H-contact manifold if and only if ξ is an eigenvector of the Ricci operator (J.C. González-Dávila and L. Vanhecke [J. Geom. 72 (2001) 65–76] proved this result for n=1). Consequently, the class of H-contact manifolds is very large. Moreover, we give some application about the topology of a compact H-contact manifold.     

On a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group

We prove that a holomorphic vector bundle over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials and , with nonnegative integral coefficients, such that the vector bundle is isomorphic to . An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group.

When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.