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\({\mathcal{W}}\)-Symmetry of the Adèlic Grassmannian

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Abstract

We give a geometric construction of the \({\mathcal{W}_{1+\infty}}\) vertex algebra as the infinitesimal form of a factorization structure on an adèlic Grassmannian. This gives a concise interpretation of the higher symmetries and Bäcklund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of “\({\mathcal{W}_{1+\infty}}\)-geometry”: the geometry of \({\mathcal{D}}\)-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Austin, Austin, TX, 78712, USA

    David Ben-Zvi

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Thomas Nevins

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  1. David Ben-Zvi
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  2. Thomas Nevins
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Correspondence to David Ben-Zvi.

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Communicated by L. Takhtajan

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Ben-Zvi, D., Nevins, T. \({\mathcal{W}}\)-Symmetry of the Adèlic Grassmannian. Commun. Math. Phys. 293, 185–204 (2010). https://doi.org/10.1007/s00220-009-0925-y

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