Representations of the Lie algebra of vector fields on a torus and the chiral de Rham complex

Billig, Yuly; Futorny, Vyacheslav

doi:10.1090/S0002-9947-2014-05959-X

Representations of the Lie algebra of vector fields on a torus and the chiral de Rham complex
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by Yuly Billig and Vyacheslav Futorny PDF

Trans. Amer. Math. Soc. 366 (2014), 4697-4731 Request permission

Abstract:

The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra – the Lie algebra $\textrm {Vect}{\mathbb {T}}^N$ of vector fields on an $N$-dimensional torus for $N > 1$. The case $N=1$ gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra $\textrm {Vect}{\mathbb {T}}^N$ has a natural class of tensor modules parametrized by finite-dimensional modules of $gl_N$. Tensor modules can be used in turn to construct bounded irreducible modules for $\textrm {Vect}{\mathbb {T}}^{N+1}$ (induced from $\textrm {Vect}{\mathbb {T}}^N$), which are the focus of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible $\Omega ^1 \left ( \mathbb {T}^{N+1} \right ) / d \Omega ^0 \left ( \mathbb {T}^{N+1} \right ) \rtimes \textrm {Vect}{\mathbb {T}}^{N+1}$-modules constructed in a paper by the first author. These modules remain irreducible when restricted to the subalgebra $\textrm {Vect}{\mathbb {T}}^{N+1}$, unless they belong to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob (1999).

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