A note on σ-model with the target Sⁿ

2020/05/13 by M. V. Movshev, Movshev, M. V.
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Black Holes and Theoretical Physics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2005.06497

openalex publication_date 2020/05/13 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

Naively the Hilbert space of a sigma model has to be defined as an L² space of functions on the space of free loops of the target. This object is not well defined. In this note we study a finite-dimensional approximations L_N(Sⁿ) of the free loops of the sphere Sⁿ. Spaces L_N(Sⁿ) are defined in terms of finite Fourier series. L_N(Sⁿ) finite-dimensional but singular. We compute Riemann and Ricci curvatures of the smooth locus of this space and study Schrödinger operator in the case of L₁(Sⁿ)

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