A note on \(\sigma\)-model with the target \(S^n\)

A note on \(\sigma\)-model with the target \(S^n\)

Naively the Hilbert space of a sigma model has to be defined as an L^2 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(S^n) of the free loops of the sphere S^n. Spaces L_N(S^n) are defined in...

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Veröffentlicht in:arXiv.org 2020-05
1. Verfasser: Movshev, M V
Format: Artikel
Sprache:eng
Schlagworte:
Fourier series
Hilbert space
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Zusammenfassung:Naively the Hilbert space of a sigma model has to be defined as an L^2 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(S^n) of the free loops of the sphere S^n. Spaces L_N(S^n) are defined in terms of finite Fourier series. L_N(S^n) finite-dimensional but singular. We compute Riemann and Ricci curvatures of the smooth locus of this space and study Schr\"odinger operator in the case of L_1(S^n)
ISSN:2331-8422