nLab complex volume

Idea

The Cheeger-Simons classes are complexified secondary invariants.

Under identifying the fundamental class of a hyperbolic 3-manifold $X$ as an element in the Bloch group, the corresponding degree-3 Cheegers-Simons invariant is the complex volume of the 3-manifold, namely the linear combination

$$CS + i vol$$

of its Chern-Simons invariant and its volume (e.g. Neumann 11, section 2.3, Garoufalidis-Thurston-Zickert 11).

This combination appears also

• as the Lagrangian density of analytically continued Chern-Simons theory.

• as the contribution of a membrane instanton wrapping a hyperbolic 3-cycle.

Properties

Volume conjecture

The volume conjecture for the Reshetikhin-Turaev construction states that in the classical limit it converges to the complex volume (MMOTY 02, Conjedtcure 1.2, see also Chen-Yang 15)

Related concepts

• Borel regulator

• SnapPy

References

• Jeff Cheeger, Jim Simons, Differential characters and geometric invariants, in Geometry and Topology, Proceedings of the Special Year, University of Maryland 1983-84, eds. J. Alexander and J. Harer, Lecture Notes in Math. 1167, Springer-Verlag, Berlin, Heidelberg, New York, 1985, pp. 50–80.

• Johan Dupont, Richard Hain, Steven Zucker?, Regulators and characteristic classes of flat bundles (arXiv:alg-geom/9202023)

• Walter Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413-474 (arXiv:math/0307092)

• Walter Neumann, Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541, Amer. Math. Soc. 2011, 233–246 (arXiv:1108.0062)

• Stavros Garoufalidis, Dylan Thurston, Christian Zickert, The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds (arXiv:1111.2828, Euclid)

The volume conjecture (Kashaev’s conjecture) for complex volume is due to

• Hitoshi Murakami, Jun Murakami, M. Okamoto, T. Takata, and Y. Yokota, Conjecture 1.2 in Kashaev’s Conjecture And The Chern-Simons Invariants Of Knots And Links, Experiment. Math. 11 (2002) 427-435 (arXiv:math/0203119)

see also

• Qingtao Chen, Tian Yang, A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary (arXiv:1503.02547)

Relation to analytic torsion is discussed in

• Varghese Mathai, section 6 of $L^2$-analytic torsion, Journal of Functional Analysis Volume 107, Issue 2, 1 August 1992, Pages 369–386

• John Lott, Heat kernels on covering spaces and topological invariants, J. Diff. Geom. 35 no 2 (1992) (pdf)