Analytical torsion

The analytic torsion , also Ray-Singer torsion (after Daniel Burrill Ray , Isadore M. Singer ), is an invariant from the mathematical sub-area of global analysis . It is defined using the regularized determinant of the Laplace operator and agrees with the Reidemeister torsion (Cheeger-Müller theorem).

definition

Let it be a Riemannian manifold and an orthogonal representation of the fundamental group , so that the chain complex defined by the action of the fundamental group on the universal superposition is acyclic . ${\displaystyle M}$ ${\displaystyle \rho \colon \pi _{1}M\to O(N)}$ ${\displaystyle C_{*}({\widetilde {M}},\mathbb {R} )\otimes _{\mathbb {R} \left[\pi _{1}M\right]}\mathbb {R} ^{N}}$

The flat bundle to be associated has a compatible metric with which one defines the Hodge-Laplace operator acting on differential forms . Let be the eigenvalues ​​of , then one defines its zeta function by ${\displaystyle \rho }$ ${\displaystyle E}$ ${\displaystyle \Lambda ^{q}(M,E)}$ ${\displaystyle \Delta _{q}}$ ${\displaystyle \lambda _{j}}$ ${\displaystyle \Delta _{q}}$

${\displaystyle \zeta _{q}(s)=\sum _{\lambda _{j}>0}\lambda _{j}^{-s}}$

for and through analytical continuation of this function for , and its regularized determinant through ${\displaystyle Re(s)>{\tfrac {N}{2}}}$ ${\displaystyle s\in \mathbb {C} }$

${\displaystyle \log \det(\Delta _{q})=-{\frac {d}{ds}}\mid _{s=0}\zeta _{q}(s)}$.

The analytical torsion is defined by ${\displaystyle T_{M}(\rho )}$

${\displaystyle \log T_{M}(\rho )={\frac {1}{2}}\sum _{q}(-1)^{q}q{\frac {d}{ds}}\mid _{s=0}\zeta _{q}(s)}$

or equivalent by

${\displaystyle T_{M}(\rho )=\Pi _{q}\det(\Delta _{q})^{-(-1)^{q}{\frac {q}{2}}}}$.

Cheeger-Müller's theorem

The Cheeger-Müller theorem (formerly the Ray-Singer conjecture ) says that analytic torsion and Reidemeister torsion are equal . It was first proven by Cheeger and Müller for orthogonal or unitary representations and later generalized by Müller to unimodular representations. The equality of the two invariants is used in the perturbative Chern-Simons theory .

literature

• Ray, DB; Singer, IM: R-torsion and the Laplacian on Riemannian manifolds. Advances in Math. 7, 145-210. (1971).
• Müller, Werner: Analytic torsion and R-torsion of Riemannian manifolds. Adv. In Math. 28 (1978), no. 3, 233-305.
• Cheeger, Jeff: Analytic torsion and the heat equation. Ann. of Math. (2) 109 (1979) no. 2, 259-322.
• Müller, Werner: Analytic torsion and R -torsion for unimodular representations. J. Amer. Math. Soc. 6, no. 3, 721-753 (1993).
• Bismuth, Jean-Michel; Lott, John: Flat vector bundles, direct images and higher real analytic torsion. J. Amer. Math. Soc. 8 (1995) no. 2, 291-363.

Web links

• Lück: Survey on analytic and topological torsion
• Bunke: Six lectures on analytic torsion