Zeta function for the Laplace operator acting on forms in a ball with gauge boundary conditions

Abstract

The Laplace operator acting on antisymmetric tensor fields in a $D$--dimensional Euclidean ball is studied. Gauge-invariant local boundary conditions (absolute and relative ones, in the language of Gilkey) are considered. The eigenfuctions of the operator are found explicitly for all values of $D$. Using in a row a number of basic techniques, as Mellin transforms, deformation and shifting of the complex integration contour, and pole compensation, the zeta function of the operator is obtained. From its expression, in particular, $\zeta (0)$ and ${\zeta }^{\prime }(0)$ are evaluated exactly. A table is given in the paper for $D=3,4,...,8$. The functional determinants and Casimir energies are obtained for $D=3,4,...,6$.

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References in corpus (4)

• Zeta function determinant of the Laplace operator on the $D$-dimensional ball
• Antisymmetric tensor fields on spheres: functional determinants and non--local counterterms
• The asymptotics of the Laplacian on a manifold with boundary II
• One-Loop Effective Action on the Four-Ball

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