On the structure of Verma module over Virasoro and Neveu-Schwarz algebras

In the paper we present a different proof of the theorem of B. L. Feigin and D. B. Fuchs about the structure of Verma modules over Virasoro algebra. We state some new results about the structure of Verma modules over Neveu-Schwarz. The proof has thwo advantages: first, it is simplier in the most interesting cases (for example in the so called minimal models), second, it can be generalized for Neveu-Schwarz algebra for some class of Verma modules.Comment: 41 pages, LaTe

Non-Local Equivariant Star Product on the Minimal Nilpotent Orbit

We construct a unique G-equivariant graded star product on the algebra $S(g)/I$ of polynomial functions on the minimal nilpotent coadjoint orbit \Omin of G where G is a complex simple Lie group and g\neq\sl_2(C). This strengthens the result of Arnal, Benamor, and Cahen. Our main result is to compute, for G classical, the star product of a momentum function $\mu_x$ with any function f. We find \mu_x\star f=\mu_xf+\half\{\mu_x,f\}t+\Lambda^x(f)t^2. For \g different from sp_n(\C), $\Lambda^x$ is not a differential operator. Instead \Lamda^x is the left quotient of an explicit order 4 algebraic differential operator $D^x$ by an order 2 invertible diagonalizable operator. Precisely, $\Lambda^x=-{1/4}\frac{1}{E'(E'+1)}D^x$ where $E'$ is a positive shift of the Euler vector field. Thus $\mu_x\star f$ is not local in f. Using $\star$ we construct a positive definite hermitian inner product on $Sg/I$. The Hilbert space completion of $Sg/I$ is then a unitary representation of $G$. This quantizes \Omin in the sense of geometric quantization and the orbit method.Comment: latex file, 13 pages. In this new version we use the star product to construct a unitary representation attached to the orbi

Exotic Differential Operators on Complex Minimal Nilpotent Orbits

Let O be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action $C^*$ on g. The algebra of differential operators on O is D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \cite{jos} and \cite{bkHam} for some results on the geometry and quantization of O. We construct an explicit subspace $A_{-1}\subset D(O)$ of commuting differential operators which are Euler homogeneous of degree -1. The space $A_{-1}$ is finite-dimensional, g-stable and carries the adjoint representation. $A_{-1}$ consists of (for $g \neq sp(2n,C)$) non-obvious order 4 differential operators obtained by quantizing symbols we obtained previously. These operators are "exotic" in that there is (apparently) no geometric or algebraic theory which explains them. The algebra generated by $A_{-1}$ is a maximal commutative subalgebra A of D(X). We find a G-equivariant algebra isomorphism R(O) to A, $f\mapsto D_f$, such that the formula $(f|g)=({constant term of}D_{\bar{g}} f)$ defines a positive-definite Hermitian inner product on R(O). We will use these operators $D_f$ to quantize O in a subsequent paper.Comment: 34 pages, corrected some typos, changed conten

Projective modules over non-commutative tori: classification of modules with constant curvature connection

We study finitely generated projective modules over noncommutative tori. We prove that for every module $E$ with constant curvature connection the corresponding element $[E]$ of the K-group is a generalized quadratic exponent and, conversely, for every positive generalized quadratic exponent $\mu$ in the K-group one can find such a module $E$ with constant curvature connection that $[E] = \mu$. In physical words we give necessary and sufficient conditions for existence of 1/2 BPS states in terms of topological numbers.Comment: Latex. Misprints correcte

On noncommutative Nahm transform

Motivated by the recently observed relation between the physics of $D$-branes in the background of $B$-field and the noncommutative geometry we study the analogue of Nahm transform for the instantons on the noncommutative torus.Comment: Latex, 22 p

Gendered Violence

This is a groundbreaking study of an important and neglected topic—the systematic use of rape as a strategic weapon of the genocidal anti-Jewish violence, known collectively as pogroms, that erupted in Ukraine in the period between 1917 and 1921, and in which at least 100,000 Jews died and undocumented numbers of Jewish women were raped. The book is based on the in-depth study of the scores of narratives of Jewish men and women who survived the pogrom violence, but were then all but forgotten for almost a century. This book deconstructs the motives of perpetrators, the experience and expression of trauma by the victimized community, and how the genocidal objectives of the pogrom perpetrators were achieved and maximized through the macabre carnival of violence