Motives: an introductory survey for physicists

@article{Rej2009MotivesAI,
  title={Motives: an introductory survey for physicists},
  author={Abhijnan Rej and Matilde Marcolli},
  journal={arXiv: High Energy Physics - Theory},
  year={2009}
}
  • A. RejM. Marcolli
  • Published 23 July 2009
  • Physics, Mathematics
  • arXiv: High Energy Physics - Theory
We survey certain accessible aspects of Grothendieck's theory of motives in arithmetic algebraic geometry for mathematical physicists, focussing on areas that have recently found applications in quantum field theory. An appendix (by Matilde Marcolli) sketches further connections between motivic theory and theoretical physics. 

Tables from this paper

Dimension, Divergence and Desingularization

I argue that consistent geometrical descriptions of the universe are far from unique even as low-energy limits and that an abstract "atomic" description of spacetime and gauge-theoretic geometry in

New and old results in resultant theory

Resultants play an increasingly important role in modern theoretical physics: they appear whenever we have nonlinear (polynomial) equations, nonquadratic forms, or non-Gaussian integrals. Being a

AP THEORY VIII: ANALOGIES WITH NCG CONCERNING (3 + 1)-QFT, F1 AND MODERN COSMOLOGY

In previous papers we pointed out many genuine similarities between the mathematically rigorous Artin Presentation Theory and some important, although still mostly heuristic, theories of modern

Dimension, Divergence and Desingularization∗ A Rough Cut on Motifs

The history of modern theoretical physics demonstrates a collective urge on behalf of its practitioners to seek beauty and simplicity in each generation’s search for answers to, what it considers,

CONFORMAL BLOCKS AS DOTSENKO–FATEEV INTEGRAL DISCRIMINANTS

As anticipated in Ref. 1, elaborated in Refs. 2–4, and explicitly formulated in Ref. 5, the Dotsenko–Fateev integral discriminant coincides with conformal blocks, thus providing an elegant approach

A second-order differential equation for the two-loop sunrise graph with arbitrary masses

We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a

AN INTRODUCTION TO ALGEBRAIC K-THEORY

These are the notes of an introductory lecture given at The 20th Winter School for Geometry and Physics, at Srni. It was meant as a leisurely exposition of classical aspects of algebraic K-theory,

Motivic measures and stable birational geometry

We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the

On Motives Associated to Graph Polynomials

The appearance of multiple zeta values in anomalous dimensions and β-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization

Noncommutative Geometry, Quantum Fields and Motives

Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil

What is motivic measure

These notes give an exposition of the theory of arithmetic motivic integration, as developed by J. Denef and F. Loeser. An appendix by M. Fried gives some historical comments on Galois

Principles of Algebraic Geometry

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications

Descent, motives and K-theory.

To an arbitrary variety over a field of characteristic zero, we associate a complex of Chow motives, which is, up to homotopy, unique and bounded. We deduce that any variety has a natural Euler

Methods of Homological Algebra

Considering homological algebra, this text is based on the systematic use of the language and ideas of derived categories and derived functors. Relations with standard cohomology theory are

Geometry on Arc Spaces of Algebraic Varieties

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and

Gauge Field Theory and Complex Geometry

Geometrical Structures in Field Theory.- 1. Grassmannians, Connections, and Integrability.- 2. The Radon-Penrose Transform.- 3. Introduction to Superalgebra.- 4. Introduction to Supergeometry.- 5.
...