Monday, June 17, 2013

1306.3420 (Li Guo et al.)

Renormalization of conical zeta values and the Euler-Maclaurin formula    [PDF]

Li Guo, Sylvie Paycha, Bin Zhang
We equip the space of convex rational cones with a connected coalgebra structure, which we further generalize to decorated cones by means of a differentiation procedure. Using the convolution product $\ast$ associated with the coproduct on cones we define an interpolator $\mu:= I^{\ast(-1)}\ast S$ as the $\ast$ quotient of an exponential discrete sum $S$ and an exponential integral $I$ on cones. A generalization of the algebraic Birkhoff decomposition to linear maps from a connected coalgebra to a space with a linear decomposition then enables us to carry out a Birkhoff-Hopf factorization $S:= S_-^{\ast (-1)}\ast S_+ $ on the map $S$ whose "holomorphic part" corresponds to $S_+$. By the uniqueness of the Birkhoff-Hopf factorization we obtain $\mu=S_+$ and $I=S_-^{\ast (-1)}$ so that this renormalization procedure \`a la Connes and Kreimer yields a new interpretation of the local Euler-Maclaurin formula on cones of Berline and Vergne. The Taylor coefficients at zero of the interpolating holomorphic function $\mu=S_+$ correspond to renormalized conical zeta values at non-positive integers. When restricted to Chen cones, this yields yet another way to renormalize multiple zeta values at non-positive integers previously investigated by the authors using other approaches. In the present approach renormalized conical multiple zeta values lie at the cross road of three a priori distinct fields, the geometry on cones with the Euler Maclaurin formula, number theory with multiple zeta values and renormalization theory with methods borrowed from quantum field theory.
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