32,775 research outputs found
On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture
For a finite-dimensional Lie algebra over a field , we deduce from the compatibility between cup products
Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an
alternative way of re-writing Kontsevich product on by means of the Alekseev--Torossian flat connection (Alekseev
and Torossian, 2010). We deduce a similar formula directly from the
Kashiwara--Vergne conjecture (Kashiwara and Vergne, 1978).Comment: 8 pages, 1 figure; notation changed; corrected many other misprints
recently noticed; comments are very welcome
The explicit equivalence between the standard and the logarithmic star product for Lie algebras
The purpose of this short note is to establish an explicit equivalence
between the two star products and on the symmetric
algebra of a finite-dimensional Lie algebra over a field of characteristic 0 associated with
the standard angular propagator and the logarithmic one: the differential
operator of infinite order with constant coefficients realizing the equivalence
is related to the incarnation of the Grothendieck-Teichm\"uller group
considered by Kontsevich.Comment: 2 figures; corrected and completed the formulation of Theorem 3.7.
Comments are very welcome
Combinatorial Quantum Gravity: Geometry from Random Bits
I propose a quantum gravity model in which geometric space emerges from
random bits in a quantum phase transition driven by the combinatorial
Ollivier-Ricci curvature and corresponding to the condensation of short cycles
in random graphs. This quantum critical point defines quantum gravity
non-perturbatively. In the ordered geometric phase at large distances the
action reduces to the standard Einstein-Hilbert term.Comment: Revised version to appear in JHE
On Relaxing Metric Information in Linear Temporal Logic
Metric LTL formulas rely on the next operator to encode time distances,
whereas qualitative LTL formulas use only the until operator. This paper shows
how to transform any metric LTL formula M into a qualitative formula Q, such
that Q is satisfiable if and only if M is satisfiable over words with
variability bounded with respect to the largest distances used in M (i.e.,
occurrences of next), but the size of Q is independent of such distances.
Besides the theoretical interest, this result can help simplify the
verification of systems with time-granularity heterogeneity, where large
distances are required to express the coarse-grain dynamics in terms of
fine-grain time units.Comment: Minor change
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