# 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} }

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.

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## 82 References

### AN INTRODUCTION TO ALGEBRAIC K-THEORY

- Mathematics
- 2005

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

- Mathematics
- 2001

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

- Mathematics
- 2006

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

- Mathematics
- 2007

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

- Physics
- 2003

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

- Mathematics
- 1978

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.

- Mathematics
- 1995

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

- Mathematics
- 1996

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

- Mathematics
- 2001

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

- Mathematics
- 1988

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