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


Authors: Yuly Billig and Vyacheslav Futorny
Journal: Trans. Amer. Math. Soc. 366 (2014), 4697-4731
MSC (2010): Primary 17B66, 17B67; Secondary 17B69
DOI: https://doi.org/10.1090/S0002-9947-2014-05959-X
Published electronically: April 25, 2014
MathSciNet review: 3217697
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Abstract: The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra - the Lie algebra $ {\rm 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 $ {\rm 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 $ {\rm Vect}{\mathbb{T}}^{N+1}$ (induced from $ {\rm 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 {\rm Vect}{\mathbb{T}}^{N+1}$-modules constructed in a paper by the first author. These modules remain irreducible when restricted to the subalgebra $ {\rm Vect}{\mathbb{T}}^{N+1}$, unless they belong to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob (1999).


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Additional Information

Yuly Billig
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
Email: billig@math.carleton.ca

Vyacheslav Futorny
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 05315-970 Brasil
Email: futorny@ime.usp.br

DOI: https://doi.org/10.1090/S0002-9947-2014-05959-X
Received by editor(s): August 31, 2011
Received by editor(s) in revised form: September 11, 2012
Published electronically: April 25, 2014
Additional Notes: The first author was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada
The second author was supported in part by the CNPq grant (301743/2007-0) and by the Fapesp grant (2010/50347-9)
Dedicated: Dedicated to Yuri Alexandrovich Bahturin
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.