arXiv subjects
Publications and source records attributed to A. Astashkevich.
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.
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.
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.