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Fyodor Malikov

Publications and source records attributed to Fyodor Malikov.

16 recordsLinked to original sources
arXiv2024

Homotopy chiral algebras

We define a notion of a homotopy chiral algebra (HCA), which means a chiral algebra up to higher homotopies, and prove that the Cech complex of a sheaf of chiral algebras admits a structure of a HCA.

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arXiv2022

Chiral Janus complexes

We propose chiral analogues of some infinite complexes appearing in the description of the coherent derived categories for projective spaces.

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arXiv2014

Chiral De Rham complex over locally complete intersections

We define a version of a derived chiral De Rham complex over a locally complete intersection, thereby "chiralizing" a result by Illusie and Bhatt. A similar construction attaches to a graded ring a dg vertex algebra, which we prove to be Morita equivalent to a dg algebra of differential operators. For example, the dg vertex algebra associated to a fat point, which also arises in the Landau-Ginzburg model, is shown to be derived rational.

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arXiv2012

Chiral differential operators on abelian varieties

The paper consists of two parts. In the first, we describe a way of getting from an algebra of chiral differential operators (cdo) on an abelian variety a cdo on the dual variety. The second is an introduction to the sigma-model on a torus and to the Wess-Zumino-Witten model from the cdo perspective.

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arXiv2009

Algebras of twisted chiral differential operators and affine localization of $\frak{g}$-modules

We propose a notion of algebra of {\it twisted} chiral differential operators over algebraic manifolds with vanishing 1st Pontrjagin class. We show that such algebras possess families of modules depending on infinitely many complex parameters, which we classify in terms of the corresponding algebra of twisted differential operators. If the underlying manifold is a flag manifold, our construction recovers modules over an affine Lie algebra parameterized by opers over the Langlands dual Lie algebra. The spaces of global sections of "smallest" such modules are irreducible $\ghat$-modules and all irreducible $\frak{g}$-integrable $\ghat$-modules at the critical level arise in this way.

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arXiv2008

Vertex Algebroids over Veronese Rings

We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.

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arXiv2000

Gerbes of chiral differential operators. III

This note is a sequel to "Gerbes of chiral differential operators. II", math.AG/0003170. We study gerbes of chiral differential operators acting on the exterior algebra $ΛE$ of a vector bundle over a smooth algebraic variety $X$. When $E=Ω^1_X$ this gerbe admits a canonical global section which coincides with the chiral de Rham complex of $X$.

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arXiv1999

Gerbes of chiral differential operators

In this note we compute the cohomological obstruction to the existence of certain sheaves of vertex algebras on smooth varieties. These sheaves have been introduced and studied in the previous work by A.Vaintrob and two of the authors. Hopefully our result clarifies to some extent the constructions of the above work.

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arXiv1999

Chiral de Rham complex

The aim of this note is to define certain sheaves of vertex algebras on smooth manifolds. For each smooth complex algebraic (or analytic) manifold $X$, we construct a sheaf $Ω^{ch}_X$, called the {\bf chiral de Rham complex} of $X$. It is a sheaf of vertex algebras in the Zarisky (or classical) topology, It comes equipped with a $\BZ$-grading by {\it fermionic charge}, and the {\it chiral de Rham differential} $d_{DR}^{ch}$, which is an endomorphism of degree 1 such that $(d_{DR}^{ch})^2=0$. One has a canonical embedding of the usual de Rham complex $(Ω_X, d_{DR})\hra (Ω_X^{ch}, d_{DR}^{ch})$ which is a quasiisomorphism. If $X$ is Calabi-Yau then this sheaf admits an N=2 supersymmetry. For some $X$ (for example, for curves or for the flag spaces $G/B$), one can construct also a purely even analogue of this sheaf, a {\it chiral structure sheaf} $\CO^{ch}_X$. For the projective line, the space of global sections of the last sheaf is the irreducible vacuum $\hsl(2)$-module on the critical level.

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arXiv1999

Chiral Poincaré duality

The aim of this note is to prove the analogue of Poincaré duality in the chiral Hodge cohomology.

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arXiv1999

Chiral de Rham complex. II

This paper is a sequel to math.AG/9803041. It consists of three parts. In the first part we give certain construction of vertex algebras which includes in particular the ones appearing in op. cit. In the second part we show how the cohomology ring $H^*(X)$ of a smooth complex variety $X$ could be restored from the correlation functions of the vertex algebra $RΓ(X;Ω^{ch}_X)$. In the third part, we prove first a useful general statement that the sheaf of loop algebras over the tangent sheaf $\Cal{T}_X$ acts naturally on $Ω^{ch}_X$ for every smooth $X$ (see §1). The Z-graded vertex algebra $H^*(X;Ω^{ch}_X)$ seems to be a quite interesting object (especially for compact $X$). In §2, we compute $H^0(CP^N;Ω^{ch}_{CP^N})$ as a module over $\hat{sl}(N+1)$.

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