Standard superfields in the harmonic formalism of supergauge theories

Scientific-Research Institute of Applied Physics; Tashkent State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 80, No. 3, pp. 391–398, September, 1989.     

Local supertwistors and N=2 conformal supergravity

Section of Theoretical Problems, USSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 79, No. 2, pp. 253–262, May, 1989.     

The holomorphic effective action inN=2D=4 supergauge theories with various gauge groups

We consider the theory of hypermultiplets in arbitrary representations of arbitrary semisimple gauge groups coupled to gauge superfields. Using the N=2 harmonic superspace formulation of these models, we find the general structure of the holomorphic effective action depending on the gauge superfield with values in the Cartan subalgebra of the gauge algebra. We find explicit expressions for the effective actions in the cases where the hypermultiplets are in the fundamental and adjoint representations of SU(n), SO(n), and Sp(2n). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 3, pp. 444–455, April, 2000.     

Chiral fermions in d=11 supergravity with additional time dimensions

Kovrov Branch, Vladimir Polytechnic Institute. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 76, No. 1, pp. 78–87, July, 1988.     

A Threefold of General Type with q 1=q 2=p g =P 2=0

One of the problems in classifying nonsingular threefolds of general type with p _g =0 lies in finding the range of the bigenus P ₂ (surfaces of general type with p _g =0 have 2P ₂10). Another problem involves finding the minimum integer m such that the m-canonical map _|mK| is birational for any threefold (m=5 in the case of surfaces). An example of a nonsingular threefold X of general type with q ₁=q ₂=0, p _g =P ₂=0,P ₃=1 is presented. In addition, the m-canonical map of X is birational if and only if m14. The threefold is obtained as a nonsingular model of a degree ten hypersurface in P ⁴ _C with the affine equation t ²=f ₁₀(x,y,z).     

Infrared finite observables in <Emphasis Type="Italic">N</Emphasis> = 8 supergravity

Using the algorithm of constructing the IR finite observables discussed in detail in our earlier papers, we study the construction of such observables in N = 8 supergravity in the first nontrivial order of perturbation theory. In general, contrary to the amplitudes defined in the presence of some IR regulator, such observables do not reveal any “simple” structure.     

On surfaces with p g =2, q=1 and K 2=5

We consider minimal surfaces of general type with p _g =2, q=1 and K ²=5. We provide a stratification of the corresponding moduli space \(\mathcal{M}\) and we give some bounds for the number and the dimensions of its irreducible components.     

TP2=Bruhat

It is shown that the Bruhat partial order on permutations is equivalent to a certain natural partial order induced by ${\mathrm{TP}}_2$ matrices and that a positive matrix is ${\mathrm{TP}}_2$ if (and only if) it satisfies certain inequalities induced by Bruhat order.     

On harmonic and 2-stein spaces of Iwasawa type

We study geometric properties of solvable metric Lie groups S of Iwasawa type; in particular harmonicity and the 2-stein condition. One restriction we obtain is that harmonic spaces of Iwasawa type have algebraic rank one, that is, the commutator subgroup of S has codimension one.We show that among Carnot solvmanifolds the only harmonic spaces are the Damek–Ricci spaces. Moreover, this rigidity result remains valid if harmonicity is replaced by the weaker 2-stein condition. As an application, we show that a harmonic Lie group of Iwasawa type with nonsingular 2-step nilpotent commutator subgroup is, up to scaling, a Damek–Ricci space.     

Bounded and L2 harmonic forms on universal covers

((Without abstract)) Submitted: April 1997, Revision: October 1997     

The Pell Equations x^2—8y^2=1 and y2—Dz^2=1

ＯｎｔｈｅＰｅｌｌｅｑｕａｔｉｏｎｓｘ２－８ｙ２＝１,　ｙ２－Ｄｚ２＝１（１）ｗｈｅｒｅＤ＞０ｉｓａｓｑｕａｒｅ－ｆｒｅｅｉｎｔｅｇｅｒ．ＣａｏＺｈｅｎｆｕ［１］ｓｈｏｗｅｄｔｈａｔｉｆＤ＝∏ｓｉ＝１Ｐｉ≡１（ｍｏｄ４）ｏｒＤ＝２∏Ｐｉ,１≤ｓ≤４,ｔｈｅｎｔｈｅｅｑｕａｔｉｏｎ（１）ｈａｓｎｏｌｙｐｏｓｉｔｉｖｅｉｎｔｅｇｅｒｓｏｌｕｔｉｏｎｚ＝６（Ｄ＝２·１７）．ＣｈｅｎｇＪｉａｎｈｕａ［２］ｓｈｏｗｅｄｔｈａｔｉｓＤ＝∏ｓｉ＝１Ｐｉ　１≤ｓ≤２,ｔｈｅｎｔｈｅｅｑｕａｔｉｏｎ（１）ｈａｓｏ…     

On Bachet's diophantine equationx 3=y 2=k

Using Eisenstein's law of cubic reciprocity we investigate cases in whichx ³=y ²+k is unsolvable in the ring of rational integers In particular we show, that for all primesp ± 1 (mod 9),p3, the equationx ³=y ²+3p(p±9) has no solutions in .     

A characterization of Inoue surfaces with $$p_g=0$$ and $$K^2=7$$K2=7

Inoue constructed the first examples of smooth minimal complex surfaces of general type with \(p_g=0\) and \(K^2=7\). These surfaces are finite Galois covers of the 4-nodal cubic surface with the Galois group, the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\). For such a surface S, the bicanonical map of S has degree 2 and it is composed with exactly one involution in the Galois group. The divisorial part of the fixed locus of this involution consists of two irreducible components: one is a genus 3 curve with self-intersection number 0 and the other is a genus 2 curve with self-intersection number \(-\,1\). Conversely, assume that S is a smooth minimal complex surface of general type with \(p_g=0\), \(K^2=7\) and having an involution \(\sigma \). We show that, if the divisorial part of the fixed locus of \(\sigma \) consists of two irreducible components \(R_1\) and \(R_2\), with \(g(R_1)=3, R_1^2=0, g(R_2)=2\) and \(R_2^2=-\,1\), then the Klein group \(\mathbb {Z}_2\times \mathbb {Z}_2\) acts faithfully on S and S is indeed an Inoue surface.     

关于Pell方程组X~2-3y~2=1与y~2-Dz~2=1的解

设D=2p_1…P_s(1≤s≤4),P_1…,P_s是互异的奇素数.证明了:Pell方程组x~2-3y~2=1,y~2-Dz~2=1除开D=2×7,2×3×5×7×13外,仅有平凡解(x,y,z)=(±2,±1,0).