Mathematics – Differential Geometry
Scientific paper
2002-04-12
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.
No associations
LandOfFree
Adiabatic limits of eta and zeta functions of elliptic operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Adiabatic limits of eta and zeta functions of elliptic operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Adiabatic limits of eta and zeta functions of elliptic operators will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-667580