The local zeta function for symmetric spaces of non-compact type

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Godoy, Tomas F.¹  Miatello, Roberto J.¹  Williams, Floyd L.² 

[1] Univ Nacl Cordoba, FaMAF CIEM, RA-5000 Cordoba, Argentina

[2] Univ Massachusetts, Dept Math & Stat, Amherst, MA 01003 USA

关键词: Local zeta function;    Symmetric spaces of noncompact type;    Casimir energy;   

DOI  :  10.1016/j.geomphys.2010.08.008

来源: Elsevier

【 摘 要 】

The Mellin transform of the heat kernel on a non-compact symmetric space X gives rise to a zeta function zeta (s; x, b) that was studied when the rank of X was 1. In this case the special values of the zeta function and of its derivative at s = 0, for example, are relevant for the quantum field effective potential in space-times modelled on X, or especially on a compact locally symmetric quotient Gamma \ X, where Gamma is a discrete group of isometries of X. Also the special value of zeta (s; x, b) at s = -1/2 determines the Casimir energy of such a space-time. In this paper we extend the study of zeta (s; x, b) to any symmetric space X of arbitrary real rank. One of our main results is Theorem 2.1, where we show that for general X and for x not equal (1) over bar, zeta (s; x, b) admits a continuation to an entire function. On the other hand, we show that under a mild condition, for x = (1) over bar, zeta (s; (1) over bar, b) has a meromorphic continuation to C with at most simple poles, all lying in the set of half-integers. In case G is complex, we give a very explicit form of the meromorphic continuation and we compute special values of the zeta function and of its derivative at s = 0 and at s = -1/2, which give a local contribution to the Casimir energy of X. To illustrate the difficulties present in the general case, we work out explicitly the meromorphic continuation for two infinite families of higher rank groups. (C) 2010 Elsevier B.V. All rights reserved.

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