The overall theme of my talk is an inversion of the overall theme of this workshop: while our collective understanding of physics has often been shaped by insights from algebraic geometry and number theory, my personal understanding of algebraic geometry and number theory has recently been shaped by insights from physics. My goal is to illustrate some challenging and subtle aspects of L-functions and secondary periods which are clarified through structural analogies with some equally challenging and subtle aspects of partition functions and anomalies. We focus on the following example:
If X is an arithmetic hyperbolic 3-manifold of finite volume with invariant trace field F, one can produce a class in K_3(F)_Q whose regulator image is equal to the complexified volume (i/2\pi^2)(vol(X)+iCS(X)) in C/Q. The volume vol(X) in R is well understood: it is related to the Dedekind zeta value \zeta_F(2) by the formula of Humbert, and can be expressed in terms of single-valued dilogarithms. The Chern-Simons invariant CS(X) in R/Q is more subtle: it is related to the eta invariant by the formula (1/2\pi^2)CS(X)=(3/2)\eta(X) mod (1/2)Z, and can be expressed in terms of multi-valued dilogarithms. One can also produce a class in K_1(F)_Q whose regulator image can be expressed in terms of logarithms and is related to the Reidemeister torsion or analytic torsion of X.
Number theorists like myself who study special values of L-functions recognize the significance of the Dedekind zeta value \zeta_F(2) as a motivic period, but the Chern-Simons/eta invariant and Reidemeister/analytic torsion are much more subtle secondary motivic periods whose arithmetic interpretation falls outside the usual conjectures about mixed motives.
Physicists who study Chern-Simons theory recognize these secondary motivic periods as phases or anomaly terms in the perturbative expansion of the partition function for 3d Chern-Simons theory with non-compact gauge group SL_2(C), with the Chern-Simons/eta invariant and Reidemeister/analytic torsion appearing at 0-loop and 1-loop respectively.
Both points of view on this example are valuable and necessary to understand this example and many others like it. The resulting reinterpretation of Beilinson's conjectures in terms of Chern-Simons theory suggests a refinement of the former and a deeper arithmetic significance of the latter, though no amount of time permits full discussion of the general story which emerges from these considerations.