Strong Non-Ultralocality of Ginsparg-Wilson Fermionic Actions

It is shown that it is impossible to construct a free theory of fermions on infinite hypercubic Euclidean lattice in even number of dimensions that: (a) is ultralocal, (b) respects the symmetries of hypercubic lattice, (c) chirally nonsymmetric part of its propagator is local, and (d) describes single species of massless Dirac fermions in the continuum limit. This establishes non-ultralocality for arbitrary doubler-free Ginsparg-Wilson fermionic action with hypercubic symmetries ("strong non-ultralocality"), and complements the earlier general result on non-ultralocality of infinitesimal Ginsparg-Wilson-Luscher symmetry transformations ("weak non-ultralocality").Comment: 21 pages, 1 figure, LATEX. Few typos corrected; few sentences reformulated; figure centere

Naive A1\mathbb{A}^1-connectedness of retract rational varieties

A smooth, proper, retract rational variety over a field $k$ is known to be $\mathbb{A}^1$-connected. We improve on this result, in the case when $k$ is infinite, showing that such varieties are naively $\mathbb{A}^1$-connected.Comment: 14 pages, comments are welcom

Geometric motivic integration on Artin n-stacks

We construct a measure on the Boolean algebra of sets of formal arcs on an Artin stack which are definable in the language of Denef-Pas. The measure takes its values in a ring that is obtained from the Grothendieck ring of Artin stacks over the residue field by a localization followed by a completion. This construction is analogous to the construction of motivic measure on varieties by Denef and Loeser. We also obtain a "change of base" formula which allows us to relate the motivic measure on different stacks

Remarks on iterations of the A1\mathbb A^1-chain connected components construction

We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $\mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $\mathbb A^1$-connected space on which the iterations of the naive $\mathbb A^1$-connected components construction do not stabilize before the $n$th stage.Comment: 8 pages, comments are welcome; v2: minor modification