Adiabatic limits of eta and zeta functions of elliptic operators

Details Adiabatic limits of eta and zeta functions of elliptic operators Adiabatic limits of eta and zeta functions of elliptic operators

2002-04-12

arxiv.org/abs/math/0204163v2

Math. Z. 246 (2004), 441-471

Mathematics

Differential Geometry

32 pages, final version

Scientific paper

10.1007/s00209-003-0578-z

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $\delta$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and $t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identify their values at $t=0$ with the holonomy of the determinant bundle, respectively with a residue trace. For invertible families of operators, the functions ${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to $t=0$ for all values of $s$. After normalizing with a Gamma factor, the zeta function satisfies in the adiabatic limit an identity reminiscent of the Riemann zeta function, while the eta function converges to the volume of the Bismut-Freed meromorphic family of connection 1-forms.

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