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The Anomaly Line Bundle of the Self-Dual Field Theory

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Abstract

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.

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Authors and Affiliations

  1. Laboratoire de Physique Théorique de l’École Normale Supérieure (Unité mixte de recherche du CNRS et de l’ENS, associée à l’Université Pierre et Marie Curie et aux fédérations de recherche FR684 et FR2687), CNRS UMR 8549, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Samuel Monnier

  2. Laboratoire de Physique Théorique et Hautes Énergies, CNRS UMR 7589 and Université Pierre et Marie Curie-Paris 6, 4 place Jussieu, 75252, Paris Cedex 05, France

    Samuel Monnier

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  1. Samuel Monnier
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Correspondence to Samuel Monnier.

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Communicated by N. A. Nekrasov

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Monnier, S. The Anomaly Line Bundle of the Self-Dual Field Theory. Commun. Math. Phys. 325, 41–72 (2014). https://doi.org/10.1007/s00220-013-1844-5

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