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The Ray-Singer torsion

John Lott

Abstract

In 1971, Ray and Singer proposed an analytic equivalent of a classical topological invariant, the R-torsion. This Ray-Singer torsion has had many ramifications in mathematics and physics. I will describe the background, the Ray-Singer papers and some subsequent work.

The Ray-Singer torsion

Abstract

In 1971, Ray and Singer proposed an analytic equivalent of a classical topological invariant, the R-torsion. This Ray-Singer torsion has had many ramifications in mathematics and physics. I will describe the background, the Ray-Singer papers and some subsequent work.
Paper Structure (19 sections, 7 theorems, 35 equations)

This paper contains 19 sections, 7 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Topological precedents
  3. Lens spaces
  4. $R$-torsion
  5. Properties of the $R$-torsion
  6. Applications of the $R$-torsion
  7. Arnold Shapiro
  8. Analytical precedents
  9. Riemann zeta function and heat conduction
  10. Minakshisundaram-Pleijel
  11. McKean-Singer
  12. The Ray-Singer papers
  13. The first Ray-Singer paper
  14. The second Ray-Singer paper
  15. Further developments
  16. ...and 4 more sections

Key Result

Theorem 3.12

Minakshisundaram-Pleijel (1949) If $x \neq y$ then $\zeta_{x,y}(s)$ extends to an analytic function of $s$. If $x = y$ then $\zeta_{x,x}(s)$ extends to a meromorphic function of $s$. If $N$ is odd then $\zeta_{x,x}(s)$ has simple poles at $\frac{N}{2} - j$ for $j = 0, 1, 2, \ldots$, while if $N$ is

Theorems & Definitions (12)

  • Definition 2.4
  • Definition 2.6
  • Theorem 3.12
  • Definition 4.4
  • Theorem 4.6
  • proof
  • Definition 4.11
  • Theorem 4.13
  • Theorem 5.4
  • Theorem 5.6
  • ...and 2 more