Zeta-Regularization of the O(N) Non-Linear Sigma Model in D dimensions

Abstract

The O(N) non-linear sigma model in a $D$-dimensional space of the form ${\mathbf{R}}^{D-M}\times {\mathbf{T}}^M$, ${\mathbf{R}}^{D-M}\times {\mathbf{S}}^M$, or ${\mathbf{T}}^M\times {\mathbf{S}}^P$ is studied, where ${\mathbf{R}}^M$, ${\mathbf{T}}^M$ and ${\mathbf{S}}^M$ correspond to flat space, a torus and a sphere, respectively. Using zeta regularization and the $1/N$ expansion, the corresponding partition functions and the gap equations are obtained. Numerical solutions of the gap equations at the critical coupling constants are given, for several values of $D$. The properties of the partition function and its asymptotic behaviour for large $D$ are discussed. In a similar way, a higher-derivative non-linear sigma model is investigated too. The physical relevance of our results is discussed.

26 pages

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• Casimir effect for a $D$-dimensional sphere
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Cited by in corpus (1)

• Report of the Detailed Calculation of the Effective Potential in Spacetimes with $S^1\times R^d$ Topology and at Finite Temperature